Subgroups — C₃₁×#33307 #21490066

Order 7936.

1502 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#497	C₂	1	3	yes	maximal, complemented
2	C₂×C₃₁×#496	C₂	1	3	yes	maximal, complemented
3	C₂×C₃₁×#495	C₂	1	3	yes	maximal, complemented
4	C₂×C₃₁×#492	C₂	1	3	yes	maximal, complemented
5	C₃₁×#2658	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2657	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	3	yes	—
9	C₂²×SD₁₆×C₃₁	C₂²	1	3	yes	—
10	C₂²×SD₁₆×C₃₁	C₂²	1	6	yes	—
11	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×D₈×C₃₁	C₂²	1	3	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
14	C₂×C₃₁×#127	C₂²	1	3	yes	—
15	C₂×C₃₁×#124	C₂²	1	3	yes	—
16	C₂×C₃₁×#124	C₂²	1	3	yes	—
17	C₃₁×#497	—	2	12	no	—
18	C₃₁×#496	—	2	12	no	—
19	C₃₁×#495	—	2	12	no	—
20	C₃₁×#492	—	2	12	no	—
21	C₂×C₃₁×#97	C₂²	1	3	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
23	C₃₁×#399	—	2	4	no	—
24	C₃₁×#392	—	2	4	no	—
25	C₃₁×#376	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
28	C₂²×D₄×C₃₁	—	2	6	no	—
29	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
31	C₂×Q₁₆×C₃₁	—	2	6	no	—
32	C₂×Q₁₆×C₃₁	—	2	12	no	—
33	C₂×SD₁₆×C₃₁	—	2	12	no	—
34	C₂×SD₁₆×C₃₁	—	2	12	no	—
35	C₂×SD₁₆×C₃₁	—	2	6	no	—
36	C₂×SD₁₆×C₃₁	—	2	6	no	—
37	C₂×SD₁₆×C₃₁	—	2	12	no	—
38	C₂×SD₁₆×C₃₁	—	2	12	no	—
39	C₂×D₈×C₃₁	—	2	6	no	—
40	C₂×D₈×C₃₁	—	2	12	no	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
43	C₃₁×C₄⋊Q₈	—	2	6	no	—
44	C₃₁×C₄⋊Q₈	—	2	6	no	—
45	C₃₁×#127	—	2	6	no	—
46	C₃₁×#127	—	2	6	no	—
47	C₃₁×#124	—	2	6	no	—
48	C₃₁×#124	—	2	6	no	—
49	C₃₁×#124	—	2	6	no	—
50	C₃₁×#124	—	2	6	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
53	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
54	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
55	C₃₁×#97	D₄	1	12	yes	complemented
56	C₄×C₈×C₃₁	D₄	1	12	yes	complemented
57	C₃₁×#95	D₄	1	4	yes	—
58	C₂⁴×C₃₁	—	2	1	no	—
59	C₂⁴×C₃₁	—	2	3	no	—
60	C₂×Q₈×C₃₁	—	2	3	no	—
61	C₂×Q₈×C₃₁	—	2	3	no	—
62	C₂×Q₈×C₃₁	—	2	3	no	—
63	C₂×Q₈×C₃₁	—	2	3	no	—
64	C₂×Q₈×C₃₁	—	2	6	no	—
65	C₂×Q₈×C₃₁	—	2	6	no	—
66	C₂×D₄×C₃₁	—	2	3	no	—
67	C₂×D₄×C₃₁	—	2	3	no	—
68	C₂×D₄×C₃₁	—	2	3	no	—
69	C₂×D₄×C₃₁	—	2	3	no	—
70	C₂×D₄×C₃₁	—	2	6	no	—
71	C₂×D₄×C₃₁	—	2	6	no	—
72	C₂×D₄×C₃₁	—	4	12	no	—
73	C₂×D₄×C₃₁	—	8	12	no	—
74	C₂²×C₄×C₃₁	—	2	1	no	—
75	C₂²×C₄×C₃₁	—	2	3	no	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₃₁	—	4	12	no	—
79	SD₁₆×C₃₁	—	4	24	no	—
80	SD₁₆×C₃₁	—	4	12	no	—
81	SD₁₆×C₃₁	—	4	4	no	—
82	D₈×C₃₁	—	4	12	no	—
83	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
84	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
85	C₂×C₈×C₃₁	C₂×D₄	1	12	yes	—
86	C₄⋊C₄×C₃₁	—	4	12	no	—
87	C₂²⋊C₄×C₃₁	—	4	12	no	—
88	C₂²⋊C₄×C₃₁	—	4	12	no	—
89	C₄²×C₃₁	C₂×D₄	1	6	yes	—
90	C₄²×C₃₁	C₂×D₄	1	6	yes	—
91	#33307	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	4	3	no	—
93	C₂³×C₃₁	—	4	3	no	—
94	C₂³×C₃₁	—	4	6	no	—
95	C₂³×C₃₁	—	4	12	no	—
96	C₂³×C₃₁	—	4	4	no	—
