Subgroups — #30772245

Order 17152.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	6	yes	maximal
2	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
4	C₄⋊(C₆₇⋊C₁₆)	—	2	12	no	—
5	C₂×#97×C₆₇	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
7	#376×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₆₇⋊C₁₆	—	2	12	no	—
9	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
12	#97×C₆₇	—	2	6	no	—
13	C₄×C₈×C₆₇	D₄	1	6	yes	—
14	C₄×C₈×C₆₇	Q₈	1	6	yes	—
15	#95×C₆₇	C₈	1	4	yes	—
16	C₆₇⋊C₁₆	—	4	4	no	—
17	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
18	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₈×C₆₇	—	2	6	no	—
20	C₂×C₈×C₆₇	C₄⋊C₄	1	6	yes	—
21	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
22	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
23	C₄²×C₆₇	C₄⋊C₄	1	6	yes	—
24	C₄²×C₆₇	M₁₆	1	6	yes	—
25	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₆₇	—	2	12	no	—
27	C₂×C₄×C₆₇	C₄⋊C₈	1	12	yes	—
28	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
30	C₈×C₆₇	—	2	6	no	—
31	C₈×C₆₇	#127	1	1	yes	characteristic
32	C₈×C₆₇	C₄⋊Q₈	1	3	yes	—
33	C₂²×C₆₇	C₂×C₄⋊C₈	1	3	yes	—
34	C₂²×C₆₇	#504	1	4	yes	—
35	C₄×C₆₇	—	2	12	no	—
36	C₄×C₆₇	#384	1	1	yes	characteristic
37	C₄×C₆₇	#423	1	3	yes	—
38	#12899	—	67	1	no	maximal, Sylow
39	C₂×C₆₇	#1842	1	1	yes	characteristic
40	C₂×C₆₇	#2151	1	6	yes	—
41	C₂×C₄⋊C₁₆	—	67	6	no	—
42	#1726	D₆₇	1	1	yes	characteristic
43	C₆₇	#12899	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₁₆	—	67	4	no	—
45	C₂×#97	Dic₆₇	1	3	yes	—
46	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
47	#376	Dic₆₇	1	1	yes	characteristic
48	C₄⋊C₁₆	—	134	12	no	—
49	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
50	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
51	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
52	C₂×C₁₆	—	134	12	no	—
53	#97	—	2	6	no	—
54	C₄×C₈	Dic₁₃₄	1	6	yes	—
55	C₄×C₈	D₂₆₈	1	6	yes	—
56	#95	C₆₇⋊C₈	1	4	yes	—
57	C₂²×C₄	C₂×C₆₇⋊C₈	1	6	yes	—
58	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
59	C₂×C₈	—	2	6	no	—
60	C₂×C₈	C₄⋊Dic₆₇	1	6	yes	—
61	C₂×C₈	C₂×D₂₆₈	1	3	yes	—
62	C₂×C₈	C₂×Dic₁₃₄	1	3	yes	—
63	C₄²	C₄⋊Dic₆₇	1	6	yes	—
64	C₄²	#11761612	1	6	yes	—
65	C₁₆	—	268	4	no	—
66	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
70	C₂×C₄	C₂×#11761612	1	3	yes	—
71	C₈	—	2	6	no	—
72	C₈	#14524040	1	1	yes	characteristic
73	C₈	#14524043	1	3	yes	—
74	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	3	yes	—
75	C₂²	#16863813	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#16863266	1	1	yes	characteristic
78	C₄	#16863420	1	3	yes	—
79	C₂	#22560361	1	1	yes	characteristic
80	C₂	#22561968	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₆₇×#1726	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#12899
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
43. C₆₇	#12899	semidirect

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
C₆₇×#1726	2
#12899	67