Subgroups — C₃₁×#3520 #16843453

Order 3968.

1124 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#554	C₂	1	2	yes	maximal, complemented
3	C₃₁×#552	C₂	1	2	yes	maximal, complemented
4	C₃₁×#551	C₂	1	4	yes	maximal, complemented
5	D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#548	C₂	1	2	yes	maximal, complemented
7	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#546	C₂	1	2	yes	maximal, complemented
10	C₃₁×#542	C₂	1	2	yes	maximal, complemented
11	C₃₁×#539	C₂	1	4	yes	maximal, complemented
12	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#525	C₂	1	2	yes	maximal, complemented
15	C₂×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
16	C₃₁×#520	C₂	1	2	yes	maximal, complemented
17	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₃₁	C₂²	1	2	yes	—
19	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
20	C₂²×Q₈×C₃₁	C₂²	1	2	yes	complemented
21	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
26	C₃₁×#126	C₂²	1	2	yes	complemented
27	C₃₁×#126	C₂²	1	4	yes	complemented
28	C₃₁×#126	C₂²	1	4	yes	complemented
29	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#125	C₂²	1	2	yes	complemented
31	C₃₁×#125	C₂²	1	4	yes	complemented
32	C₃₁×#124	C₂²	1	2	yes	—
33	C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#124	C₂²	1	4	yes	complemented
35	C₃₁×#123	C₂²	1	2	yes	—
36	C₃₁×#123	C₂²	1	4	yes	—
37	C₃₁×#123	C₂²	1	4	yes	complemented
38	C₃₁×#123	C₂²	1	4	yes	complemented
39	C₃₁×#123	C₂²	1	4	yes	complemented
40	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
41	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
42	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
43	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
44	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
45	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
46	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
47	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
48	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
49	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
50	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
51	C₃₁×#121	C₂²	1	1	yes	characteristic
52	C₃₁×#121	C₂²	1	4	yes	—
53	C₃₁×#121	C₂²	1	2	yes	—
54	C₃₁×#121	C₂²	1	2	yes	complemented
55	C₃₁×#120	C₂²	1	4	yes	—
56	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
57	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
58	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
59	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
60	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
61	C₄×D₄×C₃₁	C₂²	1	2	yes	—
62	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
63	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
64	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
65	C₄×D₄×C₃₁	C₂²	1	2	yes	—
66	C₄×D₄×C₃₁	C₂²	1	2	yes	—
67	C₄×D₄×C₃₁	C₂²	1	4	yes	—
68	C₄×D₄×C₃₁	C₂²	1	2	yes	—
69	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
70	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
71	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
72	C₃₁×#117	C₂²	1	2	yes	—
73	C₃₁×#117	C₂²	1	2	yes	complemented
74	C₃₁×#117	C₂²	1	4	yes	complemented
75	C₃₁×#117	C₂²	1	2	yes	complemented
76	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
78	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
79	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
82	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
83	C₂⁴×C₃₁	C₂³	1	2	yes	—
84	#41×C₃₁	—	2	8	no	—
85	#41×C₃₁	—	4	4	no	—
86	C₂×Q₈×C₃₁	—	4	4	no	—
87	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
88	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
90	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	C₂³	1	2	yes	—
93	C₂×D₄×C₃₁	C₂³	1	2	yes	—
94	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₃₁	C₂³	1	4	yes	—
96	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₃₁	C₂³	1	4	yes	—
98	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	—	4	4	no	—
100	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
104	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
109	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
111	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
115	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
118	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
128	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
130	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
136	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
137	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
138	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
139	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
