Subgroups — C₃₁×#33892 #21490651

Order 7936.

620 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2226	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2218	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2184	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2140	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#555	—	2	1	no	—
14	D₄²×C₃₁	—	2	1	no	—
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
17	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
18	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#476	—	2	1	no	—
22	C₃₁×#466	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	complemented
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#447	—	2	2	no	—
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#427	C₂²	1	2	yes	complemented
30	C₃₁×#367	C₂²	1	2	yes	complemented
31	C₃₁×#366	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#357	C₂²	1	1	yes	characteristic
34	C₃₁×#357	C₂²	1	2	yes	—
35	C₃₁×#357	C₂²	1	2	yes	complemented
36	C₃₁×#357	C₂²	1	2	yes	complemented
37	C₃₁×#357	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#357	C₂²	1	2	yes	complemented
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	4	1	no	—
43	C₂×D₈×C₃₁	—	2	4	no	—
44	C₂×D₈×C₃₁	—	2	2	no	—
45	C₂×D₈×C₃₁	—	2	1	no	—
46	C₂×D₈×C₃₁	—	4	2	no	—
47	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×D₈×C₃₁	C₂³	1	2	yes	—
49	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#124	—	4	2	no	—
54	C₃₁×#121	—	2	2	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₃₁×#121	—	4	1	no	—
57	C₃₁×#121	—	4	1	no	—
58	C₃₁×#120	—	4	2	no	—
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
62	C₃₁×#117	—	2	2	no	—
63	C₃₁×#117	—	4	1	no	—
64	C₃₁×D₁₆	—	2	2	no	—
65	C₃₁×D₁₆	—	4	2	no	—
66	C₃₁×M₃₂	—	4	2	no	—
67	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	C₂³	1	1	yes	characteristic
72	C₃₁×#107	C₂³	1	2	yes	—
73	C₃₁×#107	C₂³	1	1	yes	characteristic
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₃₁×#102	—	2	1	no	—
77	C₃₁×#102	—	2	1	no	—
78	C₃₁×#102	—	2	2	no	—
79	C₃₁×#102	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂⁴×C₃₁	—	4	2	no	—
83	C₂×Q₈×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	4	4	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	1	no	—
92	C₂×D₄×C₃₁	—	4	1	no	—
93	C₂×D₄×C₃₁	—	8	2	no	—
94	C₂×D₄×C₃₁	—	8	4	no	—
95	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	—	4	1	no	—
97	C₂²×C₄×C₃₁	—	4	1	no	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
99	D₈×C₃₁	—	2	1	no	—
100	D₈×C₃₁	—	2	2	no	—
101	D₈×C₃₁	—	2	1	no	—
102	D₈×C₃₁	—	2	1	no	—
103	D₈×C₃₁	—	2	2	no	—
104	D₈×C₃₁	—	4	4	no	—
105	D₈×C₃₁	—	8	1	no	—
106	D₈×C₃₁	#41	1	2	yes	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₄⋊C₄×C₃₁	—	2	1	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	4	1	no	—
116	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	1	no	—
120	C₂²⋊C₄×C₃₁	—	4	1	no	—
121	C₂²⋊C₄×C₃₁	—	4	2	no	—
122	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
123	C₄²×C₃₁	—	4	2	no	—
124	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₃₁	—	2	1	no	—
126	C₁₆×C₃₁	—	2	1	no	—
127	C₁₆×C₃₁	—	2	2	no	—
128	#33892	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	4	1	no	—
130	C₂³×C₃₁	—	4	1	no	—
131	C₂³×C₃₁	—	4	2	no	—
132	C₂³×C₃₁	—	8	4	no	—
133	C₂³×C₃₁	—	8	4	no	—
134	C₂³×C₃₁	—	16	2	no	—
135	C₂³×C₃₁	C₂×D₈	1	2	yes	—
136	Q₈×C₃₁	—	16	1	no	—
137	D₄×C₃₁	—	2	2	no	—
138	D₄×C₃₁	—	2	2	no	—
139	D₄×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	4	1	no	—
141	D₄×C₃₁	—	4	1	no	—
142	D₄×C₃₁	—	4	2	no	—
143	D₄×C₃₁	—	4	1	no	—
144	D₄×C₃₁	—	8	4	no	—
145	D₄×C₃₁	—	8	4	no	—
146	D₄×C₃₁	—	8	2	no	—
147	D₄×C₃₁	—	16	1	no	—
148	D₄×C₃₁	—	16	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	—	4	2	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	8	2	no	—
155	C₂×C₄×C₃₁	—	8	2	no	—
156	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₃₁	—	2	2	no	—
