Subgroups — C₃₁×#9623 #21466382

Order 7936.

298 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1950	C₂	1	2	yes	maximal
2	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1301	C₂	1	2	yes	maximal
6	C₃₁×#431	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#108	—	2	2	no	—
8	C₃₁×#423	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#341	—	2	2	no	—
13	C₃₁×#334	—	2	1	no	—
14	C₃₁×#334	—	2	1	no	—
15	C₃₁×#334	C₄	1	2	yes	—
16	C₃₁×#334	C₄	1	2	yes	—
17	C₃₁×#322	—	2	1	no	—
18	C₈²×C₃₁	—	2	1	no	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×#108	—	4	4	no	—
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	—	2	1	no	—
29	C₃₁×C₄⋊C₈	—	2	2	no	—
30	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₃₁×C₂²⋊C₈	Q₈	1	2	yes	—
32	C₃₁×#97	—	2	1	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₄×C₈×C₃₁	—	2	1	no	—
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₃₁	—	2	2	no	—
43	M₁₆×C₃₁	—	4	2	no	—
44	M₁₆×C₃₁	—	4	2	no	—
45	M₁₆×C₃₁	SD₁₆	1	4	yes	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	4	1	no	—
51	C₂×C₈×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	—	4	2	no	—
53	C₂×C₈×C₃₁	—	4	2	no	—
54	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₃₁	#41	1	1	yes	characteristic
61	C₄²×C₃₁	#41	1	1	yes	characteristic
62	#9623	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	1	no	—
66	C₂×C₄×C₃₁	—	4	1	no	—
67	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₃₁	#125	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₃₁	#135	1	2	yes	—
75	C₈×C₃₁	—	2	1	no	—
76	C₈×C₃₁	—	2	1	no	—
77	C₈×C₃₁	—	2	1	no	—
78	C₈×C₃₁	—	2	4	no	—
79	C₈×C₃₁	—	4	2	no	—
80	C₈×C₃₁	—	4	4	no	—
81	C₈×C₃₁	—	8	2	no	—
82	C₈×C₃₁	#108	1	2	yes	complemented
83	#1950	C₂×C₃₁	1	2	yes	—
84	#1566	C₂×C₃₁	1	1	yes	characteristic
85	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
86	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic
87	#1301	C₂×C₃₁	1	2	yes	—
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	—	2	1	no	—
90	C₂²×C₃₁	#389	1	1	yes	characteristic
91	C₂²×C₃₁	#498	1	1	yes	characteristic
92	C₂²×C₃₁	#499	1	1	yes	characteristic
93	C₄×C₃₁	—	2	1	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
97	C₄×C₃₁	#430	1	1	yes	characteristic
98	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
99	C₄×C₃₁	#465	1	2	yes	—
100	#431	C₂²×C₃₁	1	1	yes	characteristic
101	C₂×#108	—	2	2	no	—
102	#423	C₂²×C₃₁	1	2	yes	—
103	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
104	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
106	#341	—	2	2	no	—
107	#334	—	2	1	no	—
108	#334	—	2	1	no	—
109	#334	C₄×C₃₁	1	2	yes	—
110	#334	C₄×C₃₁	1	2	yes	—
111	#322	—	2	1	no	—
112	C₈²	—	2	1	no	—
113	C₂×C₃₁	—	2	1	no	—
114	C₂×C₃₁	—	2	1	no	—
115	C₂×C₃₁	#1941	1	1	yes	characteristic
116	C₂×C₃₁	#1950	1	1	yes	characteristic
117	C₂×C₃₁	#1953	1	1	yes	characteristic
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
122	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
124	#108	—	4	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
130	C₂²⋊C₈	Q₈×C₃₁	1	2	yes	—
131	#97	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
135	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
137	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
138	C₃₁	#9623	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	M₁₆	SD₁₆×C₃₁	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
155	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
156	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
158	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
159	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
160	C₄²	#41×C₃₁	1	1	yes	characteristic
161	C₄²	#41×C₃₁	1	1	yes	characteristic
162	C₂³	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
169	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
170	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
171	C₂×C₄	C₃₁×#125	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
173	C₂×C₄	C₃₁×#135	1	2	yes	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	4	no	—
178	C₈	—	4	2	no	—
179	C₈	—	4	4	no	—
180	C₈	—	8	2	no	—
181	C₈	C₃₁×#108	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	C₃₁×#389	1	1	yes	characteristic
185	C₂²	C₃₁×#498	1	1	yes	characteristic
186	C₂²	C₃₁×#499	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
191	C₄	C₃₁×#430	1	1	yes	characteristic
192	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
193	C₄	C₃₁×#465	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	C₃₁×#1941	1	1	yes	characteristic
197	C₂	C₃₁×#1950	1	1	yes	characteristic
198	C₂	C₃₁×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#389
prime radical	C₂³×C₃₁	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
62. #9623	C₃₁	direct (N × H)
82. C₈×C₃₁	#108	semidirect
138. C₃₁	#9623	direct (N × H)
181. C₈	C₃₁×#108	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#1950	2
C₃₁×#1566	2
C₂×C₃₁×#334	2
C₈×M₁₆×C₃₁	2
C₃₁×#1301	2
#9623	31

Subgroups — C31×#9623 #21466382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9623 #21466382