Subgroups — #27164040

Order 12736.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691869	C₂	1	2	yes	maximal
2	C₄²×D₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₁₉₉⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₉₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₉₉	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
9	#15691869	—	2	4	no	—
10	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
11	C₄×Dic₁₉₉	C₄	1	1	yes	characteristic
12	C₄×Dic₁₉₉	C₄	1	1	yes	characteristic
13	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
15	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂²×D₁₉₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₄×D₁₉₉	—	2	2	no	—
21	C₄×D₁₉₉	—	2	2	no	—
22	C₄×D₁₉₉	D₄	1	2	yes	—
23	C₄×D₁₉₉	Q₈	1	2	yes	—
24	C₁₉₉⋊C₈	—	2	2	no	—
25	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
28	C₈×C₁₉₉	—	2	2	no	—
29	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₁₉₉	—	2	1	no	—
31	C₂×D₁₉₉	—	2	1	no	—
32	C₂×D₁₉₉	C₄⋊C₄	1	2	yes	—
33	C₄×C₁₉₉	—	2	1	no	—
34	C₄×C₁₉₉	M₁₆	1	2	yes	complemented
35	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
37	Dic₁₉₉	—	2	1	no	—
38	Dic₁₉₉	C₄⋊C₄	1	2	yes	—
39	Dic₁₉₉	M₁₆	1	2	yes	complemented
40	C₂×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₁₉₉	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₁₉₉	C₂×M₁₆	1	1	yes	characteristic
43	D₁₉₉	—	2	2	no	—
44	C₁₉₉	#423	1	1	yes	characteristic, complemented, Sylow
45	#423	—	199	1	no	maximal, Sylow
46	C₂×M₁₆	—	199	2	no	—
47	C₂×C₄²	—	199	1	no	—
48	C₄⋊C₈	—	199	2	no	—
49	C₄⋊C₈	—	199	1	no	—
50	C₄⋊C₈	D₁₉₉	1	1	yes	characteristic, complemented
51	C₂²×C₄	—	199	1	no	—
52	C₂²×C₄	—	199	1	no	—
53	C₂²×C₄	—	199	1	no	—
54	M₁₆	—	398	4	no	—
55	C₂×C₈	—	199	2	no	—
56	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
57	C₄²	—	199	1	no	—
58	C₄²	—	199	1	no	—
59	C₄²	—	199	1	no	—
60	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
61	C₂³	—	199	1	no	—
62	C₂×C₄	—	398	2	no	—
63	C₂×C₄	—	398	2	no	—
64	C₂×C₄	—	199	1	no	—
65	C₂×C₄	—	199	1	no	—
66	C₂×C₄	—	199	1	no	—
67	C₂×C₄	—	199	2	no	—
68	C₂×C₄	—	199	2	no	—
69	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
70	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	398	2	no	—
74	C₂²	—	398	1	no	—
75	C₂²	—	398	1	no	—
76	C₂²	—	199	2	no	—
77	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	398	1	no	—
80	C₄	—	199	2	no	—
81	C₄	—	199	2	no	—
82	C₄	#15691869	1	2	yes	complemented
83	C₄	D₄×D₁₉₉	1	1	yes	characteristic
84	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
85	C₂	—	398	2	no	—
86	C₂	C₄⋊C₄×D₁₉₉	1	1	yes	characteristic
87	C₂	M₁₆×D₁₉₉	1	1	yes	characteristic
88	C₂	C₂×#15691869	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator, socle	C₂²×C₁₉₉	C₂²×C₄
Fitting	C₄⋊C₈×C₁₉₉	C₂
prime radical	C₂²×D₁₉₉	C₂×C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₄⋊C₈	D₁₉₉
hypocenter	C₁₉₉	#423
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
4. Dic₁₉₉⋊C₈	C₂	semidirect
5. C₄⋊C₈×C₁₉₉	C₂	semidirect
34. C₄×C₁₉₉	M₁₆	semidirect
39. Dic₁₉₉	M₁₆	semidirect
44. C₁₉₉	#423	semidirect
50. C₄⋊C₈	D₁₉₉	semidirect
82. C₄	#15691869	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691869	2
C₄²×D₁₉₉	2
C₄⋊(C₁₉₉⋊C₈)	2
Dic₁₉₉⋊C₈	2
C₄⋊C₈×C₁₉₉	2
#423	199