Subgroups — C₃₁×#33707 #21490466

Order 7936.

622 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2243	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2241	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2240	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2139	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	—	2	1	no	—
9	C₂×C₃₁×#135	C₂²	1	2	yes	—
10	C₃₁×#532	—	2	1	no	—
11	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₃₁×#500	—	2	1	no	—
14	C₃₁×#500	—	2	1	no	—
15	C₃₁×#500	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#495	—	2	1	no	—
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#493	C₂²	1	1	yes	characteristic
21	C₃₁×#492	—	2	1	no	—
22	C₂×C₃₁×#107	—	2	1	no	—
23	C₂×C₃₁×#107	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#358	C₂²	1	8	yes	—
26	C₃₁×#357	C₂²	1	8	yes	—
27	C₃₁×#346	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₂²×Q₈×C₃₁	—	4	1	no	—
31	C₃₁×#135	—	2	8	no	—
32	C₃₁×#135	—	4	4	no	—
33	C₂×Q₁₆×C₃₁	—	2	1	no	—
34	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₃₁	—	2	2	no	—
36	C₂×SD₁₆×C₃₁	—	2	2	no	—
37	C₂×D₈×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×C₄⋊Q₈	—	4	1	no	—
45	C₃₁×C₄⋊Q₈	—	4	1	no	—
46	C₃₁×#127	—	2	1	no	—
47	C₃₁×#124	—	2	2	no	—
48	C₃₁×#121	—	4	1	no	—
49	C₃₁×#121	—	4	1	no	—
50	C₄×D₄×C₃₁	—	4	1	no	—
51	C₄×D₄×C₃₁	—	4	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
54	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×M₃₂	—	2	4	no	—
56	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
57	C₃₁×#107	—	2	2	no	—
58	C₃₁×#107	—	2	2	no	—
59	C₃₁×#107	—	2	2	no	—
60	C₃₁×#107	C₂³	1	4	yes	—
61	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	#41×C₃₁	—	4	4	no	—
66	#41×C₃₁	—	4	4	no	—
67	C₂×Q₈×C₃₁	—	2	2	no	—
68	C₂×Q₈×C₃₁	—	4	2	no	—
69	C₂×Q₈×C₃₁	—	4	2	no	—
70	C₂×Q₈×C₃₁	—	8	4	no	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	—	4	2	no	—
76	C₂²×C₄×C₃₁	—	4	2	no	—
77	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₃₁	—	2	4	no	—
81	Q₁₆×C₃₁	—	4	2	no	—
82	SD₁₆×C₃₁	—	4	4	no	—
83	SD₁₆×C₃₁	—	4	4	no	—
84	D₈×C₃₁	—	2	4	no	—
85	D₈×C₃₁	—	4	2	no	—
86	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
87	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	#41	1	4	yes	—
91	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₄⋊C₄×C₃₁	—	4	1	no	—
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	D₈	1	1	yes	characteristic
101	C₄²×C₃₁	D₈	1	1	yes	characteristic
102	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₃₁	—	2	4	no	—
105	#33707	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	4	2	no	—
107	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₃₁	—	4	2	no	—
109	Q₈×C₃₁	—	4	2	no	—
110	Q₈×C₃₁	—	8	4	no	—
111	Q₈×C₃₁	—	8	4	no	—
112	D₄×C₃₁	—	4	2	no	—
113	D₄×C₃₁	—	4	2	no	—
114	D₄×C₃₁	—	8	4	no	—
115	D₄×C₃₁	—	8	2	no	—
116	D₄×C₃₁	—	8	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	8	4	no	—
122	C₂×C₄×C₃₁	—	8	4	no	—
123	C₂×C₄×C₃₁	#124	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₃₁	#124	1	2	yes	—
132	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
133	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
134	C₈×C₃₁	C₂×#41	1	2	yes	—
135	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#3147	C₂×C₃₁	1	1	yes	characteristic
137	#2243	C₂×C₃₁	1	4	yes	complemented
138	#2241	C₂×C₃₁	1	2	yes	complemented
139	#2240	C₂×C₃₁	1	2	yes	complemented
140	#2139	C₂×C₃₁	1	4	yes	complemented
141	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	8	2	no	—
145	C₂²×C₃₁	—	8	2	no	—
146	C₂²×C₃₁	—	8	2	no	—
147	C₂²×C₃₁	#486	1	2	yes	—
148	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₃₁	—	2	1	no	—
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	—	8	4	no	—
153	C₄×C₃₁	#486	1	2	yes	—
154	C₄×C₃₁	#509	1	2	yes	—
155	C₄×C₃₁	#510	1	2	yes	—
156	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
157	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
158	C₂×#135	—	2	1	no	—
159	C₂×#135	C₂²×C₃₁	1	2	yes	—
160	#532	—	2	1	no	—
161	C₂×C₄⋊Q₈	—	2	1	no	—
