Subgroups — C₃₁×#9823 #21466582

Order 7936.

526 conjugacy classes of subgroups.

The subgroup lattice has 234 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
2	C₃₁×#1990	C₂	1	2	yes	maximal
3	C₃₁×#1956	C₂	1	2	yes	maximal
4	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₃₁×#500	—	2	2	no	—
8	C₃₁×#500	C₄	1	4	yes	complemented
9	C₂×C₃₁×#107	—	2	2	no	—
10	C₂×C₃₁×#107	—	2	2	no	—
11	C₂×C₃₁×#103	—	2	4	no	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#384	C₂²	1	2	yes	—
16	C₃₁×#340	—	2	4	no	—
17	C₈²×C₃₁	—	2	2	no	—
18	C₂²×Q₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	D₄	1	2	yes	—
20	C₂²×C₈×C₃₁	D₄	1	2	yes	—
21	C₂²×C₈×C₃₁	D₄	1	2	yes	—
22	C₃₁×C₄⋊Q₈	—	2	2	no	—
23	C₃₁×C₄⋊Q₈	—	2	2	no	—
24	C₃₁×C₄⋊Q₈	—	2	2	no	—
25	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#107	—	2	8	no	—
33	C₃₁×#107	—	4	4	no	—
34	C₃₁×#103	—	4	8	no	—
35	C₄×C₈×C₃₁	—	2	1	no	—
36	C₄×C₈×C₃₁	—	2	1	no	—
37	C₄×C₈×C₃₁	—	2	1	no	—
38	C₄×C₈×C₃₁	—	2	1	no	—
39	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₃₁	D₄	1	2	yes	—
41	C₂×Q₈×C₃₁	—	2	4	no	—
42	C₂×Q₈×C₃₁	—	4	2	no	—
43	C₂×Q₈×C₃₁	—	4	4	no	—
44	C₂×Q₈×C₃₁	—	4	4	no	—
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	—	4	2	no	—
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	—	2	2	no	—
51	C₂×C₈×C₃₁	—	2	2	no	—
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₃₁	D₈	1	4	yes	—
56	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
57	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
58	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
59	C₄⋊C₄×C₃₁	—	2	4	no	—
60	C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₄⋊C₄×C₃₁	—	4	2	no	—
62	C₄⋊C₄×C₃₁	—	4	2	no	—
63	C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₄⋊C₄×C₃₁	—	4	4	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₃₁	#41	1	1	yes	characteristic
69	C₄²×C₃₁	#41	1	1	yes	characteristic
70	#9823	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	#127	1	1	yes	characteristic
72	Q₈×C₃₁	—	4	8	no	—
73	Q₈×C₃₁	—	8	4	no	—
74	C₂×C₄×C₃₁	—	4	4	no	—
75	C₂×C₄×C₃₁	—	8	2	no	—
76	C₂×C₄×C₃₁	—	8	2	no	—
77	C₂×C₄×C₃₁	—	8	2	no	—
78	C₂×C₄×C₃₁	—	8	2	no	—
79	C₂×C₄×C₃₁	—	8	2	no	—
80	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₃₁	#135	1	2	yes	—
91	C₂×C₄×C₃₁	#135	1	2	yes	—
92	C₈×C₃₁	—	2	4	no	—
93	C₈×C₃₁	—	2	4	no	—
94	C₈×C₃₁	#103	1	8	yes	complemented
95	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
96	#1990	C₂×C₃₁	1	2	yes	—
97	#1956	C₂×C₃₁	1	2	yes	—
98	#1845	C₂×C₃₁	1	1	yes	characteristic
99	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
100	C₂²×C₃₁	#390	1	1	yes	characteristic
101	C₂²×C₃₁	#492	1	1	yes	characteristic
102	C₂²×C₃₁	#493	1	1	yes	characteristic
103	C₂²×C₃₁	#494	1	1	yes	characteristic
104	C₂²×C₃₁	#494	1	1	yes	characteristic
105	C₂²×C₃₁	#495	1	1	yes	characteristic
106	C₂²×C₃₁	#495	1	1	yes	characteristic
107	C₄×C₃₁	—	8	4	no	—
108	C₄×C₃₁	—	16	2	no	—
109	C₄×C₃₁	C₂×#103	1	2	yes	—
110	C₄×C₃₁	C₄×D₈	1	2	yes	—
111	C₄×C₃₁	C₄×Q₁₆	1	2	yes	—
112	C₄×C₃₁	#466	1	2	yes	—
113	C₄×C₃₁	#467	1	2	yes	—
114	C₄×C₃₁	#487	1	2	yes	—
