Subgroups — C₂²×Q₈×C₆₇⋊C₈ #31720183

Order 17152.

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	C₂×Q₈×C₆₇⋊C₈	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	3	yes	maximal
3	C₂²×C₄×C₆₇⋊C₈	C₂	1	3	yes	maximal
4	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	3	yes	—
6	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	Q₈×C₆₇⋊C₈	C₂²	1	64	yes	direct product factor
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	36	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	36	yes	—
10	C₂³×Q₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₇	C₄	1	12	yes	—
12	C₂×C₄×Q₈×C₆₇	C₂²	1	12	yes	—
13	C₂²×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
15	C₂²×C₆₇⋊C₈	—	2	12	no	—
16	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	direct product factor
17	C₂²×C₆₇⋊C₈	C₂³	1	18	yes	—
18	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
19	C₄⋊(C₆₇⋊C₈)	C₂³	1	48	yes	—
20	C₄×C₆₇⋊C₈	C₂³	1	48	yes	—
21	C₂²×Q₈×C₆₇	C₈	1	16	yes	complemented
22	C₂²×Q₈×C₆₇	C₂×C₄	1	12	yes	—
23	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₆₇	C₂×C₄	1	48	yes	—
27	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	36	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	18	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	18	yes	—
31	C₂×C₆₇⋊C₈	—	2	36	no	—
32	C₂×C₆₇⋊C₈	C₂×Q₈	1	24	yes	direct product factor
33	C₂×C₆₇⋊C₈	C₂⁴	1	12	yes	—
34	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
35	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×Q₈×C₆₇	C₂×C₈	1	96	yes	complemented
37	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
38	C₂²×C₄×C₆₇	—	2	12	no	—
39	C₂²×C₄×C₆₇	C₂×C₈	1	24	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	18	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	18	yes	—
43	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₆₇	#41	1	4	yes	—
45	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
46	C₄⋊C₄×C₆₇	C₂²×C₄	1	48	yes	—
47	C₄²×C₆₇	C₂²×C₄	1	36	yes	—
48	C₄²×C₆₇	C₂⁴	1	12	yes	—
49	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
50	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
51	C₂³×C₆₇	#131	1	4	yes	—
52	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
53	Q₈×C₆₇	C₂²×C₈	1	64	yes	complemented
54	C₂×C₄×C₆₇	—	2	36	no	—
55	C₂×C₄×C₆₇	C₄×Q₈	1	12	yes	—
56	C₂×C₄×C₆₇	C₂²×C₈	1	72	yes	—
57	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
58	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
59	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
60	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
61	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
62	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
63	C₆₇⋊C₈	—	2	12	no	—
64	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	direct product factor
65	C₂²×C₆₇	C₈×Q₈	1	16	yes	complemented
66	C₂²×C₆₇	C₂×C₄×Q₈	1	6	yes	—
67	C₂²×C₆₇	C₂³×C₈	1	6	yes	—
68	C₂²×C₆₇	C₂×#131	1	6	yes	—
69	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
70	C₄×C₆₇	—	2	12	no	—
71	C₄×C₆₇	C₂×C₄×Q₈	1	6	yes	—
72	C₄×C₆₇	C₂³×C₈	1	24	yes	—
73	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
74	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
75	C₂²×C₈×Q₈	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	C₂×C₈×Q₈	1	12	yes	complemented
77	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
78	C₂×C₆₇	C₂⁴×C₈	1	1	yes	characteristic
79	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
80	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
81	C₂×C₈×Q₈	—	67	24	no	—
82	C₂²×C₄⋊C₈	—	67	3	no	—
83	C₂²×C₄×C₈	—	67	3	no	—
