Subgroups — C₃₁×#60494 #21517253

Order 7936.

2090 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3083	C₂	1	6	yes	maximal, complemented
4	C₃₁×#3081	C₂	1	6	yes	maximal, complemented
5	C₃₁×#3079	C₂	1	3	yes	maximal, complemented
6	C₃₁×#3077	C₂	1	3	yes	maximal, complemented
7	C₃₁×#3076	C₂	1	3	yes	maximal, complemented
8	C₂×C₃₁×#474	C₂	1	3	yes	maximal, complemented
9	C₃₁×#2913	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2897	C₂	1	2	yes	maximal
12	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	Q₈²×C₃₁	—	2	1	no	—
15	C₃₁×#557	—	2	1	no	—
16	C₃₁×#554	—	2	3	no	—
17	D₄×Q₈×C₃₁	—	2	3	no	—
18	C₃₁×#544	C₂²	1	6	yes	complemented
19	C₃₁×#533	C₂²	1	3	yes	complemented
20	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	complemented
21	C₃₁×#520	—	2	1	no	—
22	C₃₁×#520	C₂²	1	3	yes	—
23	C₃₁×#520	C₂²	1	2	yes	—
24	C₃₁×#520	C₂²	1	1	yes	characteristic
25	C₃₁×#518	—	2	3	no	—
26	C₃₁×#518	C₂²	1	3	yes	—
27	C₄×#41×C₃₁	—	2	1	no	—
28	C₄×#41×C₃₁	C₂²	1	3	yes	—
29	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₃₁	—	2	3	no	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
32	C₃₁×#479	C₂²	1	6	yes	complemented
33	C₃₁×#478	C₂²	1	6	yes	complemented
34	C₃₁×#478	C₂²	1	12	yes	complemented
35	C₃₁×#477	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#477	C₂²	1	6	yes	complemented
37	C₃₁×#476	C₂²	1	3	yes	complemented
38	C₃₁×#476	C₂²	1	6	yes	complemented
39	C₃₁×#475	C₂²	1	3	yes	complemented
40	C₃₁×#474	C₂²	1	3	yes	complemented
41	C₃₁×#474	C₂²	1	6	yes	complemented
42	C₃₁×#474	C₂²	1	12	yes	complemented
43	C₃₁×#428	C₂²	1	6	yes	complemented
44	C₃₁×#428	C₂²	1	6	yes	complemented
45	C₃₁×#427	C₂²	1	6	yes	complemented
46	C₂×C₃₁×#107	C₂²	1	6	yes	complemented
47	C₃₁×#424	C₂²	1	6	yes	complemented
48	C₃₁×#423	C₂²	1	3	yes	complemented
49	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
50	C₃₁×#418	C₂²	1	6	yes	—
51	C₃₁×#417	C₂²	1	6	yes	—
52	C₃₁×#416	C₂²	1	6	yes	—
53	C₂×C₃₁×#102	C₂²	1	6	yes	—
54	C₃₁×H(5,𝔽₂)	—	2	4	no	—
55	C₃₁×H(5,𝔽₂)	Q₈	1	8	yes	complemented
56	C₂×#41×C₃₁	—	2	1	no	—
57	C₂×#41×C₃₁	C₂³	1	3	yes	—
58	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₃₁	—	2	3	no	—
60	C₂²×D₄×C₃₁	—	2	3	no	—
61	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
62	C₂³×C₄×C₃₁	—	2	3	no	—
63	C₃₁×#131	—	2	8	no	—
64	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
66	C₃₁×C₄⋊Q₈	—	2	6	no	—
67	C₃₁×C₄⋊Q₈	—	2	3	no	—
68	C₃₁×C₄⋊Q₈	—	2	3	no	—
69	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
71	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₃₁×#125	—	2	6	no	—
73	C₃₁×#125	C₂³	1	6	yes	—
74	C₃₁×C₂²⋊Q₈	—	2	3	no	—
75	C₃₁×C₂²⋊Q₈	—	2	3	no	—
76	C₃₁×C₂²⋊Q₈	—	2	12	no	—
77	C₄×Q₈×C₃₁	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	2	no	—
79	C₄×Q₈×C₃₁	—	2	3	no	—
80	C₄×Q₈×C₃₁	—	2	3	no	—
81	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
83	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
85	C₄×D₄×C₃₁	—	2	3	no	—
86	C₄×D₄×C₃₁	—	2	6	no	—
87	C₄×D₄×C₃₁	—	2	3	no	—
88	C₄×D₄×C₃₁	—	2	3	no	—
89	C₄×D₄×C₃₁	—	2	3	no	—
90	C₄×D₄×C₃₁	—	2	3	no	—
91	C₄×D₄×C₃₁	—	2	6	no	—
