Subgroups — #27164210

Order 12736.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691871	C₂	1	2	yes	maximal, complemented
2	C₄×D₇₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263876	C₂	1	2	yes	maximal, complemented
5	#127×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₉₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₉₉	—	2	1	no	—
8	#15691871	—	2	4	no	—
9	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
10	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
11	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
12	C₂×D₄×C₁₉₉	—	2	1	no	—
13	C₂×D₄×C₁₉₉	—	2	1	no	—
14	C₂×D₄×C₁₉₉	C₂²	1	2	yes	—
15	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂²×D₁₉₉	—	2	1	no	—
17	D₇₉₆	—	2	1	no	—
18	D₇₉₆	D₄	1	2	yes	complemented
19	C₂×Dic₁₉₉	—	2	1	no	—
20	C₄×D₁₉₉	—	4	2	no	—
21	C₁₉₉⋊C₈	—	2	2	no	—
22	C₂³×C₁₉₉	—	2	2	no	—
23	D₄×C₁₉₉	—	2	2	no	—
24	D₄×C₁₉₉	—	2	2	no	—
25	D₄×C₁₉₉	—	4	2	no	—
26	D₄×C₁₉₉	—	4	2	no	—
27	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₂²×C₁₉₉	—	4	2	no	—
31	C₂²×C₁₉₉	—	4	2	no	—
32	C₂²×C₁₉₉	—	4	2	no	—
33	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
34	C₂×D₁₉₉	—	2	2	no	—
35	C₂×D₁₉₉	—	4	1	no	—
36	C₂×D₁₉₉	—	4	1	no	—
37	C₄×C₁₉₉	—	2	1	no	—
38	C₄×C₁₉₉	D₈	1	2	yes	complemented
39	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₉₉	#41	1	1	yes	characteristic
41	Dic₁₉₉	—	4	1	no	—
42	C₂×C₁₉₉	—	8	2	no	—
43	C₂×C₁₉₉	#121	1	1	yes	characteristic
44	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
45	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
46	D₁₉₉	—	4	2	no	—
47	C₁₉₉	#459	1	1	yes	characteristic, complemented, Sylow
48	#459	—	199	1	no	maximal, Sylow
49	C₂×D₈	—	199	2	no	—
50	#127	D₁₉₉	1	1	yes	characteristic, complemented
51	C₄×D₄	—	199	1	no	—
52	C₄⋊C₈	—	199	1	no	—
53	#102	—	199	2	no	—
54	C₂×D₄	—	2	1	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	—	199	1	no	—
57	C₂×D₄	C₂×D₁₉₉	1	2	yes	—
58	C₂²×C₄	—	398	1	no	—
59	D₈	—	398	4	no	—
60	C₂×C₈	—	199	2	no	—
61	C₄⋊C₄	—	199	1	no	—
62	C₂²⋊C₄	—	398	1	no	—
63	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
64	C₂³	—	2	2	no	—
65	C₂³	—	398	1	no	—
66	D₄	—	2	2	no	—
67	D₄	—	2	2	no	—
68	D₄	—	4	2	no	—
69	D₄	—	4	2	no	—
70	D₄	—	398	1	no	—
71	D₄	—	199	2	no	—
72	C₂×C₄	—	796	2	no	—
73	C₂×C₄	—	398	1	no	—
74	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
77	C₈	—	398	2	no	—
78	C₂²	—	4	2	no	—
79	C₂²	—	4	2	no	—
80	C₂²	—	4	2	no	—
81	C₂²	—	796	1	no	—
82	C₂²	—	796	1	no	—
83	C₂²	—	398	2	no	—
84	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	796	1	no	—
87	C₄	#15691871	1	2	yes	complemented
88	C₄	D₄×D₁₉₉	1	1	yes	characteristic
89	C₄	#15691885	1	1	yes	characteristic
90	C₂	—	8	2	no	—
91	C₂	—	796	2	no	—
92	C₂	#20263926	1	1	yes	characteristic
93	C₂	C₂×#15691871	1	1	yes	characteristic
94	C₂	#20263981	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#127×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
hypercenter	#127	D₁₉₉
hypocenter	C₁₉₉	#459
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#15691871	C₂	semidirect
2. C₄×D₇₉₆	C₂	semidirect
3. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
4. #20263876	C₂	semidirect
5. #127×C₁₉₉	C₂	semidirect
18. D₇₉₆	D₄	semidirect
38. C₄×C₁₉₉	D₈	semidirect
47. C₁₉₉	#459	semidirect
50. #127	D₁₉₉	semidirect
87. C₄	#15691871	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691871	2
C₄×D₇₉₆	2
C₄⋊(C₁₉₉⋊C₈)	2
#20263876	2
#127×C₁₉₉	2
#459	199