140	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
142	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
143	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
144	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
145	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
146	C₄²×C₃₁	C₂³	1	1	yes	characteristic
147	C₄²×C₃₁	C₂³	1	1	yes	characteristic
148	C₄²×C₃₁	C₂³	1	1	yes	characteristic
149	C₄²×C₃₁	C₂³	1	1	yes	characteristic
150	C₄²×C₃₁	C₂³	1	4	yes	—
151	C₄²×C₃₁	C₂³	1	4	yes	—
152	C₄²×C₃₁	C₂³	1	2	yes	—
153	C₂³×C₃₁	—	2	4	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	C₂⁴	1	2	yes	—
157	C₂³×C₃₁	C₂⁴	1	2	yes	—
158	C₂³×C₃₁	C₂⁴	1	2	yes	—
159	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	4	no	—
161	Q₈×C₃₁	—	2	2	no	—
162	Q₈×C₃₁	—	2	4	no	—
163	Q₈×C₃₁	—	4	4	no	—
164	D₄×C₃₁	—	2	4	no	—
165	D₄×C₃₁	—	2	8	no	—
166	D₄×C₃₁	—	2	4	no	—
167	D₄×C₃₁	—	2	2	no	—
168	D₄×C₃₁	—	2	2	no	—
169	D₄×C₃₁	—	4	4	no	—
170	D₄×C₃₁	#41	1	4	yes	complemented
171	C₂×C₄×C₃₁	—	2	8	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	4	no	—
182	C₂×C₄×C₃₁	—	4	4	no	—
183	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
187	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
191	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
192	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
193	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
194	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
196	#3520	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	2	2	no	—
207	C₂²×C₃₁	—	4	8	no	—
208	C₂²×C₃₁	C₂×#41	1	4	yes	—
209	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
210	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	—	2	1	no	—
213	C₄×C₃₁	—	2	4	no	—
214	C₄×C₃₁	—	2	1	no	—
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	2	4	no	—
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	2	2	no	—
222	C₄×C₃₁	C₂×#41	1	2	yes	—
223	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
224	#556	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#554	C₂×C₃₁	1	2	yes	complemented
226	#552	C₂×C₃₁	1	2	yes	complemented
227	#551	C₂×C₃₁	1	4	yes	complemented
228	D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#548	C₂×C₃₁	1	2	yes	complemented
230	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#546	C₂×C₃₁	1	2	yes	complemented
233	#542	C₂×C₃₁	1	2	yes	complemented
234	#539	C₂×C₃₁	1	4	yes	complemented
235	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#525	C₂×C₃₁	1	2	yes	complemented
238	C₂×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	complemented
239	#520	C₂×C₃₁	1	2	yes	complemented
240	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×C₃₁	—	2	2	no	—
242	C₂×C₃₁	—	2	4	no	—
243	C₂×C₃₁	—	4	2	no	—
244	C₂×C₃₁	—	4	2	no	—
245	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
246	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
248	C₂×#41	C₂²×C₃₁	1	2	yes	—
249	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
250	C₂²×Q₈	C₂²×C₃₁	1	2	yes	complemented
251	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
252	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
253	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
254	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
255	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
256	#126	C₂²×C₃₁	1	2	yes	complemented
257	#126	C₂²×C₃₁	1	4	yes	complemented
258	#126	C₂²×C₃₁	1	4	yes	complemented
259	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
260	#125	C₂²×C₃₁	1	2	yes	complemented
261	#125	C₂²×C₃₁	1	4	yes	complemented
262	#124	C₂²×C₃₁	1	2	yes	—
263	#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
264	#124	C₂²×C₃₁	1	4	yes	complemented
265	#123	C₂²×C₃₁	1	2	yes	—
266	#123	C₂²×C₃₁	1	4	yes	—
267	#123	C₂²×C₃₁	1	4	yes	complemented
268	#123	C₂²×C₃₁	1	4	yes	complemented
269	#123	C₂²×C₃₁	1	4	yes	complemented
270	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
272	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
273	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
274	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
275	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
276	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
277	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
278	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
279	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
280	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
281	#121	C₂²×C₃₁	1	1	yes	characteristic
282	#121	C₂²×C₃₁	1	4	yes	—
283	#121	C₂²×C₃₁	1	2	yes	—
284	#121	C₂²×C₃₁	1	2	yes	complemented
285	#120	C₂²×C₃₁	1	4	yes	—
286	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
287	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
