Subgroups — C₂×C₄⋊C₈×D₃₁ #16854260

Order 3968.

640 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₃₁	C₂	1	2	yes	maximal
2	C₂×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×D₃₁	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₁	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₃₁	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×D₃₁	—	2	8	no	—
17	C₂×C₈×D₃₁	C₂²	1	8	yes	—
18	C₄²×D₃₁	C₄	1	4	yes	—
19	C₄²×D₃₁	C₂²	1	4	yes	—
20	C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
21	Dic₃₁⋊C₈	C₂²	1	8	yes	complemented
22	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
23	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₃₁	—	2	4	no	—
31	C₂×C₄×D₃₁	C₈	1	8	yes	complemented
32	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₃₁	D₄	1	4	yes	—
36	C₂×C₄×D₃₁	Q₈	1	4	yes	—
37	C₂×C₄×D₃₁	C₂³	1	2	yes	—
38	C₂×C₈×C₃₁	—	2	4	no	—
39	C₂×C₈×C₃₁	C₂³	1	4	yes	—
40	C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₄²×C₃₁	C₂³	1	2	yes	—
42	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
43	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
44	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
45	C₄×Dic₃₁	C₂³	1	2	yes	—
46	C₂×C₃₁⋊C₈	—	2	4	no	—
47	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
48	C₈×D₃₁	—	2	16	no	—
49	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×D₃₁	C₄⋊C₄	1	4	yes	—
51	C₂²×D₃₁	C₂×C₈	1	4	yes	—
52	C₂²×D₃₁	M₁₆	1	4	yes	—
53	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₃₁	—	2	2	no	—
55	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	complemented
56	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₃₁	—	2	2	no	—
63	C₂×Dic₃₁	C₄⋊C₄	1	4	yes	—
64	C₂×Dic₃₁	C₂×C₈	1	4	yes	complemented
65	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
67	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
68	C₄×D₃₁	—	2	8	no	—
69	C₄×D₃₁	C₄⋊C₄	1	8	yes	—
70	C₄×D₃₁	C₂×C₈	1	16	yes	complemented
71	C₄×D₃₁	C₂×D₄	1	4	yes	—
72	C₄×D₃₁	C₂×Q₈	1	4	yes	—
73	C₈×C₃₁	—	2	4	no	—
74	C₃₁⋊C₈	—	2	4	no	—
75	C₂²×C₄⋊C₈	—	31	1	no	maximal, Sylow
76	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
78	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
79	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
80	C₂×D₃₁	C₄⋊C₈	1	16	yes	direct product factor
81	C₂×D₃₁	C₂×C₄⋊C₄	1	4	yes	—
82	C₂×D₃₁	C₂²×C₈	1	4	yes	—
83	C₂×D₃₁	C₂×M₁₆	1	4	yes	—
84	C₄×C₃₁	—	2	2	no	—
85	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₄×C₃₁	C₂²×C₈	1	4	yes	complemented
87	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
89	Dic₃₁	—	2	2	no	—
90	Dic₃₁	C₂×C₄⋊C₄	1	4	yes	—
91	Dic₃₁	C₂²×C₈	1	4	yes	complemented
92	C₂³×C₈	—	31	2	no	—
93	C₂²×C₄²	—	31	1	no	—
94	C₂×C₄⋊C₈	—	31	2	no	—
95	C₂×C₄⋊C₈	—	31	1	no	—
96	C₂×C₄⋊C₈	—	31	8	no	—
97	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic, direct product factor
98	C₂×C₃₁	C₂×C₄⋊C₈	1	4	yes	complemented
99	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
101	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
102	D₃₁	C₂×C₄⋊C₈	1	8	yes	direct product factor
103	C₂³×C₄	—	31	1	no	—
104	C₂³×C₄	—	31	1	no	—
105	C₂³×C₄	—	31	1	no	—
106	C₂²×C₈	—	31	2	no	—
107	C₂²×C₈	—	31	8	no	—
108	C₂²×C₈	—	62	8	no	—
