Subgroups — C₃₁×#12646 #21469405

Order 7936.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2148	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2145	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#131	—	2	1	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₃₁×#443	—	2	1	no	—
13	C₃₁×#443	—	2	1	no	—
14	C₃₁×#443	C₄	1	2	yes	complemented
15	C₃₁×#443	C₄	1	2	yes	complemented
16	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
18	C₃₁×#427	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#104	—	2	1	no	—
20	C₃₁×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#362	C₄	1	2	yes	complemented
23	C₃₁×#361	C₄	1	2	yes	complemented
24	C₃₁×#360	C₄	1	4	yes	complemented
25	C₃₁×#346	—	2	1	no	—
26	C₄×C₁₆×C₃₁	—	2	1	no	—
27	C₃₁×#339	—	2	1	no	—
28	C₂×#41×C₃₁	—	2	1	no	—
29	C₃₁×#135	—	2	1	no	—
30	C₃₁×#135	—	2	1	no	—
31	C₃₁×#135	C₂×C₄	1	2	yes	—
32	C₃₁×#135	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₃₁	—	2	1	no	—
35	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₃₁×#131	—	2	2	no	—
37	C₃₁×#131	—	2	2	no	—
38	C₃₁×#131	—	4	1	no	—
39	C₃₁×#131	—	4	1	no	—
40	C₂×M₁₆×C₃₁	—	2	1	no	—
41	C₂×M₁₆×C₃₁	—	2	1	no	—
42	C₂×M₁₆×C₃₁	—	4	1	no	—
43	C₂²×C₈×C₃₁	—	4	1	no	—
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#117	—	2	1	no	—
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₃₁×M₃₂	—	2	1	no	—
49	C₃₁×M₃₂	—	2	1	no	—
50	C₃₁×M₃₂	C₂×C₄	1	2	yes	—
51	C₃₁×M₃₂	C₂×C₄	1	2	yes	—
52	C₂×C₁₆×C₃₁	—	2	1	no	—
53	C₂×C₁₆×C₃₁	—	2	1	no	—
54	C₃₁×#108	—	2	1	no	—
55	C₃₁×#108	—	2	1	no	—
56	C₃₁×#108	C₂×C₄	1	2	yes	—
57	C₃₁×#108	C₂×C₄	1	2	yes	—
58	C₃₁×#107	—	2	1	no	—
59	C₃₁×#106	C₂×C₄	1	2	yes	—
60	C₃₁×#104	—	2	2	no	—
61	C₃₁×#104	—	2	2	no	—
62	C₃₁×#97	—	4	1	no	—
63	C₄×C₈×C₃₁	—	4	1	no	—
64	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
68	#41×C₃₁	—	2	2	no	—
69	#41×C₃₁	—	2	2	no	—
70	#41×C₃₁	—	4	1	no	—
71	#41×C₃₁	—	4	1	no	—
72	C₂×Q₈×C₃₁	—	2	1	no	—
73	C₂×D₄×C₃₁	—	2	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₃₁	—	2	1	no	—
79	Q₁₆×C₃₁	C₄²	1	2	yes	—
80	SD₁₆×C₃₁	—	2	2	no	—
81	SD₁₆×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	2	1	no	—
83	D₈×C₃₁	C₄²	1	2	yes	—
84	M₁₆×C₃₁	—	2	2	no	—
85	M₁₆×C₃₁	—	2	2	no	—
86	M₁₆×C₃₁	—	4	1	no	—
87	M₁₆×C₃₁	—	4	1	no	—
88	M₁₆×C₃₁	—	4	2	no	—
89	M₁₆×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	—	4	1	no	—
91	C₂×C₈×C₃₁	—	4	1	no	—
92	C₂×C₈×C₃₁	—	4	2	no	—
93	C₂×C₈×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
97	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
98	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
99	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
105	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂²⋊C₄×C₃₁	—	4	1	no	—
108	C₄²×C₃₁	—	2	1	no	—
109	C₄²×C₃₁	—	4	1	no	—
110	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	2	no	—
113	C₁₆×C₃₁	—	2	2	no	—
114	#12646	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	4	1	no	—
116	C₂³×C₃₁	#135	1	1	yes	characteristic
117	Q₈×C₃₁	—	2	2	no	—
118	Q₈×C₃₁	—	4	1	no	—
119	D₄×C₃₁	—	2	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	4	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	1	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
136	C₈×C₃₁	—	4	2	no	—
137	C₈×C₃₁	—	4	2	no	—
138	C₈×C₃₁	#102	1	1	yes	characteristic
139	C₈×C₃₁	#102	1	1	yes	characteristic
140	C₈×C₃₁	#102	1	1	yes	characteristic
141	C₈×C₃₁	#102	1	1	yes	characteristic
142	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
143	C₈×C₃₁	#117	1	1	yes	characteristic
144	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
145	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
146	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
147	#2148	C₂×C₃₁	1	1	yes	characteristic
148	#2145	C₂×C₃₁	1	1	yes	characteristic
149	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1389	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#1388	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂²×C₃₁	—	4	1	no	—
154	C₂²×C₃₁	—	4	2	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	#416	1	1	yes	characteristic
157	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
158	C₂²×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
159	C₄×C₃₁	—	2	2	no	—
160	C₄×C₃₁	—	4	1	no	—
161	C₄×C₃₁	—	4	2	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
164	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
165	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
