Subgroups — C₄×Dic₇₇₂ #27135355

Order 12352.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₇₂	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₈₆	C₂	1	2	yes	maximal
3	#20259833	C₂	1	1	yes	maximal, characteristic
4	#20259824	C₂	1	2	yes	maximal
5	C₄×C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₆	C₂²	1	2	yes	—
7	Dic₇₇₂	C₄	1	4	yes	direct product factor
8	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
9	#15690534	—	2	2	no	—
10	C₄×Dic₁₉₃	—	2	2	no	—
11	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
14	Dic₃₈₆	—	2	2	no	—
15	Dic₃₈₆	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₉₃	—	2	2	no	—
17	C₂×Dic₁₉₃	—	2	2	no	—
18	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
21	C₈×C₁₉₃	—	2	1	no	—
22	C₈×C₁₉₃	C₂×C₄	1	2	yes	—
23	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₉₃	—	2	1	no	—
25	C₄×C₁₉₃	Q₁₆	1	2	yes	complemented
26	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₉₃	#41	1	1	yes	characteristic
28	Dic₁₉₃	—	2	4	no	—
29	Dic₁₉₃	—	4	2	no	—
30	C₂×C₁₉₃	C₄×D₄	1	1	yes	characteristic
31	C₂×C₁₉₃	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₁₉₃	#135	1	1	yes	characteristic
33	C₁₉₃	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	193	1	no	maximal, Sylow
35	C₂×Q₁₆	—	193	1	no	—
36	C₄×Q₈	—	193	2	no	—
37	#107	—	193	1	no	—
38	#103	—	193	2	no	—
39	C₄×C₈	D₁₉₃	1	1	yes	characteristic
40	C₂×Q₈	—	193	2	no	—
41	Q₁₆	—	193	4	no	—
42	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
43	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
44	C₄⋊C₄	—	386	2	no	—
45	C₄⋊C₄	—	193	2	no	—
46	C₄²	—	386	2	no	—
47	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
48	Q₈	—	386	2	no	—
49	Q₈	—	193	4	no	—
50	C₂×C₄	—	386	2	no	—
51	C₂×C₄	—	386	2	no	—
52	C₂×C₄	D₇₇₂	1	1	yes	characteristic
53	C₂×C₄	D₇₇₂	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₄×D₁₉₃	1	2	yes	—
57	C₂²	C₂×D₇₇₂	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	772	2	no	—
60	C₄	—	386	4	no	—
61	C₄	Dic₇₇₂	1	2	yes	direct product factor
62	C₄	C₂×C₄×D₁₉₃	1	1	yes	characteristic
63	C₄	#15690555	1	1	yes	characteristic
64	C₂	C₄×D₇₇₂	1	1	yes	characteristic
65	C₂	C₂×Dic₇₇₂	1	1	yes	characteristic
66	C₂	#20259921	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇₇₂
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	C₄×C₈×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₂×D₄
cosocle	C₂×C₄×C₁₉₃	C₂³
hypercenter	C₄×C₈	D₁₉₃
hypocenter	C₁₉₃	C₄×Q₁₆
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
7. Dic₇₇₂	C₄	direct (N × H)
25. C₄×C₁₉₃	Q₁₆	semidirect
33. C₁₉₃	C₄×Q₁₆	semidirect
61. C₄	Dic₇₇₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Dic₇₇₂	2
C₄×Dic₃₈₆	2
#20259833	2
#20259824	2
C₄×C₈×C₁₉₃	2
C₄×Q₁₆	193

Subgroups — C4×Dic772 #27135355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₇₇₂ #27135355