Subgroups — #30746264

Order 17152.

414 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559955	C₂	1	2	yes	maximal
2	C₆₇⋊#1273	C₂	1	4	yes	maximal
3	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
5	#16863151	C₄	1	8	yes	—
6	C₂×#97×C₆₇	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
8	C₄³×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
10	C₄×C₆₇⋊C₈	—	2	8	no	—
11	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
15	#97×C₆₇	—	2	8	no	—
16	C₄×C₈×C₆₇	—	2	4	no	—
17	C₂×C₆₇⋊C₈	—	2	8	no	—
18	C₂×C₆₇⋊C₈	—	2	16	no	—
19	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
20	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
21	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	—	2	4	no	—
23	C₂×C₈×C₆₇	—	2	4	no	—
24	C₂×C₈×C₆₇	M₁₆	1	8	yes	—
25	C₄²×C₆₇	—	2	4	no	—
26	C₄²×C₆₇	—	2	8	no	—
27	C₄²×C₆₇	#41	1	4	yes	—
28	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	2	4	no	—
30	C₂×C₄×C₆₇	—	2	8	no	—
31	C₂×C₄×C₆₇	#117	1	4	yes	—
32	C₂×C₄×C₆₇	#117	1	2	yes	—
33	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
34	C₂×C₄×C₆₇	#131	1	8	yes	—
35	C₈×C₆₇	—	2	8	no	—
36	C₆₇⋊C₈	—	4	8	no	—
37	C₂²×C₆₇	#376	1	1	yes	characteristic
38	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
39	C₂²×C₆₇	#405	1	4	yes	—
40	C₄×C₆₇	—	4	4	no	—
41	C₄×C₆₇	#432	1	4	yes	—
42	C₄×C₆₇	#433	1	8	yes	—
43	#8134	—	67	1	no	maximal, Sylow
44	C₂×C₆₇	#1453	1	4	yes	—
45	C₂×C₆₇	#1745	1	1	yes	characteristic
46	C₂×C₆₇	#1746	1	2	yes	—
47	#1744	—	67	2	no	—
48	C₄×#97	D₆₇	1	1	yes	characteristic
49	#1273	—	67	4	no	—
50	C₆₇	#8134	1	1	yes	characteristic, complemented, Sylow
51	C₂×#97	Dic₆₇	1	4	yes	—
52	C₂×C₄×C₈	—	67	4	no	—
53	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
54	C₄³	C₂×D₆₇	1	1	yes	characteristic
55	#336	—	67	8	no	—
56	C₂²×C₈	—	67	8	no	—
57	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
58	C₂×C₄²	C₄×D₆₇	1	4	yes	—
59	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
60	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
61	#97	—	2	8	no	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	—	134	8	no	—
64	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
65	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
66	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	134	8	no	—
70	C₂×C₈	—	134	16	no	—
71	C₂×C₈	#11761612	1	8	yes	—
72	C₄²	—	2	4	no	—
73	C₄²	—	2	8	no	—
74	C₄²	#11761628	1	4	yes	—
75	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	#14524000	1	4	yes	—
79	C₂×C₄	#14524002	1	2	yes	—
80	C₂×C₄	C₂×#11761612	1	4	yes	—
81	C₂×C₄	#14524051	1	8	yes	—
82	C₈	—	2	8	no	—
83	C₈	—	268	8	no	—
84	C₂²	#16863242	1	1	yes	characteristic
85	C₂²	C₄×#11761612	1	2	yes	—
86	C₂²	#16863354	1	4	yes	—
87	C₄	—	4	4	no	—
88	C₄	#16863453	1	4	yes	—
89	C₄	#16863458	1	8	yes	—
90	C₂	#22558506	1	4	yes	—
91	C₂	#22559958	1	1	yes	characteristic
92	C₂	#22559962	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂×C₄²
Fitting	C₄×#97×C₆₇	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#97	D₆₇
hypocenter	C₆₇	#8134
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
50. C₆₇	#8134	semidirect

Maximal subgroups

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

subgroup	index
#22559955	2
C₆₇⋊#1273	2
C₄×#97×C₆₇	2
#8134	67