92	C₄×D₄×C₃₁	—	2	3	no	—
93	C₄×D₄×C₃₁	—	2	3	no	—
94	C₄×D₄×C₃₁	D₄	1	3	yes	—
95	C₄×D₄×C₃₁	D₄	1	3	yes	—
96	C₄×D₄×C₃₁	C₂³	1	3	yes	—
97	C₄×D₄×C₃₁	C₂³	1	6	yes	—
98	C₄×D₄×C₃₁	C₂³	1	3	yes	—
99	C₃₁×#117	—	2	3	no	—
100	C₃₁×#117	—	2	3	no	—
101	C₃₁×#117	C₂³	1	6	yes	—
102	C₃₁×#117	C₂³	1	3	yes	—
103	C₃₁×#117	C₂³	1	3	yes	—
104	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
108	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
109	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
110	C₂×C₄²×C₃₁	—	2	3	no	—
111	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
112	C₃₁×#107	C₂³	1	6	yes	—
113	C₃₁×#107	C₂³	1	12	yes	—
114	C₃₁×#107	C₂³	1	2	yes	—
115	C₃₁×#106	C₂³	1	6	yes	—
116	C₃₁×#106	C₂³	1	6	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
119	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
120	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
121	C₃₁×#103	C₂³	1	2	yes	—
122	C₃₁×#103	C₂³	1	6	yes	—
123	C₃₁×#102	C₂³	1	6	yes	—
124	C₃₁×#102	C₂³	1	12	yes	—
125	C₃₁×#102	C₂³	1	6	yes	—
126	C₂⁴×C₃₁	—	2	3	no	—
127	#41×C₃₁	—	2	8	no	—
128	#41×C₃₁	—	2	6	no	—
129	#41×C₃₁	C₂×Q₈	1	4	yes	—
130	#41×C₃₁	C₂×Q₈	1	12	yes	complemented
131	#41×C₃₁	#41	1	4	yes	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	3	no	—
134	C₂×Q₈×C₃₁	—	2	6	no	—
135	C₂×Q₈×C₃₁	—	8	6	no	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₃₁	—	2	3	no	—
139	C₂×D₄×C₃₁	—	2	24	no	—
140	C₂×D₄×C₃₁	—	2	3	no	—
141	C₂×D₄×C₃₁	—	2	6	no	—
142	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
143	C₂×D₄×C₃₁	C₂×Q₈	1	12	yes	complemented
144	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
145	C₂²×C₄×C₃₁	—	2	3	no	—
146	C₂²×C₄×C₃₁	—	2	3	no	—
147	C₂²×C₄×C₃₁	—	2	6	no	—
148	C₂²×C₄×C₃₁	—	2	3	no	—
149	C₂²×C₄×C₃₁	—	2	3	no	—
150	C₂²×C₄×C₃₁	—	2	6	no	—
151	C₂²×C₄×C₃₁	—	8	6	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
155	M₁₆×C₃₁	—	2	12	no	—
156	C₂×C₈×C₃₁	—	2	12	no	—
157	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
158	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
159	C₄⋊C₄×C₃₁	—	2	3	no	—
160	C₄⋊C₄×C₃₁	—	2	3	no	—
161	C₄⋊C₄×C₃₁	—	2	3	no	—
162	C₄⋊C₄×C₃₁	—	2	6	no	—
163	C₄⋊C₄×C₃₁	—	2	6	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	6	no	—
167	C₄⋊C₄×C₃₁	—	2	6	no	—
168	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
170	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
171	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
172	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
174	C₂²⋊C₄×C₃₁	—	2	3	no	—
175	C₂²⋊C₄×C₃₁	—	2	6	no	—
176	C₂²⋊C₄×C₃₁	—	2	3	no	—
177	C₂²⋊C₄×C₃₁	—	2	6	no	—
178	C₂²⋊C₄×C₃₁	—	2	3	no	—
179	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
180	C₄²×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	2	3	no	—
182	C₄²×C₃₁	—	2	3	no	—
183	C₄²×C₃₁	—	2	6	no	—
184	C₄²×C₃₁	C₂×D₄	1	3	yes	—
185	C₄²×C₃₁	C₂×D₄	1	3	yes	—
186	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
187	C₄²×C₃₁	C₂⁴	1	3	yes	—
188	#60494	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₃₁	—	2	3	no	—
190	C₂³×C₃₁	—	2	12	no	—
191	C₂³×C₃₁	—	8	3	no	—
