Subgroups — C₃₁×#3558 #16843491

Order 3968.

1160 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#562	C₂	1	2	yes	maximal, complemented
2	C₃₁×H(3,𝔽₄)	C₂	1	2	yes	maximal, complemented
3	C₃₁×#559	C₂	1	2	yes	maximal, complemented
4	C₃₁×#548	C₂	1	2	yes	maximal, complemented
5	C₃₁×#546	C₂	1	2	yes	maximal, complemented
6	C₃₁×#546	C₂	1	4	yes	maximal, complemented
7	D₄²×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#542	C₂	1	2	yes	maximal, complemented
9	C₃₁×#538	C₂	1	2	yes	maximal, complemented
10	C₃₁×#538	C₂	1	4	yes	maximal, complemented
11	C₃₁×#535	C₂	1	2	yes	maximal, complemented
12	C₃₁×#534	C₂	1	2	yes	maximal, complemented
13	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
17	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
18	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₃₁	C₂²	1	4	yes	complemented
21	C₂²×D₄×C₃₁	C₂²	1	2	yes	complemented
22	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#127	C₂²	1	2	yes	—
24	C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#126	C₂²	1	2	yes	complemented
26	C₃₁×#126	C₂²	1	4	yes	complemented
27	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#124	C₂²	1	2	yes	complemented
29	C₃₁×#124	C₂²	1	2	yes	complemented
30	C₃₁×#124	C₂²	1	2	yes	complemented
31	C₃₁×#124	C₂²	1	2	yes	complemented
32	C₃₁×#124	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#124	C₂²	1	2	yes	complemented
34	C₃₁×#124	C₂²	1	2	yes	complemented
35	C₃₁×#124	C₂²	1	4	yes	complemented
36	C₃₁×#124	C₂²	1	4	yes	complemented
37	C₃₁×#124	C₂²	1	4	yes	complemented
38	C₃₁×#123	C₂²	1	2	yes	complemented
39	C₃₁×#123	C₂²	1	4	yes	complemented
40	C₃₁×#123	C₂²	1	4	yes	complemented
41	C₃₁×#123	C₂²	1	4	yes	complemented
42	C₃₁×#123	C₂²	1	4	yes	complemented
43	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
45	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
46	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
47	C₃₁×#121	C₂²	1	1	yes	characteristic
48	C₃₁×#121	C₂²	1	2	yes	—
49	C₃₁×#121	C₂²	1	2	yes	complemented
50	C₃₁×#121	C₂²	1	2	yes	complemented
51	C₃₁×#121	C₂²	1	4	yes	complemented
52	C₃₁×#121	C₂²	1	4	yes	complemented
53	C₃₁×#121	C₂²	1	4	yes	complemented
54	C₃₁×#121	C₂²	1	4	yes	complemented
55	C₃₁×#121	C₂²	1	4	yes	complemented
56	C₃₁×#120	C₂²	1	4	yes	complemented
57	C₃₁×#120	C₂²	1	4	yes	complemented
58	C₃₁×#120	C₂²	1	4	yes	complemented
59	C₃₁×#120	C₂²	1	4	yes	complemented
60	C₃₁×#120	C₂²	1	4	yes	complemented
61	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
62	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
63	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
64	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
65	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
66	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
67	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
68	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
69	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
70	C₃₁×#117	C₂²	1	2	yes	complemented
71	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
72	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
73	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
74	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
75	C₂⁴×C₃₁	C₂³	1	2	yes	—
76	C₂⁴×C₃₁	C₂³	1	4	yes	—
77	#41×C₃₁	—	4	4	no	—
78	#41×C₃₁	—	4	4	no	—
79	C₂×Q₈×C₃₁	—	4	2	no	—
80	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
83	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	—	4	4	no	—
86	C₂×D₄×C₃₁	—	4	8	no	—
87	C₂×D₄×C₃₁	C₂³	1	4	yes	—
88	C₂×D₄×C₃₁	C₂³	1	2	yes	—
89	C₂×D₄×C₃₁	C₂³	1	2	yes	—
90	C₂×D₄×C₃₁	C₂³	1	2	yes	—
91	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₃₁	C₂³	1	4	yes	—
93	C₂×D₄×C₃₁	C₂³	1	2	yes	—