288	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
289	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
290	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
291	C₄×D₄	C₂²×C₃₁	1	2	yes	—
292	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
293	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
294	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
295	C₄×D₄	C₂²×C₃₁	1	2	yes	—
296	C₄×D₄	C₂²×C₃₁	1	2	yes	—
297	C₄×D₄	C₂²×C₃₁	1	4	yes	—
298	C₄×D₄	C₂²×C₃₁	1	2	yes	—
299	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
300	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
301	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
302	#117	C₂²×C₃₁	1	2	yes	—
303	#117	C₂²×C₃₁	1	2	yes	complemented
304	#117	C₂²×C₃₁	1	4	yes	complemented
305	#117	C₂²×C₃₁	1	2	yes	complemented
306	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
308	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
309	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
310	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
311	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
312	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
313	C₃₁	#3520	1	1	yes	characteristic, direct product factor, Sylow
314	C₂⁴	C₂³×C₃₁	1	2	yes	—
315	#41	—	2	8	no	—
316	#41	—	4	4	no	—
317	C₂×Q₈	—	4	4	no	—
318	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
319	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
321	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	C₂³×C₃₁	1	2	yes	—
324	C₂×D₄	C₂³×C₃₁	1	2	yes	—
325	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×D₄	C₂³×C₃₁	1	4	yes	—
327	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×D₄	C₂³×C₃₁	1	4	yes	—
329	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
335	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
337	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
338	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
339	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
340	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
341	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
342	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
343	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
346	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
348	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
349	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
351	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
354	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
355	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
356	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
357	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
359	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
360	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
361	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
362	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
363	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
364	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
365	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
366	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
368	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
371	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
372	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
374	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
375	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
376	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₄²	C₂³×C₃₁	1	1	yes	characteristic
378	C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	C₄²	C₂³×C₃₁	1	1	yes	characteristic
380	C₄²	C₂³×C₃₁	1	1	yes	characteristic
381	C₄²	C₂³×C₃₁	1	4	yes	—
382	C₄²	C₂³×C₃₁	1	4	yes	—
383	C₄²	C₂³×C₃₁	1	2	yes	—
384	C₂³	—	2	4	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	4	2	no	—
387	C₂³	C₂⁴×C₃₁	1	2	yes	—
388	C₂³	C₂⁴×C₃₁	1	2	yes	—
389	C₂³	C₂⁴×C₃₁	1	2	yes	—
390	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
391	Q₈	—	2	4	no	—
392	Q₈	—	2	2	no	—
393	Q₈	—	2	4	no	—
394	Q₈	—	4	4	no	—
395	D₄	—	2	4	no	—
396	D₄	—	2	8	no	—
397	D₄	—	2	4	no	—
398	D₄	—	2	2	no	—
399	D₄	—	2	2	no	—
400	D₄	—	4	4	no	—
401	D₄	#41×C₃₁	1	4	yes	complemented
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	4	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
416	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
418	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
421	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
422	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
423	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
424	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