192	C₂³×C₃₁	—	8	3	no	—
193	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
194	Q₈×C₃₁	—	2	1	no	—
195	Q₈×C₃₁	—	4	2	no	—
196	Q₈×C₃₁	—	4	6	no	—
197	Q₈×C₃₁	—	8	6	no	—
198	Q₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
199	Q₈×C₃₁	C₂²×Q₈	1	2	yes	—
200	D₄×C₃₁	—	2	12	no	—
201	D₄×C₃₁	—	2	12	no	—
202	D₄×C₃₁	—	2	3	no	—
203	D₄×C₃₁	C₂²⋊Q₈	1	12	yes	—
204	D₄×C₃₁	C₂²×Q₈	1	6	yes	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	12	no	—
207	C₂×C₄×C₃₁	—	2	6	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	3	no	—
211	C₂×C₄×C₃₁	—	2	6	no	—
212	C₂×C₄×C₃₁	—	4	3	no	—
213	C₂×C₄×C₃₁	—	4	6	no	—
214	C₂×C₄×C₃₁	—	8	3	no	—
215	C₂×C₄×C₃₁	—	8	6	no	—
216	C₂×C₄×C₃₁	—	8	3	no	—
217	C₂×C₄×C₃₁	—	8	6	no	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×Q₈	1	6	yes	—
224	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
225	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	2	6	no	—
228	#3503	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₄×H(5,𝔽₂)	C₂×C₃₁	1	1	yes	characteristic
230	#3083	C₂×C₃₁	1	6	yes	complemented
231	#3081	C₂×C₃₁	1	6	yes	complemented
232	#3079	C₂×C₃₁	1	3	yes	complemented
233	#3077	C₂×C₃₁	1	3	yes	complemented
234	#3076	C₂×C₃₁	1	3	yes	complemented
235	C₂×#474	C₂×C₃₁	1	3	yes	complemented
236	#2913	C₂×C₃₁	1	2	yes	complemented
237	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#2897	C₂×C₃₁	1	2	yes	—
239	C₂²×C₃₁	—	2	3	no	—
240	C₂²×C₃₁	—	2	3	no	—
241	C₂²×C₃₁	—	2	6	no	—
242	C₂²×C₃₁	—	4	3	no	—
243	C₂²×C₃₁	—	4	3	no	—
244	C₂²×C₃₁	—	8	12	no	—
245	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	6	yes	—
246	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
247	C₄×C₃₁	—	2	1	no	—
248	C₄×C₃₁	—	2	3	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	3	no	—
251	C₄×C₃₁	—	4	1	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	3	no	—
254	C₄×C₃₁	—	4	6	no	—
255	C₄×C₃₁	C₂×C₂²⋊Q₈	1	6	yes	—
256	C₄×C₃₁	#576	1	2	yes	—
257	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
258	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
259	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#131	C₂²×C₃₁	1	2	yes	—
261	Q₈²	—	2	1	no	—
262	#557	—	2	1	no	—
263	#554	—	2	3	no	—
264	D₄×Q₈	—	2	3	no	—
265	#544	C₂²×C₃₁	1	6	yes	complemented
266	#533	C₂²×C₃₁	1	3	yes	complemented
267	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	complemented
268	#520	—	2	1	no	—
269	#520	C₂²×C₃₁	1	3	yes	—
270	#520	C₂²×C₃₁	1	2	yes	—
271	#520	C₂²×C₃₁	1	1	yes	characteristic
272	#518	—	2	3	no	—
273	#518	C₂²×C₃₁	1	3	yes	—
274	C₄×#41	—	2	1	no	—
275	C₄×#41	C₂²×C₃₁	1	3	yes	—
276	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄×D₄	—	2	3	no	—
278	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
279	#479	C₂²×C₃₁	1	6	yes	complemented
280	#478	C₂²×C₃₁	1	6	yes	complemented
281	#478	C₂²×C₃₁	1	12	yes	complemented
282	#477	C₂²×C₃₁	1	1	yes	characteristic, complemented
283	#477	C₂²×C₃₁	1	6	yes	complemented
284	#476	C₂²×C₃₁	1	3	yes	complemented
285	#476	C₂²×C₃₁	1	6	yes	complemented
286	#475	C₂²×C₃₁	1	3	yes	complemented
287	#474	C₂²×C₃₁	1	3	yes	complemented
