Subgroups — C₂²×C₇×C₈×Q₈ #13237401

Order 1792.

2478 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×Q₈	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₇×C₄⋊C₈	C₂	1	3	yes	maximal
4	C₂²×C₄×C₇×C₈	C₂	1	3	yes	maximal
5	C₂³×C₇×Q₈	C₄	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₂²	1	3	yes	—
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇×Q₈	C₄	1	12	yes	—
9	C₂×C₄×C₇×Q₈	C₂²	1	12	yes	—
10	C₂²×C₇×C₄⋊C₄	C₄	1	3	yes	—
11	C₂²×C₄²×C₇	C₂²	1	3	yes	—
12	C₇×C₈×Q₈	C₂²	1	64	yes	direct product factor
13	C₂×C₇×C₄⋊C₈	C₂²	1	36	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	36	yes	—
15	C₂²×C₈×Q₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×Q₈	C₈	1	16	yes	direct product factor
17	C₂²×C₇×Q₈	C₂×C₄	1	12	yes	—
18	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂²×C₇×C₈	—	2	12	no	—
22	C₂²×C₇×C₈	Q₈	1	8	yes	direct product factor
23	C₂²×C₇×C₈	C₂³	1	18	yes	—
24	C₂²×C₇×C₈	C₂³	1	6	yes	—
25	C₄×C₇×Q₈	C₂×C₄	1	48	yes	—
26	C₄×C₇×Q₈	C₂³	1	16	yes	—
27	C₂×C₇×C₄⋊C₄	C₂×C₄	1	36	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	18	yes	—
29	C₂×C₄²×C₇	C₂³	1	18	yes	—
30	C₇×C₄⋊C₈	C₂³	1	48	yes	—
31	C₄×C₇×C₈	C₂³	1	48	yes	—
32	C₂²×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
33	C₂×C₈×Q₈	C₂×C₇	1	24	yes	direct product factor
34	C₂²×C₄⋊C₈	C₂×C₇	1	3	yes	—
35	C₂²×C₄×C₈	C₂×C₇	1	3	yes	—
36	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₇×Q₈	C₂×C₈	1	96	yes	direct product factor
38	C₂×C₇×Q₈	C₂²×C₄	1	16	yes	—
39	C₂²×C₄×C₇	—	2	12	no	—
40	C₂²×C₄×C₇	C₂×C₈	1	24	yes	—
41	C₂²×C₄×C₇	C₂²×C₄	1	18	yes	—
42	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₇	C₂²×C₄	1	18	yes	—
44	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₇	#41	1	4	yes	—
46	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
47	C₂×C₇×C₈	—	2	36	no	—
48	C₂×C₇×C₈	C₂×Q₈	1	24	yes	direct product factor
49	C₂×C₇×C₈	C₂⁴	1	12	yes	—
50	C₂×C₇×C₈	C₂⁴	1	4	yes	—
51	C₇×C₄⋊C₄	C₂²×C₄	1	48	yes	—
52	C₄²×C₇	C₂²×C₄	1	36	yes	—
53	C₄²×C₇	C₂⁴	1	12	yes	—
54	C₂³×Q₈	C₄×C₇	1	1	yes	characteristic
55	C₂³×C₈	C₂²×C₇	1	3	yes	—
56	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
57	C₂×C₄×Q₈	C₄×C₇	1	12	yes	—
58	C₂×C₄×Q₈	C₂²×C₇	1	12	yes	—
59	C₂²×C₄⋊C₄	C₄×C₇	1	3	yes	—
60	C₂²×C₄²	C₂²×C₇	1	3	yes	—
61	C₈×Q₈	C₂²×C₇	1	64	yes	direct product factor
62	C₂×C₄⋊C₈	C₂²×C₇	1	36	yes	—