425	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₂²	—	2	1	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	4	8	no	—
438	C₂²	C₂×#41×C₃₁	1	4	yes	—
439	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
440	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
441	C₄	—	2	2	no	—
442	C₄	—	2	1	no	—
443	C₄	—	2	4	no	—
444	C₄	—	2	1	no	—
445	C₄	—	2	2	no	—
446	C₄	—	2	2	no	—
447	C₄	—	2	4	no	—
448	C₄	—	2	2	no	—
449	C₄	—	2	1	no	—
450	C₄	—	2	1	no	—
451	C₄	—	2	2	no	—
452	C₄	C₂×#41×C₃₁	1	2	yes	—
453	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
454	C₂	—	2	2	no	—
455	C₂	—	2	4	no	—
456	C₂	—	4	2	no	—
457	C₂	—	4	2	no	—
458	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
459	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
460	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#556	C₂	semidirect
2. C₃₁×#554	C₂	semidirect
3. C₃₁×#552	C₂	semidirect
4. C₃₁×#551	C₂	semidirect
5. D₄×Q₈×C₃₁	C₂	semidirect
6. C₃₁×#548	C₂	semidirect
7. C₃₁×#547	C₂	semidirect
8. C₃₁×#547	C₂	semidirect
9. C₃₁×#546	C₂	semidirect
10. C₃₁×#542	C₂	semidirect
11. C₃₁×#539	C₂	semidirect
12. C₃₁×#533	C₂	semidirect
13. C₃₁×#529	C₂	semidirect
14. C₃₁×#525	C₂	semidirect
15. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
16. C₃₁×#520	C₂	semidirect
17. C₂×C₄×D₄×C₃₁	C₂	semidirect
19. C₂×#41×C₃₁	C₂²	semidirect
20. C₂²×Q₈×C₃₁	C₂²	semidirect
23. C₃₁×C₄⋊Q₈	C₂²	semidirect
24. C₃₁×C₄⋊Q₈	C₂²	semidirect
25. C₃₁×C₄⋊Q₈	C₂²	semidirect
26. C₃₁×#126	C₂²	semidirect
27. C₃₁×#126	C₂²	semidirect
28. C₃₁×#126	C₂²	semidirect
29. C₃₁×#125	C₂²	semidirect
30. C₃₁×#125	C₂²	semidirect
31. C₃₁×#125	C₂²	semidirect
33. C₃₁×#124	C₂²	semidirect
34. C₃₁×#124	C₂²	semidirect
37. C₃₁×#123	C₂²	semidirect
38. C₃₁×#123	C₂²	semidirect
39. C₃₁×#123	C₂²	semidirect
44. C₃₁×C₂²⋊Q₈	C₂²	semidirect
45. C₃₁×C₂²⋊Q₈	C₂²	semidirect
46. C₃₁×C₂²⋊Q₈	C₂²	semidirect
47. C₃₁×C₂²⋊Q₈	C₂²	semidirect
48. C₃₁×C₂²⋊Q₈	C₂²	semidirect
49. C₃₁×C₂²⋊Q₈	C₂²	semidirect
50. C₃₁×C₂²⋊Q₈	C₂²	semidirect
54. C₃₁×#121	C₂²	semidirect
56. C₄×Q₈×C₃₁	C₂²	semidirect
57. C₄×Q₈×C₃₁	C₂²	semidirect
58. C₄×Q₈×C₃₁	C₂²	semidirect
59. C₄×Q₈×C₃₁	C₂²	semidirect
69. C₄×D₄×C₃₁	C₂²	semidirect
70. C₄×D₄×C₃₁	C₂²	semidirect
71. C₄×D₄×C₃₁	C₂²	semidirect
73. C₃₁×#117	C₂²	semidirect
74. C₃₁×#117	C₂²	semidirect
75. C₃₁×#117	C₂²	semidirect
77. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
78. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
79. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
170. D₄×C₃₁	#41	semidirect
196. #3520	C₃₁	direct (N × H)
224. #556	C₂×C₃₁	semidirect
225. #554	C₂×C₃₁	semidirect
226. #552	C₂×C₃₁	semidirect
227. #551	C₂×C₃₁	semidirect
228. D₄×Q₈	C₂×C₃₁	semidirect
229. #548	C₂×C₃₁	semidirect
230. #547	C₂×C₃₁	semidirect
231. #547	C₂×C₃₁	semidirect
232. #546	C₂×C₃₁	semidirect
233. #542	C₂×C₃₁	semidirect
234. #539	C₂×C₃₁	semidirect
235. #533	C₂×C₃₁	semidirect
236. #529	C₂×C₃₁	semidirect
237. #525	C₂×C₃₁	semidirect
238. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
239. #520	C₂×C₃₁	semidirect
240. C₂×C₄×D₄	C₂×C₃₁	semidirect
249. C₂×#41	C₂²×C₃₁	semidirect
250. C₂²×Q₈	C₂²×C₃₁	semidirect
253. C₄⋊Q₈	C₂²×C₃₁	semidirect
254. C₄⋊Q₈	C₂²×C₃₁	semidirect
255. C₄⋊Q₈	C₂²×C₃₁	semidirect
256. #126	C₂²×C₃₁	semidirect
257. #126	C₂²×C₃₁	semidirect
258. #126	C₂²×C₃₁	semidirect
259. #125	C₂²×C₃₁	semidirect
260. #125	C₂²×C₃₁	semidirect
261. #125	C₂²×C₃₁	semidirect
263. #124	C₂²×C₃₁	semidirect
264. #124	C₂²×C₃₁	semidirect
267. #123	C₂²×C₃₁	semidirect
268. #123	C₂²×C₃₁	semidirect
269. #123	C₂²×C₃₁	semidirect
274. C₂²⋊Q₈	C₂²×C₃₁	semidirect
275. C₂²⋊Q₈	C₂²×C₃₁	semidirect
276. C₂²⋊Q₈	C₂²×C₃₁	semidirect
277. C₂²⋊Q₈	C₂²×C₃₁	semidirect
278. C₂²⋊Q₈	C₂²×C₃₁	semidirect
279. C₂²⋊Q₈	C₂²×C₃₁	semidirect
280. C₂²⋊Q₈	C₂²×C₃₁	semidirect
284. #121	C₂²×C₃₁	semidirect
286. C₄×Q₈	C₂²×C₃₁	semidirect
287. C₄×Q₈	C₂²×C₃₁	semidirect
288. C₄×Q₈	C₂²×C₃₁	semidirect
289. C₄×Q₈	C₂²×C₃₁	semidirect
299. C₄×D₄	C₂²×C₃₁	semidirect
300. C₄×D₄	C₂²×C₃₁	semidirect
301. C₄×D₄	C₂²×C₃₁	semidirect
303. #117	C₂²×C₃₁	semidirect
304. #117	C₂²×C₃₁	semidirect
305. #117	C₂²×C₃₁	semidirect
307. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
308. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
309. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
313. C₃₁	#3520	direct (N × H)
401. D₄	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#556	2
C₃₁×#554	2
C₃₁×#552	2
C₃₁×#551	2
D₄×Q₈×C₃₁	2
C₃₁×#548	2
C₃₁×#547	2
C₃₁×#546	2
C₃₁×#542	2
C₃₁×#539	2
C₃₁×#533	2
C₃₁×#529	2
C₃₁×#525	2
C₂×C₃₁×C₂²⋊Q₈	2
C₃₁×#520	2
C₂×C₄×D₄×C₃₁	2
#3520	31

Subgroups — C31×#3520 #16843453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3520 #16843453