84	C₆₇	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
85	C₂³×Q₈	Dic₆₇	1	1	yes	characteristic
86	C₂³×C₈	—	67	1	no	—
87	C₂³×C₈	—	67	3	no	—
88	C₂×C₄×Q₈	Dic₆₇	1	12	yes	—
89	C₂×C₄×Q₈	C₂×D₆₇	1	12	yes	—
90	C₂²×C₄⋊C₄	Dic₆₇	1	3	yes	—
91	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
92	C₈×Q₈	—	67	64	no	—
93	C₂×C₄⋊C₈	—	67	36	no	—
94	C₂×C₄×C₈	—	67	36	no	—
95	C₂²×Q₈	C₆₇⋊C₈	1	16	yes	direct product factor
96	C₂²×Q₈	C₂×Dic₆₇	1	12	yes	—
97	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
98	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
99	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	C₂²×C₈	—	67	8	no	—
101	C₂²×C₈	—	67	6	no	—
102	C₂²×C₈	—	67	18	no	—
103	C₂²×C₈	—	134	12	no	—
104	C₄×Q₈	C₂×Dic₆₇	1	48	yes	—
105	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
106	C₂×C₄⋊C₄	C₂×Dic₆₇	1	36	yes	—
107	C₂×C₄²	C₂×Dic₆₇	1	18	yes	—
108	C₂×C₄²	C₂²×D₆₇	1	18	yes	—
109	C₄⋊C₈	—	67	48	no	—
110	C₄×C₈	—	67	48	no	—
111	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
112	C₂×Q₈	C₂×C₆₇⋊C₈	1	96	yes	direct product factor
113	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	C₂×C₆₇⋊C₈	1	24	yes	—
116	C₂²×C₄	C₂²×Dic₆₇	1	18	yes	—
117	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
118	C₂²×C₄	C₂²×Dic₆₇	1	18	yes	—
119	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
120	C₂²×C₄	#11761629	1	4	yes	—
121	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
122	C₂×C₈	—	67	24	no	—
123	C₂×C₈	—	67	4	no	—
124	C₂×C₈	—	67	12	no	—
125	C₂×C₈	—	134	36	no	—
126	C₄⋊C₄	C₂²×Dic₆₇	1	48	yes	—
127	C₄²	C₂²×Dic₆₇	1	36	yes	—
128	C₄²	C₂³×D₆₇	1	12	yes	—
129	C₂³	Q₈×Dic₆₇	1	4	yes	—
130	C₂³	C₂²×C₆₇⋊C₈	1	4	yes	—
131	C₂³	#14524050	1	4	yes	—
132	C₂³	C₂³×Dic₆₇	1	3	yes	—
133	Q₈	C₂²×C₆₇⋊C₈	1	64	yes	direct product factor
134	C₂×C₄	—	2	36	no	—
135	C₂×C₄	Q₈×Dic₆₇	1	12	yes	—
136	C₂×C₄	C₂²×C₆₇⋊C₈	1	72	yes	—
137	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
138	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
139	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
140	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
141	C₂×C₄	C₂×#11761629	1	6	yes	—
142	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
143	C₈	—	67	8	no	—
144	C₈	—	134	12	no	—
145	C₂²	Q₈×C₆₇⋊C₈	1	16	yes	direct product factor
146	C₂²	C₂×Q₈×Dic₆₇	1	6	yes	—
147	C₂²	C₂³×C₆₇⋊C₈	1	6	yes	—
148	C₂²	C₂×#14524050	1	6	yes	—
149	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
150	C₄	—	2	12	no	—
151	C₄	C₂×Q₈×Dic₆₇	1	6	yes	—
152	C₄	C₂³×C₆₇⋊C₈	1	24	yes	—
153	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
154	C₄	C₂²×#11761629	1	1	yes	characteristic
155	C₂	C₂×Q₈×C₆₇⋊C₈	1	12	yes	direct product factor
156	C₂	C₂²×Q₈×Dic₆₇	1	1	yes	characteristic
157	C₂	C₂⁴×C₆₇⋊C₈	1	1	yes	characteristic
158	C₂	C₂²×#14524050	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₆₇⋊C₈	C₂	direct (N × H)
7. Q₈×C₆₇⋊C₈	C₂²	direct (N × H)
16. C₂²×C₆₇⋊C₈	Q₈	direct (N × H)
21. C₂²×Q₈×C₆₇	C₈	semidirect
32. C₂×C₆₇⋊C₈	C₂×Q₈	direct (N × H)
36. C₂×Q₈×C₆₇	C₂×C₈	semidirect
53. Q₈×C₆₇	C₂²×C₈	semidirect
64. C₆₇⋊C₈	C₂²×Q₈	direct (N × H)
65. C₂²×C₆₇	C₈×Q₈	semidirect
76. C₂×C₆₇	C₂×C₈×Q₈	semidirect
84. C₆₇	C₂²×C₈×Q₈	semidirect
95. C₂²×Q₈	C₆₇⋊C₈	direct (N × H)
112. C₂×Q₈	C₂×C₆₇⋊C₈	direct (N × H)
133. Q₈	C₂²×C₆₇⋊C₈	direct (N × H)
145. C₂²	Q₈×C₆₇⋊C₈	direct (N × H)
155. C₂	C₂×Q₈×C₆₇⋊C₈	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×Q₈×C₆₇⋊C₈	2
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂²×C₄×C₆₇⋊C₈	2
C₂²×C₄×Q₈×C₆₇	2
C₂²×C₈×Q₈	67

Subgroups — C22×Q8×C67⋊C8 #31720183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×C₆₇⋊C₈ #31720183