Subgroups — #27164690

Order 12736.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691887	C₂	1	2	yes	maximal, complemented
2	#20263952	C₂	1	3	yes	maximal, complemented
3	#20263930	C₂	1	6	yes	maximal, complemented
4	C₄×D₇₉₆	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691887	—	2	8	no	—
7	C₂×D₇₉₆	C₂²	1	3	yes	—
8	C₂×D₇₉₆	C₂²	1	6	yes	—
9	C₂×D₇₉₆	C₂²	1	6	yes	—
10	C₂×C₄×D₁₉₉	C₂²	1	6	yes	—
11	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
12	(C₂×D₁₉₉)⋊C₄	C₂²	1	6	yes	—
13	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₉	C₂²	1	3	yes	—
15	C₄²×C₁₉₉	C₂²	1	3	yes	—
16	C₂²×D₁₉₉	C₂³	1	6	yes	—
17	D₇₉₆	—	2	12	no	—
18	D₇₉₆	—	2	6	no	—
19	D₇₉₆	—	4	6	no	—
20	C₂×Dic₁₉₉	C₂³	1	2	yes	—
21	C₄×D₁₉₉	—	2	12	no	—
22	Q₈×C₁₉₉	D₄	1	4	yes	complemented
23	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
24	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
25	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₉₉	—	2	6	no	—
28	C₂×D₁₉₉	—	2	6	no	—
29	C₂×D₁₉₉	—	2	6	no	—
30	C₄×C₁₉₉	—	2	3	no	—
31	C₄×C₁₉₉	C₂×D₄	1	6	yes	—
32	C₄×C₁₉₉	#41	1	2	yes	complemented
33	Dic₁₉₉	—	2	2	no	—
34	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₁₉₉	C₂×#41	1	1	yes	characteristic
36	C₂×C₁₉₉	H(5,𝔽₂)	1	1	yes	characteristic
37	D₁₉₉	—	4	6	no	—
38	C₁₉₉	#550	1	1	yes	characteristic, complemented, Sylow
39	#550	—	199	1	no	maximal, Sylow
40	C₂×#41	—	199	2	no	—
41	#127	—	199	3	no	—
42	#121	—	199	6	no	—
43	C₄×Q₈	D₁₉₉	1	1	yes	characteristic, complemented
44	C₄×D₄	—	199	3	no	—
45	#41	—	398	8	no	—
46	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
47	C₂×D₄	—	199	6	no	—
48	C₂×D₄	—	199	3	no	—
49	C₂×D₄	—	199	6	no	—
50	C₂²×C₄	—	199	6	no	—
51	C₄⋊C₄	—	199	1	no	—
52	C₄⋊C₄	C₂×D₁₉₉	1	3	yes	—
53	C₂²⋊C₄	—	199	6	no	—
54	C₄²	C₂×D₁₉₉	1	3	yes	—
55	C₂³	—	199	6	no	—
56	Q₈	D₇₉₆	1	4	yes	complemented
57	D₄	—	796	6	no	—
58	D₄	—	398	12	no	—
59	D₄	—	398	6	no	—
60	C₂×C₄	—	398	12	no	—
61	C₂×C₄	—	199	2	no	—
62	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
63	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
64	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
65	C₂²	—	398	6	no	—
66	C₂²	—	398	6	no	—
67	C₂²	—	398	6	no	—
68	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	398	2	no	—
71	C₄	C₂×D₇₉₆	1	6	yes	—
72	C₄	#15691887	1	2	yes	complemented
73	C₂	—	796	6	no	—
74	C₂	C₂²×D₇₉₆	1	1	yes	characteristic
75	C₂	C₂×#15691887	1	1	yes	characteristic
76	C₂	#20264001	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	C₄×Q₈×C₁₉₉	C₂
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	#550
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15691887	C₂	semidirect
2. #20263952	C₂	semidirect
3. #20263930	C₂	semidirect
4. C₄×D₇₉₆	C₂	semidirect
5. C₄×Q₈×C₁₉₉	C₂	semidirect
22. Q₈×C₁₉₉	D₄	semidirect
32. C₄×C₁₉₉	#41	semidirect
38. C₁₉₉	#550	semidirect
43. C₄×Q₈	D₁₉₉	semidirect
56. Q₈	D₇₉₆	semidirect
72. C₄	#15691887	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691887	2
#20263952	2
#20263930	2
C₄×D₇₉₆	2
C₄×Q₈×C₁₉₉	2
#550	199