161	C₈×C₃₁	—	2	1	no	—
162	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
163	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
164	#3341	C₂×C₃₁	1	1	yes	characteristic, complemented
165	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#2234	C₂×C₃₁	1	2	yes	complemented
168	#2226	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#2218	C₂×C₃₁	1	2	yes	complemented
170	#2206	C₂×C₃₁	1	1	yes	characteristic, complemented
171	#2184	C₂×C₃₁	1	2	yes	complemented
172	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
173	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#2140	C₂×C₃₁	1	2	yes	complemented
175	C₂²×C₃₁	—	2	2	no	—
176	C₂²×C₃₁	—	2	2	no	—
177	C₂²×C₃₁	—	2	2	no	—
178	C₂²×C₃₁	—	4	2	no	—
179	C₂²×C₃₁	—	8	1	no	—
180	C₂²×C₃₁	—	8	1	no	—
181	C₂²×C₃₁	—	8	1	no	—
182	C₂²×C₃₁	—	8	2	no	—
183	C₂²×C₃₁	—	8	1	no	—
184	C₂²×C₃₁	—	8	2	no	—
185	C₂²×C₃₁	—	8	2	no	—
186	C₂²×C₃₁	—	8	1	no	—
187	C₂²×C₃₁	—	16	8	no	—
188	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
189	C₄×C₃₁	—	2	1	no	—
190	C₄×C₃₁	—	2	2	no	—
191	C₄×C₃₁	—	8	1	no	—
192	C₄×C₃₁	—	8	1	no	—
193	C₄×C₃₁	—	8	2	no	—
194	C₄×C₃₁	#509	1	2	yes	—
195	C₄×C₃₁	#546	1	1	yes	characteristic
196	C₄×C₃₁	#577	1	1	yes	characteristic
197	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
198	#555	—	2	1	no	—
199	D₄²	—	2	1	no	—
200	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
202	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
203	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
204	#493	C₂²×C₃₁	1	1	yes	characteristic
205	#482	—	2	2	no	—
206	#476	—	2	1	no	—
207	#466	C₂²×C₃₁	1	2	yes	—
208	#466	C₂²×C₃₁	1	2	yes	complemented
209	#459	—	2	1	no	—
210	#447	—	2	2	no	—
211	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
213	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
214	#427	C₂²×C₃₁	1	2	yes	complemented
215	#367	C₂²×C₃₁	1	2	yes	complemented
216	#366	C₂²×C₃₁	1	1	yes	characteristic, complemented
217	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
218	#357	C₂²×C₃₁	1	1	yes	characteristic
219	#357	C₂²×C₃₁	1	2	yes	—
220	#357	C₂²×C₃₁	1	2	yes	complemented
221	#357	C₂²×C₃₁	1	2	yes	complemented
222	#357	C₂²×C₃₁	1	1	yes	characteristic, complemented
223	#357	C₂²×C₃₁	1	2	yes	complemented
224	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
225	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
226	C₂×C₃₁	—	4	2	no	—
227	C₂×C₃₁	—	8	2	no	—
228	C₂×C₃₁	—	16	1	no	—
229	C₂×C₃₁	—	16	2	no	—
230	C₂×C₃₁	#3294	1	1	yes	characteristic
231	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
232	C₂×C₃₁	#3415	1	1	yes	characteristic
233	C₂²×D₄	—	2	2	no	—
234	C₂²×D₄	—	4	1	no	—
235	C₂×D₈	—	2	4	no	—
236	C₂×D₈	—	2	2	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	4	2	no	—
239	C₂×D₈	C₂³×C₃₁	1	2	yes	—
240	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
243	C₄⋊Q₈	—	2	1	no	—
244	#127	—	2	1	no	—
245	#124	—	4	2	no	—
246	#121	—	2	2	no	—
247	#121	—	2	2	no	—
248	#121	—	4	1	no	—
249	#121	—	4	1	no	—
250	#120	—	4	2	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
254	#117	—	2	2	no	—
255	#117	—	4	1	no	—
256	D₁₆	—	2	2	no	—
257	D₁₆	—	4	2	no	—
258	M₃₂	—	4	2	no	—
259	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
260	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
261	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
262	#107	—	2	1	no	—
263	#107	C₂³×C₃₁	1	1	yes	characteristic
264	#107	C₂³×C₃₁	1	2	yes	—
265	#107	C₂³×C₃₁	1	1	yes	characteristic
266	#106	—	2	2	no	—
267	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
268	#102	—	2	1	no	—
269	#102	—	2	1	no	—
270	#102	—	2	2	no	—
271	#102	—	2	2	no	—
272	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
273	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₃₁	#33892	1	1	yes	characteristic, direct product factor, Sylow