288	#474	C₂²×C₃₁	1	6	yes	complemented
289	#474	C₂²×C₃₁	1	12	yes	complemented
290	#428	C₂²×C₃₁	1	6	yes	complemented
291	#428	C₂²×C₃₁	1	6	yes	complemented
292	#427	C₂²×C₃₁	1	6	yes	complemented
293	C₂×#107	C₂²×C₃₁	1	6	yes	complemented
294	#424	C₂²×C₃₁	1	6	yes	complemented
295	#423	C₂²×C₃₁	1	3	yes	complemented
296	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
297	#418	C₂²×C₃₁	1	6	yes	—
298	#417	C₂²×C₃₁	1	6	yes	—
299	#416	C₂²×C₃₁	1	6	yes	—
300	C₂×#102	C₂²×C₃₁	1	6	yes	—
301	C₂×C₃₁	—	2	6	no	—
302	C₂×C₃₁	—	4	6	no	—
303	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
305	C₂×C₃₁	#3586	1	1	yes	characteristic
306	H(5,𝔽₂)	—	2	4	no	—
307	H(5,𝔽₂)	Q₈×C₃₁	1	8	yes	complemented
308	C₂×#41	—	2	1	no	—
309	C₂×#41	C₂³×C₃₁	1	3	yes	—
310	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×Q₈	—	2	3	no	—
312	C₂²×D₄	—	2	3	no	—
313	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
314	C₂³×C₄	—	2	3	no	—
315	#131	—	2	8	no	—
316	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
317	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
318	C₄⋊Q₈	—	2	6	no	—
319	C₄⋊Q₈	—	2	3	no	—
320	C₄⋊Q₈	—	2	3	no	—
321	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
322	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
323	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
324	#125	—	2	6	no	—
325	#125	C₂³×C₃₁	1	6	yes	—
326	C₂²⋊Q₈	—	2	3	no	—
327	C₂²⋊Q₈	—	2	3	no	—
328	C₂²⋊Q₈	—	2	12	no	—
329	C₄×Q₈	—	2	1	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	—	2	3	no	—
332	C₄×Q₈	—	2	3	no	—
333	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
334	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
335	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
337	C₄×D₄	—	2	3	no	—
338	C₄×D₄	—	2	6	no	—
339	C₄×D₄	—	2	3	no	—
340	C₄×D₄	—	2	3	no	—
341	C₄×D₄	—	2	3	no	—
342	C₄×D₄	—	2	3	no	—
343	C₄×D₄	—	2	6	no	—
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	D₄×C₃₁	1	3	yes	—
347	C₄×D₄	D₄×C₃₁	1	3	yes	—
348	C₄×D₄	C₂³×C₃₁	1	3	yes	—
349	C₄×D₄	C₂³×C₃₁	1	6	yes	—
350	C₄×D₄	C₂³×C₃₁	1	3	yes	—
351	#117	—	2	3	no	—
352	#117	—	2	3	no	—
353	#117	C₂³×C₃₁	1	3	yes	—
354	#117	C₂³×C₃₁	1	6	yes	—
355	#117	C₂³×C₃₁	1	3	yes	—
356	C₂×C₄⋊C₄	—	2	3	no	—
357	C₂×C₄⋊C₄	—	2	3	no	—
358	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
360	C₂×C₂²⋊C₄	—	2	3	no	—
361	C₂×C₂²⋊C₄	—	2	6	no	—
362	C₂×C₄²	—	2	3	no	—
363	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
364	#107	C₂³×C₃₁	1	6	yes	—
365	#107	C₂³×C₃₁	1	12	yes	—
366	#107	C₂³×C₃₁	1	2	yes	—
367	#106	C₂³×C₃₁	1	6	yes	—
368	#106	C₂³×C₃₁	1	6	yes	—
369	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
371	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
372	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
373	#103	C₂³×C₃₁	1	2	yes	—
374	#103	C₂³×C₃₁	1	6	yes	—
375	#102	C₂³×C₃₁	1	6	yes	—
376	#102	C₂³×C₃₁	1	12	yes	—
377	#102	C₂³×C₃₁	1	6	yes	—
378	C₃₁	#60494	1	1	yes	characteristic, direct product factor, Sylow
379	C₂⁴	—	2	3	no	—
380	#41	—	2	8	no	—
381	#41	—	2	6	no	—
382	#41	C₂×Q₈×C₃₁	1	4	yes	—
383	#41	C₂×Q₈×C₃₁	1	12	yes	complemented