109	C₂²×C₈	C₂×D₃₁	1	2	yes	—
110	C₂×C₄²	—	31	1	no	—
111	C₂×C₄²	—	31	1	no	—
112	C₂×C₄²	—	31	1	no	—
113	C₂×C₄²	—	31	4	no	—
114	C₂×C₄²	—	31	4	no	—
115	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
116	C₄⋊C₈	—	31	8	no	—
117	C₄⋊C₈	—	31	4	no	—
118	C₄⋊C₈	C₂×D₃₁	1	4	yes	direct product factor
119	C₃₁	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
120	C₂⁴	—	31	1	no	—
121	C₂²×C₄	—	31	1	no	—
122	C₂²×C₄	—	31	1	no	—
123	C₂²×C₄	—	31	8	no	—
124	C₂²×C₄	—	31	2	no	—
125	C₂²×C₄	—	31	4	no	—
126	C₂²×C₄	—	31	4	no	—
127	C₂²×C₄	—	31	4	no	—
128	C₂²×C₄	—	31	2	no	—
129	C₂²×C₄	—	31	1	no	—
130	C₂²×C₄	—	31	4	no	—
131	C₂²×C₄	—	62	4	no	—
132	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	31	4	no	—
137	C₂×C₈	—	62	16	no	—
138	C₂×C₈	—	62	4	no	—
139	C₂×C₈	C₂²×D₃₁	1	4	yes	—
140	C₄²	—	31	4	no	—
141	C₄²	—	31	2	no	—
142	C₄²	—	31	4	no	—
143	C₄²	—	31	2	no	—
144	C₄²	C₄×D₃₁	1	2	yes	—
145	C₄²	C₂²×D₃₁	1	2	yes	—
146	C₂³	—	31	4	no	—
147	C₂³	—	31	4	no	—
148	C₂³	—	31	4	no	—
149	C₂³	—	31	2	no	—
150	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	31	4	no	—
153	C₂×C₄	—	31	4	no	—
154	C₂×C₄	—	31	2	no	—
155	C₂×C₄	—	31	2	no	—
156	C₂×C₄	—	31	8	no	—
157	C₂×C₄	—	31	16	no	—
158	C₂×C₄	—	31	4	no	—
159	C₂×C₄	—	31	4	no	—
160	C₂×C₄	—	31	2	no	—
161	C₂×C₄	—	62	8	no	—
162	C₂×C₄	—	62	2	no	—
163	C₂×C₄	C₈×D₃₁	1	4	yes	complemented
164	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
167	C₂×C₄	D₄×D₃₁	1	2	yes	—
168	C₂×C₄	Q₈×D₃₁	1	2	yes	—
169	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	62	4	no	—
172	C₂²	—	31	4	no	—
173	C₂²	—	31	4	no	—
174	C₂²	—	31	4	no	—
175	C₂²	—	31	16	no	—
176	C₂²	C₄⋊C₄×D₃₁	1	2	yes	—
177	C₂²	C₂×C₈×D₃₁	1	2	yes	—
178	C₂²	M₁₆×D₃₁	1	2	yes	—
179	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	31	4	no	—
182	C₄	—	31	4	no	—
183	C₄	—	62	2	no	—
184	C₄	C₄⋊C₄×D₃₁	1	2	yes	—
185	C₄	C₂×C₈×D₃₁	1	4	yes	complemented
186	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
187	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
188	C₂	—	31	8	no	—
189	C₂	C₄⋊C₈×D₃₁	1	4	yes	direct product factor
190	C₂	C₂×C₄⋊C₄×D₃₁	1	1	yes	characteristic
191	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic
192	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
4. C₂×Dic₃₁⋊C₈	C₂	semidirect
5. C₄⋊C₈×D₃₁	C₂	direct (N × H)
6. C₂×C₃₁×C₄⋊C₈	C₂	semidirect
20. C₃₁×C₄⋊C₈	C₂²	semidirect
21. Dic₃₁⋊C₈	C₂²	semidirect
22. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
31. C₂×C₄×D₃₁	C₈	semidirect
55. C₂×C₄×C₃₁	C₂×C₈	semidirect
64. C₂×Dic₃₁	C₂×C₈	semidirect
70. C₄×D₃₁	C₂×C₈	semidirect
80. C₂×D₃₁	C₄⋊C₈	direct (N × H)
86. C₄×C₃₁	C₂²×C₈	semidirect
91. Dic₃₁	C₂²×C₈	semidirect
97. C₂×C₄⋊C₈	D₃₁	direct (N × H)
98. C₂×C₃₁	C₂×C₄⋊C₈	semidirect
102. D₃₁	C₂×C₄⋊C₈	direct (N × H)
118. C₄⋊C₈	C₂×D₃₁	direct (N × H)
119. C₃₁	C₂²×C₄⋊C₈	semidirect
163. C₂×C₄	C₈×D₃₁	semidirect
185. C₄	C₂×C₈×D₃₁	semidirect
189. C₂	C₄⋊C₈×D₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₃₁	2
C₂×C₄²×D₃₁	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×Dic₃₁⋊C₈	2
C₄⋊C₈×D₃₁	2
C₂×C₃₁×C₄⋊C₈	2
C₂²×C₄⋊C₈	31

Subgroups — C2×C4⋊C8×D31 #16854260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₈×D₃₁ #16854260