Subgroups — C₃₁×#33921 #21490680

Order 7936.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 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	C₃₁×#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2233	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2218	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×#2159	C₂	1	2	yes	maximal, complemented
12	C₃₁×#555	—	2	1	no	—
13	C₃₁×#547	—	2	1	no	—
14	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#482	—	2	2	no	—
19	C₃₁×#480	—	2	2	no	—
20	C₃₁×#476	—	2	1	no	—
21	C₃₁×#474	—	2	1	no	—
22	C₃₁×#466	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	—
24	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#427	C₂²	1	2	yes	—
28	C₂×C₃₁×#107	C₂²	1	2	yes	—
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#367	C₂²	1	2	yes	—
31	C₃₁×#367	C₂²	1	2	yes	—
32	C₃₁×#366	C₂²	1	1	yes	characteristic
33	C₃₁×#366	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₃₁×#357	C₂²	1	1	yes	characteristic
36	C₃₁×#357	C₂²	1	1	yes	characteristic
37	C₃₁×#357	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₃₁	—	4	1	no	—
42	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×C₄⋊Q₈	—	2	1	no	—
47	C₃₁×#124	—	4	2	no	—
48	C₃₁×#123	—	4	2	no	—
49	C₃₁×C₂²⋊Q₈	—	4	1	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₃₁×#121	—	4	1	no	—
53	C₄×D₄×C₃₁	—	2	1	no	—
54	C₄×D₄×C₃₁	—	2	1	no	—
55	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
56	C₃₁×#117	—	2	2	no	—
57	C₃₁×#117	—	4	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₃₁×D₁₆	—	2	2	no	—
60	C₃₁×M₃₂	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#107	—	2	2	no	—
65	C₃₁×#107	—	2	1	no	—
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	2	yes	—
69	C₃₁×#107	C₂³	1	1	yes	characteristic
70	C₃₁×#107	C₂³	1	2	yes	—
71	C₃₁×#106	—	2	2	no	—
72	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₃₁×#102	—	2	1	no	—
74	C₃₁×#102	—	2	1	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	#41×C₃₁	—	8	4	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	—	4	2	no	—
86	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
91	D₈×C₃₁	—	2	1	no	—
92	D₈×C₃₁	—	2	1	no	—
93	D₈×C₃₁	—	2	1	no	—
94	D₈×C₃₁	—	2	1	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	1	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
114	C₄²×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₃₁	—	2	2	no	—
117	C₁₆×C₃₁	—	2	1	no	—
118	C₁₆×C₃₁	#41	1	2	yes	complemented
119	#33921	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	4	1	no	—
121	C₂³×C₃₁	—	4	1	no	—
122	C₂³×C₃₁	C₂×D₈	1	2	yes	—
123	Q₈×C₃₁	—	8	2	no	—
124	Q₈×C₃₁	—	16	1	no	—
125	D₄×C₃₁	—	2	2	no	—
126	D₄×C₃₁	—	4	1	no	—
127	D₄×C₃₁	—	4	1	no	—
128	D₄×C₃₁	—	4	1	no	—
129	D₄×C₃₁	—	4	1	no	—
130	D₄×C₃₁	—	8	4	no	—
131	D₄×C₃₁	—	16	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	4	1	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	1	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	8	4	no	—
139	C₂×C₄×C₃₁	—	8	2	no	—
140	C₂×C₄×C₃₁	—	8	2	no	—
141	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₃₁	—	2	2	no	—
146	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
149	#3341	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#3334	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2234	C₂×C₃₁	1	2	yes	complemented
153	#2233	C₂×C₃₁	1	2	yes	complemented
154	#2226	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#2218	C₂×C₃₁	1	2	yes	complemented
157	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#2159	C₂×C₃₁	1	2	yes	complemented
160	C₂²×C₃₁	—	2	2	no	—
161	C₂²×C₃₁	—	2	2	no	—
162	C₂²×C₃₁	—	2	2	no	—
163	C₂²×C₃₁	—	8	1	no	—
164	C₂²×C₃₁	—	8	1	no	—
165	C₂²×C₃₁	—	8	1	no	—
166	C₂²×C₃₁	—	8	1	no	—
167	C₂²×C₃₁	—	8	1	no	—
168	C₂²×C₃₁	—	8	1	no	—
169	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₃₁	—	2	2	no	—
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	8	1	no	—
173	C₄×C₃₁	—	8	1	no	—
174	C₄×C₃₁	—	8	2	no	—
175	C₄×C₃₁	—	8	2	no	—
176	C₄×C₃₁	#509	1	2	yes	—
177	C₄×C₃₁	#547	1	1	yes	characteristic
178	C₄×C₃₁	#578	1	1	yes	characteristic
179	#555	—	2	1	no	—
180	#547	—	2	1	no	—
181	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
183	#500	C₂²×C₃₁	1	1	yes	characteristic
