Subgroups — #27164571

Order 12736.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263954	C₂	1	2	yes	maximal, complemented
2	#20263946	C₂	1	2	yes	maximal, complemented
3	#20263932	C₂	1	4	yes	maximal
4	C₄×Dic₃₉₈	C₂	1	4	yes	maximal, complemented
5	#20263914	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
7	#117×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₈	C₂²	1	4	yes	—
9	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
10	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
11	C₄⋊Dic₁₉₉	C₂²	1	4	yes	—
12	Dic₁₉₉⋊C₄	C₂²	1	4	yes	—
13	Dic₁₉₉⋊C₄	C₂²	1	4	yes	—
14	C₂²⋊Dic₁₉₉	C₂²	1	2	yes	—
15	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
16	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₉₉	C₂²	1	2	yes	—
18	C₄×Dic₁₉₉	C₂²	1	2	yes	—
19	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₁₉₉	C₂²	1	2	yes	—
22	C₄²×C₁₉₉	C₂²	1	2	yes	—
23	Dic₃₉₈	—	2	8	no	—
24	C₂×Dic₁₉₉	—	2	4	no	—
25	C₂×Dic₁₉₉	C₂³	1	2	yes	—
26	C₂×Dic₁₉₉	C₂³	1	4	yes	—
27	C₂×Dic₁₉₉	C₂³	1	2	yes	—
28	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	Q₈	1	4	yes	complemented
30	C₂×C₄×C₁₉₉	C₂³	1	4	yes	—
31	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
33	C₂²×C₁₉₉	—	2	2	no	—
34	C₂²×C₁₉₉	C₂×Q₈	1	2	yes	—
35	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
36	C₄×C₁₉₉	—	2	4	no	—
37	C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
38	C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
39	Dic₁₉₉	—	2	2	no	—
40	Dic₁₉₉	—	2	4	no	—
41	Dic₁₉₉	#41	1	4	yes	—
42	C₂×C₁₉₉	—	2	2	no	—
43	C₂×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₁₉₉	C₂×#41	1	2	yes	—
45	C₁₉₉	#533	1	1	yes	characteristic, complemented, Sylow
46	#533	—	199	1	no	maximal, Sylow
47	C₄⋊Q₈	—	199	2	no	—
48	#125	—	199	2	no	—
49	C₂²⋊Q₈	—	199	4	no	—
50	C₄×Q₈	—	199	4	no	—
51	#117	—	199	1	no	—
52	#117	D₁₉₉	1	1	yes	characteristic
53	C₂×C₄²	—	199	1	no	—
54	C₂×Q₈	—	199	4	no	—
55	C₂²×C₄	—	199	2	no	—
56	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
57	C₄⋊C₄	—	199	4	no	—
58	C₄⋊C₄	—	199	4	no	—
59	C₄⋊C₄	—	199	2	no	—
60	C₄⋊C₄	—	199	4	no	—
61	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
62	C₂²⋊C₄	—	199	2	no	—
63	C₂²⋊C₄	C₂×D₁₉₉	1	2	yes	—
64	C₄²	—	199	1	no	—
65	C₄²	—	199	1	no	—
66	C₄²	—	199	2	no	—
67	C₄²	—	199	2	no	—
68	C₄²	C₂×D₁₉₉	1	2	yes	—
69	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
70	Q₈	—	398	8	no	—
71	C₂×C₄	—	398	4	no	—
72	C₂×C₄	—	199	2	no	—
73	C₂×C₄	—	199	4	no	—
74	C₂×C₄	—	199	2	no	—
75	C₂×C₄	Dic₃₉₈	1	4	yes	complemented
76	C₂×C₄	C₂²×D₁₉₉	1	4	yes	—
77	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	C₂×Dic₃₉₈	1	2	yes	—
81	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	398	2	no	—
84	C₄	—	398	4	no	—
85	C₄	—	199	4	no	—
86	C₄	C₂×Dic₃₉₈	1	2	yes	—
87	C₄	C₂×Dic₃₉₈	1	2	yes	—
88	C₂	—	2	2	no	—
89	C₂	C₂²×Dic₃₉₈	1	1	yes	characteristic
90	C₂	#41×D₁₉₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Frattini	C₂²	C₂³×D₁₉₉
Fitting	#117×C₁₉₉	C₂
prime radical	C₂³×C₁₉₉	C₂³
hypercenter	#117	D₁₉₉
hypocenter	C₁₉₉	#533
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #20263954	C₂	semidirect
2. #20263946	C₂	semidirect
4. C₄×Dic₃₉₈	C₂	semidirect
29. C₂×C₄×C₁₉₉	Q₈	semidirect
45. C₁₉₉	#533	semidirect
75. C₂×C₄	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
#20263946	2
#20263932	2
C₄×Dic₃₉₈	2
#20263914	2
C₂×C₄×Dic₁₉₉	2
#117×C₁₉₉	2
#533	199