Subgroups — #27164223

Order 12736.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691874	C₂	1	2	yes	maximal
2	#20263955	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263878	C₂	1	2	yes	maximal, complemented
5	C₄×D₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₂²	1	2	yes	—
7	#15691874	—	2	4	no	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₉₉	—	2	1	no	—
10	C₄⋊Dic₁₉₉	—	2	1	no	—
11	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₉₉	—	2	1	no	—
13	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
15	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	Dic₃₉₈	—	2	2	no	—
17	Dic₃₉₈	—	2	2	no	—
18	C₂×Dic₁₉₉	—	2	2	no	—
19	C₁₉₉⋊C₈	—	2	2	no	—
20	C₂³×C₁₉₉	—	2	1	no	—
21	D₄×C₁₉₉	—	2	1	no	—
22	D₄×C₁₉₉	D₄	1	2	yes	—
23	C₂×C₄×C₁₉₉	—	2	1	no	—
24	C₂×C₄×C₁₉₉	—	4	2	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	C₂²×C₁₉₉	—	2	2	no	—
29	C₂²×C₁₉₉	—	4	1	no	—
30	C₂²×C₁₉₉	—	4	1	no	—
31	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₉₉	—	2	1	no	—
33	C₄×C₁₉₉	—	4	1	no	—
34	C₄×C₁₉₉	SD₁₆	1	2	yes	complemented
35	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₉₉	#41	1	1	yes	characteristic
37	Dic₁₉₉	—	4	2	no	—
38	C₂×C₁₉₉	—	4	2	no	—
39	C₂×C₁₉₉	#121	1	1	yes	characteristic
40	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₁₉₉	#137	1	1	yes	characteristic
42	C₁₉₉	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	199	1	no	maximal, Sylow
44	C₂×SD₁₆	—	199	2	no	—
45	C₄⋊Q₈	—	199	1	no	—
46	C₄×D₄	D₁₉₉	1	1	yes	characteristic
47	C₄⋊C₈	—	199	1	no	—
48	#103	—	199	2	no	—
49	C₂×Q₈	—	199	2	no	—
50	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	398	4	no	—
53	C₂×C₈	—	199	2	no	—
54	C₄⋊C₄	—	398	1	no	—
55	C₄⋊C₄	—	398	1	no	—
56	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	398	2	no	—
61	Q₈	—	398	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₇₉₆	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	398	2	no	—
67	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
70	C₈	—	398	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	796	2	no	—
78	C₄	#15691874	1	2	yes	complemented
79	C₄	C₂×D₇₉₆	1	1	yes	characteristic
80	C₄	#15691886	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#20263929	1	1	yes	characteristic
83	C₂	C₂×#15691874	1	1	yes	characteristic
84	C₂	#20263985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄×D₄×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂×D₄×C₁₉₉	C₂²
hypercenter	C₄×D₄	D₁₉₉
hypocenter	C₁₉₉	#461
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #20263955	C₂	semidirect
3. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
4. #20263878	C₂	semidirect
34. C₄×C₁₉₉	SD₁₆	semidirect
42. C₁₉₉	#461	semidirect
78. C₄	#15691874	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691874	2
#20263955	2
C₄⋊(C₁₉₉⋊C₈)	2
#20263878	2
C₄×D₄×C₁₉₉	2
#461	199