94	C₂×D₄×C₃₁	C₂³	1	4	yes	—
95	C₂×D₄×C₃₁	C₂³	1	4	yes	—
96	C₂×D₄×C₃₁	C₂³	1	2	yes	—
97	C₂×D₄×C₃₁	C₂³	1	2	yes	—
98	C₂×D₄×C₃₁	C₂³	1	4	yes	—
99	C₂×D₄×C₃₁	C₂³	1	4	yes	—
100	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
106	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
107	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
108	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
116	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
117	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
118	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
124	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
129	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
131	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₄²×C₃₁	C₂³	1	1	yes	characteristic
135	C₄²×C₃₁	C₂³	1	1	yes	characteristic
136	C₄²×C₃₁	C₂³	1	1	yes	characteristic
137	C₄²×C₃₁	C₂³	1	1	yes	characteristic
138	C₄²×C₃₁	C₂³	1	2	yes	—
139	C₄²×C₃₁	C₂³	1	4	yes	—
140	C₂³×C₃₁	—	2	4	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	4	4	no	—
143	C₂³×C₃₁	—	4	4	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	4	no	—
146	C₂³×C₃₁	C₂⁴	1	2	yes	—
147	C₂³×C₃₁	C₂⁴	1	4	yes	—
148	C₂³×C₃₁	C₂⁴	1	4	yes	—
149	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
150	Q₈×C₃₁	—	2	4	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	1	no	—
153	Q₈×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	2	4	no	—
155	D₄×C₃₁	—	2	8	no	—
156	D₄×C₃₁	—	2	4	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	4	4	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	4	no	—
161	D₄×C₃₁	—	4	4	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	1	no	—
164	D₄×C₃₁	—	4	4	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	4	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	4	no	—
174	C₂×C₄×C₃₁	—	4	4	no	—
175	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
182	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	#3558	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	4	no	—
191	C₂²×C₃₁	—	2	4	no	—
192	C₂²×C₃₁	—	2	4	no	—
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	2	4	no	—
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	2	4	no	—
198	C₂²×C₃₁	—	8	4	no	—
199	C₂²×C₃₁	—	8	8	no	—
200	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
201	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
202	C₄×C₃₁	—	2	1	no	—
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	4	no	—
208	C₄×C₃₁	—	2	4	no	—
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	2	1	no	—
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
213	#562	C₂×C₃₁	1	2	yes	complemented
214	H(3,𝔽₄)	C₂×C₃₁	1	2	yes	complemented
215	#559	C₂×C₃₁	1	2	yes	complemented
216	#548	C₂×C₃₁	1	2	yes	complemented
217	#546	C₂×C₃₁	1	2	yes	complemented
218	#546	C₂×C₃₁	1	4	yes	complemented
219	D₄²	C₂×C₃₁	1	2	yes	complemented
220	#542	C₂×C₃₁	1	2	yes	complemented
221	#538	C₂×C₃₁	1	2	yes	complemented
222	#538	C₂×C₃₁	1	4	yes	complemented
223	#535	C₂×C₃₁	1	2	yes	complemented
224	#534	C₂×C₃₁	1	2	yes	complemented
225	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂×#124	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂×C₃₁	—	2	2	no	—
229	C₂×C₃₁	—	4	4	no	—
230	C₂×C₃₁	—	4	2	no	—
231	C₂×C₃₁	—	4	4	no	—
232	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₃₁	#585	1	1	yes	characteristic
235	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
236	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
237	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
239	C₂²×D₄	C₂²×C₃₁	1	4	yes	complemented
240	C₂²×D₄	C₂²×C₃₁	1	2	yes	complemented
241	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
242	#127	C₂²×C₃₁	1	2	yes	—
243	#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
