Subgroups — #30749565

Order 17152.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558081	C₂	1	3	yes	maximal
2	#22558074	C₂	1	3	yes	maximal
3	C₆₇×#1455	C₂	1	1	yes	maximal, characteristic
4	#16863453	C₂²	1	3	yes	—
5	#16863264	C₂²	1	1	yes	characteristic
6	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
7	#322×C₆₇	C₄	1	3	yes	complemented
8	C₈²×C₆₇	C₄	1	1	yes	characteristic, complemented
9	C₄⋊(C₆₇⋊C₈)	—	2	3	no	—
10	C₂²⋊(C₆₇⋊C₈)	—	2	3	no	—
11	#14523954	—	2	3	no	—
12	C₆₇⋊#95	—	2	3	no	—
13	C₂×M₁₆×C₆₇	—	2	3	no	—
14	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
15	#97×C₆₇	—	2	3	no	—
16	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
17	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
18	C₂²×Dic₆₇	—	4	1	no	—
19	C₂×C₆₇⋊C₈	—	4	3	no	—
20	C₂²×C₄×C₆₇	#41	1	3	yes	—
21	M₁₆×C₆₇	—	4	6	no	—
22	C₂×C₈×C₆₇	—	2	3	no	—
23	C₂×C₈×C₆₇	—	2	3	no	—
24	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
25	C₄²×C₆₇	#41	1	3	yes	—
26	C₂³×C₆₇	#126	1	1	yes	characteristic
27	C₂×C₄×C₆₇	—	4	3	no	—
28	C₂×C₄×C₆₇	#117	1	3	yes	—
29	C₂×C₄×C₆₇	#126	1	3	yes	—
30	C₂×Dic₆₇	—	8	3	no	—
31	C₈×C₆₇	—	2	6	no	—
32	C₈×C₆₇	—	4	3	no	—
33	C₆₇⋊C₈	—	8	3	no	—
34	C₂²×C₆₇	—	2	3	no	—
35	C₂²×C₆₇	#383	1	1	yes	characteristic
36	C₄×C₆₇	—	2	3	no	—
37	C₄×C₆₇	—	2	3	no	—
38	Dic₆₇	—	16	1	no	—
39	#8804	—	67	1	no	maximal, Sylow
40	C₂×C₆₇	—	4	1	no	—
41	C₂×C₆₇	#1841	1	3	yes	—
42	#1455	D₆₇	1	1	yes	characteristic
43	#1291	—	67	3	no	—
44	#1288	—	67	3	no	—
45	C₆₇	#8804	1	1	yes	characteristic, complemented, Sylow
46	#432	—	67	3	no	—
47	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
48	#383	—	67	1	no	—
49	#322	Dic₆₇	1	3	yes	complemented
50	C₈²	Dic₆₇	1	1	yes	characteristic, complemented
51	C₂×M₁₆	—	2	3	no	—
52	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
53	C₄⋊C₈	—	134	3	no	—
54	C₂²⋊C₈	—	134	3	no	—
55	#97	—	2	3	no	—
56	#97	—	134	3	no	—
57	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
58	C₄×C₈	C₂×Dic₆₇	1	3	yes	—
59	#95	—	134	3	no	—
60	C₂²×C₄	—	268	1	no	—
61	C₂²×C₄	#11761628	1	3	yes	—
62	M₁₆	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	—	268	3	no	—
66	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
67	C₄²	#11761628	1	3	yes	—
68	C₂³	#14524037	1	1	yes	characteristic
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	536	3	no	—
71	C₂×C₄	#14524002	1	3	yes	—
72	C₂×C₄	#14524037	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₈	—	4	3	no	—
75	C₈	—	536	3	no	—
76	C₂²	—	2	3	no	—
77	C₂²	#16863264	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	1072	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#22560358	1	3	yes	—

	subgroup	G/H
center	C₂²	#16863264
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1455	C₂
socle	C₂²×C₆₇	#383
prime radical	C₂³×C₆₇	#126
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1455	D₆₇
hypocenter	C₆₇	#8804
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
7. #322×C₆₇	C₄	semidirect
8. C₈²×C₆₇	C₄	semidirect
45. C₆₇	#8804	semidirect
49. #322	Dic₆₇	semidirect
50. C₈²	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22558081	2
#22558074	2
C₆₇×#1455	2
#8804	67