Subgroups — #30772253

Order 17152.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561975	C₂	1	3	yes	maximal
2	#22561968	C₂	1	3	yes	maximal
3	C₆₇×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	C₂²	1	3	yes	—
5	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
6	#16863217	—	2	6	no	—
7	C₄⋊(C₆₇⋊C₁₆)	—	2	6	no	—
8	#405×C₆₇	C₂²	1	3	yes	—
9	#338×C₆₇	C₄	1	1	yes	characteristic
10	H(3,ℤ/4)×C₆₇	C₄	1	3	yes	—
11	#14523983	—	2	6	no	—
12	C₂×C₆₇⋊C₁₆	—	2	3	no	—
13	C₂×C₆₇⋊C₁₆	—	2	3	no	—
14	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
16	#117×C₆₇	C₂×C₄	1	3	yes	—
17	#97×C₆₇	Q₈	1	6	yes	—
18	C₄×C₈×C₆₇	D₄	1	6	yes	—
19	C₆₇⋊C₁₆	—	4	1	no	—
20	C₆₇⋊C₁₆	—	4	3	no	—
21	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
22	M₁₆×C₆₇	—	2	6	no	—
23	C₂×C₈×C₆₇	C₄⋊C₄	1	6	yes	—
24	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
25	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
26	C₄⋊C₄×C₆₇	—	2	3	no	—
27	C₂²⋊C₄×C₆₇	—	2	3	no	—
28	C₄²×C₆₇	C₄⋊C₄	1	6	yes	—
29	C₂³×C₆₇	#131	1	1	yes	characteristic
30	C₂×C₄×C₆₇	—	2	6	no	—
31	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
32	C₂×C₄×C₆₇	#131	1	3	yes	—
33	C₈×C₆₇	—	2	6	no	—
34	C₈×C₆₇	#127	1	1	yes	characteristic
35	C₈×C₆₇	C₄⋊Q₈	1	3	yes	—
36	C₂²×C₆₇	—	4	1	no	—
37	C₂²×C₆₇	#424	1	3	yes	—
38	C₄×C₆₇	—	4	6	no	—
39	C₄×C₆₇	#384	1	1	yes	characteristic
40	C₄×C₆₇	#424	1	3	yes	—
41	#12902	—	67	1	no	maximal, Sylow
42	C₂×C₆₇	—	2	3	no	—
43	C₂×C₆₇	#1843	1	1	yes	characteristic
44	#2153	—	67	3	no	—
45	#2151	—	67	3	no	—
46	#1747	D₆₇	1	1	yes	characteristic
47	C₆₇	#12902	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	—	67	3	no	—
49	C₂²×C₁₆	—	67	1	no	—
50	#405	C₂×D₆₇	1	3	yes	—
51	#364	—	134	6	no	—
52	C₄⋊C₁₆	—	134	6	no	—
53	#338	Dic₆₇	1	1	yes	characteristic
54	H(3,ℤ/4)	Dic₆₇	1	3	yes	—
55	C₂×M₁₆	C₂×Dic₆₇	1	3	yes	—
56	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
57	#117	C₂×Dic₆₇	1	3	yes	—
58	M₃₂	—	134	6	no	—
59	C₂×C₁₆	—	134	3	no	—
60	C₂×C₁₆	—	134	3	no	—
61	#97	Dic₁₃₄	1	6	yes	—
62	C₄×C₈	D₂₆₈	1	6	yes	—
63	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	C₄⋊Dic₆₇	1	6	yes	—
66	C₂×C₈	C₂×D₂₆₈	1	3	yes	—
67	C₂×C₈	C₂×Dic₁₃₄	1	3	yes	—
68	C₄⋊C₄	—	2	3	no	—
69	C₂²⋊C₄	—	2	3	no	—
70	C₄²	C₄⋊Dic₆₇	1	6	yes	—
71	C₁₆	—	268	1	no	—
72	C₁₆	—	268	3	no	—
73	C₂³	#14524050	1	1	yes	characteristic
74	C₂×C₄	—	2	6	no	—
75	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
76	C₂×C₄	#14524050	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	#14524040	1	1	yes	characteristic
79	C₈	#14524043	1	3	yes	—
80	C₂²	—	4	1	no	—
81	C₂²	#16863423	1	3	yes	—
82	C₄	—	4	6	no	—
83	C₄	#16863266	1	1	yes	characteristic
84	C₄	#16863423	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#22560363	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14524040
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1747	C₂
socle	C₂×C₆₇	#1843
prime radical	C₂³×C₆₇	#131
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1747	D₆₇
hypocenter	C₆₇	#12902
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
47. C₆₇	#12902	semidirect

Maximal subgroups

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

subgroup	index
#22561975	2
#22561968	2
C₆₇×#1747	2
#12902	67