244	#126	C₂²×C₃₁	1	2	yes	complemented
245	#126	C₂²×C₃₁	1	4	yes	complemented
246	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
247	#124	C₂²×C₃₁	1	2	yes	complemented
248	#124	C₂²×C₃₁	1	2	yes	complemented
249	#124	C₂²×C₃₁	1	2	yes	complemented
250	#124	C₂²×C₃₁	1	2	yes	complemented
251	#124	C₂²×C₃₁	1	1	yes	characteristic, complemented
252	#124	C₂²×C₃₁	1	2	yes	complemented
253	#124	C₂²×C₃₁	1	2	yes	complemented
254	#124	C₂²×C₃₁	1	4	yes	complemented
255	#124	C₂²×C₃₁	1	4	yes	complemented
256	#124	C₂²×C₃₁	1	4	yes	complemented
257	#123	C₂²×C₃₁	1	2	yes	complemented
258	#123	C₂²×C₃₁	1	4	yes	complemented
259	#123	C₂²×C₃₁	1	4	yes	complemented
260	#123	C₂²×C₃₁	1	4	yes	complemented
261	#123	C₂²×C₃₁	1	4	yes	complemented
262	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
263	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
264	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
265	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
266	#121	C₂²×C₃₁	1	1	yes	characteristic
267	#121	C₂²×C₃₁	1	2	yes	—
268	#121	C₂²×C₃₁	1	2	yes	complemented
269	#121	C₂²×C₃₁	1	2	yes	complemented
270	#121	C₂²×C₃₁	1	4	yes	complemented
271	#121	C₂²×C₃₁	1	4	yes	complemented
272	#121	C₂²×C₃₁	1	4	yes	complemented
273	#121	C₂²×C₃₁	1	4	yes	complemented
274	#121	C₂²×C₃₁	1	4	yes	complemented
275	#120	C₂²×C₃₁	1	4	yes	complemented
276	#120	C₂²×C₃₁	1	4	yes	complemented
277	#120	C₂²×C₃₁	1	4	yes	complemented
278	#120	C₂²×C₃₁	1	4	yes	complemented
279	#120	C₂²×C₃₁	1	4	yes	complemented
280	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
282	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
283	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
284	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
285	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
286	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
287	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
288	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
289	#117	C₂²×C₃₁	1	2	yes	complemented
290	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
291	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
292	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
293	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
294	C₃₁	#3558	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	C₂³×C₃₁	1	2	yes	—
296	C₂⁴	C₂³×C₃₁	1	4	yes	—
297	#41	—	4	4	no	—
298	#41	—	4	4	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
301	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
303	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	4	4	no	—
306	C₂×D₄	—	4	8	no	—
307	C₂×D₄	C₂³×C₃₁	1	4	yes	—
308	C₂×D₄	C₂³×C₃₁	1	2	yes	—
309	C₂×D₄	C₂³×C₃₁	1	2	yes	—
310	C₂×D₄	C₂³×C₃₁	1	2	yes	—
311	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×D₄	C₂³×C₃₁	1	4	yes	—
313	C₂×D₄	C₂³×C₃₁	1	2	yes	—
314	C₂×D₄	C₂³×C₃₁	1	4	yes	—
315	C₂×D₄	C₂³×C₃₁	1	4	yes	—
316	C₂×D₄	C₂³×C₃₁	1	2	yes	—
317	C₂×D₄	C₂³×C₃₁	1	2	yes	—
318	C₂×D₄	C₂³×C₃₁	1	4	yes	—
319	C₂×D₄	C₂³×C₃₁	1	4	yes	—
320	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
323	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
325	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
326	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
327	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
328	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
329	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
330	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
332	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
333	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
334	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
335	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