184	#500	C₂²×C₃₁	1	1	yes	characteristic
185	#482	—	2	2	no	—
186	#480	—	2	2	no	—
187	#476	—	2	1	no	—
188	#474	—	2	1	no	—
189	#466	C₂²×C₃₁	1	2	yes	—
190	#466	C₂²×C₃₁	1	2	yes	—
191	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
193	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
194	#427	C₂²×C₃₁	1	2	yes	—
195	C₂×#107	C₂²×C₃₁	1	2	yes	—
196	#367	C₂²×C₃₁	1	2	yes	—
197	#367	C₂²×C₃₁	1	2	yes	—
198	#367	C₂²×C₃₁	1	2	yes	—
199	#366	C₂²×C₃₁	1	1	yes	characteristic
200	#366	C₂²×C₃₁	1	2	yes	—
201	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	#357	C₂²×C₃₁	1	1	yes	characteristic
203	#357	C₂²×C₃₁	1	1	yes	characteristic
204	#357	C₂²×C₃₁	1	2	yes	—
205	#357	C₂²×C₃₁	1	2	yes	—
206	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
207	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₃₁	—	4	2	no	—
209	C₂×C₃₁	—	16	1	no	—
210	C₂×C₃₁	—	16	1	no	—
211	C₂×C₃₁	#3334	1	1	yes	characteristic
212	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
213	C₂×C₃₁	#3416	1	1	yes	characteristic
214	C₂×#41	—	4	1	no	—
215	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
216	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#124	—	4	2	no	—
221	#123	—	4	2	no	—
222	C₂²⋊Q₈	—	4	1	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	#121	—	4	1	no	—
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
229	#117	—	2	2	no	—
230	#117	—	4	1	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	D₁₆	—	2	2	no	—
233	M₃₂	—	2	4	no	—
234	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
235	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
236	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
237	#107	—	2	2	no	—
238	#107	—	2	1	no	—
239	#107	—	2	1	no	—
240	#107	C₂³×C₃₁	1	1	yes	characteristic
241	#107	C₂³×C₃₁	1	1	yes	characteristic
242	#107	C₂³×C₃₁	1	2	yes	—
243	#107	C₂³×C₃₁	1	2	yes	—
244	#106	—	2	2	no	—
245	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
246	#102	—	2	1	no	—
247	#102	—	2	1	no	—
248	#102	—	2	2	no	—
249	#102	—	2	2	no	—
250	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
251	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₃₁	#33921	1	1	yes	characteristic, direct product factor, Sylow
253	#41	—	8	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
273	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
288	C₄²	—	4	2	no	—
289	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	#41×C₃₁	1	2	yes	complemented
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	C₂×D₈×C₃₁	1	2	yes	—
296	Q₈	—	8	2	no	—
297	Q₈	—	16	1	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	8	4	no	—
304	D₄	—	16	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
315	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
316	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
321	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	2	no	—
338	C₄	C₃₁×#509	1	2	yes	—
339	C₄	C₃₁×#547	1	1	yes	characteristic
340	C₄	C₃₁×#578	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	16	1	no	—
343	C₂	—	16	1	no	—
344	C₂	C₃₁×#3334	1	1	yes	characteristic
345	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
346	C₂	C₃₁×#3416	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. C₃₁×#3334	C₂	semidirect
3. C₃₁×#2254	C₂	semidirect
4. C₃₁×#2234	C₂	semidirect
5. C₃₁×#2233	C₂	semidirect
6. C₃₁×#2226	C₂	semidirect
7. C₃₁×#2224	C₂	semidirect
8. C₃₁×#2218	C₂	semidirect
9. C₄×C₃₁×D₁₆	C₂	semidirect
10. C₃₁×#2169	C₂	semidirect
11. C₃₁×#2159	C₂	semidirect
118. C₁₆×C₃₁	#41	semidirect
119. #33921	C₃₁	direct (N × H)
149. #3341	C₂×C₃₁	semidirect
150. #3334	C₂×C₃₁	semidirect
151. #2254	C₂×C₃₁	semidirect
152. #2234	C₂×C₃₁	semidirect
153. #2233	C₂×C₃₁	semidirect
154. #2226	C₂×C₃₁	semidirect
155. #2224	C₂×C₃₁	semidirect
156. #2218	C₂×C₃₁	semidirect
157. C₄×D₁₆	C₂×C₃₁	semidirect
158. #2169	C₂×C₃₁	semidirect
159. #2159	C₂×C₃₁	semidirect
252. C₃₁	#33921	direct (N × H)
292. C₁₆	#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×#3341	2
C₃₁×#3334	2
C₃₁×#2254	2
C₃₁×#2234	2
C₃₁×#2233	2
C₃₁×#2226	2
C₃₁×#2224	2
C₃₁×#2218	2
C₄×C₃₁×D₁₆	2
C₃₁×#2169	2
C₃₁×#2159	2
#33921	31

Subgroups — C31×#33921 #21490680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33921 #21490680