Subgroups — C₃₁×#11493 #21468252

Order 7936.

342 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2162	C₂	1	2	yes	maximal
3	C₃₁×#1322	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
5	C₈×Q₈×C₃₁	—	2	2	no	—
6	C₈×Q₈×C₃₁	—	2	2	no	—
7	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×C₄⋊C₁₆	—	2	2	no	—
12	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
13	C₄×C₁₆×C₃₁	—	2	2	no	—
14	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
18	C₄×Q₈×C₃₁	—	2	2	no	—
19	C₄×Q₈×C₃₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₃₁	C₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
22	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₃₁	—	2	4	no	—
25	C₃₁×C₄⋊C₈	—	2	4	no	—
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₄×C₈×C₃₁	—	2	2	no	—
28	C₄×C₈×C₃₁	—	2	2	no	—
29	C₄×C₈×C₃₁	D₄	1	2	yes	—
30	C₄×C₈×C₃₁	D₄	1	2	yes	—
31	C₂×Q₈×C₃₁	—	2	2	no	—
32	C₂×Q₈×C₃₁	—	2	2	no	—
33	C₂×Q₈×C₃₁	C₁₆	1	4	yes	complemented
34	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	—	2	2	no	—
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂×C₈×C₃₁	—	4	1	no	—
42	C₂×C₈×C₃₁	—	4	2	no	—
43	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
46	C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₄²×C₃₁	—	2	2	no	—
49	C₄²×C₃₁	—	2	2	no	—
50	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₃₁	M₁₆	1	2	yes	—
52	C₁₆×C₃₁	—	4	4	no	—
53	#11493	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
55	Q₈×C₃₁	—	2	8	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	—	2	1	no	—
59	C₂×C₄×C₃₁	—	2	1	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	4	1	no	—
62	C₂×C₄×C₃₁	—	4	2	no	—
63	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×C₁₆	1	4	yes	—
66	C₂×C₄×C₃₁	C₂×C₁₆	1	2	yes	—
67	C₂×C₄×C₃₁	M₃₂	1	2	yes	—
68	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₃₁	—	2	2	no	—
71	C₈×C₃₁	—	2	1	no	—
72	C₈×C₃₁	—	4	2	no	—
73	C₈×C₃₁	#101	1	2	yes	—
74	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
75	#2162	C₂×C₃₁	1	2	yes	—
76	#1322	C₂×C₃₁	1	4	yes	—
77	C₂²×C₃₁	—	2	1	no	—
78	C₂²×C₃₁	—	2	1	no	—
79	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂²×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
81	C₂²×C₃₁	C₂×M₃₂	1	1	yes	characteristic
82	C₄×C₃₁	—	2	4	no	—
83	C₄×C₃₁	—	2	2	no	—
84	C₄×C₃₁	—	2	1	no	—
85	C₄×C₃₁	—	4	2	no	—
86	C₄×C₃₁	C₂²⋊C₁₆	1	4	yes	—
87	C₄×C₃₁	#410	1	1	yes	characteristic
88	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
89	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
90	C₈×Q₈	—	2	2	no	—
91	C₈×Q₈	—	2	2	no	—
92	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
93	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
94	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
95	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
96	C₄⋊C₁₆	—	2	2	no	—
97	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
98	C₄×C₁₆	—	2	2	no	—
99	C₂×C₃₁	—	2	1	no	—
100	C₂×C₃₁	—	2	1	no	—
101	C₂×C₃₁	#1461	1	1	yes	characteristic
102	C₂×C₃₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
103	C₂×C₃₁	#2115	1	1	yes	characteristic
104	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
105	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
106	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
107	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₂×C₄⋊C₄	C₈×C₃₁	1	2	yes	—
111	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
113	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
114	C₂×C₁₆	—	2	4	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	D₄×C₃₁	1	2	yes	—
120	C₄×C₈	D₄×C₃₁	1	2	yes	—
121	C₃₁	#11493	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	C₁₆×C₃₁	1	4	yes	complemented
125	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
126	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
127	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
129	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
136	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
142	C₄²	M₁₆×C₃₁	1	2	yes	—
143	C₁₆	—	4	4	no	—
144	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
154	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
155	C₂×C₄	C₂×C₁₆×C₃₁	1	4	yes	—
156	C₂×C₄	C₂×C₁₆×C₃₁	1	2	yes	—
157	C₂×C₄	C₃₁×M₃₂	1	2	yes	—
158	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	4	2	no	—
163	C₈	C₃₁×#101	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
167	C₂²	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
168	C₂²	C₂×C₃₁×M₃₂	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	C₃₁×C₂²⋊C₁₆	1	4	yes	—
174	C₄	C₃₁×#410	1	1	yes	characteristic
175	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	C₃₁×#1461	1	1	yes	characteristic
179	C₂	C₂×C₃₁×C₂²⋊C₁₆	1	1	yes	characteristic
180	C₂	C₃₁×#2115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₃₁	C₂²×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
33. C₂×Q₈×C₃₁	C₁₆	semidirect
53. #11493	C₃₁	direct (N × H)
121. C₃₁	#11493	direct (N × H)
124. C₂×Q₈	C₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
C₃₁×#2162	2
C₃₁×#1322	2
#11493	31

Subgroups — C31×#11493 #21468252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11493 #21468252