Subgroups — C₃₁×#60496 #21517255

Order 7936.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3084	C₂	1	3	yes	maximal, complemented
4	C₃₁×#3083	C₂	1	3	yes	maximal, complemented
5	C₃₁×#3081	C₂	1	3	yes	maximal, complemented
6	C₃₁×#3081	C₂	1	3	yes	maximal, complemented
7	C₃₁×#3080	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3077	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3076	C₂	1	3	yes	maximal, complemented
10	C₂×C₃₁×#478	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2912	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2897	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#2897	C₂	1	1	yes	maximal, characteristic
16	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₃₁×#557	—	2	1	no	—
20	C₃₁×#557	—	2	1	no	—
21	C₃₁×#554	—	2	3	no	—
22	C₃₁×#554	—	2	3	no	—
23	C₃₁×#544	C₂²	1	3	yes	complemented
24	C₃₁×#544	C₂²	1	3	yes	complemented
25	C₃₁×#533	C₂²	1	3	yes	complemented
26	C₂×C₃₁×#125	C₂²	1	3	yes	complemented
27	C₃₁×#520	—	2	1	no	—
28	C₃₁×#520	C₂²	1	3	yes	—
29	C₃₁×#520	C₂²	1	1	yes	characteristic
30	C₃₁×#520	C₂²	1	1	yes	characteristic
31	C₃₁×#520	C₂²	1	1	yes	characteristic
32	C₃₁×#518	—	2	3	no	—
33	C₃₁×#518	C₂²	1	3	yes	—
34	C₄×#41×C₃₁	—	2	1	no	—
35	C₄×#41×C₃₁	C₂²	1	3	yes	—
36	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₃₁	—	2	3	no	—
38	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
39	C₃₁×#479	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#479	C₂²	1	3	yes	complemented
41	C₃₁×#479	C₂²	1	6	yes	complemented
42	C₃₁×#478	C₂²	1	3	yes	complemented
43	C₃₁×#478	C₂²	1	3	yes	complemented
44	C₃₁×#478	C₂²	1	6	yes	complemented
45	C₃₁×#478	C₂²	1	6	yes	complemented
46	C₃₁×#478	C₂²	1	6	yes	complemented
47	C₃₁×#478	C₂²	1	6	yes	complemented
48	C₃₁×#477	C₂²	1	3	yes	complemented
49	C₃₁×#476	C₂²	1	3	yes	complemented
50	C₃₁×#476	C₂²	1	6	yes	complemented
51	C₃₁×#475	C₂²	1	3	yes	complemented
52	C₃₁×#474	C₂²	1	3	yes	complemented
53	C₃₁×#474	C₂²	1	6	yes	complemented
54	C₃₁×#428	C₂²	1	3	yes	complemented
55	C₃₁×#428	C₂²	1	3	yes	complemented
56	C₃₁×#428	C₂²	1	3	yes	complemented
57	C₃₁×#428	C₂²	1	3	yes	complemented
58	C₃₁×#427	C₂²	1	3	yes	complemented
59	C₃₁×#427	C₂²	1	3	yes	complemented
60	C₂×C₃₁×#107	C₂²	1	3	yes	complemented
61	C₂×C₃₁×#106	C₂²	1	3	yes	complemented
62	C₃₁×#424	C₂²	1	3	yes	complemented
63	C₃₁×#424	C₂²	1	3	yes	complemented
64	C₃₁×#423	C₂²	1	3	yes	complemented
65	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
66	C₃₁×#418	C₂²	1	3	yes	—
67	C₃₁×#418	C₂²	1	3	yes	—
68	C₃₁×#417	C₂²	1	3	yes	—
69	C₃₁×#417	C₂²	1	3	yes	—
70	C₃₁×#416	C₂²	1	3	yes	—
71	C₃₁×#416	C₂²	1	3	yes	—
72	C₂×C₃₁×#102	C₂²	1	3	yes	—
73	C₂×C₃₁×#102	C₂²	1	3	yes	—
74	C₃₁×H(5,𝔽₂)	—	2	4	no	—
75	C₃₁×H(5,𝔽₂)	Q₈	1	8	yes	complemented
76	C₂×#41×C₃₁	—	2	1	no	—
77	C₂×#41×C₃₁	C₂³	1	3	yes	—
78	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₃₁	—	2	3	no	—
80	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
81	C₂³×C₄×C₃₁	—	2	3	no	—
82	C₃₁×#131	—	2	4	no	—
