Subgroups — #47981139

Order 35456.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31800120	C₂	1	1	yes	maximal, characteristic
2	C₈×#14555660	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22579902	C₄	1	1	yes	characteristic
5	#22579886	—	2	1	no	—
6	#22579816	C₄	1	1	yes	characteristic
7	#22579808	—	2	1	no	—
8	C₄×#14555660	—	2	1	no	—
9	M₃₂×C₂₇₇	—	2	1	no	—
10	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
11	#17984563	—	2	1	no	—
12	Dic₁₁₀₈	C₈	1	1	yes	characteristic, complemented
13	#17984552	—	2	1	no	—
14	D₂₂₁₆	C₈	1	1	yes	characteristic, complemented
15	#17984547	—	2	1	no	—
16	#17984546	—	4	1	no	—
17	C₂×#14555660	—	2	2	no	—
18	C₈×D₂₇₇	—	4	1	no	—
19	C₂×C₈×C₂₇₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₇₇	D₄	1	1	yes	characteristic
21	C₄²×C₂₇₇	D₄	1	1	yes	characteristic
22	C₁₆×C₂₇₇	—	4	1	no	—
23	Dic₅₅₄	—	2	1	no	—
24	D₁₁₀₈	—	2	1	no	—
25	C₂²⋊D₂₇₇	—	4	1	no	—
26	C₄×D₂₇₇	—	4	1	no	—
27	#14555660	—	2	4	no	—
28	#14555660	—	4	1	no	—
29	C₂×C₄×C₂₇₇	—	2	1	no	—
30	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₂₇₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₇₇	M₁₆	1	1	yes	characteristic
33	C₈×C₂₇₇	D₈	1	1	yes	characteristic
34	C₈×C₂₇₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₇₇	#104	1	1	yes	characteristic
36	C₂×D₂₇₇	—	4	1	no	—
37	C₄×C₂₇₇	—	2	2	no	—
38	C₄×C₂₇₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₂₇₇	#102	1	1	yes	characteristic
40	Dic₂₇₇	—	4	1	no	—
41	C₂×C₂₇₇	—	2	1	no	—
42	C₂×C₂₇₇	D₄⋊C₈	1	1	yes	characteristic
43	D₂₇₇	—	8	1	no	—
44	C₂₇₇	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	277	1	no	maximal, Sylow
46	#443	—	277	1	no	—
47	#364	D₂₇₇	1	1	yes	characteristic, complemented
48	C₈²	—	277	1	no	—
49	#135	—	277	1	no	—
50	#131	—	554	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	277	1	no	—
53	#104	—	554	1	no	—
54	C₄×C₈	—	554	1	no	—
55	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
56	#41	—	554	1	no	—
57	Q₁₆	—	277	1	no	—
58	SD₁₆	—	554	1	no	—
59	D₈	—	277	1	no	—
60	M₁₆	—	554	1	no	—
61	M₁₆	—	1108	1	no	—
62	C₂×C₈	—	554	2	no	—
63	C₂×C₈	—	1108	1	no	—
64	C₂×C₈	C₄×D₂₇₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₇₇	1	1	yes	characteristic
66	C₄²	D₁₁₀₈	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	554	1	no	—
69	D₄	—	554	1	no	—
70	D₄	—	1108	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	1108	1	no	—
73	C₂×C₄	#17984540	1	1	yes	characteristic
74	C₈	—	554	4	no	—
75	C₈	—	1108	1	no	—
76	C₈	C₈×D₂₇₇	1	1	yes	characteristic
77	C₈	#17984546	1	1	yes	characteristic
78	C₈	#17984548	1	1	yes	characteristic
79	C₈	#17984550	1	1	yes	characteristic
80	C₂²	—	1108	1	no	—
81	C₂²	#22579806	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	1108	1	no	—
84	C₄	#22579792	1	1	yes	characteristic
85	C₄	#22579800	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2216	1	no	—
88	C₂	#31799729	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#17984548
commutator	C₈×C₂₇₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₇₇
Fitting	#364×C₂₇₇	C₂
socle	C₂×C₂₇₇	D₄⋊C₈
prime radical	#22579902	C₄
cosocle	C₄×C₈×C₂₇₇	C₂²
hypercenter	#364	D₂₇₇
hypocenter	C₂₇₇	#1335
LCS	G ▷ C₈×C₂₇₇ ▷ C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×#14555660	C₂	semidirect
3. #364×C₂₇₇	C₂	semidirect
12. Dic₁₁₀₈	C₈	semidirect
14. D₂₂₁₆	C₈	semidirect
44. C₂₇₇	#1335	semidirect
47. #364	D₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
#31800120	2
C₈×#14555660	2
#364×C₂₇₇	2
#1335	277