Subgroups — C₃₁×#33706 #21490465

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2243	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2242	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2141	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2140	C₂	1	2	yes	maximal
7	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₃₁	—	2	1	no	—
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#533	—	2	1	no	—
12	C₂×C₃₁×#124	—	2	1	no	—
13	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	2	yes	complemented
15	C₃₁×#499	—	2	1	no	—
16	C₃₁×#499	C₂²	1	2	yes	complemented
17	C₃₁×#498	—	2	1	no	—
18	C₃₁×#495	—	2	1	no	—
19	C₃₁×#495	—	2	1	no	—
20	C₃₁×#495	C₂²	1	2	yes	—
21	C₃₁×#495	C₂²	1	2	yes	—
22	C₃₁×#427	—	2	1	no	—
23	C₃₁×#427	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#358	C₂²	1	8	yes	complemented
26	C₃₁×#357	C₂²	1	8	yes	—
27	C₃₁×#346	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
29	C₂²×Q₈×C₃₁	—	2	1	no	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
31	C₂×Q₁₆×C₃₁	—	2	1	no	—
32	C₂×Q₁₆×C₃₁	—	2	4	no	—
33	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
34	C₂×SD₁₆×C₃₁	—	2	2	no	—
35	C₂×SD₁₆×C₃₁	—	2	2	no	—
36	C₂×SD₁₆×C₃₁	—	4	4	no	—
37	C₂×D₈×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	—	2	4	no	—
39	C₂×D₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×#125	—	2	2	no	—
44	C₃₁×#124	—	2	2	no	—
45	C₃₁×#124	—	2	2	no	—
46	C₃₁×#124	—	4	2	no	—
47	C₃₁×C₂²⋊Q₈	—	4	2	no	—
48	C₄×Q₈×C₃₁	—	4	2	no	—
49	C₃₁×#117	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
51	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
52	C₃₁×M₃₂	—	2	4	no	—
53	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
54	C₃₁×#107	—	2	2	no	—
55	C₃₁×#107	C₂³	1	4	yes	—
56	C₃₁×#106	—	2	2	no	—
57	C₃₁×#106	—	2	2	no	—
58	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂⁴×C₃₁	—	4	1	no	—
63	C₂×Q₈×C₃₁	—	2	2	no	—
64	C₂×Q₈×C₃₁	—	4	2	no	—
65	C₂×Q₈×C₃₁	—	4	1	no	—
66	C₂×Q₈×C₃₁	—	4	2	no	—
67	C₂×Q₈×C₃₁	—	4	2	no	—
68	C₂×D₄×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	4	2	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂×D₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	—	4	1	no	—
73	C₂²×C₄×C₃₁	D₈	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	Q₁₆×C₃₁	—	2	4	no	—
76	Q₁₆×C₃₁	—	4	2	no	—
77	SD₁₆×C₃₁	—	4	4	no	—
78	SD₁₆×C₃₁	—	4	4	no	—
79	D₈×C₃₁	—	2	4	no	—
80	D₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
82	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	#41	1	4	yes	—
86	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	4	4	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₂²⋊C₄×C₃₁	—	4	2	no	—
91	C₂²⋊C₄×C₃₁	—	4	4	no	—
92	C₂²⋊C₄×C₃₁	—	4	4	no	—
93	C₄²×C₃₁	—	4	2	no	—
94	C₄²×C₃₁	D₈	1	2	yes	—
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₁₆×C₃₁	—	2	4	no	—
98	#33706	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	4	2	no	—
100	C₂³×C₃₁	—	8	4	no	—
101	C₂³×C₃₁	—	8	2	no	—
102	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
103	Q₈×C₃₁	—	4	2	no	—
104	Q₈×C₃₁	—	4	2	no	—
105	Q₈×C₃₁	—	8	4	no	—
106	Q₈×C₃₁	—	8	2	no	—
107	D₄×C₃₁	—	4	2	no	—
108	D₄×C₃₁	—	4	2	no	—
109	D₄×C₃₁	—	8	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	8	2	no	—
114	C₂×C₄×C₃₁	#124	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₃₁	#124	1	2	yes	—
121	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
122	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
123	C₈×C₃₁	C₂×#41	1	2	yes	—
124	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
126	#2243	C₂×C₃₁	1	4	yes	complemented
127	#2242	C₂×C₃₁	1	4	yes	complemented
128	#2141	C₂×C₃₁	1	2	yes	complemented
129	#2140	C₂×C₃₁	1	2	yes	—
130	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	8	2	no	—
133	C₂²×C₃₁	—	8	4	no	—
134	C₂²×C₃₁	—	16	4	no	—
135	C₂²×C₃₁	#486	1	2	yes	—
136	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
137	C₄×C₃₁	—	2	2	no	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	8	2	no	—
140	C₄×C₃₁	—	8	4	no	—
141	C₄×C₃₁	#486	1	2	yes	—
142	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
143	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
145	C₂²×SD₁₆	—	2	1	no	—
146	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
