Subgroups — C₃₁×#33633 #21490392

Order 7936.

534 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×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2951	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2228	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2227	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2224	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2139	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#2138	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₃₁×#544	—	2	1	no	—
14	C₃₁×#520	—	2	1	no	—
15	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#501	—	2	2	no	—
18	C₃₁×#500	C₂²	1	2	yes	—
19	C₃₁×#499	C₂²	1	2	yes	—
20	C₃₁×#440	—	2	2	no	—
21	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
22	C₄×D₈×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#428	—	2	1	no	—
24	C₃₁×#427	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₃₁×#417	—	2	1	no	—
28	C₃₁×#405	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×#366	C₂²	1	2	yes	—
32	C₃₁×#366	C₂²	1	2	yes	—
33	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	1	yes	characteristic
36	C₃₁×#358	C₂²	1	1	yes	characteristic
37	C₃₁×#358	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	1	yes	characteristic
39	C₃₁×#357	C₂²	1	1	yes	characteristic
40	C₃₁×#357	C₂²	1	2	yes	—
41	C₂×#41×C₃₁	—	2	1	no	—
42	C₃₁×#135	—	2	2	no	—
43	C₃₁×#135	Q₈	1	4	yes	complemented
44	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×SD₁₆×C₃₁	—	2	1	no	—
46	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	—	2	2	no	—
50	C₃₁×C₄⋊Q₈	—	4	1	no	—
51	C₃₁×#125	—	2	2	no	—
52	C₃₁×#125	—	4	1	no	—
53	C₃₁×C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	2	2	no	—
56	C₄×D₄×C₃₁	—	4	2	no	—
57	C₃₁×#117	—	2	1	no	—
58	C₃₁×#117	—	4	1	no	—
59	C₃₁×#117	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
63	C₃₁×M₃₂	—	2	2	no	—
64	C₂×C₁₆×C₃₁	—	2	2	no	—
65	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#107	—	2	1	no	—
69	C₃₁×#107	—	2	1	no	—
70	C₃₁×#107	—	2	2	no	—
71	C₃₁×#107	C₂³	1	2	yes	—
72	C₃₁×#107	C₂³	1	2	yes	—
73	C₃₁×#107	C₂³	1	1	yes	characteristic
74	C₃₁×#107	C₂³	1	1	yes	characteristic
75	C₃₁×#106	—	2	1	no	—
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×#103	—	2	2	no	—
78	C₃₁×#102	—	2	2	no	—
79	C₃₁×#97	D₄	1	2	yes	—
80	C₄×C₈×C₃₁	C₂³	1	2	yes	—
81	#41×C₃₁	—	2	4	no	—
82	#41×C₃₁	—	4	2	no	—
83	C₂×Q₈×C₃₁	—	2	1	no	—
84	C₂×Q₈×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₃₁	—	2	1	no	—
93	Q₁₆×C₃₁	C₂×Q₈	1	2	yes	—
94	SD₁₆×C₃₁	—	2	4	no	—
95	D₈×C₃₁	—	2	1	no	—
96	D₈×C₃₁	C₂×Q₈	1	2	yes	—
97	M₁₆×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₃₁	#41	1	2	yes	—
102	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	—	4	1	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	—	4	2	no	—
112	C₄⋊C₄×C₃₁	—	4	2	no	—
113	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
115	C₂²⋊C₄×C₃₁	—	4	1	no	—
116	C₂²⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
118	C₄²×C₃₁	—	4	1	no	—
119	C₄²×C₃₁	—	4	2	no	—
120	C₄²×C₃₁	C₂×D₄	1	2	yes	—
121	C₁₆×C₃₁	—	2	1	no	—
122	C₁₆×C₃₁	—	2	1	no	—
123	C₁₆×C₃₁	—	2	2	no	—
124	#33633	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₃₁	—	4	1	no	—
126	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₃₁	—	2	2	no	—
128	Q₈×C₃₁	—	4	1	no	—
129	Q₈×C₃₁	—	16	2	no	—
130	D₄×C₃₁	—	2	2	no	—
131	D₄×C₃₁	—	4	4	no	—
132	D₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	—	4	4	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	16	2	no	—
140	C₂×C₄×C₃₁	—	16	1	no	—
141	C₂×C₄×C₃₁	—	16	1	no	—
142	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₃₁	#137	1	2	yes	—
147	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₃₁	—	2	2	no	—
149	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
150	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
151	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
152	#3164	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2951	C₂×C₃₁	1	1	yes	characteristic
154	#2229	C₂×C₃₁	1	2	yes	complemented
155	#2228	C₂×C₃₁	1	2	yes	complemented
156	#2227	C₂×C₃₁	1	2	yes	complemented
157	#2224	C₂×C₃₁	1	2	yes	complemented
158	#2159	C₂×C₃₁	1	1	yes	characteristic, complemented
159	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#2151	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#2139	C₂×C₃₁	1	1	yes	characteristic
162	#2138	C₂×C₃₁	1	1	yes	characteristic
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	4	2	no	—
166	C₂²×C₃₁	—	8	1	no	—
167	C₂²×C₃₁	—	8	1	no	—
168	C₂²×C₃₁	#474	1	2	yes	—
169	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₃₁	—	2	2	no	—
171	C₄×C₃₁	—	2	2	no	—
172	C₄×C₃₁	—	4	2	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	—	8	2	no	—