83	C₃₁×#131	—	2	4	no	—
84	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
85	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
86	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
87	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
88	C₃₁×C₄⋊Q₈	—	2	3	no	—
89	C₃₁×C₄⋊Q₈	—	2	3	no	—
90	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
91	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
92	C₃₁×#125	—	2	3	no	—
93	C₃₁×#125	—	2	3	no	—
94	C₃₁×#125	—	2	3	no	—
95	C₃₁×#125	—	2	3	no	—
96	C₃₁×#125	C₂³	1	3	yes	—
97	C₃₁×#125	C₂³	1	6	yes	—
98	C₃₁×#125	C₂³	1	1	yes	characteristic
99	C₃₁×C₂²⋊Q₈	—	2	3	no	—
100	C₃₁×C₂²⋊Q₈	—	2	3	no	—
101	C₃₁×C₂²⋊Q₈	—	2	6	no	—
102	C₃₁×C₂²⋊Q₈	—	2	6	no	—
103	C₄×Q₈×C₃₁	—	2	1	no	—
104	C₄×Q₈×C₃₁	—	2	1	no	—
105	C₄×Q₈×C₃₁	—	2	1	no	—
106	C₄×Q₈×C₃₁	—	2	3	no	—
107	C₄×Q₈×C₃₁	—	2	3	no	—
108	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
109	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
110	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
111	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
112	C₄×D₄×C₃₁	—	2	3	no	—
113	C₄×D₄×C₃₁	—	2	3	no	—
114	C₄×D₄×C₃₁	—	2	3	no	—
115	C₄×D₄×C₃₁	—	2	3	no	—
116	C₄×D₄×C₃₁	—	2	3	no	—
117	C₄×D₄×C₃₁	—	2	3	no	—
118	C₄×D₄×C₃₁	—	2	3	no	—
119	C₄×D₄×C₃₁	—	2	6	no	—
120	C₄×D₄×C₃₁	—	2	3	no	—
121	C₄×D₄×C₃₁	—	2	3	no	—
122	C₄×D₄×C₃₁	D₄	1	3	yes	—
123	C₄×D₄×C₃₁	D₄	1	3	yes	—
124	C₄×D₄×C₃₁	C₂³	1	6	yes	—
125	C₄×D₄×C₃₁	C₂³	1	3	yes	—
126	C₄×D₄×C₃₁	C₂³	1	3	yes	—
127	C₃₁×#117	—	2	3	no	—
128	C₃₁×#117	—	2	3	no	—
129	C₃₁×#117	C₂³	1	3	yes	—
130	C₃₁×#117	C₂³	1	3	yes	—
131	C₃₁×#117	C₂³	1	3	yes	—
132	C₃₁×#117	C₂³	1	3	yes	—
133	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
134	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
135	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
136	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
137	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
138	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
139	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
140	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
141	C₂×C₄²×C₃₁	—	2	3	no	—
142	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
143	C₃₁×#107	C₂³	1	3	yes	—
144	C₃₁×#107	C₂³	1	6	yes	—
145	C₃₁×#107	C₂³	1	1	yes	characteristic
146	C₃₁×#107	C₂³	1	3	yes	—
147	C₃₁×#107	C₂³	1	3	yes	—
148	C₃₁×#106	C₂³	1	3	yes	—
149	C₃₁×#106	C₂³	1	6	yes	—
150	C₃₁×#106	C₂³	1	1	yes	characteristic
151	C₃₁×#106	C₂³	1	3	yes	—
152	C₃₁×#106	C₂³	1	3	yes	—
153	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
154	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
155	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
156	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
157	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
158	C₃₁×#103	C₂³	1	1	yes	characteristic
159	C₃₁×#103	C₂³	1	1	yes	characteristic
160	C₃₁×#103	C₂³	1	3	yes	—
161	C₃₁×#103	C₂³	1	3	yes	—