63	C₂×C₄×C₈	C₂²×C₇	1	36	yes	—
64	C₂³×C₇	C₄×Q₈	1	4	yes	—
65	C₂³×C₇	C₂²×C₈	1	4	yes	—
66	C₂³×C₇	#131	1	4	yes	—
67	C₂³×C₇	C₂³×C₄	1	3	yes	—
68	C₇×Q₈	C₂²×C₈	1	64	yes	direct product factor
69	C₂×C₄×C₇	—	2	36	no	—
70	C₂×C₄×C₇	C₄×Q₈	1	12	yes	—
71	C₂×C₄×C₇	C₂²×C₈	1	72	yes	—
72	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
73	C₂×C₄×C₇	C₂³×C₄	1	3	yes	—
74	C₂×C₄×C₇	C₂³×C₄	1	12	yes	—
75	C₂×C₄×C₇	C₂²×Q₈	1	6	yes	—
76	C₂×C₄×C₇	C₂×#41	1	6	yes	—
77	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
78	C₇×C₈	—	2	12	no	—
79	C₇×C₈	C₂²×Q₈	1	8	yes	direct product factor
80	C₂²×Q₈	C₇×C₈	1	16	yes	direct product factor
81	C₂²×Q₈	C₂×C₄×C₇	1	12	yes	—
82	C₂³×C₄	C₂×C₄×C₇	1	3	yes	—
83	C₂³×C₄	C₂×C₄×C₇	1	3	yes	—
84	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
85	C₂²×C₈	—	2	12	no	—
86	C₂²×C₈	C₇×Q₈	1	8	yes	direct product factor
87	C₂²×C₈	C₂³×C₇	1	18	yes	—
88	C₂²×C₈	C₂³×C₇	1	6	yes	—
89	C₄×Q₈	C₂×C₄×C₇	1	48	yes	—
90	C₄×Q₈	C₂³×C₇	1	16	yes	—
91	C₂×C₄⋊C₄	C₂×C₄×C₇	1	36	yes	—
92	C₂×C₄²	C₂×C₄×C₇	1	18	yes	—
93	C₂×C₄²	C₂³×C₇	1	18	yes	—
94	C₄⋊C₈	C₂³×C₇	1	48	yes	—
95	C₄×C₈	C₂³×C₇	1	48	yes	—
96	C₂²×C₇	C₈×Q₈	1	16	yes	direct product factor
97	C₂²×C₇	C₂×C₄×Q₈	1	6	yes	—
98	C₂²×C₇	C₂³×C₈	1	6	yes	—
99	C₂²×C₇	C₂×#131	1	6	yes	—
100	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₇	—	2	12	no	—
102	C₄×C₇	C₂×C₄×Q₈	1	6	yes	—
103	C₄×C₇	C₂³×C₈	1	24	yes	—
104	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
105	C₄×C₇	C₂²×#41	1	1	yes	characteristic
106	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
107	C₂×Q₈	C₂×C₇×C₈	1	96	yes	direct product factor
108	C₂×Q₈	C₂²×C₄×C₇	1	16	yes	—
109	C₂²×C₄	—	2	12	no	—
110	C₂²×C₄	C₂×C₇×C₈	1	24	yes	—
111	C₂²×C₄	C₂²×C₄×C₇	1	18	yes	—
112	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
113	C₂²×C₄	C₂²×C₄×C₇	1	18	yes	—
114	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
115	C₂²×C₄	C₇×#41	1	4	yes	—
116	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
117	C₂×C₈	—	2	36	no	—
118	C₂×C₈	C₂×C₇×Q₈	1	24	yes	direct product factor
119	C₂×C₈	C₂⁴×C₇	1	12	yes	—
120	C₂×C₈	C₂⁴×C₇	1	4	yes	—
121	C₄⋊C₄	C₂²×C₄×C₇	1	48	yes	—
122	C₄²	C₂²×C₄×C₇	1	36	yes	—
123	C₄²	C₂⁴×C₇	1	12	yes	—
124	C₂×C₇	C₂×C₈×Q₈	1	12	yes	direct product factor
125	C₂×C₇	C₂²×C₄×Q₈	1	1	yes	characteristic
126	C₂×C₇	C₂⁴×C₈	1	1	yes	characteristic
