Subgroups — #27164408

Order 12736.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691873	C₂	1	2	yes	maximal, complemented
2	C₂×#15691871	C₂	1	2	yes	maximal, complemented
3	#20263952	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263866	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₉₆	—	2	2	no	—
7	C₂×D₇₉₆	C₂²	1	2	yes	—
8	#15691873	—	2	4	no	—
9	#15691871	—	2	4	no	—
10	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₉	—	2	2	no	—
16	D₇₉₆	—	2	2	no	—
17	D₇₉₆	—	2	2	no	—
18	D₇₉₆	—	4	4	no	—
19	C₁₉₉⋊C₈	D₄	1	4	yes	complemented
20	C₂³×C₁₉₉	—	2	1	no	—
21	Q₈×C₁₉₉	—	2	2	no	—
22	D₄×C₁₉₉	—	2	2	no	—
23	C₂×C₄×C₁₉₉	—	2	1	no	—
24	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
25	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₉₉	—	4	1	no	—
27	C₂²×C₁₉₉	—	4	2	no	—
28	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	C₂×D₁₉₉	—	4	2	no	—
30	C₂×D₁₉₉	—	4	4	no	—
31	C₄×C₁₉₉	—	2	2	no	—
32	C₄×C₁₉₉	—	4	1	no	—
33	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
34	C₂×C₁₉₉	—	8	1	no	—
35	C₂×C₁₉₉	#127	1	1	yes	characteristic
36	C₂×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
37	D₁₉₉	—	8	2	no	—
38	C₁₉₉	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	199	1	no	maximal, Sylow
40	C₂×SD₁₆	—	199	2	no	—
41	C₂×D₈	—	199	2	no	—
42	#127	—	199	1	no	—
43	#124	D₁₉₉	1	1	yes	characteristic, complemented
44	#97	—	199	1	no	—
45	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
46	C₂×D₄	—	398	2	no	—
47	C₂×D₄	—	199	2	no	—
48	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
49	SD₁₆	—	398	4	no	—
50	D₈	—	398	4	no	—
51	C₂×C₈	—	199	2	no	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	398	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	796	4	no	—
59	D₄	—	398	2	no	—
60	D₄	—	398	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
63	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
64	C₈	—	199	4	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	796	2	no	—
68	C₂²	—	796	4	no	—
69	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₁₉₉	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	1592	2	no	—
75	C₂	#20263951	1	1	yes	characteristic
76	C₂	#20263980	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#124×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
hypercenter	#124	D₁₉₉
hypocenter	C₁₉₉	#496
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#15691873	C₂	semidirect
2. C₂×#15691871	C₂	semidirect
3. #20263952	C₂	semidirect
4. #20263866	C₂	semidirect
5. #124×C₁₉₉	C₂	semidirect
19. C₁₉₉⋊C₈	D₄	semidirect
38. C₁₉₉	#496	semidirect
43. #124	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691873	2
C₂×#15691871	2
#20263952	2
#20263866	2
#124×C₁₉₉	2
#496	199