Subgroups — C₆₇×#33302 #30712553

Order 17152.

1422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₆₇	C₂	1	12	yes	maximal
2	C₆₇×#2868	C₂	1	2	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆×C₆₇	C₂²	1	16	yes	—
5	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	—
6	#494×C₆₇	—	2	48	no	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	6	yes	—
8	#384×C₆₇	—	2	8	no	—
9	C₄³×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
11	C₂×Q₁₆×C₆₇	—	2	96	no	—
12	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
13	C₄⋊Q₈×C₆₇	—	2	48	no	—
14	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
15	C₂×C₄²×C₆₇	D₄	1	4	yes	—
16	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
17	C₄×C₈×C₆₇	D₄	1	24	yes	—
18	C₂×Q₈×C₆₇	—	2	48	no	—
19	C₂²×C₄×C₆₇	—	2	8	no	—
20	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
21	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
22	Q₁₆×C₆₇	—	4	64	no	—
23	C₂×C₈×C₆₇	C₂×D₄	1	24	yes	—
24	C₄⋊C₄×C₆₇	—	4	48	no	—
25	C₄²×C₆₇	Q₁₆	1	16	yes	complemented
26	C₄²×C₆₇	C₂×D₄	1	12	yes	—
27	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
28	Q₈×C₆₇	—	4	32	no	—
29	C₂×C₄×C₆₇	—	4	24	no	—
30	C₂×C₄×C₆₇	#127	1	12	yes	—
31	C₂×C₄×C₆₇	C₂×Q₁₆	1	24	yes	—
32	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
33	C₈×C₆₇	#127	1	16	yes	—
34	C₂²×C₆₇	C₂×#127	1	3	yes	—
35	C₂²×C₆₇	C₂²×Q₁₆	1	4	yes	—
36	C₄×C₆₇	—	8	8	no	—
37	C₄×C₆₇	#494	1	24	yes	complemented
38	C₄×C₆₇	C₂×#127	1	4	yes	—
39	#33302	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₆₇	#2867	1	1	yes	characteristic
41	C₂×C₆₇	C₂×#494	1	6	yes	—
42	C₂×#494	C₂×C₆₇	1	12	yes	—
43	#2868	C₂×C₆₇	1	2	yes	—
44	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic
45	C₆₇	#33302	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²×Q₁₆	C₂²×C₆₇	1	16	yes	—
47	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	—
48	#494	—	2	48	no	—
49	C₂×C₄×C₈	C₂²×C₆₇	1	6	yes	—
50	#384	—	2	8	no	—
51	C₄³	C₂²×C₆₇	1	1	yes	characteristic
52	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
53	C₂×Q₁₆	—	2	96	no	—
54	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
55	C₄⋊Q₈	—	2	48	no	—
56	C₂×C₄⋊C₄	—	2	24	no	—
57	C₂×C₄²	D₄×C₆₇	1	4	yes	—
58	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
59	C₄×C₈	D₄×C₆₇	1	24	yes	—
60	C₂×Q₈	—	2	48	no	—
61	C₂²×C₄	—	2	8	no	—
62	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
63	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
64	Q₁₆	—	4	64	no	—
65	C₂×C₈	C₂×D₄×C₆₇	1	24	yes	—
66	C₄⋊C₄	—	4	48	no	—
67	C₄²	Q₁₆×C₆₇	1	16	yes	complemented
68	C₄²	C₂×D₄×C₆₇	1	12	yes	—
69	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
70	Q₈	—	4	32	no	—
71	C₂×C₄	—	4	24	no	—
72	C₂×C₄	#127×C₆₇	1	12	yes	—
73	C₂×C₄	C₂×Q₁₆×C₆₇	1	24	yes	—
74	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
75	C₈	#127×C₆₇	1	16	yes	—
76	C₂²	C₂×#127×C₆₇	1	3	yes	—
77	C₂²	C₂²×Q₁₆×C₆₇	1	4	yes	—
78	C₄	—	8	8	no	—
79	C₄	#494×C₆₇	1	24	yes	complemented
80	C₄	C₂×#127×C₆₇	1	4	yes	—
81	C₂	C₆₇×#2867	1	1	yes	characteristic
82	C₂	C₂×#494×C₆₇	1	6	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₄²×C₆₇	Q₁₆	semidirect
37. C₄×C₆₇	#494	semidirect
39. #33302	C₆₇	direct (N × H)
45. C₆₇	#33302	direct (N × H)
67. C₄²	Q₁₆×C₆₇	semidirect
79. C₄	#494×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₆₇	2
C₆₇×#2868	2
C₄²×C₈×C₆₇	2
#33302	67

Subgroups — C67×#33302 #30712553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33302 #30712553