Subgroups — C₃₁×#33246 #21490005

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2642	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2641	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1973	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1968	C₂	1	4	yes	maximal, complemented
9	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
16	C₃₁×#499	—	2	4	no	—
17	C₃₁×#492	—	2	4	no	—
18	C₃₁×#491	—	2	4	no	—
19	C₃₁×#490	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	4	yes	—
23	C₂×C₃₁×#102	C₂²	1	4	yes	—
24	C₂×C₃₁×#97	C₂²	1	2	yes	—
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#400	—	2	4	no	—
29	C₃₁×#394	—	2	4	no	—
30	C₃₁×#391	C₂²	1	4	yes	—
31	C₃₁×#390	C₂²	1	4	yes	—
32	C₃₁×#376	C₂²	1	1	yes	characteristic
33	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×D₄×C₃₁	—	2	2	no	—
35	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₃₁	—	2	4	no	—
37	C₂×SD₁₆×C₃₁	—	2	4	no	—
38	C₂×SD₁₆×C₃₁	—	2	8	no	—
39	C₂×SD₁₆×C₃₁	—	2	8	no	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×#127	—	2	2	no	—
46	C₃₁×#127	—	2	2	no	—
47	C₃₁×#125	—	2	2	no	—
48	C₃₁×#125	—	2	2	no	—
49	C₃₁×#125	—	2	2	no	—
50	C₃₁×#125	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
57	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#107	—	2	4	no	—
60	C₃₁×#106	—	2	4	no	—
61	C₃₁×#103	—	2	8	no	—
62	C₃₁×#102	—	2	8	no	—
63	C₃₁×#97	—	2	4	no	—
64	C₃₁×#97	D₄	1	4	yes	—
65	C₄×C₈×C₃₁	—	2	4	no	—
66	C₄×C₈×C₃₁	D₄	1	4	yes	—
67	C₃₁×#95	—	2	8	no	—
68	C₃₁×#95	D₄	1	4	yes	—
69	C₂⁴×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	4	no	—
73	C₂×Q₈×C₃₁	—	2	4	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	2	4	no	—
77	C₂×D₄×C₃₁	—	2	4	no	—
78	C₂×D₄×C₃₁	—	4	4	no	—
79	C₂×D₄×C₃₁	—	8	4	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	SD₁₆×C₃₁	—	4	8	no	—
87	SD₁₆×C₃₁	—	4	8	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₁	#41	1	4	yes	—
94	C₂×C₈×C₃₁	#41	1	4	yes	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	4	4	no	—
100	C₄⋊C₄×C₃₁	—	4	4	no	—
101	C₄⋊C₄×C₃₁	—	4	4	no	—
102	C₂²⋊C₄×C₃₁	—	8	8	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	2	no	—
105	C₄²×C₃₁	C₂×D₄	1	2	yes	—
106	C₄²×C₃₁	C₂×D₄	1	2	yes	—
107	#33246	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	4	2	no	—
109	C₂³×C₃₁	—	4	4	no	—
110	C₂³×C₃₁	—	4	8	no	—
111	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₃₁	—	4	2	no	—
113	Q₈×C₃₁	—	4	2	no	—
114	Q₈×C₃₁	—	4	4	no	—
115	D₄×C₃₁	—	4	2	no	—
116	D₄×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	4	4	no	—
118	D₄×C₃₁	—	8	8	no	—
119	C₂×C₄×C₃₁	—	2	4	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	8	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	4	no	—
128	C₂×C₄×C₃₁	#127	1	4	yes	—
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
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	C₈×C₃₁	—	2	4	no	—
134	C₈×C₃₁	#124	1	4	yes	—
135	C₈×C₃₁	#124	1	4	yes	—
136	C₂×#499	C₂×C₃₁	1	1	yes	characteristic, complemented
137	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×#490	C₂×C₃₁	1	1	yes	characteristic
140	#2642	C₂×C₃₁	1	1	yes	characteristic, complemented
141	#2641	C₂×C₃₁	1	1	yes	characteristic
142	#1973	C₂×C₃₁	1	4	yes	complemented
143	#1968	C₂×C₃₁	1	4	yes	complemented
144	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂²×C₃₁	—	8	2	no	—
146	C₂²×C₃₁	—	8	4	no	—
147	C₂²×C₃₁	—	8	8	no	—
148	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
149	C₂²×C₃₁	#532	1	2	yes	—
150	C₂²×C₃₁	#577	1	4	yes	—
151	C₄×C₃₁	—	2	4	no	—
152	C₄×C₃₁	—	4	4	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	—	8	2	no	—
155	C₄×C₃₁	—	8	2	no	—
156	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