147	#533	—	2	1	no	—
148	C₂×#124	—	2	1	no	—
149	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
150	#500	C₂²×C₃₁	1	2	yes	complemented
151	#499	—	2	1	no	—
152	#499	C₂²×C₃₁	1	2	yes	complemented
153	#498	—	2	1	no	—
154	#495	—	2	1	no	—
155	#495	—	2	1	no	—
156	#495	C₂²×C₃₁	1	2	yes	—
157	#495	C₂²×C₃₁	1	2	yes	—
158	#427	—	2	1	no	—
159	#427	C₂²×C₃₁	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
161	#358	C₂²×C₃₁	1	8	yes	complemented
162	#357	C₂²×C₃₁	1	8	yes	—
163	#346	C₂²×C₃₁	1	2	yes	complemented
164	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
165	C₂×C₃₁	—	2	2	no	—
166	C₂×C₃₁	—	16	2	no	—
167	C₂×C₃₁	C₂×#486	1	1	yes	characteristic
168	C₂×C₃₁	#3414	1	2	yes	—
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂×Q₁₆	—	2	1	no	—
172	C₂×Q₁₆	—	2	4	no	—
173	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
174	C₂×SD₁₆	—	2	2	no	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	4	4	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	4	no	—
179	C₂×D₈	C₂³×C₃₁	1	2	yes	—
180	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
182	C₄⋊Q₈	—	2	2	no	—
183	#125	—	2	2	no	—
184	#124	—	2	2	no	—
185	#124	—	2	2	no	—
186	#124	—	4	2	no	—
187	C₂²⋊Q₈	—	4	2	no	—
188	C₄×Q₈	—	4	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₂²⋊C₄	—	4	2	no	—
191	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
192	M₃₂	—	2	4	no	—
193	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
194	#107	—	2	2	no	—
195	#107	C₂³×C₃₁	1	4	yes	—
196	#106	—	2	2	no	—
197	#106	—	2	2	no	—
198	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
199	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
202	C₃₁	#33706	1	1	yes	characteristic, direct product factor, Sylow
203	C₂⁴	—	4	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	D₈×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	Q₁₆	—	2	4	no	—
217	Q₁₆	—	4	2	no	—
218	SD₁₆	—	4	4	no	—
219	SD₁₆	—	4	4	no	—
220	D₈	—	2	4	no	—
221	D₈	—	4	2	no	—
222	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
223	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₈	#41×C₃₁	1	4	yes	—
227	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	4	2	no	—
235	C₄²	D₈×C₃₁	1	2	yes	—
236	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₁₆	—	2	4	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	2	no	—
242	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	4	no	—
246	Q₈	—	8	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₃₁×#124	1	2	yes	—
255	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
256	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
257	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
259	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
260	C₈	C₃₁×#124	1	2	yes	—
261	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
262	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
263	C₈	C₂×#41×C₃₁	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	4	no	—
268	C₂²	C₃₁×#486	1	2	yes	—
269	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₃₁×#486	1	2	yes	—
275	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
276	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×C₃₁×#486	1	1	yes	characteristic
280	C₂	C₃₁×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
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. C₃₁×#3160	C₂	semidirect
3. C₃₁×#2243	C₂	semidirect
4. C₃₁×#2242	C₂	semidirect
5. C₃₁×#2141	C₂	semidirect
7. C₄×C₃₁×M₃₂	C₂	semidirect
14. C₃₁×#500	C₂²	semidirect
16. C₃₁×#499	C₂²	semidirect
25. C₃₁×#358	C₂²	semidirect
27. C₃₁×#346	C₂²	semidirect
28. C₄×C₁₆×C₃₁	C₂²	semidirect
98. #33706	C₃₁	direct (N × H)
124. #3160	C₂×C₃₁	semidirect
126. #2243	C₂×C₃₁	semidirect
127. #2242	C₂×C₃₁	semidirect
128. #2141	C₂×C₃₁	semidirect
130. C₄×M₃₂	C₂×C₃₁	semidirect
150. #500	C₂²×C₃₁	semidirect
152. #499	C₂²×C₃₁	semidirect
161. #358	C₂²×C₃₁	semidirect
163. #346	C₂²×C₃₁	semidirect
164. C₄×C₁₆	C₂²×C₃₁	semidirect
202. C₃₁	#33706	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×#3160	2
C₂×C₃₁×#495	2
C₃₁×#2243	2
C₃₁×#2242	2
C₃₁×#2141	2
C₃₁×#2140	2
C₄×C₃₁×M₃₂	2
#33706	31

Subgroups — C31×#33706 #21490465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33706 #21490465