Subgroups — #27164294

Order 12736.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263988	C₂	1	2	yes	maximal
2	C₂×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263961	C₂	1	1	yes	maximal, characteristic
4	#20263873	C₂	1	2	yes	maximal, complemented
5	#108×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691886	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉₈	C₂²	1	2	yes	—
8	Dic₇₉₆	—	2	2	no	—
9	Dic₇₉₆	—	2	1	no	—
10	Dic₇₉₆	—	2	1	no	—
11	#15691875	—	2	2	no	—
12	#15691870	C₂²	1	1	yes	characteristic
13	#15691869	—	2	1	no	—
14	C₈×D₁₉₉	—	2	1	no	—
15	M₁₆×C₁₉₉	C₂²	1	2	yes	—
16	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	Dic₃₉₈	—	2	2	no	—
18	Dic₃₉₈	—	2	2	no	—
19	Dic₃₉₈	D₄	1	1	yes	characteristic
20	D₇₉₆	D₄	1	1	yes	characteristic
21	C₂²⋊D₁₉₉	—	2	1	no	—
22	C₂×Dic₁₉₉	—	2	2	no	—
23	C₄×D₁₉₉	—	2	1	no	—
24	C₁₉₉⋊C₈	—	2	1	no	—
25	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
26	C₈×C₁₉₉	—	2	2	no	—
27	C₈×C₁₉₉	D₄	1	1	yes	characteristic
28	C₈×C₁₉₉	D₄	1	1	yes	characteristic
29	C₂²×C₁₉₉	#41	1	1	yes	characteristic
30	C₂×D₁₉₉	—	2	1	no	—
31	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
33	Dic₁₉₉	—	2	1	no	—
34	Dic₁₉₉	—	4	2	no	—
35	C₂×C₁₉₉	—	2	1	no	—
36	C₂×C₁₉₉	#121	1	1	yes	characteristic
37	D₁₉₉	—	4	1	no	—
38	C₁₉₉	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	199	1	no	maximal, Sylow
40	#137	—	199	2	no	—
41	C₂×Q₁₆	—	199	1	no	—
42	#131	—	199	1	no	—
43	#108	D₁₉₉	1	1	yes	characteristic, complemented
44	#101	—	199	2	no	—
45	#41	—	199	1	no	—
46	C₂×Q₈	—	199	2	no	—
47	Q₁₆	—	398	2	no	—
48	Q₁₆	—	398	1	no	—
49	Q₁₆	—	398	1	no	—
50	SD₁₆	—	398	2	no	—
51	M₁₆	—	398	1	no	—
52	M₁₆	—	199	1	no	—
53	M₁₆	C₂×D₁₉₉	1	2	yes	—
54	C₂×C₈	—	398	1	no	—
55	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
56	Q₈	—	398	2	no	—
57	Q₈	—	398	2	no	—
58	Q₈	—	199	1	no	—
59	D₄	—	398	1	no	—
60	D₄	—	199	1	no	—
61	C₂×C₄	—	398	2	no	—
62	C₂×C₄	—	398	1	no	—
63	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	398	1	no	—
66	C₈	D₇₉₆	1	1	yes	characteristic
67	C₈	D₇₉₆	1	1	yes	characteristic
68	C₂²	—	398	1	no	—
69	C₂²	#15691887	1	1	yes	characteristic
70	C₄	—	796	2	no	—
71	C₄	—	398	1	no	—
72	C₄	D₄×D₁₉₉	1	1	yes	characteristic
73	C₄	C₂×D₇₉₆	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	796	1	no	—
76	C₂	#20263930	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20263930
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#108×C₁₉₉	C₂
socle	C₂×C₁₉₉	#121
prime radical	#15691886	C₂²
hypercenter	#108	D₁₉₉
hypocenter	C₁₉₉	#473
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₉₆	C₂	semidirect
4. #20263873	C₂	semidirect
5. #108×C₁₉₉	C₂	semidirect
38. C₁₉₉	#473	semidirect
43. #108	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263988	2
C₂×Dic₇₉₆	2
#20263961	2
#20263873	2
#108×C₁₉₉	2
#473	199