127	C₂×C₇	C₂²×#131	1	1	yes	characteristic
128	C₂³	C₄×C₇×Q₈	1	4	yes	—
129	C₂³	C₂²×C₇×C₈	1	4	yes	—
130	C₂³	C₇×#131	1	4	yes	—
131	C₂³	C₂³×C₄×C₇	1	3	yes	—
132	Q₈	C₂²×C₇×C₈	1	64	yes	direct product factor
133	C₂×C₄	—	2	36	no	—
134	C₂×C₄	C₄×C₇×Q₈	1	12	yes	—
135	C₂×C₄	C₂²×C₇×C₈	1	72	yes	—
136	C₂×C₄	C₂³×C₄×C₇	1	12	yes	—
137	C₂×C₄	C₂³×C₄×C₇	1	3	yes	—
138	C₂×C₄	C₂³×C₄×C₇	1	12	yes	—
139	C₂×C₄	C₂²×C₇×Q₈	1	6	yes	—
140	C₂×C₄	C₂×C₇×#41	1	6	yes	—
141	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
142	C₈	—	2	12	no	—
143	C₈	C₂²×C₇×Q₈	1	8	yes	direct product factor
144	C₇	C₂²×C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²	C₇×C₈×Q₈	1	16	yes	direct product factor
146	C₂²	C₂×C₄×C₇×Q₈	1	6	yes	—
147	C₂²	C₂³×C₇×C₈	1	6	yes	—
148	C₂²	C₂×C₇×#131	1	6	yes	—
149	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂×C₄×C₇×Q₈	1	6	yes	—
152	C₄	C₂³×C₇×C₈	1	24	yes	—
153	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
154	C₄	C₂²×C₇×#41	1	1	yes	characteristic
155	C₂	C₂×C₇×C₈×Q₈	1	12	yes	direct product factor
156	C₂	C₂²×C₄×C₇×Q₈	1	1	yes	characteristic
157	C₂	C₂⁴×C₇×C₈	1	1	yes	characteristic
158	C₂	C₂²×C₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₇×C₈	C₂²
commutator	C₂	C₂⁴×C₇×C₈
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle, prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₈×Q₈	C₂	direct (N × H)
12. C₇×C₈×Q₈	C₂²	direct (N × H)
15. C₂²×C₈×Q₈	C₇	direct (N × H)
16. C₂²×C₇×Q₈	C₈	direct (N × H)
22. C₂²×C₇×C₈	Q₈	direct (N × H)
33. C₂×C₈×Q₈	C₂×C₇	direct (N × H)
37. C₂×C₇×Q₈	C₂×C₈	direct (N × H)
48. C₂×C₇×C₈	C₂×Q₈	direct (N × H)
61. C₈×Q₈	C₂²×C₇	direct (N × H)
68. C₇×Q₈	C₂²×C₈	direct (N × H)
79. C₇×C₈	C₂²×Q₈	direct (N × H)
80. C₂²×Q₈	C₇×C₈	direct (N × H)
86. C₂²×C₈	C₇×Q₈	direct (N × H)
96. C₂²×C₇	C₈×Q₈	direct (N × H)
107. C₂×Q₈	C₂×C₇×C₈	direct (N × H)
118. C₂×C₈	C₂×C₇×Q₈	direct (N × H)
124. C₂×C₇	C₂×C₈×Q₈	direct (N × H)
132. Q₈	C₂²×C₇×C₈	direct (N × H)
143. C₈	C₂²×C₇×Q₈	direct (N × H)
144. C₇	C₂²×C₈×Q₈	direct (N × H)
145. C₂²	C₇×C₈×Q₈	direct (N × H)
155. C₂	C₂×C₇×C₈×Q₈	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×C₄×C₇×Q₈	2
C₂×C₇×C₈×Q₈	2
C₂²×C₇×C₄⋊C₈	2
C₂²×C₄×C₇×C₈	2
C₂²×C₈×Q₈	7

Subgroups — C22×C7×C8×Q8 #13237401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×C₈×Q₈ #13237401