Subgroups — C₃₁×#33133 #21489892

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2625	C₂	1	2	yes	maximal
4	C₃₁×#1971	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1944	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1933	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1894	C₂	1	2	yes	maximal
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₃₁×#467	—	2	4	no	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	C₂²	1	2	yes	—
17	C₂×C₃₁×#103	C₂²	1	4	yes	—
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₃₁×#405	—	2	4	no	—
20	C₂×C₃₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#400	—	2	2	no	—
23	C₃₁×#398	—	2	2	no	—
24	C₃₁×#397	—	2	4	no	—
25	C₃₁×#391	C₂²	1	4	yes	—
26	C₃₁×#389	C₂²	1	4	yes	—
27	C₃₁×#380	C₂²	1	2	yes	—
28	C₃₁×#340	C₂²	1	2	yes	—
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×Q₁₆×C₃₁	—	2	2	no	—
33	C₂×Q₁₆×C₃₁	—	2	4	no	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	D₄	1	4	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₂²⋊Q₈	—	2	4	no	—
42	C₃₁×C₂²⋊Q₈	—	2	4	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×#117	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₃₁×#107	—	2	2	no	—
56	C₃₁×#106	—	2	4	no	—
57	C₃₁×#103	—	2	4	no	—
58	C₃₁×#103	—	2	8	no	—
59	C₃₁×#97	—	2	4	no	—
60	C₄×C₈×C₃₁	—	2	4	no	—
61	C₃₁×#95	—	2	4	no	—
62	C₃₁×#95	—	2	4	no	—
63	C₃₁×#95	—	2	4	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₃₁	—	2	2	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	2	4	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	4	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	#41	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₃₁	—	4	4	no	—
82	M₁₆×C₃₁	—	2	8	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	8	no	—
87	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₁	#41	1	4	yes	—
90	C₂×C₈×C₃₁	#41	1	4	yes	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	8	4	no	—
99	C₂²⋊C₄×C₃₁	—	2	4	no	—
100	C₂²⋊C₄×C₃₁	—	4	4	no	—
101	C₂²⋊C₄×C₃₁	—	8	2	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	#33133	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	2	2	no	—
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	—	2	2	no	—
108	C₂³×C₃₁	#121	1	2	yes	—
109	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₃₁	—	4	2	no	—
111	Q₈×C₃₁	—	4	2	no	—
112	Q₈×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	4	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	#121	1	2	yes	—
130	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
132	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
133	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
134	C₈×C₃₁	—	2	2	no	—
135	C₈×C₃₁	—	2	4	no	—
136	C₈×C₃₁	#123	1	4	yes	—
137	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
139	#2625	C₂×C₃₁	1	2	yes	—
140	#1971	C₂×C₃₁	1	2	yes	complemented
141	#1944	C₂×C₃₁	1	2	yes	complemented
142	#1933	C₂×C₃₁	1	4	yes	complemented
143	#1894	C₂×C₃₁	1	2	yes	—
144	#1807	C₂×C₃₁	1	1	yes	characteristic
145	C₂²×C₃₁	—	2	2	no	—
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	4	4	no	—
149	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
150	C₂²×C₃₁	#532	1	2	yes	—
151	C₂²×C₃₁	#577	1	2	yes	—
152	C₂²×C₃₁	#578	1	2	yes	—
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	4	2	no	—
155	C₄×C₃₁	—	4	2	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	—	8	4	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
160	C₄×C₃₁	#525	1	1	yes	characteristic
161	C₄×C₃₁	#529	1	2	yes	—
162	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
163	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
164	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
166	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
167	#467	—	2	4	no	—
168	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×#106	C₂²×C₃₁	1	2	yes	—
170	C₂×#103	C₂²×C₃₁	1	4	yes	—
171	C₂×#103	C₂²×C₃₁	1	2	yes	—
172	#405	—	2	4	no	—
173	C₂×#97	C₂²×C₃₁	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
175	#400	—	2	2	no	—
176	#398	—	2	2	no	—
177	#397	—	2	4	no	—
178	#391	C₂²×C₃₁	1	4	yes	—
179	#389	C₂²×C₃₁	1	4	yes	—
180	#380	C₂²×C₃₁	1	2	yes	—
181	#340	C₂²×C₃₁	1	2	yes	—
182	#340	C₂²×C₃₁	1	4	yes	—
183	C₂×C₃₁	—	4	2	no	—
184	C₂×C₃₁	#2854	1	1	yes	characteristic
185	C₂×C₃₁	#3059	1	1	yes	characteristic
186	C₂×C₃₁	#3060	1	1	yes	characteristic
187	C₂×C₃₁	#3178	1	4	yes	—
188	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
189	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	4	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	D₄×C₃₁	1	4	yes	—
196	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₂²⋊Q₈	—	2	4	no	—
200	C₂²⋊Q₈	—	2	4	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
206	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
212	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
213	#107	—	2	2	no	—
214	#106	—	2	4	no	—
215	#103	—	2	4	no	—
216	#103	—	2	8	no	—
217	#97	—	2	4	no	—
218	C₄×C₈	—	2	4	no	—
219	#95	—	2	2	no	—
220	#95	—	2	4	no	—
221	#95	—	2	4	no	—
222	#95	—	2	4	no	—
223	C₃₁	#33133	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
238	C₂²×C₄	#41×C₃₁	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
240	Q₁₆	—	4	4	no	—
241	M₁₆	—	2	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	8	no	—
246	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
248	C₂×C₈	#41×C₃₁	1	4	yes	—
249	C₂×C₈	#41×C₃₁	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₂²⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	4	4	no	—
260	C₂²⋊C₄	—	8	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	C₃₁×#121	1	2	yes	—
267	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	C₃₁×#121	1	2	yes	—
288	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
290	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
291	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	2	4	no	—
294	C₈	C₃₁×#123	1	4	yes	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	4	no	—
299	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
300	C₂²	C₃₁×#532	1	2	yes	—
301	C₂²	C₃₁×#577	1	2	yes	—
302	C₂²	C₃₁×#578	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	4	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
310	C₄	C₃₁×#525	1	1	yes	characteristic
311	C₄	C₃₁×#529	1	2	yes	—
312	C₂	—	4	2	no	—
313	C₂	C₃₁×#2854	1	1	yes	characteristic
314	C₂	C₃₁×#3059	1	1	yes	characteristic
315	C₂	C₃₁×#3060	1	1	yes	characteristic
316	C₂	C₃₁×#3178	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1971	C₂	semidirect
5. C₃₁×#1944	C₂	semidirect
6. C₃₁×#1933	C₂	semidirect
104. #33133	C₃₁	direct (N × H)
140. #1971	C₂×C₃₁	semidirect
141. #1944	C₂×C₃₁	semidirect
142. #1933	C₂×C₃₁	semidirect
223. C₃₁	#33133	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×#467	2
C₂×C₃₁×#405	2
C₃₁×#2625	2
C₃₁×#1971	2
C₃₁×#1944	2
C₃₁×#1933	2
C₃₁×#1894	2
C₃₁×#1807	2
#33133	31

Subgroups — C31×#33133 #21489892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33133 #21489892