157	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
158	C₄×C₃₁	#532	1	2	yes	—
159	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
160	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
165	#499	—	2	4	no	—
166	#492	—	2	4	no	—
167	#491	—	2	4	no	—
168	#490	—	2	4	no	—
169	C₂×#107	C₂²×C₃₁	1	2	yes	—
170	C₂×#106	C₂²×C₃₁	1	2	yes	—
171	C₂×#103	C₂²×C₃₁	1	4	yes	—
172	C₂×#102	C₂²×C₃₁	1	4	yes	—
173	C₂×#97	C₂²×C₃₁	1	2	yes	—
174	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
177	#400	—	2	4	no	—
178	#394	—	2	4	no	—
179	#391	C₂²×C₃₁	1	4	yes	—
180	#390	C₂²×C₃₁	1	4	yes	—
181	#376	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₃₁	—	16	2	no	—
183	C₂×C₃₁	#2858	1	1	yes	characteristic
184	C₂×C₃₁	#3154	1	2	yes	—
185	C₂×C₃₁	#3174	1	4	yes	—
186	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
187	C₂²×D₄	—	2	2	no	—
188	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
189	C₂×SD₁₆	—	2	4	no	—
190	C₂×SD₁₆	—	2	4	no	—
191	C₂×SD₁₆	—	2	8	no	—
192	C₂×SD₁₆	—	2	8	no	—
193	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	2	2	no	—
198	#127	—	2	2	no	—
199	#127	—	2	2	no	—
200	#125	—	2	2	no	—
201	#125	—	2	2	no	—
202	#125	—	2	2	no	—
203	#125	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	8	no	—
210	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
211	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
212	#107	—	2	4	no	—
213	#106	—	2	4	no	—
214	#103	—	2	8	no	—
215	#102	—	2	8	no	—
216	#97	—	2	4	no	—
217	#97	D₄×C₃₁	1	4	yes	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	D₄×C₃₁	1	4	yes	—
220	#95	—	2	8	no	—
221	#95	D₄×C₃₁	1	4	yes	—
222	C₃₁	#33246	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
238	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
240	SD₁₆	—	4	8	no	—
241	SD₁₆	—	4	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
247	C₂×C₈	#41×C₃₁	1	4	yes	—
248	C₂×C₈	#41×C₃₁	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	8	8	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	C₂×D₄×C₃₁	1	2	yes	—
260	C₄²	C₂×D₄×C₃₁	1	2	yes	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	8	no	—
264	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	Q₈	—	4	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₃₁×#127	1	4	yes	—
282	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
285	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
286	C₈	—	2	4	no	—
287	C₈	C₃₁×#124	1	4	yes	—
288	C₈	C₃₁×#124	1	4	yes	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	4	no	—
291	C₂²	—	8	8	no	—
292	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
293	C₂²	C₃₁×#532	1	2	yes	—
294	C₂²	C₃₁×#577	1	4	yes	—
295	C₄	—	2	4	no	—
296	C₄	—	4	4	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
301	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
302	C₄	C₃₁×#532	1	2	yes	—
303	C₂	—	16	2	no	—
304	C₂	C₃₁×#2858	1	1	yes	characteristic
305	C₂	C₃₁×#3154	1	2	yes	—
306	C₂	C₃₁×#3174	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#499	C₂	semidirect
3. C₂×C₃₁×#491	C₂	semidirect
5. C₃₁×#2642	C₂	semidirect
7. C₃₁×#1973	C₂	semidirect
8. C₃₁×#1968	C₂	semidirect
9. C₃₁×#1726	C₂	semidirect
107. #33246	C₃₁	direct (N × H)
136. C₂×#499	C₂×C₃₁	semidirect
138. C₂×#491	C₂×C₃₁	semidirect
140. #2642	C₂×C₃₁	semidirect
142. #1973	C₂×C₃₁	semidirect
143. #1968	C₂×C₃₁	semidirect
144. #1726	C₂×C₃₁	semidirect
222. C₃₁	#33246	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×#499	2
C₂×C₃₁×#492	2
C₂×C₃₁×#491	2
C₂×C₃₁×#490	2
C₃₁×#2642	2
C₃₁×#2641	2
C₃₁×#1973	2
C₃₁×#1968	2
C₃₁×#1726	2
#33246	31

Subgroups — C31×#33246 #21490005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33246 #21490005