Subgroups — #38766683

Order 25472.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27164071	C₂	1	2	yes	maximal
2	C₄³×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₉₉⋊C₈)	C₄	1	4	yes	complemented
4	C₂²⋊(C₁₉₉⋊C₈)	—	2	2	no	—
5	C₄×C₁₉₉⋊C₈	—	2	2	no	—
6	C₂×C₄²×C₁₉₉	—	2	2	no	—
7	C₂×C₄²×C₁₉₉	C₄	1	2	yes	—
8	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₉₉⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₁₉₉	—	2	2	no	—
11	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
13	C₄²×C₁₉₉	—	2	8	no	—
14	C₄²×C₁₉₉	—	2	2	no	—
15	C₄²×C₁₉₉	C₈	1	4	yes	complemented
16	C₄²×C₁₉₉	C₂×C₄	1	2	yes	—
17	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
18	C₄²×C₁₉₉	Q₈	1	1	yes	characteristic
19	C₁₉₉⋊C₈	—	4	4	no	—
20	C₂³×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	—	2	8	no	—
22	C₂×C₄×C₁₉₉	—	2	4	no	—
23	C₂×C₄×C₁₉₉	—	2	4	no	—
24	C₂×C₄×C₁₉₉	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₉₉	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₉₉	M₁₆	1	2	yes	—
30	C₂²×C₁₉₉	—	2	2	no	—
31	C₂²×C₁₉₉	#95	1	1	yes	characteristic
32	C₂²×C₁₉₉	C₂²⋊C₈	1	2	yes	—
33	C₄×C₁₉₉	—	2	8	no	—
34	C₄×C₁₉₉	—	2	2	no	—
35	C₄×C₁₉₉	C₄×C₈	1	1	yes	characteristic
36	C₄×C₁₉₉	#97	1	1	yes	characteristic
37	C₄×C₁₉₉	#104	1	4	yes	—
38	C₄×C₁₉₉	C₄⋊C₈	1	2	yes	—
39	C₂×C₁₉₉	—	2	2	no	—
40	C₂×C₁₉₉	#336	1	1	yes	characteristic
41	C₂×C₁₉₉	#339	1	2	yes	—
42	C₁₉₉	#1276	1	1	yes	characteristic, complemented, Sylow
43	#1276	—	199	1	no	maximal, Sylow
44	#431	—	199	2	no	—
45	C₄³	D₁₉₉	1	1	yes	characteristic
46	C₂×C₄²	—	2	2	no	—
47	C₂×C₄²	Dic₁₉₉	1	2	yes	—
48	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
49	C₄⋊C₈	—	199	4	no	—
50	C₂²⋊C₈	—	398	2	no	—
51	C₄×C₈	—	398	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
55	C₂×C₈	—	398	4	no	—
56	C₄²	—	2	8	no	—
57	C₄²	—	2	2	no	—
58	C₄²	C₁₉₉⋊C₈	1	4	yes	complemented
59	C₄²	C₄×D₁₉₉	1	2	yes	—
60	C₄²	D₇₉₆	1	1	yes	characteristic
61	C₄²	Dic₃₉₈	1	1	yes	characteristic
62	C₂³	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
63	C₂×C₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
67	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
68	C₂×C₄	Dic₁₉₉⋊C₄	1	2	yes	—
69	C₂×C₄	C₄⋊Dic₁₉₉	1	1	yes	characteristic
70	C₂×C₄	C₂×C₁₉₉⋊C₈	1	2	yes	—
71	C₂×C₄	#15691870	1	2	yes	—
72	C₈	—	796	4	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₁₉₉⋊#95	1	1	yes	characteristic
75	C₂²	C₂²⋊(C₁₉₉⋊C₈)	1	2	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	2	no	—
78	C₄	C₄×C₁₉₉⋊C₈	1	1	yes	characteristic
79	C₄	#20263866	1	1	yes	characteristic
80	C₄	#20263883	1	4	yes	—
81	C₄	C₄⋊(C₁₉₉⋊C₈)	1	2	yes	—
82	C₂	—	2	2	no	—
83	C₂	#27163773	1	1	yes	characteristic
84	C₂	#27163778	1	2	yes	—

	subgroup	G/H
center	C₄²	D₇₉₆
commutator	C₄×C₁₉₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₉₉
Fitting	C₄³×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#95
prime radical	C₂³×C₁₉₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉₉	C₂²
hypercenter	C₄³	D₁₉₉
hypocenter	C₁₉₉	#1276
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₁₉₉⋊C₈)	C₄	semidirect
15. C₄²×C₁₉₉	C₈	semidirect
42. C₁₉₉	#1276	semidirect
58. C₄²	C₁₉₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#27164071	2
C₄³×C₁₉₉	2
#1276	199