Subgroups — #38766821

Order 25472.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic
2	#27163769	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₉₆	C₄	1	1	yes	characteristic
5	C₄×Dic₃₉₈	—	2	1	no	—
6	#20263889	C₄	1	1	yes	characteristic
7	C₄⋊(C₁₉₉⋊C₈)	—	2	1	no	—
8	#20263880	—	2	1	no	—
9	C₂×C₁₆×C₁₉₉	—	2	1	no	—
10	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₉₈	—	2	1	no	—
12	Dic₇₉₆	—	2	1	no	—
13	Dic₇₉₆	C₈	1	2	yes	complemented
14	C₂×C₁₉₉⋊C₈	—	4	1	no	—
15	C₄⋊Dic₁₉₉	—	2	1	no	—
16	Dic₁₉₉⋊C₄	—	4	1	no	—
17	C₄×Dic₁₉₉	—	4	1	no	—
18	C₂×C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
20	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
21	C₁₆×C₁₉₉	—	2	2	no	—
22	Dic₃₉₈	—	2	2	no	—
23	Dic₃₉₈	—	4	1	no	—
24	C₂×Dic₁₉₉	—	4	1	no	—
25	C₂×Dic₁₉₉	—	4	1	no	—
26	C₁₉₉⋊C₈	—	8	1	no	—
27	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₉₉	—	2	1	no	—
31	C₈×C₁₉₉	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₉₉	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₉₉	#102	1	1	yes	characteristic
34	C₄×C₁₉₉	—	2	1	no	—
35	C₄×C₁₉₉	—	2	1	no	—
36	C₄×C₁₉₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₁₉₉	#104	1	1	yes	characteristic
38	Dic₁₉₉	—	4	2	no	—
39	Dic₁₉₉	—	8	1	no	—
40	C₂×C₁₉₉	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₁₉₉	#360	1	1	yes	characteristic
42	C₂×C₁₉₉	#362	1	1	yes	characteristic
43	C₁₉₉	#1334	1	1	yes	characteristic, complemented, Sylow
44	#1334	—	199	1	no	maximal, Sylow
45	C₄×Q₁₆	—	199	1	no	—
46	#346	D₁₉₉	1	1	yes	characteristic
47	#334	—	199	1	no	—
48	C₂×Q₁₆	—	199	1	no	—
49	C₄×Q₈	—	398	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	199	1	no	—
52	C₄⋊C₈	—	398	1	no	—
53	#103	—	398	1	no	—
54	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
55	C₂×Q₈	—	398	1	no	—
56	Q₁₆	—	398	1	no	—
57	Q₁₆	—	199	2	no	—
58	C₂×C₈	—	796	1	no	—
59	C₂×C₈	C₄×D₁₉₉	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
61	C₄⋊C₄	—	796	1	no	—
62	C₄⋊C₄	—	398	1	no	—
63	C₄²	—	796	1	no	—
64	C₄²	D₇₉₆	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	Q₈	—	796	1	no	—
67	Q₈	—	398	2	no	—
68	C₂×C₄	—	796	1	no	—
69	C₂×C₄	—	796	1	no	—
70	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
71	C₂×C₄	D₁₅₉₂	1	1	yes	characteristic
72	C₂×C₄	#15691875	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	1592	1	no	—
75	C₈	C₈×D₁₉₉	1	1	yes	characteristic
76	C₈	#15691869	1	1	yes	characteristic
77	C₂²	#20263877	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	796	2	no	—
81	C₄	—	1592	1	no	—
82	C₄	(C₂×D₁₉₉)⋊C₈	1	1	yes	characteristic
83	C₄	#20263883	1	1	yes	characteristic
84	C₂	#27163747	1	1	yes	characteristic
85	C₂	#27163829	1	1	yes	characteristic
86	C₂	#27163835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263877
commutator	C₈×C₁₉₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₉₉
Fitting	#346×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	#102
cosocle	C₄×C₈×C₁₉₉	C₂²
hypercenter	#346	D₁₉₉
hypocenter	C₁₉₉	#1334
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₇₉₆	C₈	semidirect
43. C₁₉₉	#1334	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₇₉₆	2
#27163769	2
#346×C₁₉₉	2
#1334	199