162	C₃₁×#102	C₂³	1	3	yes	—
163	C₃₁×#102	C₂³	1	3	yes	—
164	C₃₁×#102	C₂³	1	6	yes	—
165	C₃₁×#102	C₂³	1	6	yes	—
166	C₃₁×#102	C₂³	1	3	yes	—
167	C₃₁×#102	C₂³	1	3	yes	—
168	C₂⁴×C₃₁	—	2	3	no	—
169	#41×C₃₁	—	2	8	no	—
170	#41×C₃₁	—	2	6	no	—
171	#41×C₃₁	C₂×Q₈	1	12	yes	—
172	#41×C₃₁	C₂×Q₈	1	4	yes	—
173	#41×C₃₁	#41	1	4	yes	—
174	C₂×Q₈×C₃₁	—	2	3	no	—
175	C₂×Q₈×C₃₁	—	2	3	no	—
176	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂×D₄×C₃₁	—	2	3	no	—
179	C₂×D₄×C₃₁	—	2	24	no	—
180	C₂×D₄×C₃₁	—	2	3	no	—
181	C₂×D₄×C₃₁	—	2	6	no	—
182	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
183	C₂×D₄×C₃₁	C₂×Q₈	1	12	yes	—
184	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
185	C₂²×C₄×C₃₁	—	2	3	no	—
186	C₂²×C₄×C₃₁	—	2	3	no	—
187	C₂²×C₄×C₃₁	—	2	3	no	—
188	C₂²×C₄×C₃₁	—	2	3	no	—
189	C₂²×C₄×C₃₁	—	2	3	no	—
190	C₂²×C₄×C₃₁	—	2	3	no	—
191	C₂²×C₄×C₃₁	—	2	6	no	—
192	C₂²×C₄×C₃₁	—	8	6	no	—
193	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
194	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
195	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
196	M₁₆×C₃₁	—	2	6	no	—
197	M₁₆×C₃₁	—	2	6	no	—
198	C₂×C₈×C₃₁	—	2	6	no	—
199	C₂×C₈×C₃₁	—	2	6	no	—
200	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
201	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
202	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
203	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
204	C₄⋊C₄×C₃₁	—	2	3	no	—
205	C₄⋊C₄×C₃₁	—	2	3	no	—
206	C₄⋊C₄×C₃₁	—	2	3	no	—
207	C₄⋊C₄×C₃₁	—	2	3	no	—
208	C₄⋊C₄×C₃₁	—	2	6	no	—
209	C₄⋊C₄×C₃₁	—	2	6	no	—
210	C₄⋊C₄×C₃₁	—	2	3	no	—
211	C₄⋊C₄×C₃₁	—	2	3	no	—
212	C₄⋊C₄×C₃₁	—	2	1	no	—
213	C₄⋊C₄×C₃₁	—	2	1	no	—
214	C₄⋊C₄×C₃₁	—	2	1	no	—
215	C₄⋊C₄×C₃₁	—	2	3	no	—
216	C₄⋊C₄×C₃₁	—	2	3	no	—
217	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
218	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
219	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
222	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
223	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
224	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
225	C₂²⋊C₄×C₃₁	—	2	3	no	—
226	C₂²⋊C₄×C₃₁	—	2	3	no	—
227	C₂²⋊C₄×C₃₁	—	2	3	no	—
228	C₂²⋊C₄×C₃₁	—	2	3	no	—
229	C₂²⋊C₄×C₃₁	—	2	6	no	—
230	C₂²⋊C₄×C₃₁	—	2	3	no	—
231	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
232	C₄²×C₃₁	—	2	1	no	—
233	C₄²×C₃₁	—	2	3	no	—
234	C₄²×C₃₁	—	2	3	no	—
235	C₄²×C₃₁	—	2	3	no	—
236	C₄²×C₃₁	—	2	3	no	—
237	C₄²×C₃₁	C₂×D₄	1	3	yes	—
238	C₄²×C₃₁	C₂×D₄	1	3	yes	—
239	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
240	C₄²×C₃₁	C₂⁴	1	3	yes	—
241	#60496	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂³×C₃₁	—	2	3	no	—
243	C₂³×C₃₁	—	2	12	no	—
244	C₂³×C₃₁	—	8	3	no	—
245	C₂³×C₃₁	—	8	3	no	—
246	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
247	Q₈×C₃₁	—	2	1	no	—
248	Q₈×C₃₁	—	8	3	no	—
249	Q₈×C₃₁	—	8	3	no	—
250	Q₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
251	Q₈×C₃₁	C₂²×Q₈	1	2	yes	—
252	D₄×C₃₁	—	2	12	no	—
