Subgroups — #27164326

Order 12736.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263947	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₉₈	C₂	1	1	yes	maximal, characteristic
3	#20263889	C₂	1	1	yes	maximal, characteristic
4	#20263887	C₂	1	1	yes	maximal, characteristic
5	#20263880	C₂	1	1	yes	maximal, characteristic
6	#20263880	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
12	Dic₁₉₉⋊C₄	—	2	1	no	—
13	Dic₁₉₉⋊C₄	—	2	1	no	—
14	Dic₁₉₉⋊C₄	—	2	1	no	—
15	C₄×Dic₁₉₉	—	2	1	no	—
16	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
19	Dic₃₉₈	—	2	1	no	—
20	Dic₃₉₈	Q₈	1	2	yes	—
21	C₂×Dic₁₉₉	—	2	1	no	—
22	C₂×Dic₁₉₉	—	2	1	no	—
23	C₂×Dic₁₉₉	—	2	1	no	—
24	C₂×Dic₁₉₉	—	2	1	no	—
25	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
28	C₈×C₁₉₉	—	2	1	no	—
29	C₈×C₁₉₉	—	2	1	no	—
30	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₁₉₉	—	2	1	no	—
32	C₄×C₁₉₉	—	2	1	no	—
33	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₉₉	#41	1	1	yes	characteristic
35	Dic₁₉₉	—	2	2	no	—
36	Dic₁₉₉	—	4	1	no	—
37	Dic₁₉₉	—	4	1	no	—
38	Dic₁₉₉	—	4	1	no	—
39	C₂×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₁₉₉	#135	1	1	yes	characteristic
41	C₂×C₁₉₉	#137	1	1	yes	characteristic
42	C₁₉₉	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	199	1	no	maximal, Sylow
44	#125	—	199	1	no	—
45	C₄×Q₈	—	199	1	no	—
46	#107	—	199	1	no	—
47	#106	—	199	1	no	—
48	C₄⋊C₈	D₁₉₉	1	1	yes	characteristic
49	#103	—	199	1	no	—
50	#103	—	199	1	no	—
51	C₂×Q₈	—	199	1	no	—
52	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
53	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
54	C₄⋊C₄	—	398	1	no	—
55	C₄⋊C₄	—	398	1	no	—
56	C₄⋊C₄	—	398	1	no	—
57	C₄⋊C₄	—	199	1	no	—
58	C₄⋊C₄	—	199	1	no	—
59	C₄⋊C₄	—	199	1	no	—
60	C₄²	—	398	1	no	—
61	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
62	Q₈	—	398	1	no	—
63	Q₈	—	199	2	no	—
64	C₂×C₄	—	398	1	no	—
65	C₂×C₄	—	398	1	no	—
66	C₂×C₄	—	398	1	no	—
67	C₂×C₄	—	398	1	no	—
68	C₂×C₄	D₇₉₆	1	1	yes	characteristic
69	C₂×C₄	D₇₉₆	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	796	1	no	—
77	C₄	—	796	1	no	—
78	C₄	—	796	1	no	—
79	C₄	—	398	2	no	—
80	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
81	C₄	#15691885	1	1	yes	characteristic
82	C₂	#20263935	1	1	yes	characteristic
83	C₂	#20263977	1	1	yes	characteristic
84	C₂	#20263988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₉₆
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄⋊C₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	C₄⋊C₈	D₁₉₉
hypocenter	C₁₉₉	#479
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
42. C₁₉₉	#479	semidirect

Maximal subgroups

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

subgroup	index
#20263947	2
C₄×Dic₃₉₈	2
#20263889	2
#20263887	2
#20263880	2
C₄⋊C₈×C₁₉₉	2
#479	199