Subgroups — C₂×Q₈×C₆₇⋊C₁₆ #31009313

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₆₇⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	3	yes	maximal
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	3	yes	maximal
4	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	3	yes	—
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	12	yes	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	12	yes	—
9	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
10	C₈×Q₈×C₆₇	C₄	1	4	yes	—
11	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₄	1	3	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
14	C₂×C₆₇⋊C₁₆	—	2	6	no	—
15	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	direct product factor
16	C₂×C₆₇⋊C₁₆	C₂³	1	6	yes	—
17	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
18	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
19	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
20	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
21	C₄×Q₈×C₆₇	C₈	1	4	yes	—
22	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₆₇	C₈	1	3	yes	—
24	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
25	C₄⋊C₈×C₆₇	C₂×C₄	1	12	yes	—
26	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
27	C₄×C₈×C₆₇	C₂³	1	6	yes	—
28	C₆₇⋊C₁₆	—	2	6	no	—
29	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	direct product factor
30	C₂×Q₈×C₆₇	C₁₆	1	8	yes	complemented
31	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
32	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
33	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	—	2	6	no	—
36	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
37	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
39	C₂×C₈×C₆₇	#41	1	2	yes	—
40	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
41	C₄⋊C₄×C₆₇	C₂×C₈	1	12	yes	—
42	C₄²×C₆₇	C₂×C₈	1	6	yes	—
43	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
44	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
45	Q₈×C₆₇	C₂×C₁₆	1	16	yes	complemented
46	C₂×C₄×C₆₇	—	2	6	no	—
47	C₂×C₄×C₆₇	C₂×C₁₆	1	12	yes	—
48	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
49	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
50	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
52	C₂×C₄×C₆₇	#131	1	2	yes	—
53	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
54	C₈×C₆₇	—	2	6	no	—
55	C₈×C₆₇	C₄×Q₈	1	2	yes	—
56	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
57	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
58	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
59	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
60	C₂²×C₆₇	#504	1	2	yes	—
61	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
62	C₄×C₆₇	—	2	6	no	—
63	C₄×C₆₇	C₈×Q₈	1	2	yes	—
64	C₄×C₆₇	C₂²×C₁₆	1	12	yes	—
65	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
67	C₂×Q₈×C₁₆	—	67	1	no	maximal, Sylow
68	C₂×C₆₇	Q₈×C₁₆	1	4	yes	complemented
69	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
70	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
71	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
72	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
73	Q₈×C₁₆	—	67	8	no	—
74	C₂×C₄⋊C₁₆	—	67	3	no	—
75	C₂×C₄×C₁₆	—	67	3	no	—
76	C₆₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
78	C₂²×C₁₆	—	67	1	no	—
79	C₂²×C₁₆	—	67	3	no	—
80	C₈×Q₈	Dic₆₇	1	4	yes	—
81	C₈×Q₈	C₂×D₆₇	1	4	yes	—
82	C₂×C₄⋊C₈	Dic₆₇	1	3	yes	—
83	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
84	C₄⋊C₁₆	—	67	12	no	—
85	C₄×C₁₆	—	67	12	no	—
86	C₂²×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
87	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
88	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
89	C₄×Q₈	C₆₇⋊C₈	1	4	yes	—
90	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
91	C₂×C₄⋊C₄	C₆₇⋊C₈	1	3	yes	—
92	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
93	C₂×C₁₆	—	67	4	no	—
94	C₂×C₁₆	—	67	2	no	—
95	C₂×C₁₆	—	67	6	no	—
96	C₂×C₁₆	—	134	6	no	—
97	C₄⋊C₈	C₂×Dic₆₇	1	12	yes	—
98	C₄×C₈	C₂×Dic₆₇	1	6	yes	—
99	C₄×C₈	C₂²×D₆₇	1	6	yes	—
100	C₂×Q₈	C₆₇⋊C₁₆	1	8	yes	direct product factor
101	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
102	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
103	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
104	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
105	C₂×C₈	—	2	6	no	—
106	C₂×C₈	C₂²×Dic₆₇	1	6	yes	—
107	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
108	C₂×C₈	Q₈×D₆₇	1	2	yes	—
109	C₂×C₈	#11761629	1	2	yes	—
110	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
111	C₄⋊C₄	C₂×C₆₇⋊C₈	1	12	yes	—
112	C₄²	C₂×C₆₇⋊C₈	1	6	yes	—
113	C₄²	C₂²×Dic₆₇	1	6	yes	—
114	C₁₆	—	67	4	no	—
115	C₁₆	—	134	6	no	—
116	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
117	Q₈	C₂×C₆₇⋊C₁₆	1	16	yes	direct product factor
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	C₂×C₆₇⋊C₁₆	1	12	yes	—
120	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
121	C₂×C₄	C₂²×C₆₇⋊C₈	1	6	yes	—
122	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
123	C₂×C₄	C₂²×C₆₇⋊C₈	1	6	yes	—
124	C₂×C₄	#14524050	1	2	yes	—
125	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
126	C₈	—	2	6	no	—
127	C₈	Q₈×Dic₆₇	1	2	yes	—
128	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
129	C₈	C₂×#11761629	1	1	yes	characteristic
130	C₂²	Q₈×C₆₇⋊C₈	1	2	yes	—
131	C₂²	C₂²×C₆₇⋊C₁₆	1	2	yes	—
132	C₂²	#16863813	1	2	yes	—
133	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
134	C₄	—	2	6	no	—
135	C₄	Q₈×C₆₇⋊C₈	1	2	yes	—
136	C₄	C₂²×C₆₇⋊C₁₆	1	12	yes	—
137	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
138	C₄	C₂×#14524050	1	1	yes	characteristic
139	C₂	Q₈×C₆₇⋊C₁₆	1	4	yes	direct product factor
140	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
141	C₂	C₂³×C₆₇⋊C₁₆	1	1	yes	characteristic
142	C₂	C₂×#16863813	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×C₆₇⋊C₁₆	C₂	direct (N × H)
15. C₂×C₆₇⋊C₁₆	Q₈	direct (N × H)
29. C₆₇⋊C₁₆	C₂×Q₈	direct (N × H)
30. C₂×Q₈×C₆₇	C₁₆	semidirect
45. Q₈×C₆₇	C₂×C₁₆	semidirect
68. C₂×C₆₇	Q₈×C₁₆	semidirect
76. C₆₇	C₂×Q₈×C₁₆	semidirect
100. C₂×Q₈	C₆₇⋊C₁₆	direct (N × H)
117. Q₈	C₂×C₆₇⋊C₁₆	direct (N × H)
139. C₂	Q₈×C₆₇⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×C₆₇⋊C₁₆	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×C₈×Q₈×C₆₇	2
C₂×Q₈×C₁₆	67

Subgroups — C2×Q8×C67⋊C16 #31009313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₆₇⋊C₁₆ #31009313