253	D₄×C₃₁	—	2	12	no	—
254	D₄×C₃₁	—	2	3	no	—
255	D₄×C₃₁	C₂²⋊Q₈	1	12	yes	—
256	D₄×C₃₁	C₂²×Q₈	1	6	yes	—
257	C₂×C₄×C₃₁	—	2	1	no	—
258	C₂×C₄×C₃₁	—	2	12	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	2	3	no	—
263	C₂×C₄×C₃₁	—	2	3	no	—
264	C₂×C₄×C₃₁	—	2	3	no	—
265	C₂×C₄×C₃₁	—	4	3	no	—
266	C₂×C₄×C₃₁	—	4	3	no	—
267	C₂×C₄×C₃₁	—	4	6	no	—
268	C₂×C₄×C₃₁	—	8	3	no	—
269	C₂×C₄×C₃₁	—	8	6	no	—
270	C₂×C₄×C₃₁	—	8	3	no	—
271	C₂×C₄×C₃₁	—	8	3	no	—
272	C₂×C₄×C₃₁	—	8	3	no	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
277	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
278	C₂×C₄×C₃₁	C₂²×Q₈	1	6	yes	—
279	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
280	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
281	C₈×C₃₁	—	2	1	no	—
282	C₈×C₃₁	—	2	1	no	—
283	C₈×C₃₁	—	2	3	no	—
284	C₈×C₃₁	—	2	3	no	—
285	#3504	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₄×H(5,𝔽₂)	C₂×C₃₁	1	1	yes	characteristic
287	#3084	C₂×C₃₁	1	3	yes	complemented
288	#3083	C₂×C₃₁	1	3	yes	complemented
289	#3081	C₂×C₃₁	1	3	yes	complemented
290	#3081	C₂×C₃₁	1	3	yes	complemented
291	#3080	C₂×C₃₁	1	3	yes	complemented
292	#3077	C₂×C₃₁	1	3	yes	complemented
293	#3076	C₂×C₃₁	1	3	yes	complemented
294	C₂×#478	C₂×C₃₁	1	3	yes	complemented
295	#2913	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#2912	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#2897	C₂×C₃₁	1	1	yes	characteristic
299	#2897	C₂×C₃₁	1	1	yes	characteristic
300	C₂²×C₃₁	—	2	3	no	—
301	C₂²×C₃₁	—	2	3	no	—
302	C₂²×C₃₁	—	2	6	no	—
303	C₂²×C₃₁	—	4	3	no	—
304	C₂²×C₃₁	—	4	3	no	—
305	C₂²×C₃₁	—	8	12	no	—
306	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	6	yes	—
307	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
308	C₄×C₃₁	—	2	1	no	—
309	C₄×C₃₁	—	2	3	no	—
310	C₄×C₃₁	—	2	1	no	—
311	C₄×C₃₁	—	2	3	no	—
312	C₄×C₃₁	—	2	2	no	—
313	C₄×C₃₁	—	4	1	no	—
314	C₄×C₃₁	—	4	3	no	—
315	C₄×C₃₁	—	4	1	no	—
316	C₄×C₃₁	—	4	3	no	—
317	C₄×C₃₁	—	4	3	no	—
318	C₄×C₃₁	—	4	1	no	—
319	C₄×C₃₁	C₂×C₂²⋊Q₈	1	6	yes	—
320	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
321	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
322	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
325	#557	—	2	1	no	—
326	#557	—	2	1	no	—
327	#554	—	2	3	no	—
328	#554	—	2	3	no	—
329	#544	C₂²×C₃₁	1	3	yes	complemented
330	#544	C₂²×C₃₁	1	3	yes	complemented
331	#533	C₂²×C₃₁	1	3	yes	complemented
332	C₂×#125	C₂²×C₃₁	1	3	yes	complemented
333	#520	—	2	1	no	—
334	#520	C₂²×C₃₁	1	3	yes	—
335	#520	C₂²×C₃₁	1	1	yes	characteristic
336	#520	C₂²×C₃₁	1	1	yes	characteristic
337	#520	C₂²×C₃₁	1	1	yes	characteristic
338	#518	—	2	3	no	—
339	#518	C₂²×C₃₁	1	3	yes	—
340	C₄×#41	—	2	1	no	—
341	C₄×#41	C₂²×C₃₁	1	3	yes	—
342	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
343	C₂×C₄×D₄	—	2	3	no	—
344	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
345	#479	C₂²×C₃₁	1	1	yes	characteristic, complemented
346	#479	C₂²×C₃₁	1	3	yes	complemented
347	#479	C₂²×C₃₁	1	6	yes	complemented
