Subgroups — C₂×C₁₆×C₆₇⋊C₈ #30764802

Order 17152.

398 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₁₆×C₆₇⋊C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
7	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
10	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₆₇	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
14	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₆₇⋊C₈	C₈	1	4	yes	—
16	C₂×C₆₇⋊C₁₆	C₈	1	8	yes	complemented
17	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₄×C₆₇⋊C₈	C₈	1	4	yes	—
19	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₇	C₈	1	8	yes	complemented
24	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₆₇⋊C₈	C₁₆	1	16	yes	direct product factor
30	C₂×C₆₇⋊C₈	C₂×C₈	1	8	yes	—
31	C₆₇⋊C₁₆	C₂×C₈	1	8	yes	complemented
32	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₆₇	C₄²	1	1	yes	characteristic
41	C₄²×C₆₇	C₂×C₈	1	2	yes	—
42	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₆₇	C₂×C₈	1	8	yes	complemented
44	C₂³×C₆₇	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
48	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×C₁₆	1	8	yes	—
50	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
52	C₈×C₆₇	C₄×C₈	1	2	yes	—
53	C₈×C₆₇	C₄×C₈	1	2	yes	—
54	C₈×C₆₇	C₂²×C₈	1	2	yes	—
55	C₈×C₆₇	C₂²×C₈	1	2	yes	—
56	C₆₇⋊C₈	C₂×C₁₆	1	16	yes	direct product factor
57	C₂²×C₆₇	C₈²	1	2	yes	—
58	C₂²×C₆₇	C₄×C₁₆	1	4	yes	—
59	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₆₇	C₈²	1	2	yes	—
61	C₄×C₆₇	C₄×C₁₆	1	4	yes	—
62	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₆₇	C₂²×C₁₆	1	4	yes	—
65	C₂×C₈×C₁₆	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	C₈×C₁₆	1	4	yes	complemented
67	C₂×C₆₇	C₂×C₈²	1	1	yes	characteristic
68	C₂×C₆₇	C₂×C₄×C₁₆	1	2	yes	—
69	C₂×C₄×C₁₆	—	67	1	no	—
70	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
71	C₂×C₈²	—	67	1	no	—
72	C₈×C₁₆	—	67	4	no	—
73	C₆₇	C₂×C₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	—	67	2	no	—
75	C₂²×C₁₆	Dic₆₇	1	2	yes	—
76	C₂×C₄×C₈	—	67	2	no	—
77	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
78	C₄×C₁₆	—	67	2	no	—
79	C₄×C₁₆	—	67	2	no	—
80	C₄×C₁₆	Dic₆₇	1	2	yes	—
81	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
82	C₈²	—	67	2	no	—
83	C₈²	—	67	2	no	—
84	C₂²×C₈	—	67	4	no	—
85	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
87	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
88	C₂×C₁₆	—	67	8	no	—
89	C₂×C₁₆	—	67	4	no	—
90	C₂×C₁₆	C₆₇⋊C₈	1	8	yes	direct product factor
91	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
92	C₄×C₈	—	67	4	no	—
93	C₄×C₈	—	67	4	no	—
94	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
95	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
99	C₂²×C₄	C₈×D₆₇	1	2	yes	—
100	C₂×C₈	—	67	16	no	—
101	C₂×C₈	—	67	8	no	—
102	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
104	C₂×C₈	C₈×D₆₇	1	4	yes	—
105	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
106	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
107	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
108	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
109	C₄²	C₈×D₆₇	1	2	yes	—
110	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
111	C₁₆	—	67	8	no	—
112	C₁₆	C₂×C₆₇⋊C₈	1	8	yes	direct product factor
113	C₂³	C₈×Dic₆₇	1	1	yes	characteristic
114	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
115	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
116	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
117	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
118	C₂×C₄	C₁₆×D₆₇	1	8	yes	—
119	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
120	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
121	C₈	—	67	16	no	—
122	C₈	C₈×Dic₆₇	1	2	yes	—
123	C₈	C₄×C₆₇⋊C₈	1	2	yes	—
124	C₈	C₂×C₈×D₆₇	1	2	yes	—
125	C₈	C₂²×C₆₇⋊C₈	1	2	yes	—
126	C₂²	C₈×C₆₇⋊C₈	1	2	yes	—
127	C₂²	C₁₆×Dic₆₇	1	4	yes	—
128	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
129	C₄	C₈×C₆₇⋊C₈	1	2	yes	—
130	C₄	C₁₆×Dic₆₇	1	4	yes	—
131	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
132	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
133	C₄	C₂×C₁₆×D₆₇	1	4	yes	—
134	C₂	C₁₆×C₆₇⋊C₈	1	4	yes	direct product factor
135	C₂	C₂×C₈×C₆₇⋊C₈	1	1	yes	characteristic
136	C₂	C₂×C₁₆×Dic₆₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₆₇
commutator, hypocenter	C₆₇	C₂×C₈×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₁₆×C₆₇⋊C₈	C₂	direct (N × H)
16. C₂×C₆₇⋊C₁₆	C₈	semidirect
23. C₂×C₁₆×C₆₇	C₈	semidirect
29. C₂×C₆₇⋊C₈	C₁₆	direct (N × H)
31. C₆₇⋊C₁₆	C₂×C₈	semidirect
43. C₁₆×C₆₇	C₂×C₈	semidirect
56. C₆₇⋊C₈	C₂×C₁₆	direct (N × H)
66. C₂×C₆₇	C₈×C₁₆	semidirect
73. C₆₇	C₂×C₈×C₁₆	semidirect
90. C₂×C₁₆	C₆₇⋊C₈	direct (N × H)
112. C₁₆	C₂×C₆₇⋊C₈	direct (N × H)
134. C₂	C₁₆×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×C₄×C₆₇⋊C₁₆	2
C₂×C₈×C₆₇⋊C₈	2
C₁₆×C₆₇⋊C₈	2
C₂×C₄×C₁₆×C₆₇	2
C₂×C₈×C₁₆	67

Subgroups — C2×C16×C67⋊C8 #30764802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₆×C₆₇⋊C₈ #30764802