175	C₄×C₃₁	—	8	1	no	—
176	C₄×C₃₁	—	8	1	no	—
177	C₄×C₃₁	#474	1	2	yes	—
178	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
180	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
181	#544	—	2	1	no	—
182	#520	—	2	1	no	—
183	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
184	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
185	#501	—	2	2	no	—
186	#500	C₂²×C₃₁	1	2	yes	—
187	#499	C₂²×C₃₁	1	2	yes	—
188	#440	—	2	2	no	—
189	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
190	C₄×D₈	C₂²×C₃₁	1	2	yes	—
191	#428	—	2	1	no	—
192	#427	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
195	#417	—	2	1	no	—
196	#405	C₂²×C₃₁	1	1	yes	characteristic
197	#367	C₂²×C₃₁	1	2	yes	—
198	#366	C₂²×C₃₁	1	2	yes	—
199	#366	C₂²×C₃₁	1	2	yes	—
200	#366	C₂²×C₃₁	1	2	yes	—
201	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
202	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
203	#358	C₂²×C₃₁	1	1	yes	characteristic
204	#358	C₂²×C₃₁	1	1	yes	characteristic
205	#358	C₂²×C₃₁	1	2	yes	—
206	#357	C₂²×C₃₁	1	1	yes	characteristic
207	#357	C₂²×C₃₁	1	1	yes	characteristic
208	#357	C₂²×C₃₁	1	2	yes	—
209	C₂×C₃₁	—	2	2	no	—
210	C₂×C₃₁	—	8	2	no	—
211	C₂×C₃₁	C₂×#474	1	1	yes	characteristic
212	C₂×C₃₁	#3415	1	1	yes	characteristic
213	C₂×C₃₁	#3417	1	1	yes	characteristic
214	C₂×#41	—	2	1	no	—
215	#135	—	2	2	no	—
216	#135	Q₈×C₃₁	1	4	yes	complemented
217	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
218	C₂×SD₁₆	—	2	1	no	—
219	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
222	C₄⋊Q₈	—	2	2	no	—
223	C₄⋊Q₈	—	4	1	no	—
224	#125	—	2	2	no	—
225	#125	—	4	1	no	—
226	C₂²⋊Q₈	—	4	2	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	4	2	no	—
230	#117	—	2	1	no	—
231	#117	—	4	1	no	—
232	#117	D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	4	1	no	—
236	M₃₂	—	2	2	no	—
237	C₂×C₁₆	—	2	2	no	—
238	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
239	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
240	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
241	#107	—	2	1	no	—
242	#107	—	2	1	no	—
243	#107	—	2	2	no	—
244	#107	C₂³×C₃₁	1	2	yes	—
245	#107	C₂³×C₃₁	1	1	yes	characteristic
246	#107	C₂³×C₃₁	1	1	yes	characteristic
247	#107	C₂³×C₃₁	1	2	yes	—
248	#106	—	2	1	no	—
249	#106	—	2	2	no	—
250	#103	—	2	2	no	—
251	#102	—	2	2	no	—
252	#97	D₄×C₃₁	1	2	yes	—
253	C₄×C₈	C₂³×C₃₁	1	2	yes	—
254	C₃₁	#33633	1	1	yes	characteristic, direct product factor, Sylow
255	#41	—	2	4	no	—
256	#41	—	4	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
266	Q₁₆	—	2	1	no	—
267	Q₁₆	C₂×Q₈×C₃₁	1	2	yes	—
268	SD₁₆	—	2	4	no	—
269	D₈	—	2	1	no	—
270	D₈	C₂×Q₈×C₃₁	1	2	yes	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
275	C₂×C₈	#41×C₃₁	1	2	yes	—
276	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
288	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂×D₄×C₃₁	1	2	yes	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₂³	—	4	1	no	—
299	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	16	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	16	2	no	—
313	C₂×C₄	—	16	1	no	—
314	C₂×C₄	—	16	1	no	—
315	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
316	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
317	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
319	C₂×C₄	C₃₁×#137	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
323	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
324	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	C₃₁×#474	1	2	yes	—
331	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	C₃₁×#474	1	2	yes	—
340	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
341	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂×C₃₁×#474	1	1	yes	characteristic
345	C₂	C₃₁×#3415	1	1	yes	characteristic
346	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3164	C₂	semidirect
3. C₃₁×#2229	C₂	semidirect
4. C₃₁×#2228	C₂	semidirect
5. C₃₁×#2227	C₂	semidirect
6. C₃₁×#2224	C₂	semidirect
7. C₃₁×#2159	C₂	semidirect
8. C₂×C₃₁×#366	C₂	semidirect
9. C₃₁×#2151	C₂	semidirect
43. C₃₁×#135	Q₈	semidirect
124. #33633	C₃₁	direct (N × H)
152. #3164	C₂×C₃₁	semidirect
154. #2229	C₂×C₃₁	semidirect
155. #2228	C₂×C₃₁	semidirect
156. #2227	C₂×C₃₁	semidirect
157. #2224	C₂×C₃₁	semidirect
158. #2159	C₂×C₃₁	semidirect
159. C₂×#366	C₂×C₃₁	semidirect
160. #2151	C₂×C₃₁	semidirect
216. #135	Q₈×C₃₁	semidirect
254. C₃₁	#33633	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×#3164	2
C₃₁×#2951	2
C₃₁×#2229	2
C₃₁×#2228	2
C₃₁×#2227	2
C₃₁×#2224	2
C₃₁×#2159	2
C₂×C₃₁×#366	2
C₃₁×#2151	2
C₃₁×#2139	2
C₃₁×#2138	2
#33633	31

Subgroups — C31×#33633 #21490392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33633 #21490392