Subgroups — C₂×C₃₁×#1367 #21468754

Order 7936.

286 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1367	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₃₁	—	2	2	no	—
5	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×C₄⋊C₁₆	—	2	2	no	—
9	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	—
10	C₈²×C₃₁	C₄	1	2	yes	—
11	C₈²×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₈×C₃₁	—	2	1	no	—
13	C₂²×C₈×C₃₁	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₃₁	—	2	4	no	—
19	C₂×C₁₆×C₃₁	—	4	4	no	—
20	C₄×C₈×C₃₁	—	2	1	no	—
21	C₄×C₈×C₃₁	C₈	1	2	yes	—
22	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₃₁	—	2	4	no	—
32	C₂×C₈×C₃₁	—	2	2	no	—
33	C₂×C₈×C₃₁	—	2	2	no	—
34	C₂×C₈×C₃₁	—	2	2	no	—
35	C₂×C₈×C₃₁	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
41	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
46	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₃₁	—	4	4	no	—
48	C₂×#1367	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₃₁	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₃₁	—	2	2	no	—
51	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₃₁	#106	1	2	yes	—
55	C₂×C₄×C₃₁	#108	1	2	yes	—
56	C₂×C₄×C₃₁	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₃₁	M₃₂	1	2	yes	—
58	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₃₁	—	2	4	no	—
62	C₈×C₃₁	—	2	1	no	—
63	C₈×C₃₁	—	2	1	no	—
64	C₈×C₃₁	—	2	2	no	—
65	C₈×C₃₁	#106	1	2	yes	—
66	C₈×C₃₁	C₂×C₁₆	1	4	yes	complemented
67	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
68	C₂×C₄⋊C₁₆	C₂×C₃₁	1	2	yes	—
69	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic
70	#1367	C₂×C₃₁	1	4	yes	direct product factor
71	C₂²×C₃₁	#334	1	2	yes	—
72	C₂²×C₃₁	C₄⋊C₁₆	1	2	yes	—
73	C₂²×C₃₁	#364	1	2	yes	—
74	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₄×C₃₁	—	2	1	no	—
76	C₄×C₃₁	—	2	1	no	—
77	C₄×C₃₁	#334	1	2	yes	—
78	C₄×C₃₁	C₄⋊C₁₆	1	2	yes	—
79	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
80	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
81	C₄×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
82	C₄×C₃₁	C₂×M₃₂	1	1	yes	characteristic
83	C₂²×C₁₆	—	2	2	no	—
84	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
85	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	2	no	—
88	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	—
89	C₈²	C₄×C₃₁	1	2	yes	—
90	C₈²	C₂²×C₃₁	1	2	yes	—
91	C₂×C₃₁	#1367	1	4	yes	direct product factor
92	C₂×C₃₁	C₂×#334	1	1	yes	characteristic
93	C₂×C₃₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
94	C₂×C₃₁	C₂×#364	1	1	yes	characteristic
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	C₈×C₃₁	1	1	yes	characteristic
97	C₂²×C₈	C₈×C₃₁	1	1	yes	characteristic
98	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
99	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	4	4	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	C₈×C₃₁	1	2	yes	—
105	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
106	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
107	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
108	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
109	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
110	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
111	C₃₁	C₂×#1367	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
114	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₁₆×C₃₁	1	4	yes	complemented
120	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
121	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
122	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
123	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
124	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
125	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
126	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
127	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
128	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
129	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
130	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
131	C₁₆	—	4	4	no	—
132	C₂³	C₃₁×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₄	C₃₁×C₄⋊C₈	1	1	yes	characteristic
137	C₂×C₄	C₃₁×#106	1	2	yes	—
138	C₂×C₄	C₃₁×#108	1	2	yes	—
139	C₂×C₄	C₂×C₁₆×C₃₁	1	2	yes	—
140	C₂×C₄	C₃₁×M₃₂	1	2	yes	—
141	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
143	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	2	no	—
148	C₈	C₃₁×#106	1	2	yes	—
149	C₈	C₂×C₁₆×C₃₁	1	4	yes	complemented
150	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
151	C₂²	C₃₁×#334	1	2	yes	—
152	C₂²	C₃₁×C₄⋊C₁₆	1	2	yes	—
153	C₂²	C₃₁×#364	1	2	yes	—
154	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	C₃₁×#334	1	2	yes	—
158	C₄	C₃₁×C₄⋊C₁₆	1	2	yes	—
159	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
160	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
161	C₄	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
162	C₄	C₂×C₃₁×M₃₂	1	1	yes	characteristic
163	C₂	C₃₁×#1367	1	4	yes	direct product factor
164	C₂	C₂×C₃₁×#334	1	1	yes	characteristic
165	C₂	C₂×C₃₁×C₄⋊C₁₆	1	1	yes	characteristic
166	C₂	C₂×C₃₁×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₃₁	D₄
commutator	C₄	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1367	C₂	direct (N × H)
35. C₂×C₈×C₃₁	C₁₆	semidirect
48. C₂×#1367	C₃₁	direct (N × H)
66. C₈×C₃₁	C₂×C₁₆	semidirect
70. #1367	C₂×C₃₁	direct (N × H)
91. C₂×C₃₁	#1367	direct (N × H)
111. C₃₁	C₂×#1367	direct (N × H)
119. C₂×C₈	C₁₆×C₃₁	semidirect
149. C₈	C₂×C₁₆×C₃₁	semidirect
163. C₂	C₃₁×#1367	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×C₄⋊C₁₆	2
C₂×C₈²×C₃₁	2
C₃₁×#1367	2
C₂×#1367	31

Subgroups — C2×C31×#1367 #21468754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1367 #21468754