97	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₃₁	—	4	6	no	—
99	Q₈×C₃₁	—	4	1	no	—
100	Q₈×C₃₁	—	4	3	no	—
101	Q₈×C₃₁	—	4	3	no	—
102	Q₈×C₃₁	—	4	3	no	—
103	D₄×C₃₁	—	4	3	no	—
104	D₄×C₃₁	—	4	3	no	—
105	D₄×C₃₁	—	4	1	no	—
106	D₄×C₃₁	—	4	3	no	—
107	D₄×C₃₁	—	4	6	no	—
108	D₄×C₃₁	—	8	24	no	—
109	C₂×C₄×C₃₁	—	2	12	no	—
110	C₂×C₄×C₃₁	—	4	3	no	—
111	C₂×C₄×C₃₁	—	4	3	no	—
112	C₂×C₄×C₃₁	—	4	6	no	—
113	C₂×C₄×C₃₁	#127	1	12	yes	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
116	C₈×C₃₁	#127	1	12	yes	—
117	C₈×C₃₁	#127	1	4	yes	—
118	C₂×#497	C₂×C₃₁	1	3	yes	complemented
119	C₂×#496	C₂×C₃₁	1	3	yes	complemented
120	C₂×#495	C₂×C₃₁	1	3	yes	complemented
121	C₂×#492	C₂×C₃₁	1	3	yes	complemented
122	#2658	C₂×C₃₁	1	1	yes	characteristic, complemented
123	#2657	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂²×C₃₁	—	8	1	no	—
126	C₂²×C₃₁	—	8	3	no	—
127	C₂²×C₃₁	—	8	12	no	—
128	C₂²×C₃₁	—	8	6	no	—
129	C₂²×C₃₁	—	8	6	no	—
130	C₂²×C₃₁	C₂×#127	1	3	yes	—
131	C₂²×C₃₁	#577	1	4	yes	—
132	C₄×C₃₁	—	2	12	no	—
133	C₄×C₃₁	—	8	1	no	—
134	C₄×C₃₁	—	8	3	no	—
135	C₄×C₃₁	C₂×#127	1	3	yes	—
136	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
137	C₂²×Q₁₆	C₂²×C₃₁	1	3	yes	—
138	C₂²×SD₁₆	C₂²×C₃₁	1	3	yes	—
139	C₂²×SD₁₆	C₂²×C₃₁	1	6	yes	—
140	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
141	C₂²×D₈	C₂²×C₃₁	1	3	yes	—
142	C₂×C₄⋊Q₈	C₂²×C₃₁	1	3	yes	—
143	C₂×#127	C₂²×C₃₁	1	3	yes	—
144	C₂×#124	C₂²×C₃₁	1	3	yes	—
145	C₂×#124	C₂²×C₃₁	1	3	yes	—
146	#497	—	2	12	no	—
147	#496	—	2	12	no	—
148	#495	—	2	12	no	—
149	#492	—	2	12	no	—
150	C₂×#97	C₂²×C₃₁	1	3	yes	—
151	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
152	#399	—	2	4	no	—
153	#392	—	2	4	no	—
154	#376	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₃₁	—	16	3	no	—
156	C₂×C₃₁	—	16	1	no	—
157	C₂×C₃₁	#2867	1	1	yes	characteristic
158	C₂×C₃₁	#3154	1	6	yes	—
159	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
160	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
161	C₂²×D₄	—	2	6	no	—
162	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
163	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
164	C₂×Q₁₆	—	2	6	no	—
165	C₂×Q₁₆	—	2	12	no	—
166	C₂×SD₁₆	—	2	12	no	—
167	C₂×SD₁₆	—	2	12	no	—
168	C₂×SD₁₆	—	2	6	no	—
169	C₂×SD₁₆	—	2	6	no	—
170	C₂×SD₁₆	—	2	12	no	—
171	C₂×SD₁₆	—	2	12	no	—
172	C₂×D₈	—	2	6	no	—
173	C₂×D₈	—	2	12	no	—
174	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
175	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
176	C₄⋊Q₈	—	2	6	no	—
177	C₄⋊Q₈	—	2	6	no	—
178	#127	—	2	6	no	—
179	#127	—	2	6	no	—
180	#124	—	2	6	no	—
181	#124	—	2	6	no	—
182	#124	—	2	6	no	—
183	#124	—	2	6	no	—
184	C₂×C₄⋊C₄	—	2	6	no	—
185	C₂×C₂²⋊C₄	—	2	6	no	—
186	C₂×C₂²⋊C₄	—	2	6	no	—
187	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
188	#97	D₄×C₃₁	1	12	yes	complemented
189	C₄×C₈	D₄×C₃₁	1	12	yes	complemented
190	#95	D₄×C₃₁	1	4	yes	—
191	C₃₁	#33307	1	1	yes	characteristic, direct product factor, Sylow
