Subgroups — #28383319

Order 14272.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297292	C₂	1	1	yes	maximal, characteristic
2	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
3	#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₄₄₆	C₄	1	1	yes	characteristic
5	C₂×D₈₉₂	C₄	1	1	yes	characteristic
6	C₂×C₂₂₃⋊C₈	—	2	1	no	—
7	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
8	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
9	C₂×C₈×C₂₂₃	—	2	1	no	—
10	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₂₂₃	—	2	1	no	—
12	Dic₄₄₆	—	4	1	no	—
13	D₈₉₂	—	4	1	no	—
14	C₂×Dic₂₂₃	—	2	1	no	—
15	C₂₂₃⋊C₈	—	2	2	no	—
16	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
19	C₈×C₂₂₃	—	2	2	no	—
20	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×D₂₂₃	—	4	1	no	—
22	C₂×D₂₂₃	—	4	1	no	—
23	C₂×D₂₂₃	—	4	1	no	—
24	C₄×C₂₂₃	—	2	1	no	—
25	C₄×C₂₂₃	—	2	1	no	—
26	C₄×C₂₂₃	—	2	1	no	—
27	Dic₂₂₃	—	4	1	no	—
28	C₂×C₂₂₃	#100	1	1	yes	characteristic
29	C₂×C₂₂₃	#104	1	1	yes	characteristic
30	C₂×C₂₂₃	#104	1	1	yes	characteristic
31	D₂₂₃	—	8	1	no	—
32	C₂₂₃	#329	1	1	yes	characteristic, complemented, Sylow
33	#329	—	223	1	no	maximal, Sylow
34	#124	—	223	1	no	—
35	#97	—	223	1	no	—
36	#97	D₂₂₃	1	1	yes	characteristic, complemented
37	C₂×Q₈	—	223	1	no	—
38	C₂×D₄	—	223	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	446	1	no	—
41	C₂²⋊C₄	—	446	1	no	—
42	C₂²⋊C₄	—	446	1	no	—
43	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
44	C₂³	—	446	1	no	—
45	Q₈	—	892	1	no	—
46	D₄	—	892	1	no	—
47	C₂×C₄	—	446	1	no	—
48	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
50	C₂×C₄	D₈₉₂	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	446	2	no	—
53	C₂²	—	892	1	no	—
54	C₂²	—	892	1	no	—
55	C₂²	—	892	1	no	—
56	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	892	1	no	—
61	C₂	—	1784	1	no	—
62	C₂	#20297220	1	1	yes	characteristic
63	C₂	#20297230	1	1	yes	characteristic
64	C₂	#20297232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₂₃)⋊C₄
commutator	C₂×C₄×C₂₂₃	C₂×C₄
Frattini	C₄²	C₂×D₂₂₃
Fitting	#97×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	C₂×D₈₉₂	C₄
cosocle	C₄²×C₂₂₃	C₂²
hypercenter	#97	D₂₂₃
hypocenter	C₂₂₃	#329
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. #20297215	C₂	semidirect
3. #97×C₂₂₃	C₂	semidirect
32. C₂₂₃	#329	semidirect
36. #97	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297292	2
#20297215	2
#97×C₂₂₃	2
#329	223