162	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
163	#500	—	2	1	no	—
164	#500	—	2	1	no	—
165	#500	C₂²×C₃₁	1	2	yes	—
166	#500	C₂²×C₃₁	1	2	yes	—
167	#495	—	2	1	no	—
168	#495	C₂²×C₃₁	1	2	yes	—
169	#494	C₂²×C₃₁	1	1	yes	characteristic
170	#493	C₂²×C₃₁	1	1	yes	characteristic
171	#492	—	2	1	no	—
172	C₂×#107	—	2	1	no	—
173	C₂×#107	C₂²×C₃₁	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
175	#358	C₂²×C₃₁	1	8	yes	—
176	#357	C₂²×C₃₁	1	8	yes	—
177	#346	C₂²×C₃₁	1	2	yes	—
178	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
179	C₂×C₃₁	—	2	2	no	—
180	C₂×C₃₁	—	16	2	no	—
181	C₂×C₃₁	C₂×#486	1	1	yes	characteristic
182	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
183	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
184	C₂×#41	—	2	2	no	—
185	C₂²×Q₈	—	4	1	no	—
186	#135	—	2	8	no	—
187	#135	—	4	4	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
190	C₂×SD₁₆	—	2	2	no	—
191	C₂×SD₁₆	—	2	2	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₈	D₄×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	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	4	1	no	—
200	C₄⋊Q₈	—	4	1	no	—
201	#127	—	2	1	no	—
202	#124	—	2	2	no	—
203	#121	—	4	1	no	—
204	#121	—	4	1	no	—
205	C₄×D₄	—	4	1	no	—
206	C₄×D₄	—	4	1	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	4	1	no	—
209	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
210	M₃₂	—	2	4	no	—
211	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
212	#107	—	2	2	no	—
213	#107	—	2	2	no	—
214	#107	—	2	2	no	—
215	#107	C₂³×C₃₁	1	4	yes	—
216	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
217	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
218	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
220	C₃₁	#33707	1	1	yes	characteristic, direct product factor, Sylow
221	#41	—	4	4	no	—
222	#41	—	4	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	8	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	Q₁₆	—	2	4	no	—
237	Q₁₆	—	4	2	no	—
238	SD₁₆	—	4	4	no	—
239	SD₁₆	—	4	4	no	—
240	D₈	—	2	4	no	—
241	D₈	—	4	2	no	—
242	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
243	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₈	#41×C₃₁	1	4	yes	—
247	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	D₈×C₃₁	1	1	yes	characteristic
257	C₄²	D₈×C₃₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₁₆	—	2	4	no	—
261	C₂³	—	4	2	no	—
262	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	4	no	—
266	Q₈	—	8	4	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	8	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	C₃₁×#124	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
285	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
286	C₈	C₃₁×#124	1	2	yes	—
287	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
288	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
289	C₈	C₂×#41×C₃₁	1	2	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	C₃₁×#486	1	2	yes	—
296	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	4	no	—
301	C₄	C₃₁×#486	1	2	yes	—
302	C₄	C₃₁×#509	1	2	yes	—
303	C₄	C₃₁×#510	1	2	yes	—
304	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
305	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₂×C₃₁×#486	1	1	yes	characteristic
309	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
310	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#500	C₂	semidirect
3. C₃₁×#2243	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
5. C₃₁×#2240	C₂	semidirect
6. C₃₁×#2139	C₂	semidirect
7. C₄×C₃₁×M₃₂	C₂	semidirect
105. #33707	C₃₁	direct (N × H)
135. C₂×#500	C₂×C₃₁	semidirect
137. #2243	C₂×C₃₁	semidirect
138. #2241	C₂×C₃₁	semidirect
139. #2240	C₂×C₃₁	semidirect
140. #2139	C₂×C₃₁	semidirect
141. C₄×M₃₂	C₂×C₃₁	semidirect
220. C₃₁	#33707	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×#500	2
C₃₁×#3147	2
C₃₁×#2243	2
C₃₁×#2241	2
C₃₁×#2240	2
C₃₁×#2139	2
C₄×C₃₁×M₃₂	2
#33707	31

Subgroups — C31×#33707 #21490466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33707 #21490466