348	#478	C₂²×C₃₁	1	3	yes	complemented
349	#478	C₂²×C₃₁	1	6	yes	complemented
350	#478	C₂²×C₃₁	1	6	yes	complemented
351	#478	C₂²×C₃₁	1	6	yes	complemented
352	#478	C₂²×C₃₁	1	3	yes	complemented
353	#478	C₂²×C₃₁	1	6	yes	complemented
354	#477	C₂²×C₃₁	1	3	yes	complemented
355	#476	C₂²×C₃₁	1	3	yes	complemented
356	#476	C₂²×C₃₁	1	6	yes	complemented
357	#475	C₂²×C₃₁	1	3	yes	complemented
358	#474	C₂²×C₃₁	1	3	yes	complemented
359	#474	C₂²×C₃₁	1	6	yes	complemented
360	#428	C₂²×C₃₁	1	3	yes	complemented
361	#428	C₂²×C₃₁	1	3	yes	complemented
362	#428	C₂²×C₃₁	1	3	yes	complemented
363	#428	C₂²×C₃₁	1	3	yes	complemented
364	#427	C₂²×C₃₁	1	3	yes	complemented
365	#427	C₂²×C₃₁	1	3	yes	complemented
366	C₂×#107	C₂²×C₃₁	1	3	yes	complemented
367	C₂×#106	C₂²×C₃₁	1	3	yes	complemented
368	#424	C₂²×C₃₁	1	3	yes	complemented
369	#424	C₂²×C₃₁	1	3	yes	complemented
370	#423	C₂²×C₃₁	1	3	yes	complemented
371	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
372	#418	C₂²×C₃₁	1	3	yes	—
373	#418	C₂²×C₃₁	1	3	yes	—
374	#417	C₂²×C₃₁	1	3	yes	—
375	#417	C₂²×C₃₁	1	3	yes	—
376	#416	C₂²×C₃₁	1	3	yes	—
377	#416	C₂²×C₃₁	1	3	yes	—
378	C₂×#102	C₂²×C₃₁	1	3	yes	—
379	C₂×#102	C₂²×C₃₁	1	3	yes	—
380	C₂×C₃₁	—	2	6	no	—
381	C₂×C₃₁	—	4	6	no	—
382	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
383	C₂×C₃₁	#3584	1	1	yes	characteristic
384	C₂×C₃₁	#3585	1	1	yes	characteristic
385	H(5,𝔽₂)	—	2	4	no	—
386	H(5,𝔽₂)	Q₈×C₃₁	1	8	yes	complemented
387	C₂×#41	—	2	1	no	—
388	C₂×#41	C₂³×C₃₁	1	3	yes	—
389	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
390	C₂²×D₄	—	2	3	no	—
391	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
392	C₂³×C₄	—	2	3	no	—
393	#131	—	2	4	no	—
394	#131	—	2	4	no	—
395	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
396	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
397	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
398	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
399	C₄⋊Q₈	—	2	3	no	—
400	C₄⋊Q₈	—	2	3	no	—
401	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
402	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
403	#125	—	2	3	no	—
404	#125	—	2	3	no	—
405	#125	—	2	3	no	—
406	#125	—	2	3	no	—
407	#125	C₂³×C₃₁	1	3	yes	—
408	#125	C₂³×C₃₁	1	6	yes	—
409	#125	C₂³×C₃₁	1	1	yes	characteristic
410	C₂²⋊Q₈	—	2	3	no	—
411	C₂²⋊Q₈	—	2	3	no	—
412	C₂²⋊Q₈	—	2	6	no	—
413	C₂²⋊Q₈	—	2	6	no	—
414	C₄×Q₈	—	2	1	no	—
415	C₄×Q₈	—	2	1	no	—
416	C₄×Q₈	—	2	1	no	—
417	C₄×Q₈	—	2	3	no	—
418	C₄×Q₈	—	2	3	no	—
419	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
420	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
421	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
423	C₄×D₄	—	2	3	no	—
424	C₄×D₄	—	2	3	no	—
425	C₄×D₄	—	2	3	no	—
426	C₄×D₄	—	2	3	no	—
427	C₄×D₄	—	2	3	no	—
428	C₄×D₄	—	2	3	no	—
429	C₄×D₄	—	2	3	no	—
430	C₄×D₄	—	2	6	no	—
431	C₄×D₄	—	2	3	no	—
432	C₄×D₄	—	2	3	no	—
433	C₄×D₄	D₄×C₃₁	1	3	yes	—
434	C₄×D₄	D₄×C₃₁	1	3	yes	—
435	C₄×D₄	C₂³×C₃₁	1	3	yes	—
