Subgroups — #27135002

Order 12352.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259896	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13095393	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₇₂	—	2	1	no	—
5	C₂×D₇₇₂	—	2	1	no	—
6	C₂×D₇₇₂	C₄	1	2	yes	—
7	C₂×#13095393	—	2	1	no	—
8	C₂×C₈×C₁₉₃	—	2	1	no	—
9	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
10	C₂²×D₁₉₃	—	2	2	no	—
11	D₇₇₂	—	4	2	no	—
12	D₇₇₂	—	4	1	no	—
13	D₇₇₂	—	4	1	no	—
14	D₇₇₂	—	4	1	no	—
15	D₇₇₂	—	4	1	no	—
16	#13095393	—	4	1	no	—
17	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
20	C₈×C₁₉₃	—	4	1	no	—
21	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₁₉₃	—	4	2	no	—
23	C₂×D₁₉₃	—	4	2	no	—
24	C₂×D₁₉₃	—	4	2	no	—
25	C₄×C₁₉₃	—	2	1	no	—
26	C₄×C₁₉₃	D₈	1	1	yes	characteristic
27	C₄×C₁₉₃	D₈	1	1	yes	characteristic
28	C₄×C₁₉₃	SD₁₆	1	1	yes	characteristic
29	C₄×C₁₉₃	SD₁₆	1	1	yes	characteristic
30	C₂×C₁₉₃	#100	1	1	yes	characteristic
31	C₂×C₁₉₃	#102	1	1	yes	characteristic
32	C₂×C₁₉₃	#102	1	1	yes	characteristic
33	D₁₉₃	—	8	2	no	—
34	C₁₉₃	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	193	1	no	maximal, Sylow
36	#127	—	193	1	no	—
37	C₄⋊C₈	—	193	1	no	—
38	C₄⋊C₈	D₁₉₃	1	1	yes	characteristic, complemented
39	C₂×D₄	—	386	1	no	—
40	C₂×D₄	—	386	1	no	—
41	C₂×D₄	—	193	2	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	386	1	no	—
44	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
45	C₂³	—	386	2	no	—
46	D₄	—	772	2	no	—
47	D₄	—	772	1	no	—
48	D₄	—	772	1	no	—
49	D₄	—	772	1	no	—
50	D₄	—	772	1	no	—
51	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
53	C₂×C₄	D₇₇₂	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	772	1	no	—
56	C₂²	—	772	2	no	—
57	C₂²	—	772	2	no	—
58	C₂²	—	772	2	no	—
59	C₂²	#15690532	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#15690540	1	1	yes	characteristic
62	C₄	D₁₅₄₄	1	1	yes	characteristic
63	C₄	#15690542	1	1	yes	characteristic
64	C₄	#15690544	1	1	yes	characteristic
65	C₂	—	1544	2	no	—
66	C₂	#20259815	1	1	yes	characteristic
67	C₂	#20259819	1	1	yes	characteristic
68	C₂	#20259821	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690532
commutator	C₂×C₄×C₁₉₃	C₂×C₄
Frattini	C₄²	C₂×D₁₉₃
Fitting	C₄⋊C₈×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂²⋊C₄
prime radical	#20259896	C₂
cosocle	C₄²×C₁₉₃	C₂²
hypercenter	C₄⋊C₈	D₁₉₃
hypocenter	C₁₉₃	#331
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13095393	C₂	semidirect
3. C₄⋊C₈×C₁₉₃	C₂	semidirect
34. C₁₉₃	#331	semidirect
38. C₄⋊C₈	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259896	2
C₄⋊#13095393	2
C₄⋊C₈×C₁₉₃	2
#331	193