192	C₂⁴	—	2	1	no	—
193	C₂⁴	—	2	3	no	—
194	C₂×Q₈	—	2	3	no	—
195	C₂×Q₈	—	2	3	no	—
196	C₂×Q₈	—	2	3	no	—
197	C₂×Q₈	—	2	3	no	—
198	C₂×Q₈	—	2	6	no	—
199	C₂×Q₈	—	2	6	no	—
200	C₂×D₄	—	2	3	no	—
201	C₂×D₄	—	2	3	no	—
202	C₂×D₄	—	2	3	no	—
203	C₂×D₄	—	2	3	no	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	—	2	6	no	—
206	C₂×D₄	—	4	12	no	—
207	C₂×D₄	—	8	12	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	3	no	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
211	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
212	Q₁₆	—	4	12	no	—
213	SD₁₆	—	4	24	no	—
214	SD₁₆	—	4	12	no	—
215	SD₁₆	—	4	4	no	—
216	D₈	—	4	12	no	—
217	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
218	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
219	C₂×C₈	C₂×D₄×C₃₁	1	12	yes	—
220	C₄⋊C₄	—	4	12	no	—
221	C₂²⋊C₄	—	4	12	no	—
222	C₂²⋊C₄	—	4	12	no	—
223	C₄²	C₂×D₄×C₃₁	1	6	yes	—
224	C₄²	C₂×D₄×C₃₁	1	6	yes	—
225	C₂³	—	4	3	no	—
226	C₂³	—	4	3	no	—
227	C₂³	—	4	6	no	—
228	C₂³	—	4	12	no	—
229	C₂³	—	4	4	no	—
230	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
231	Q₈	—	4	6	no	—
232	Q₈	—	4	1	no	—
233	Q₈	—	4	3	no	—
234	Q₈	—	4	3	no	—
235	Q₈	—	4	3	no	—
236	D₄	—	4	3	no	—
237	D₄	—	4	3	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	3	no	—
240	D₄	—	4	6	no	—
241	D₄	—	8	24	no	—
242	C₂×C₄	—	2	12	no	—
243	C₂×C₄	—	4	3	no	—
244	C₂×C₄	—	4	3	no	—
245	C₂×C₄	—	4	6	no	—
246	C₂×C₄	C₃₁×#127	1	12	yes	—
247	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
248	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
249	C₈	C₃₁×#127	1	12	yes	—
250	C₈	C₃₁×#127	1	4	yes	—
251	C₂²	—	8	1	no	—
252	C₂²	—	8	3	no	—
253	C₂²	—	8	12	no	—
254	C₂²	—	8	6	no	—
255	C₂²	—	8	6	no	—
256	C₂²	C₂×C₃₁×#127	1	3	yes	—
257	C₂²	C₃₁×#577	1	4	yes	—
258	C₄	—	2	12	no	—
259	C₄	—	8	1	no	—
260	C₄	—	8	3	no	—
261	C₄	C₂×C₃₁×#127	1	3	yes	—
262	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
263	C₂	—	16	3	no	—
264	C₂	—	16	1	no	—
265	C₂	C₃₁×#2867	1	1	yes	characteristic
266	C₂	C₃₁×#3154	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
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×#497	C₂	semidirect
2. C₂×C₃₁×#496	C₂	semidirect
3. C₂×C₃₁×#495	C₂	semidirect
4. C₂×C₃₁×#492	C₂	semidirect
5. C₃₁×#2658	C₂	semidirect
6. C₃₁×#2657	C₂	semidirect
7. C₃₁×#1726	C₂	semidirect
55. C₃₁×#97	D₄	semidirect
56. C₄×C₈×C₃₁	D₄	semidirect
91. #33307	C₃₁	direct (N × H)
118. C₂×#497	C₂×C₃₁	semidirect
119. C₂×#496	C₂×C₃₁	semidirect
120. C₂×#495	C₂×C₃₁	semidirect
121. C₂×#492	C₂×C₃₁	semidirect
122. #2658	C₂×C₃₁	semidirect
123. #2657	C₂×C₃₁	semidirect
124. #1726	C₂×C₃₁	semidirect
188. #97	D₄×C₃₁	semidirect
189. C₄×C₈	D₄×C₃₁	semidirect
191. C₃₁	#33307	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×#497	2
C₂×C₃₁×#496	2
C₂×C₃₁×#495	2
C₂×C₃₁×#492	2
C₃₁×#2658	2
C₃₁×#2657	2
C₃₁×#1726	2
#33307	31

Subgroups — C31×#33307 #21490066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33307 #21490066