Subgroups — #27164366

Order 12736.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263952	C₂	1	1	yes	maximal, characteristic
2	#20263875	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₉₆	—	2	1	no	—
6	C₂×D₇₉₆	—	2	1	no	—
7	C₂×D₇₉₆	C₂²	1	2	yes	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₉₉	—	2	1	no	—
10	C₄⋊C₄×C₁₉₉	—	2	1	no	—
11	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
12	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂²×D₁₉₉	—	2	2	no	—
14	D₇₉₆	—	2	4	no	—
15	D₇₉₆	—	4	2	no	—
16	D₇₉₆	—	4	2	no	—
17	C₁₉₉⋊C₈	—	2	2	no	—
18	Q₈×C₁₉₉	—	4	2	no	—
19	C₂×C₄×C₁₉₉	—	2	2	no	—
20	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₉₉	—	4	2	no	—
25	C₂×D₁₉₉	—	4	2	no	—
26	C₂×D₁₉₉	—	4	2	no	—
27	C₄×C₁₉₉	—	4	2	no	—
28	C₄×C₁₉₉	D₈	1	2	yes	—
29	C₄×C₁₉₉	SD₁₆	1	2	yes	—
30	C₄×C₁₉₉	#41	1	2	yes	—
31	C₂×C₁₉₉	#124	1	1	yes	characteristic
32	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
33	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
34	D₁₉₉	—	8	2	no	—
35	C₁₉₉	#486	1	1	yes	characteristic, complemented, Sylow
36	#486	—	199	1	no	maximal, Sylow
37	C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic, complemented
38	#127	—	199	1	no	—
39	#102	—	199	4	no	—
40	C₄×C₈	—	199	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	398	1	no	—
43	C₂×D₄	—	398	1	no	—
44	C₂×D₄	—	199	2	no	—
45	C₂×C₈	—	199	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
48	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
49	C₂³	—	398	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	796	2	no	—
52	D₄	—	796	2	no	—
53	D₄	—	398	4	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
58	C₈	—	398	2	no	—
59	C₂²	—	796	2	no	—
60	C₂²	—	796	2	no	—
61	C₂²	—	796	2	no	—
62	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#15691871	1	2	yes	—
65	C₄	#15691873	1	2	yes	—
66	C₄	#15691887	1	2	yes	—
67	C₂	—	1592	2	no	—
68	C₂	#20263944	1	1	yes	characteristic
69	C₂	C₂×#15691871	1	1	yes	characteristic
70	C₂	C₂×#15691873	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⋊Q₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	#20263952	C₂
hypercenter	C₄⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#486
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20263875	C₂	semidirect
3. C₄×C₁₉₉⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₁₉₉	C₂	semidirect
35. C₁₉₉	#486	semidirect
37. C₄⋊Q₈	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263952	2
#20263875	2
C₄×C₁₉₉⋊C₈	2
C₄⋊Q₈×C₁₉₉	2
#486	199