384	#41	#41×C₃₁	1	4	yes	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	2	3	no	—
387	C₂×Q₈	—	2	6	no	—
388	C₂×Q₈	—	8	6	no	—
389	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
391	C₂×D₄	—	2	3	no	—
392	C₂×D₄	—	2	24	no	—
393	C₂×D₄	—	2	3	no	—
394	C₂×D₄	—	2	6	no	—
395	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
396	C₂×D₄	C₂×Q₈×C₃₁	1	12	yes	complemented
397	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
398	C₂²×C₄	—	2	3	no	—
399	C₂²×C₄	—	2	3	no	—
400	C₂²×C₄	—	2	6	no	—
401	C₂²×C₄	—	2	3	no	—
402	C₂²×C₄	—	2	3	no	—
403	C₂²×C₄	—	2	6	no	—
404	C₂²×C₄	—	8	6	no	—
405	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
406	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
408	M₁₆	—	2	12	no	—
409	C₂×C₈	—	2	12	no	—
410	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
411	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
412	C₄⋊C₄	—	2	3	no	—
413	C₄⋊C₄	—	2	3	no	—
414	C₄⋊C₄	—	2	3	no	—
415	C₄⋊C₄	—	2	6	no	—
416	C₄⋊C₄	—	2	6	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	6	no	—
420	C₄⋊C₄	—	2	6	no	—
421	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
422	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
423	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
424	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
425	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
427	C₂²⋊C₄	—	2	3	no	—
428	C₂²⋊C₄	—	2	6	no	—
429	C₂²⋊C₄	—	2	3	no	—
430	C₂²⋊C₄	—	2	6	no	—
431	C₂²⋊C₄	—	2	3	no	—
432	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	3	no	—
435	C₄²	—	2	3	no	—
436	C₄²	—	2	6	no	—
437	C₄²	C₂×D₄×C₃₁	1	3	yes	—
438	C₄²	C₂×D₄×C₃₁	1	3	yes	—
439	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
440	C₄²	C₂⁴×C₃₁	1	3	yes	—
441	C₂³	—	2	3	no	—
442	C₂³	—	2	12	no	—
443	C₂³	—	8	3	no	—
444	C₂³	—	8	3	no	—
445	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
446	Q₈	—	2	1	no	—
447	Q₈	—	4	2	no	—
448	Q₈	—	4	6	no	—
449	Q₈	—	8	6	no	—
450	Q₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
451	Q₈	C₂²×Q₈×C₃₁	1	2	yes	—
452	D₄	—	2	12	no	—
453	D₄	—	2	12	no	—
454	D₄	—	2	3	no	—
455	D₄	C₃₁×C₂²⋊Q₈	1	12	yes	—
456	D₄	C₂²×Q₈×C₃₁	1	6	yes	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	12	no	—
459	C₂×C₄	—	2	6	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	3	no	—
463	C₂×C₄	—	2	6	no	—
464	C₂×C₄	—	4	3	no	—
465	C₂×C₄	—	4	6	no	—
466	C₂×C₄	—	8	3	no	—
467	C₂×C₄	—	8	6	no	—
468	C₂×C₄	—	8	3	no	—
469	C₂×C₄	—	8	6	no	—
470	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
471	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
473	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
474	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
475	C₂×C₄	C₂²×Q₈×C₃₁	1	6	yes	—
476	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
477	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
478	C₈	—	2	2	no	—
479	C₈	—	2	6	no	—
480	C₂²	—	2	3	no	—
481	C₂²	—	2	3	no	—
482	C₂²	—	2	6	no	—
483	C₂²	—	4	3	no	—
484	C₂²	—	4	3	no	—
485	C₂²	—	8	12	no	—
486	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	6	yes	—
