Subgroups — #30770545

Order 17152.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562461	C₂	1	4	yes	maximal
2	#22559827	C₂	1	1	yes	maximal, characteristic
3	C₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1713	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523989	—	2	8	no	—
6	#16863801	—	2	2	no	—
7	#16863776	C₂²	1	2	yes	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₆₇	—	2	2	no	—
10	#494×C₆₇	C₂²	1	2	yes	—
11	C₈²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₆₈	—	2	4	no	—
13	#14524043	—	2	2	no	—
14	#14523989	—	4	16	no	—
15	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
16	#14523977	—	4	4	no	—
17	C₂×Q₁₆×C₆₇	—	2	4	no	—
18	C₄⋊Q₈×C₆₇	—	2	2	no	—
19	#107×C₆₇	—	4	4	no	—
20	C₄×C₈×C₆₇	D₄	1	2	yes	—
21	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₃₄	—	4	2	no	—
23	Dic₂₆₈	—	4	8	no	—
24	C₄⋊Dic₆₇	—	4	4	no	—
25	C₆₇⋊C₁₆	D₈	1	8	yes	—
26	C₂×Q₈×C₆₇	—	4	2	no	—
27	Q₁₆×C₆₇	—	4	8	no	—
28	C₂×C₈×C₆₇	D₈	1	4	yes	—
29	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
30	C₄⋊C₄×C₆₇	—	4	4	no	—
31	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₆₇	—	8	4	no	—
33	C₂×C₄×C₆₇	—	8	2	no	—
34	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
36	C₂×Dic₆₇	—	8	2	no	—
37	Dic₁₃₄	—	8	4	no	—
38	C₈×C₆₇	Q₃₂	1	8	yes	complemented
39	C₈×C₆₇	C₂×D₈	1	4	yes	—
40	C₂²×C₆₇	#493	1	1	yes	characteristic
41	C₄×C₆₇	—	16	2	no	—
42	C₄×C₆₇	#493	1	2	yes	—
43	C₄×C₆₇	C₂×Q₃₂	1	4	yes	—
44	Dic₆₇	—	16	2	no	—
45	#12516	—	67	1	no	maximal, Sylow
46	C₂×C₆₇	#1712	1	1	yes	characteristic
47	C₂×C₆₇	#2247	1	2	yes	—
48	#2247	—	67	4	no	—
49	#1713	—	67	1	no	—
50	#1713	D₆₇	1	1	yes	characteristic
51	C₈×C₁₆	—	67	1	no	—
52	C₆₇	#12516	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₃₂	—	134	8	no	—
54	#500	—	2	2	no	—
55	#500	—	134	2	no	—
56	#494	—	67	2	no	—
57	#494	C₂×D₆₇	1	2	yes	—
58	C₄×C₁₆	—	67	2	no	—
59	C₈²	C₂×D₆₇	1	1	yes	characteristic
60	C₂×Q₁₆	—	2	4	no	—
61	C₂×Q₁₆	—	134	4	no	—
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	134	2	no	—
64	Q₃₂	—	268	16	no	—
65	C₂×C₁₆	—	67	4	no	—
66	#107	—	4	4	no	—
67	#107	—	268	4	no	—
68	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
69	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
70	C₂×Q₈	—	4	2	no	—
71	C₂×Q₈	—	268	2	no	—
72	Q₁₆	—	4	8	no	—
73	Q₁₆	—	268	8	no	—
74	C₂×C₈	#11761613	1	4	yes	—
75	C₂×C₈	D₄×D₆₇	1	2	yes	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	268	4	no	—
78	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
79	C₁₆	—	67	8	no	—
80	Q₈	—	8	4	no	—
81	Q₈	—	536	4	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	536	2	no	—
84	C₂×C₄	#14524039	1	1	yes	characteristic
85	C₂×C₄	C₂×#11761613	1	2	yes	—
86	C₈	#14523989	1	8	yes	complemented
87	C₈	D₈×D₆₇	1	4	yes	—
88	C₂²	#16863772	1	1	yes	characteristic
89	C₄	—	16	2	no	—
90	C₄	—	1072	2	no	—
91	C₄	#16863771	1	2	yes	—
92	C₄	C₂×#14523989	1	4	yes	—
93	C₂	#22559823	1	1	yes	characteristic
94	C₂	#22562461	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1713	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1713	D₆₇
hypocenter	C₆₇	#12516
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
38. C₈×C₆₇	Q₃₂	semidirect
52. C₆₇	#12516	semidirect
86. C₈	#14523989	semidirect

Maximal subgroups

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

subgroup	index
#22562461	2
#22559827	2
C₈×C₆₇⋊C₁₆	2
C₆₇×#1713	2
#12516	67