337	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
338	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
339	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
340	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
341	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
343	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
345	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
348	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
349	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
351	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
354	C₄²	C₂³×C₃₁	1	1	yes	characteristic
355	C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	C₄²	C₂³×C₃₁	1	1	yes	characteristic
357	C₄²	C₂³×C₃₁	1	1	yes	characteristic
358	C₄²	C₂³×C₃₁	1	2	yes	—
359	C₄²	C₂³×C₃₁	1	4	yes	—
360	C₂³	—	2	4	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	4	no	—
363	C₂³	—	4	4	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	4	no	—
366	C₂³	C₂⁴×C₃₁	1	2	yes	—
367	C₂³	C₂⁴×C₃₁	1	4	yes	—
368	C₂³	C₂⁴×C₃₁	1	4	yes	—
369	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
370	Q₈	—	2	4	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	4	1	no	—
373	Q₈	—	4	2	no	—
374	D₄	—	2	4	no	—
375	D₄	—	2	8	no	—
376	D₄	—	2	4	no	—
377	D₄	—	2	4	no	—
378	D₄	—	4	4	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	4	no	—
381	D₄	—	4	4	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	1	no	—
384	D₄	—	4	4	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	4	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	4	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	4	no	—
394	C₂×C₄	—	4	4	no	—
395	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
396	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
402	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
403	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
404	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
405	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	2	4	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	8	4	no	—
418	C₂²	—	8	8	no	—
419	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
420	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
421	C₄	—	2	1	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	1	no	—
424	C₄	—	2	1	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	4	no	—
427	C₄	—	2	4	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	2	no	—
431	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
432	C₂	—	2	2	no	—
433	C₂	—	4	2	no	—
434	C₂	—	4	4	no	—
435	C₂	—	4	4	no	—
436	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
437	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
438	C₂	C₃₁×#585	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×#562	C₂	semidirect
2. C₃₁×H(3,𝔽₄)	C₂	semidirect
3. C₃₁×#559	C₂	semidirect
4. C₃₁×#548	C₂	semidirect
5. C₃₁×#546	C₂	semidirect
6. C₃₁×#546	C₂	semidirect
7. D₄²×C₃₁	C₂	semidirect
8. C₃₁×#542	C₂	semidirect
9. C₃₁×#538	C₂	semidirect
10. C₃₁×#538	C₂	semidirect
11. C₃₁×#535	C₂	semidirect
12. C₃₁×#534	C₂	semidirect
13. C₃₁×#532	C₂	semidirect
14. C₃₁×#529	C₂	semidirect
15. C₂×C₃₁×#124	C₂	semidirect
16. C₂×#41×C₃₁	C₂²	semidirect
17. C₂×#41×C₃₁	C₂²	semidirect
18. C₂²×Q₈×C₃₁	C₂²	semidirect
20. C₂²×D₄×C₃₁	C₂²	semidirect
21. C₂²×D₄×C₃₁	C₂²	semidirect
22. C₃₁×C₄⋊Q₈	C₂²	semidirect
24. C₃₁×#127	C₂²	semidirect
25. C₃₁×#126	C₂²	semidirect
26. C₃₁×#126	C₂²	semidirect
27. C₃₁×#125	C₂²	semidirect
28. C₃₁×#124	C₂²	semidirect
29. C₃₁×#124	C₂²	semidirect
30. C₃₁×#124	C₂²	semidirect
31. C₃₁×#124	C₂²	semidirect
32. C₃₁×#124	C₂²	semidirect
33. C₃₁×#124	C₂²	semidirect
34. C₃₁×#124	C₂²	semidirect
35. C₃₁×#124	C₂²	semidirect
