Subgroups — #22561608

Order 8576.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863444	C₂	1	3	yes	maximal, complemented
2	C₂×#14523959	C₂	1	3	yes	maximal
3	#338×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
5	C₂×#11761612	C₂²	1	3	yes	—
6	#14523978	—	2	6	no	—
7	#14523959	—	2	6	no	—
8	C₂×M₁₆×C₆₇	C₂²	1	3	yes	—
9	C₂³×D₆₇	Q₈	1	2	yes	—
10	C₂×D₂₆₈	—	2	3	no	—
11	C₂×D₂₆₈	—	2	3	no	—
12	#11761612	—	2	3	no	—
13	#11761612	D₄	1	6	yes	—
14	C₂×C₆₇⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
16	M₁₆×C₆₇	—	2	3	no	—
17	M₁₆×C₆₇	—	2	3	no	—
18	C₂×C₈×C₆₇	—	2	3	no	—
19	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
20	C₂²×D₆₇	—	2	6	no	—
21	C₂²×D₆₇	—	8	2	no	—
22	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₆₇	#41	1	3	yes	—
24	D₂₆₈	—	4	3	no	—
25	D₂₆₈	—	4	1	no	—
26	C₈×C₆₇	—	4	3	no	—
27	C₆₇⋊C₈	—	2	6	no	—
28	C₂²×C₆₇	—	4	1	no	—
29	C₂²×C₆₇	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₆₇	—	4	2	no	—
31	C₂×D₆₇	—	8	6	no	—
32	C₄×C₆₇	#121	1	3	yes	—
33	C₄×C₆₇	#126	1	1	yes	characteristic
34	C₂×C₆₇	—	2	3	no	—
35	C₂×C₆₇	#396	1	1	yes	characteristic
36	D₆₇	—	8	2	no	—
37	#2069	—	67	1	no	maximal, Sylow
38	C₆₇	#2069	1	1	yes	characteristic, complemented, Sylow
39	#430	—	67	3	no	—
40	C₂×#100	—	67	3	no	—
41	#338	D₆₇	1	1	yes	characteristic, complemented
42	C₂²×D₄	—	67	1	no	—
43	C₂×M₁₆	—	67	3	no	—
44	C₂×M₁₆	C₂×D₆₇	1	3	yes	—
45	#108	—	134	6	no	—
46	#100	—	134	6	no	—
47	C₂⁴	—	67	2	no	—
48	C₂×D₄	—	134	3	no	—
49	C₂×D₄	—	134	3	no	—
50	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	67	6	no	—
54	M₁₆	—	134	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	134	3	no	—
57	C₂³	—	536	2	no	—
58	C₂³	—	134	6	no	—
59	C₂³	Q₈×D₆₇	1	1	yes	characteristic
60	D₄	—	268	3	no	—
61	D₄	—	268	1	no	—
62	C₂×C₄	D₄×D₆₇	1	3	yes	—
63	C₂×C₄	#11761628	1	3	yes	—
64	C₈	—	4	3	no	—
65	C₈	—	134	6	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	268	2	no	—
68	C₂²	—	536	6	no	—
69	C₂²	#14524021	1	3	yes	—
70	C₄	#14524016	1	3	yes	—
71	C₄	#14524037	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	536	2	no	—
74	C₂	#16863320	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16863320
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#338×C₆₇	C₂
socle	C₂×C₆₇	#396
hypercenter	#338	D₆₇
hypocenter	C₆₇	#2069
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #16863444	C₂	semidirect
3. #338×C₆₇	C₂	semidirect
38. C₆₇	#2069	semidirect
41. #338	D₆₇	semidirect