487	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
488	C₄	—	2	2	no	—
489	C₄	—	2	3	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	3	no	—
492	C₄	—	4	3	no	—
493	C₄	—	4	6	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	2	no	—
496	C₄	C₂×C₃₁×C₂²⋊Q₈	1	6	yes	—
497	C₄	C₃₁×#576	1	2	yes	—
498	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
499	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
500	C₂	—	2	6	no	—
501	C₂	—	4	6	no	—
502	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
503	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
504	C₂	C₃₁×#3586	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×#3503	C₂	semidirect
3. C₃₁×#3083	C₂	semidirect
4. C₃₁×#3081	C₂	semidirect
5. C₃₁×#3079	C₂	semidirect
6. C₃₁×#3077	C₂	semidirect
7. C₃₁×#3076	C₂	semidirect
8. C₂×C₃₁×#474	C₂	semidirect
9. C₃₁×#2913	C₂	semidirect
10. C₃₁×#2906	C₂	semidirect
18. C₃₁×#544	C₂²	semidirect
19. C₃₁×#533	C₂²	semidirect
20. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
32. C₃₁×#479	C₂²	semidirect
33. C₃₁×#478	C₂²	semidirect
34. C₃₁×#478	C₂²	semidirect
35. C₃₁×#477	C₂²	semidirect
36. C₃₁×#477	C₂²	semidirect
37. C₃₁×#476	C₂²	semidirect
38. C₃₁×#476	C₂²	semidirect
39. C₃₁×#475	C₂²	semidirect
40. C₃₁×#474	C₂²	semidirect
41. C₃₁×#474	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
43. C₃₁×#428	C₂²	semidirect
44. C₃₁×#428	C₂²	semidirect
45. C₃₁×#427	C₂²	semidirect
46. C₂×C₃₁×#107	C₂²	semidirect
47. C₃₁×#424	C₂²	semidirect
48. C₃₁×#423	C₂²	semidirect
49. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
55. C₃₁×H(5,𝔽₂)	Q₈	semidirect
130. #41×C₃₁	C₂×Q₈	semidirect
143. C₂×D₄×C₃₁	C₂×Q₈	semidirect
188. #60494	C₃₁	direct (N × H)
228. #3503	C₂×C₃₁	semidirect
230. #3083	C₂×C₃₁	semidirect
231. #3081	C₂×C₃₁	semidirect
232. #3079	C₂×C₃₁	semidirect
233. #3077	C₂×C₃₁	semidirect
234. #3076	C₂×C₃₁	semidirect
235. C₂×#474	C₂×C₃₁	semidirect
236. #2913	C₂×C₃₁	semidirect
237. #2906	C₂×C₃₁	semidirect
265. #544	C₂²×C₃₁	semidirect
266. #533	C₂²×C₃₁	semidirect
267. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
279. #479	C₂²×C₃₁	semidirect
280. #478	C₂²×C₃₁	semidirect
281. #478	C₂²×C₃₁	semidirect
282. #477	C₂²×C₃₁	semidirect
283. #477	C₂²×C₃₁	semidirect
284. #476	C₂²×C₃₁	semidirect
285. #476	C₂²×C₃₁	semidirect
286. #475	C₂²×C₃₁	semidirect
287. #474	C₂²×C₃₁	semidirect
288. #474	C₂²×C₃₁	semidirect
289. #474	C₂²×C₃₁	semidirect
290. #428	C₂²×C₃₁	semidirect
291. #428	C₂²×C₃₁	semidirect
292. #427	C₂²×C₃₁	semidirect
293. C₂×#107	C₂²×C₃₁	semidirect
294. #424	C₂²×C₃₁	semidirect
295. #423	C₂²×C₃₁	semidirect
296. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
307. H(5,𝔽₂)	Q₈×C₃₁	semidirect
378. C₃₁	#60494	direct (N × H)
383. #41	C₂×Q₈×C₃₁	semidirect
396. C₂×D₄	C₂×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3503	2
C₄×C₃₁×H(5,𝔽₂)	2
C₃₁×#3083	2
C₃₁×#3081	2
C₃₁×#3079	2
C₃₁×#3077	2
C₃₁×#3076	2
C₂×C₃₁×#474	2
C₃₁×#2913	2
C₃₁×#2906	2
C₃₁×#2897	2
#60494	31

Subgroups — C31×#60494 #21517253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60494 #21517253