Subgroups — #27164306

Order 12736.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263954	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263887	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263886	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263876	C₂	1	1	yes	maximal, characteristic
7	#102×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₈	—	2	1	no	—
9	C₂²×Dic₁₉₉	—	2	1	no	—
10	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
13	Dic₁₉₉⋊C₄	—	2	1	no	—
14	C₂²⋊Dic₁₉₉	—	2	1	no	—
15	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
19	Dic₃₉₈	—	4	2	no	—
20	C₂×Dic₁₉₉	—	2	1	no	—
21	C₂×Dic₁₉₉	—	2	1	no	—
22	C₂×Dic₁₉₉	—	4	2	no	—
23	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
24	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
25	C₁₉₉⋊C₈	—	2	1	no	—
26	C₂³×C₁₉₉	—	2	1	no	—
27	D₄×C₁₉₉	—	2	1	no	—
28	D₄×C₁₉₉	Q₈	1	2	yes	complemented
29	C₂×C₄×C₁₉₉	—	2	1	no	—
30	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
31	C₈×C₁₉₉	—	2	1	no	—
32	C₂²×C₁₉₉	—	2	2	no	—
33	C₂²×C₁₉₉	—	4	1	no	—
34	C₂²×C₁₉₉	—	4	1	no	—
35	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₉₉	—	4	1	no	—
37	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₁₉₉	#41	1	1	yes	characteristic
39	Dic₁₉₉	—	2	1	no	—
40	Dic₁₉₉	—	4	1	no	—
41	Dic₁₉₉	—	4	1	no	—
42	Dic₁₉₉	SD₁₆	1	2	yes	—
43	C₂×C₁₉₉	—	4	2	no	—
44	C₂×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₁₉₉	#476	1	1	yes	characteristic, complemented, Sylow
48	#476	—	199	1	no	maximal, Sylow
49	C₄⋊Q₈	—	199	1	no	—
50	C₄×D₄	—	199	1	no	—
51	#106	—	199	1	no	—
52	#106	—	199	1	no	—
53	C₄⋊C₈	—	199	1	no	—
54	#102	—	199	1	no	—
55	#102	D₁₉₉	1	1	yes	characteristic
56	C₂×Q₈	—	398	1	no	—
57	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
58	C₂²×C₄	—	398	1	no	—
59	C₂×C₈	—	199	1	no	—
60	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
61	C₄⋊C₄	—	398	1	no	—
62	C₄⋊C₄	—	199	1	no	—
63	C₄⋊C₄	—	199	1	no	—
64	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
65	C₂²⋊C₄	—	398	1	no	—
66	C₄²	—	199	1	no	—
67	C₂³	—	2	1	no	—
68	Q₈	—	796	2	no	—
69	D₄	—	2	1	no	—
70	D₄	Dic₃₉₈	1	2	yes	complemented
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	796	2	no	—
73	C₂×C₄	—	398	1	no	—
74	C₂×C₄	—	398	1	no	—
75	C₂×C₄	—	199	1	no	—
76	C₂×C₄	—	199	1	no	—
77	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	398	1	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	796	1	no	—
86	C₄	—	796	1	no	—
87	C₄	—	398	1	no	—
88	C₄	—	199	2	no	—
89	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
90	C₄	#15691885	1	1	yes	characteristic
91	C₂	—	4	2	no	—
92	C₂	#20263932	1	1	yes	characteristic
93	C₂	SD₁₆×D₁₉₉	1	1	yes	characteristic
94	C₂	#20263979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#102×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂×D₄×C₁₉₉	C₂²
hypercenter	#102	D₁₉₉
hypocenter	C₁₉₉	#476
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #20263954	C₂	semidirect
3. #20263887	C₂	semidirect
4. #20263886	C₂	semidirect
5. Dic₁₉₉⋊C₈	C₂	semidirect
28. D₄×C₁₉₉	Q₈	semidirect
47. C₁₉₉	#476	semidirect
70. D₄	Dic₃₉₈	semidirect

Maximal subgroups

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

subgroup	index
#20263954	2
D₄×Dic₁₉₉	2
#20263887	2
#20263886	2
Dic₁₉₉⋊C₈	2
#20263876	2
#102×C₁₉₉	2
#476	199