115	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
116	#500	—	2	2	no	—
117	#500	C₄×C₃₁	1	4	yes	complemented
118	C₂×#107	—	2	2	no	—
119	C₂×#107	—	2	2	no	—
120	C₂×#103	—	2	4	no	—
121	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
124	#384	C₂²×C₃₁	1	2	yes	—
125	#340	—	2	4	no	—
126	C₈²	—	2	2	no	—
127	C₂×C₃₁	#1706	1	1	yes	characteristic
128	C₂×C₃₁	#1707	1	1	yes	characteristic
129	C₂×C₃₁	#1710	1	1	yes	characteristic
130	C₂×C₃₁	#1713	1	1	yes	characteristic
131	C₂×C₃₁	#1956	1	1	yes	characteristic
132	C₂×C₃₁	#1969	1	1	yes	characteristic
133	C₂×C₃₁	#1970	1	1	yes	characteristic
134	C₂²×Q₈	—	2	2	no	—
135	C₂²×C₈	D₄×C₃₁	1	2	yes	—
136	C₂²×C₈	D₄×C₃₁	1	2	yes	—
137	C₂²×C₈	D₄×C₃₁	1	2	yes	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
148	#107	—	2	8	no	—
149	#107	—	4	4	no	—
150	#103	—	4	8	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
156	C₄×C₈	D₄×C₃₁	1	2	yes	—
157	C₃₁	#9823	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
172	C₂×C₈	D₈×C₃₁	1	4	yes	—
173	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
174	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
175	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
184	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
185	C₄²	#41×C₃₁	1	1	yes	characteristic
186	C₄²	#41×C₃₁	1	1	yes	characteristic
187	C₂³	C₃₁×#127	1	1	yes	characteristic
188	Q₈	—	4	8	no	—
189	Q₈	—	8	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
200	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
201	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
203	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
204	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
205	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
206	C₂×C₄	C₃₁×#135	1	2	yes	—
207	C₂×C₄	C₃₁×#135	1	2	yes	—
208	C₈	—	2	4	no	—
209	C₈	—	2	4	no	—
210	C₈	C₃₁×#103	1	8	yes	complemented
211	C₂²	C₃₁×#390	1	1	yes	characteristic
212	C₂²	C₃₁×#492	1	1	yes	characteristic
213	C₂²	C₃₁×#493	1	1	yes	characteristic
214	C₂²	C₃₁×#494	1	1	yes	characteristic
215	C₂²	C₃₁×#494	1	1	yes	characteristic
216	C₂²	C₃₁×#495	1	1	yes	characteristic
217	C₂²	C₃₁×#495	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	—	16	2	no	—
220	C₄	C₂×C₃₁×#103	1	2	yes	—
221	C₄	C₄×D₈×C₃₁	1	2	yes	—
222	C₄	C₄×Q₁₆×C₃₁	1	2	yes	—
223	C₄	C₃₁×#466	1	2	yes	—
224	C₄	C₃₁×#467	1	2	yes	—
225	C₄	C₃₁×#487	1	2	yes	—
226	C₂	C₃₁×#1706	1	1	yes	characteristic
227	C₂	C₃₁×#1707	1	1	yes	characteristic
228	C₂	C₃₁×#1710	1	1	yes	characteristic
229	C₂	C₃₁×#1713	1	1	yes	characteristic
230	C₂	C₃₁×#1956	1	1	yes	characteristic
231	C₂	C₃₁×#1969	1	1	yes	characteristic
232	C₂	C₃₁×#1970	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₃₁×#500	C₄	semidirect
70. #9823	C₃₁	direct (N × H)
94. C₈×C₃₁	#103	semidirect
117. #500	C₄×C₃₁	semidirect
157. C₃₁	#9823	direct (N × H)
210. C₈	C₃₁×#103	semidirect

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×#1990	2
C₃₁×#1956	2
C₃₁×#1845	2
C₂×C₈²×C₃₁	2
#9823	31

Subgroups — C31×#9823 #21466582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9823 #21466582