Subgroups — #467×C₆₇ #16862995

Order 4288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal
4	#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#103×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×Q₈×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄×C₆₇	C₂²	1	1	yes	characteristic
8	Q₁₆×C₆₇	—	2	2	no	—
9	C₂×C₈×C₆₇	—	2	2	no	—
10	C₂×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₆₇	—	2	2	no	—
13	C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₆₇	—	2	2	no	—
15	C₂³×C₆₇	D₄	1	1	yes	characteristic
16	Q₈×C₆₇	—	2	2	no	—
17	Q₈×C₆₇	—	2	2	no	—
18	C₂×C₄×C₆₇	—	2	1	no	—
19	C₂×C₄×C₆₇	—	2	1	no	—
20	C₂×C₄×C₆₇	—	2	2	no	—
21	C₂×C₄×C₆₇	—	2	2	no	—
22	C₂×C₄×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₈×C₆₇	—	2	1	no	—
25	C₈×C₆₇	D₄	1	2	yes	—
26	C₂²×C₆₇	—	2	1	no	—
27	C₂²×C₆₇	—	2	1	no	—
28	C₂²×C₆₇	Q₁₆	1	2	yes	complemented
29	C₂²×C₆₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₆₇	—	2	1	no	—
31	C₄×C₆₇	—	4	2	no	—
32	C₄×C₆₇	—	4	2	no	—
33	C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₆₇	#41	1	1	yes	characteristic
35	C₂×C₆₇	—	2	2	no	—
36	C₂×C₆₇	#121	1	1	yes	characteristic
37	C₂×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
38	C₂×C₆₇	#135	1	1	yes	characteristic
39	C₆₇	#467	1	1	yes	characteristic, direct product factor, Sylow
40	#467	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
42	C₂²×C₈	C₂×C₆₇	1	1	yes	characteristic
43	C₂²⋊Q₈	C₂×C₆₇	1	2	yes	—
44	#107	C₂×C₆₇	1	1	yes	characteristic, complemented
45	#103	C₂×C₆₇	1	2	yes	complemented
46	C₂×Q₈	C₂²×C₆₇	1	2	yes	—
47	C₂²×C₄	C₂²×C₆₇	1	1	yes	characteristic
48	Q₁₆	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	C₂²×C₆₇	1	1	yes	characteristic
51	C₂×C₈	C₂²×C₆₇	1	1	yes	characteristic
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
54	C₂²⋊C₄	—	2	2	no	—
55	C₂³	D₄×C₆₇	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	D₄×C₆₇	1	1	yes	characteristic
63	C₂×C₄	C₂³×C₆₇	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	D₄×C₆₇	1	2	yes	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	Q₁₆×C₆₇	1	2	yes	complemented
69	C₂²	C₂×D₄×C₆₇	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	2	no	—
72	C₄	—	4	2	no	—
73	C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
74	C₄	#41×C₆₇	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	#121×C₆₇	1	1	yes	characteristic
77	C₂	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
78	C₂	#135×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₆₇
prime radical	C₂³×C₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×C₆₇	C₂	semidirect
4. #107×C₆₇	C₂	semidirect
5. #103×C₆₇	C₂	semidirect
28. C₂²×C₆₇	Q₁₆	semidirect
39. C₆₇	#467	direct (N × H)
40. #467	C₆₇	direct (N × H)
41. C₂×Q₁₆	C₂×C₆₇	semidirect
44. #107	C₂×C₆₇	semidirect
45. #103	C₂×C₆₇	semidirect
68. C₂²	Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×C₆₇	2
C₂²×C₈×C₆₇	2
C₂²⋊Q₈×C₆₇	2
#107×C₆₇	2
#103×C₆₇	2
#467	67

Subgroups — #467×C67 #16862995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #467×C₆₇ #16862995