36. C₃₁×#124	C₂²	semidirect
37. C₃₁×#124	C₂²	semidirect
38. C₃₁×#123	C₂²	semidirect
39. C₃₁×#123	C₂²	semidirect
40. C₃₁×#123	C₂²	semidirect
41. C₃₁×#123	C₂²	semidirect
42. C₃₁×#123	C₂²	semidirect
43. C₃₁×C₂²⋊Q₈	C₂²	semidirect
44. C₃₁×C₂²⋊Q₈	C₂²	semidirect
45. C₃₁×C₂²⋊Q₈	C₂²	semidirect
46. C₃₁×C₂²⋊Q₈	C₂²	semidirect
49. C₃₁×#121	C₂²	semidirect
50. C₃₁×#121	C₂²	semidirect
51. C₃₁×#121	C₂²	semidirect
52. C₃₁×#121	C₂²	semidirect
53. C₃₁×#121	C₂²	semidirect
54. C₃₁×#121	C₂²	semidirect
55. C₃₁×#121	C₂²	semidirect
56. C₃₁×#120	C₂²	semidirect
57. C₃₁×#120	C₂²	semidirect
58. C₃₁×#120	C₂²	semidirect
59. C₃₁×#120	C₂²	semidirect
60. C₃₁×#120	C₂²	semidirect
61. C₄×Q₈×C₃₁	C₂²	semidirect
62. C₄×Q₈×C₃₁	C₂²	semidirect
63. C₄×D₄×C₃₁	C₂²	semidirect
64. C₄×D₄×C₃₁	C₂²	semidirect
65. C₄×D₄×C₃₁	C₂²	semidirect
66. C₄×D₄×C₃₁	C₂²	semidirect
67. C₄×D₄×C₃₁	C₂²	semidirect
68. C₄×D₄×C₃₁	C₂²	semidirect
69. C₄×D₄×C₃₁	C₂²	semidirect
70. C₃₁×#117	C₂²	semidirect
71. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
72. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
73. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
74. C₂×C₄²×C₃₁	C₂²	semidirect
186. #3558	C₃₁	direct (N × H)
213. #562	C₂×C₃₁	semidirect
214. H(3,𝔽₄)	C₂×C₃₁	semidirect
215. #559	C₂×C₃₁	semidirect
216. #548	C₂×C₃₁	semidirect
217. #546	C₂×C₃₁	semidirect
218. #546	C₂×C₃₁	semidirect
219. D₄²	C₂×C₃₁	semidirect
220. #542	C₂×C₃₁	semidirect
221. #538	C₂×C₃₁	semidirect
222. #538	C₂×C₃₁	semidirect
223. #535	C₂×C₃₁	semidirect
224. #534	C₂×C₃₁	semidirect
225. #532	C₂×C₃₁	semidirect
226. #529	C₂×C₃₁	semidirect
227. C₂×#124	C₂×C₃₁	semidirect
235. C₂×#41	C₂²×C₃₁	semidirect
236. C₂×#41	C₂²×C₃₁	semidirect
237. C₂²×Q₈	C₂²×C₃₁	semidirect
239. C₂²×D₄	C₂²×C₃₁	semidirect
240. C₂²×D₄	C₂²×C₃₁	semidirect
241. C₄⋊Q₈	C₂²×C₃₁	semidirect
243. #127	C₂²×C₃₁	semidirect
244. #126	C₂²×C₃₁	semidirect
245. #126	C₂²×C₃₁	semidirect
246. #125	C₂²×C₃₁	semidirect
247. #124	C₂²×C₃₁	semidirect
248. #124	C₂²×C₃₁	semidirect
249. #124	C₂²×C₃₁	semidirect
250. #124	C₂²×C₃₁	semidirect
251. #124	C₂²×C₃₁	semidirect
252. #124	C₂²×C₃₁	semidirect
253. #124	C₂²×C₃₁	semidirect
254. #124	C₂²×C₃₁	semidirect
255. #124	C₂²×C₃₁	semidirect
256. #124	C₂²×C₃₁	semidirect
257. #123	C₂²×C₃₁	semidirect
258. #123	C₂²×C₃₁	semidirect
259. #123	C₂²×C₃₁	semidirect
260. #123	C₂²×C₃₁	semidirect
261. #123	C₂²×C₃₁	semidirect
262. C₂²⋊Q₈	C₂²×C₃₁	semidirect
263. C₂²⋊Q₈	C₂²×C₃₁	semidirect
264. C₂²⋊Q₈	C₂²×C₃₁	semidirect
265. C₂²⋊Q₈	C₂²×C₃₁	semidirect
268. #121	C₂²×C₃₁	semidirect
269. #121	C₂²×C₃₁	semidirect
270. #121	C₂²×C₃₁	semidirect
271. #121	C₂²×C₃₁	semidirect
272. #121	C₂²×C₃₁	semidirect
273. #121	C₂²×C₃₁	semidirect
274. #121	C₂²×C₃₁	semidirect
275. #120	C₂²×C₃₁	semidirect
276. #120	C₂²×C₃₁	semidirect
277. #120	C₂²×C₃₁	semidirect
278. #120	C₂²×C₃₁	semidirect
279. #120	C₂²×C₃₁	semidirect
280. C₄×Q₈	C₂²×C₃₁	semidirect
281. C₄×Q₈	C₂²×C₃₁	semidirect
282. C₄×D₄	C₂²×C₃₁	semidirect
283. C₄×D₄	C₂²×C₃₁	semidirect
284. C₄×D₄	C₂²×C₃₁	semidirect
285. C₄×D₄	C₂²×C₃₁	semidirect
286. C₄×D₄	C₂²×C₃₁	semidirect
287. C₄×D₄	C₂²×C₃₁	semidirect
288. C₄×D₄	C₂²×C₃₁	semidirect
289. #117	C₂²×C₃₁	semidirect
290. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
291. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
292. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
293. C₂×C₄²	C₂²×C₃₁	semidirect
294. C₃₁	#3558	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×#562	2
C₃₁×H(3,𝔽₄)	2
C₃₁×#559	2
C₃₁×#548	2
C₃₁×#546	2
D₄²×C₃₁	2
C₃₁×#542	2
C₃₁×#538	2
C₃₁×#535	2
C₃₁×#534	2
C₃₁×#532	2
C₃₁×#529	2
C₂×C₃₁×#124	2
#3558	31

Subgroups — C31×#3558 #16843491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3558 #16843491