436	C₄×D₄	C₂³×C₃₁	1	6	yes	—
437	C₄×D₄	C₂³×C₃₁	1	3	yes	—
438	#117	—	2	3	no	—
439	#117	—	2	3	no	—
440	#117	C₂³×C₃₁	1	3	yes	—
441	#117	C₂³×C₃₁	1	3	yes	—
442	#117	C₂³×C₃₁	1	3	yes	—
443	#117	C₂³×C₃₁	1	3	yes	—
444	C₂×C₄⋊C₄	—	2	3	no	—
445	C₂×C₄⋊C₄	—	2	3	no	—
446	C₂×C₄⋊C₄	—	2	3	no	—
447	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
448	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
449	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
450	C₂×C₂²⋊C₄	—	2	3	no	—
451	C₂×C₂²⋊C₄	—	2	6	no	—
452	C₂×C₄²	—	2	3	no	—
453	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
454	#107	C₂³×C₃₁	1	3	yes	—
455	#107	C₂³×C₃₁	1	6	yes	—
456	#107	C₂³×C₃₁	1	1	yes	characteristic
457	#107	C₂³×C₃₁	1	3	yes	—
458	#107	C₂³×C₃₁	1	3	yes	—
459	#106	C₂³×C₃₁	1	3	yes	—
460	#106	C₂³×C₃₁	1	6	yes	—
461	#106	C₂³×C₃₁	1	1	yes	characteristic
462	#106	C₂³×C₃₁	1	3	yes	—
463	#106	C₂³×C₃₁	1	3	yes	—
464	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
465	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
466	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
467	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
468	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
469	#103	C₂³×C₃₁	1	1	yes	characteristic
470	#103	C₂³×C₃₁	1	1	yes	characteristic
471	#103	C₂³×C₃₁	1	3	yes	—
472	#103	C₂³×C₃₁	1	3	yes	—
473	#102	C₂³×C₃₁	1	3	yes	—
474	#102	C₂³×C₃₁	1	3	yes	—
475	#102	C₂³×C₃₁	1	6	yes	—
476	#102	C₂³×C₃₁	1	6	yes	—
477	#102	C₂³×C₃₁	1	3	yes	—
478	#102	C₂³×C₃₁	1	3	yes	—
479	C₃₁	#60496	1	1	yes	characteristic, direct product factor, Sylow
480	C₂⁴	—	2	3	no	—
481	#41	—	2	8	no	—
482	#41	—	2	6	no	—
483	#41	C₂×Q₈×C₃₁	1	12	yes	—
484	#41	C₂×Q₈×C₃₁	1	4	yes	—
485	#41	#41×C₃₁	1	4	yes	—
486	C₂×Q₈	—	2	3	no	—
487	C₂×Q₈	—	2	3	no	—
488	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
490	C₂×D₄	—	2	3	no	—
491	C₂×D₄	—	2	24	no	—
492	C₂×D₄	—	2	3	no	—
493	C₂×D₄	—	2	6	no	—
494	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
495	C₂×D₄	C₂×Q₈×C₃₁	1	12	yes	—
496	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
497	C₂²×C₄	—	2	3	no	—
498	C₂²×C₄	—	2	3	no	—
499	C₂²×C₄	—	2	3	no	—
500	C₂²×C₄	—	2	3	no	—
501	C₂²×C₄	—	2	3	no	—
502	C₂²×C₄	—	2	3	no	—
503	C₂²×C₄	—	2	6	no	—
504	C₂²×C₄	—	8	6	no	—
505	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
506	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
507	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
508	M₁₆	—	2	6	no	—
509	M₁₆	—	2	6	no	—
510	C₂×C₈	—	2	6	no	—
511	C₂×C₈	—	2	6	no	—
512	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
513	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
514	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
515	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
516	C₄⋊C₄	—	2	3	no	—
517	C₄⋊C₄	—	2	3	no	—
518	C₄⋊C₄	—	2	3	no	—
519	C₄⋊C₄	—	2	3	no	—
520	C₄⋊C₄	—	2	6	no	—
521	C₄⋊C₄	—	2	6	no	—
522	C₄⋊C₄	—	2	3	no	—
523	C₄⋊C₄	—	2	3	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	3	no	—