166	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
167	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×#131	—	2	1	no	—
169	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
171	#443	—	2	1	no	—
172	#443	—	2	1	no	—
173	#443	C₄×C₃₁	1	2	yes	complemented
174	#443	C₄×C₃₁	1	2	yes	complemented
175	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
177	#427	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×#104	—	2	1	no	—
179	#405	—	2	1	no	—
180	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
181	#362	C₄×C₃₁	1	2	yes	complemented
182	#361	C₄×C₃₁	1	2	yes	complemented
183	#360	C₄×C₃₁	1	4	yes	complemented
184	#346	—	2	1	no	—
185	C₄×C₁₆	—	2	1	no	—
186	#339	—	2	1	no	—
187	C₂×C₃₁	—	2	1	no	—
188	C₂×C₃₁	—	2	1	no	—
189	C₂×C₃₁	—	2	1	no	—
190	C₂×C₃₁	—	8	2	no	—
191	C₂×C₃₁	C₄×#102	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	C₂×C₄×C₃₁	1	2	yes	—
196	#135	C₂×C₄×C₃₁	1	2	yes	—
197	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	4	1	no	—
203	#131	—	4	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	4	1	no	—
207	C₂²×C₈	—	4	1	no	—
208	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
210	#117	—	2	1	no	—
211	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
212	M₃₂	—	2	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	C₂×C₄×C₃₁	1	2	yes	—
215	M₃₂	C₂×C₄×C₃₁	1	2	yes	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	—	2	1	no	—
220	#108	C₂×C₄×C₃₁	1	2	yes	—
221	#108	C₂×C₄×C₃₁	1	2	yes	—
222	#107	—	2	1	no	—
223	#106	C₂×C₄×C₃₁	1	2	yes	—
224	#104	—	2	2	no	—
225	#104	—	2	2	no	—
226	#97	—	4	1	no	—
227	C₄×C₈	—	4	1	no	—
228	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
229	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
231	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
232	C₃₁	#12646	1	1	yes	characteristic, direct product factor, Sylow
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	4	1	no	—
236	#41	—	4	1	no	—
237	C₂×Q₈	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	C₄²×C₃₁	1	2	yes	—
245	SD₁₆	—	2	2	no	—
246	SD₁₆	—	2	2	no	—
247	D₈	—	2	1	no	—
248	D₈	C₄²×C₃₁	1	2	yes	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
262	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
263	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
264	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
270	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
276	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	2	2	no	—
279	C₂³	—	4	1	no	—
280	C₂³	C₃₁×#135	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	4	1	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
300	C₈	—	4	2	no	—
301	C₈	—	4	2	no	—
302	C₈	C₃₁×#102	1	1	yes	characteristic
303	C₈	C₃₁×#102	1	1	yes	characteristic
304	C₈	C₃₁×#102	1	1	yes	characteristic
305	C₈	C₃₁×#102	1	1	yes	characteristic
306	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
307	C₈	C₃₁×#117	1	1	yes	characteristic
308	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
309	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
310	C₂²	—	4	1	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	C₃₁×#416	1	1	yes	characteristic
314	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
315	C₂²	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
321	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
322	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
323	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	8	2	no	—
328	C₂	C₄×C₃₁×#102	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₃₁	#102
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	C₄×#102
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₃₁×M₃₂	C₂	semidirect
5. C₃₁×#1772	C₂	semidirect
6. C₃₁×#1389	C₂	semidirect
7. C₃₁×#1388	C₂	semidirect
14. C₃₁×#443	C₄	semidirect
15. C₃₁×#443	C₄	semidirect
22. C₃₁×#362	C₄	semidirect
23. C₃₁×#361	C₄	semidirect
24. C₃₁×#360	C₄	semidirect
114. #12646	C₃₁	direct (N × H)
149. C₄×M₃₂	C₂×C₃₁	semidirect
150. #1772	C₂×C₃₁	semidirect
151. #1389	C₂×C₃₁	semidirect
152. #1388	C₂×C₃₁	semidirect
173. #443	C₄×C₃₁	semidirect
174. #443	C₄×C₃₁	semidirect
181. #362	C₄×C₃₁	semidirect
182. #361	C₄×C₃₁	semidirect
183. #360	C₄×C₃₁	semidirect
232. C₃₁	#12646	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#443	2
C₃₁×#2148	2
C₃₁×#2145	2
C₄×C₃₁×M₃₂	2
C₃₁×#1772	2
C₃₁×#1389	2
C₃₁×#1388	2
#12646	31

Subgroups — C31×#12646 #21469405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12646 #21469405