275	C₂⁴	—	4	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	4	4	no	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	4	1	no	—
286	C₂×D₄	—	8	2	no	—
287	C₂×D₄	—	8	4	no	—
288	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
292	D₈	—	2	1	no	—
293	D₈	—	2	2	no	—
294	D₈	—	2	1	no	—
295	D₈	—	2	1	no	—
296	D₈	—	2	2	no	—
297	D₈	—	4	4	no	—
298	D₈	—	8	1	no	—
299	D₈	#41×C₃₁	1	2	yes	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
304	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
310	C₂²⋊C₄	—	4	2	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	1	no	—
313	C₂²⋊C₄	—	4	1	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	8	4	no	—
325	C₂³	—	8	4	no	—
326	C₂³	—	16	2	no	—
327	C₂³	C₂×D₈×C₃₁	1	2	yes	—
328	Q₈	—	16	1	no	—
329	D₄	—	2	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	1	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	1	no	—
336	D₄	—	8	4	no	—
337	D₄	—	8	4	no	—
338	D₄	—	8	2	no	—
339	D₄	—	16	1	no	—
340	D₄	—	16	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
349	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
350	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
351	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	1	no	—
354	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
355	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
356	C₂²	—	2	2	no	—
357	C₂²	—	2	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	8	1	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	1	no	—
368	C₂²	—	16	8	no	—
369	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	2	no	—
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	2	no	—
375	C₄	C₃₁×#509	1	2	yes	—
376	C₄	C₃₁×#546	1	1	yes	characteristic
377	C₄	C₃₁×#577	1	1	yes	characteristic
378	C₂	—	4	2	no	—
379	C₂	—	8	2	no	—
380	C₂	—	16	2	no	—
381	C₂	—	16	1	no	—
382	C₂	C₃₁×#3294	1	1	yes	characteristic
383	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
384	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3341	C₂	semidirect
2. D₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2240	C₂	semidirect
4. C₃₁×#2234	C₂	semidirect
5. C₃₁×#2226	C₂	semidirect
6. C₃₁×#2218	C₂	semidirect
7. C₃₁×#2206	C₂	semidirect
8. C₃₁×#2184	C₂	semidirect
9. C₄×C₃₁×D₁₆	C₂	semidirect
10. C₃₁×#2169	C₂	semidirect
11. C₃₁×#2140	C₂	semidirect
16. C₂×C₃₁×D₁₆	C₂²	semidirect
17. C₂×C₃₁×M₃₂	C₂²	semidirect
18. C₃₁×#500	C₂²	semidirect
23. C₃₁×#466	C₂²	semidirect
27. C₄×D₈×C₃₁	C₂²	semidirect
29. C₃₁×#427	C₂²	semidirect
30. C₃₁×#367	C₂²	semidirect
31. C₃₁×#366	C₂²	semidirect
32. C₃₁×C₄⋊C₁₆	C₂²	semidirect
35. C₃₁×#357	C₂²	semidirect
36. C₃₁×#357	C₂²	semidirect
37. C₃₁×#357	C₂²	semidirect
38. C₃₁×#357	C₂²	semidirect
39. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
40. C₄×C₁₆×C₃₁	C₂²	semidirect
128. #33892	C₃₁	direct (N × H)
164. #3341	C₂×C₃₁	semidirect
165. D₄×D₈	C₂×C₃₁	semidirect
166. #2240	C₂×C₃₁	semidirect
167. #2234	C₂×C₃₁	semidirect
168. #2226	C₂×C₃₁	semidirect
169. #2218	C₂×C₃₁	semidirect
170. #2206	C₂×C₃₁	semidirect
171. #2184	C₂×C₃₁	semidirect
172. C₄×D₁₆	C₂×C₃₁	semidirect
173. #2169	C₂×C₃₁	semidirect
174. #2140	C₂×C₃₁	semidirect
201. C₂×D₁₆	C₂²×C₃₁	semidirect
202. C₂×M₃₂	C₂²×C₃₁	semidirect
203. #500	C₂²×C₃₁	semidirect
208. #466	C₂²×C₃₁	semidirect
212. C₄×D₈	C₂²×C₃₁	semidirect
214. #427	C₂²×C₃₁	semidirect
215. #367	C₂²×C₃₁	semidirect
216. #366	C₂²×C₃₁	semidirect
217. C₄⋊C₁₆	C₂²×C₃₁	semidirect
220. #357	C₂²×C₃₁	semidirect
221. #357	C₂²×C₃₁	semidirect
222. #357	C₂²×C₃₁	semidirect
223. #357	C₂²×C₃₁	semidirect
224. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
225. C₄×C₁₆	C₂²×C₃₁	semidirect
274. C₃₁	#33892	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×#3341	2
D₄×D₈×C₃₁	2
C₃₁×#2240	2
C₃₁×#2234	2
C₃₁×#2226	2
C₃₁×#2218	2
C₃₁×#2206	2
C₃₁×#2184	2
C₄×C₃₁×D₁₆	2
C₃₁×#2169	2
C₃₁×#2140	2
#33892	31

Subgroups — C31×#33892 #21490651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33892 #21490651