528	C₄⋊C₄	—	2	3	no	—
529	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
530	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
531	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
532	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
533	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
534	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
535	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
536	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
537	C₂²⋊C₄	—	2	3	no	—
538	C₂²⋊C₄	—	2	3	no	—
539	C₂²⋊C₄	—	2	3	no	—
540	C₂²⋊C₄	—	2	3	no	—
541	C₂²⋊C₄	—	2	6	no	—
542	C₂²⋊C₄	—	2	3	no	—
543	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	3	no	—
546	C₄²	—	2	3	no	—
547	C₄²	—	2	3	no	—
548	C₄²	—	2	3	no	—
549	C₄²	C₂×D₄×C₃₁	1	3	yes	—
550	C₄²	C₂×D₄×C₃₁	1	3	yes	—
551	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
552	C₄²	C₂⁴×C₃₁	1	3	yes	—
553	C₂³	—	2	3	no	—
554	C₂³	—	2	12	no	—
555	C₂³	—	8	3	no	—
556	C₂³	—	8	3	no	—
557	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
558	Q₈	—	2	1	no	—
559	Q₈	—	8	3	no	—
560	Q₈	—	8	3	no	—
561	Q₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
562	Q₈	C₂²×Q₈×C₃₁	1	2	yes	—
563	D₄	—	2	12	no	—
564	D₄	—	2	12	no	—
565	D₄	—	2	3	no	—
566	D₄	C₃₁×C₂²⋊Q₈	1	12	yes	—
567	D₄	C₂²×Q₈×C₃₁	1	6	yes	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	12	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	3	no	—
574	C₂×C₄	—	2	3	no	—
575	C₂×C₄	—	2	3	no	—
576	C₂×C₄	—	4	3	no	—
577	C₂×C₄	—	4	3	no	—
578	C₂×C₄	—	4	6	no	—
579	C₂×C₄	—	8	3	no	—
580	C₂×C₄	—	8	6	no	—
581	C₂×C₄	—	8	3	no	—
582	C₂×C₄	—	8	3	no	—
583	C₂×C₄	—	8	3	no	—
584	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
586	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
588	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
589	C₂×C₄	C₂²×Q₈×C₃₁	1	6	yes	—
590	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
591	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
592	C₈	—	2	1	no	—
593	C₈	—	2	1	no	—
594	C₈	—	2	3	no	—
595	C₈	—	2	3	no	—
596	C₂²	—	2	3	no	—
597	C₂²	—	2	3	no	—
598	C₂²	—	2	6	no	—
599	C₂²	—	4	3	no	—
600	C₂²	—	4	3	no	—
601	C₂²	—	8	12	no	—
602	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	6	yes	—
603	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
604	C₄	—	2	2	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	3	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	3	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	3	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	3	no	—
613	C₄	—	4	3	no	—
614	C₄	—	4	1	no	—
615	C₄	C₂×C₃₁×C₂²⋊Q₈	1	6	yes	—
616	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
617	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
618	C₂	—	2	6	no	—
619	C₂	—	4	6	no	—
620	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
621	C₂	C₃₁×#3584	1	1	yes	characteristic
622	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3504	C₂	semidirect
3. C₃₁×#3084	C₂	semidirect
