Subgroups — #30772231

Order 17152.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558287	C₂	1	6	yes	maximal
2	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₆₇	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
6	C₄³×C₆₇	C₄	1	1	yes	characteristic
7	C₂×C₆₇⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₆₇	C₈	1	3	yes	—
11	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₆₇	—	2	6	no	—
14	C₄×C₈×C₆₇	—	2	6	no	—
15	C₆₇⋊C₁₆	—	4	4	no	—
16	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
18	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₈×C₆₇	—	2	3	no	—
20	C₂×C₈×C₆₇	—	2	3	no	—
21	C₂×C₈×C₆₇	M₁₆	1	6	yes	—
22	C₂×C₈×C₆₇	#41	1	6	yes	—
23	C₄²×C₆₇	—	2	6	no	—
24	C₄²×C₆₇	—	2	3	no	—
25	C₄²×C₆₇	—	2	3	no	—
26	C₄²×C₆₇	C₁₆	1	4	yes	complemented
27	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₆₇	—	2	3	no	—
29	C₂×C₄×C₆₇	—	2	3	no	—
30	C₂×C₄×C₆₇	—	2	3	no	—
31	C₂×C₄×C₆₇	—	2	3	no	—
32	C₂×C₄×C₆₇	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₆₇	M₃₂	1	6	yes	—
34	C₂×C₄×C₆₇	#117	1	3	yes	—
35	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
36	C₈×C₆₇	—	2	6	no	—
37	C₈×C₆₇	#126	1	4	yes	—
38	C₂²×C₆₇	#431	1	3	yes	—
39	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
40	C₂²×C₆₇	C₂×M₃₂	1	3	yes	—
41	C₄×C₆₇	—	2	6	no	—
42	C₄×C₆₇	—	2	6	no	—
43	C₄×C₆₇	#383	1	1	yes	characteristic
44	C₄×C₆₇	#432	1	3	yes	—
45	#12894	—	67	1	no	maximal, Sylow
46	C₂×C₆₇	#1839	1	1	yes	characteristic
47	C₂×C₆₇	#2162	1	3	yes	—
48	C₂×C₆₇	#2163	1	3	yes	—
49	C₄²×C₈	D₆₇	1	1	yes	characteristic
50	#1374	—	67	6	no	—
51	C₆₇	#12894	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	67	4	no	—
53	C₂×C₄×C₈	Dic₆₇	1	3	yes	—
54	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
55	C₄³	Dic₆₇	1	1	yes	characteristic
56	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
57	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
58	C₂×C₄²	C₆₇⋊C₈	1	3	yes	—
59	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
61	C₂×C₁₆	—	134	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
65	C₂²×C₄	C₂×C₆₇⋊C₈	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#11761612	1	6	yes	—
70	C₂×C₈	#11761628	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	C₆₇⋊C₁₆	1	4	yes	complemented
75	C₁₆	—	268	4	no	—
76	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×C₆₇⋊C₁₆	1	6	yes	—
82	C₂×C₄	#14523983	1	6	yes	—
83	C₂×C₄	#14524002	1	3	yes	—
84	C₂×C₄	C₂×#11761612	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#14524037	1	4	yes	—
87	C₂²	#16863449	1	3	yes	—
88	C₂²	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
89	C₂²	C₂×#14523983	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#16863264	1	1	yes	characteristic
93	C₄	#16863453	1	3	yes	—
94	C₂	#22560352	1	1	yes	characteristic
95	C₂	#22562008	1	3	yes	—
96	C₂	#22562012	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#12894
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
26. C₄²×C₆₇	C₁₆	semidirect
51. C₆₇	#12894	semidirect
74. C₄²	C₆₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22558287	2
C₄²×C₈×C₆₇	2
#12894	67