Subgroups — #27164276

Order 12736.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263934	C₂	1	2	yes	maximal
3	#20263887	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263880	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₂²	1	2	yes	—
7	Dic₇₉₆	—	2	2	no	—
8	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	Dic₁₉₉⋊C₄	—	2	2	no	—
10	C₂²⋊Dic₁₉₉	—	2	2	no	—
11	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	M₁₆×C₁₉₉	—	2	2	no	—
13	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	Dic₃₉₈	—	2	2	no	—
16	Dic₃₉₈	—	2	2	no	—
17	C₂×Dic₁₉₉	—	2	2	no	—
18	C₂×Dic₁₉₉	—	2	2	no	—
19	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	—	2	1	no	—
21	C₂×C₄×C₁₉₉	—	2	1	no	—
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₉	—	2	1	no	—
25	C₈×C₁₉₉	D₄	1	2	yes	—
26	C₂²×C₁₉₉	—	2	1	no	—
27	C₂²×C₁₉₉	—	2	2	no	—
28	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₉₉	—	2	1	no	—
30	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₉₉	#41	1	1	yes	characteristic
32	Dic₁₉₉	—	4	2	no	—
33	Dic₁₉₉	—	4	2	no	—
34	C₂×C₁₉₉	—	4	1	no	—
35	C₂×C₁₉₉	#121	1	1	yes	characteristic
36	C₂×C₁₉₉	#137	1	2	yes	—
37	C₁₉₉	#470	1	1	yes	characteristic, complemented, Sylow
38	#470	—	199	1	no	maximal, Sylow
39	C₂×Q₁₆	—	199	1	no	—
40	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic
41	C₂²⋊Q₈	—	199	2	no	—
42	#106	—	199	1	no	—
43	#103	—	199	2	no	—
44	C₂×Q₈	—	199	2	no	—
45	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
46	Q₁₆	—	398	2	no	—
47	M₁₆	—	2	2	no	—
48	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
49	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
50	C₄⋊C₄	—	398	2	no	—
51	C₄⋊C₄	—	199	2	no	—
52	C₂²⋊C₄	—	398	2	no	—
53	C₂³	D₇₉₆	1	1	yes	characteristic
54	Q₈	—	398	2	no	—
55	Q₈	—	398	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	398	2	no	—
59	C₂×C₄	—	398	2	no	—
60	C₂×C₄	D₇₉₆	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	C₂²⋊D₁₉₉	1	2	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	796	2	no	—
69	C₄	—	796	2	no	—
70	C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
71	C₄	#15691886	1	1	yes	characteristic
72	C₂	—	4	1	no	—
73	C₂	#20263929	1	1	yes	characteristic
74	C₂	#20263988	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₉₆
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₂×M₁₆×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂³×C₁₉₉	D₄
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	#470
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₇₉₆	C₂	semidirect
3. #20263887	C₂	semidirect
4. #20263880	C₂	semidirect
37. C₁₉₉	#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₉₆	2
#20263934	2
#20263887	2
#20263880	2
C₂×M₁₆×C₁₉₉	2
#470	199