4. C₃₁×#3083	C₂	semidirect
5. C₃₁×#3081	C₂	semidirect
6. C₃₁×#3081	C₂	semidirect
7. C₃₁×#3080	C₂	semidirect
8. C₃₁×#3077	C₂	semidirect
9. C₃₁×#3076	C₂	semidirect
10. C₂×C₃₁×#478	C₂	semidirect
11. C₃₁×#2913	C₂	semidirect
12. C₃₁×#2912	C₂	semidirect
13. C₃₁×#2906	C₂	semidirect
23. C₃₁×#544	C₂²	semidirect
24. C₃₁×#544	C₂²	semidirect
25. C₃₁×#533	C₂²	semidirect
26. C₂×C₃₁×#125	C₂²	semidirect
39. C₃₁×#479	C₂²	semidirect
40. C₃₁×#479	C₂²	semidirect
41. C₃₁×#479	C₂²	semidirect
42. C₃₁×#478	C₂²	semidirect
43. C₃₁×#478	C₂²	semidirect
44. C₃₁×#478	C₂²	semidirect
45. C₃₁×#478	C₂²	semidirect
46. C₃₁×#478	C₂²	semidirect
47. C₃₁×#478	C₂²	semidirect
48. C₃₁×#477	C₂²	semidirect
49. C₃₁×#476	C₂²	semidirect
50. C₃₁×#476	C₂²	semidirect
51. C₃₁×#475	C₂²	semidirect
52. C₃₁×#474	C₂²	semidirect
53. C₃₁×#474	C₂²	semidirect
54. C₃₁×#428	C₂²	semidirect
55. C₃₁×#428	C₂²	semidirect
56. C₃₁×#428	C₂²	semidirect
57. C₃₁×#428	C₂²	semidirect
58. C₃₁×#427	C₂²	semidirect
59. C₃₁×#427	C₂²	semidirect
60. C₂×C₃₁×#107	C₂²	semidirect
61. C₂×C₃₁×#106	C₂²	semidirect
62. C₃₁×#424	C₂²	semidirect
63. C₃₁×#424	C₂²	semidirect
64. C₃₁×#423	C₂²	semidirect
65. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
75. C₃₁×H(5,𝔽₂)	Q₈	semidirect
241. #60496	C₃₁	direct (N × H)
285. #3504	C₂×C₃₁	semidirect
287. #3084	C₂×C₃₁	semidirect
288. #3083	C₂×C₃₁	semidirect
289. #3081	C₂×C₃₁	semidirect
290. #3081	C₂×C₃₁	semidirect
291. #3080	C₂×C₃₁	semidirect
292. #3077	C₂×C₃₁	semidirect
293. #3076	C₂×C₃₁	semidirect
294. C₂×#478	C₂×C₃₁	semidirect
295. #2913	C₂×C₃₁	semidirect
296. #2912	C₂×C₃₁	semidirect
297. #2906	C₂×C₃₁	semidirect
329. #544	C₂²×C₃₁	semidirect
330. #544	C₂²×C₃₁	semidirect
331. #533	C₂²×C₃₁	semidirect
332. C₂×#125	C₂²×C₃₁	semidirect
345. #479	C₂²×C₃₁	semidirect
346. #479	C₂²×C₃₁	semidirect
347. #479	C₂²×C₃₁	semidirect
348. #478	C₂²×C₃₁	semidirect
349. #478	C₂²×C₃₁	semidirect
350. #478	C₂²×C₃₁	semidirect
351. #478	C₂²×C₃₁	semidirect
352. #478	C₂²×C₃₁	semidirect
353. #478	C₂²×C₃₁	semidirect
354. #477	C₂²×C₃₁	semidirect
355. #476	C₂²×C₃₁	semidirect
356. #476	C₂²×C₃₁	semidirect
357. #475	C₂²×C₃₁	semidirect
358. #474	C₂²×C₃₁	semidirect
359. #474	C₂²×C₃₁	semidirect
360. #428	C₂²×C₃₁	semidirect
361. #428	C₂²×C₃₁	semidirect
362. #428	C₂²×C₃₁	semidirect
363. #428	C₂²×C₃₁	semidirect
364. #427	C₂²×C₃₁	semidirect
365. #427	C₂²×C₃₁	semidirect
366. C₂×#107	C₂²×C₃₁	semidirect
367. C₂×#106	C₂²×C₃₁	semidirect
368. #424	C₂²×C₃₁	semidirect
369. #424	C₂²×C₃₁	semidirect
370. #423	C₂²×C₃₁	semidirect
371. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
386. H(5,𝔽₂)	Q₈×C₃₁	semidirect
479. C₃₁	#60496	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3504	2
C₄×C₃₁×H(5,𝔽₂)	2
C₃₁×#3084	2
C₃₁×#3083	2
C₃₁×#3081	2
C₃₁×#3080	2
C₃₁×#3077	2
C₃₁×#3076	2
C₂×C₃₁×#478	2
C₃₁×#2913	2
C₃₁×#2912	2
C₃₁×#2906	2
C₃₁×#2897	2
#60496	31

Subgroups — C31×#60496 #21517255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60496 #21517255