Subgroups — #555×C₂₂₃ #28383267

Order 14272.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#124×C₂₂₃	C₂	1	4	yes	maximal, complemented
3	#121×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	C₄×D₄×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	#117×C₂₂₃	C₂	1	4	yes	maximal, complemented
6	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
7	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
8	C₂×D₄×C₂₂₃	C₂²	1	4	yes	—
9	C₂²×C₄×C₂₂₃	C₂²	1	4	yes	—
10	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
11	C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
12	C₂²⋊C₄×C₂₂₃	C₂²	1	8	yes	—
13	C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	—
14	C₄²×C₂₂₃	C₂²	1	4	yes	—
15	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₂₂₃	C₂³	1	4	yes	—
17	Q₈×C₂₂₃	—	4	2	no	—
18	D₄×C₂₂₃	—	2	4	no	—
19	D₄×C₂₂₃	—	4	4	no	—
20	C₂×C₄×C₂₂₃	—	2	8	no	—
21	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
22	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
24	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
25	C₂²×C₂₂₃	—	2	4	no	—
26	C₂²×C₂₂₃	—	2	4	no	—
27	C₂²×C₂₂₃	—	2	4	no	—
28	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₂₂₃	—	2	4	no	—
30	C₄×C₂₂₃	—	2	4	no	—
31	C₄×C₂₂₃	—	2	1	no	—
32	C₄×C₂₂₃	#41	1	4	yes	—
33	C₂×C₂₂₃	—	4	4	no	—
34	C₂×C₂₂₃	C₂×#41	1	2	yes	—
35	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
36	C₂₂₃	#555	1	1	yes	characteristic, direct product factor, Sylow
37	#555	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₄⋊Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
39	#124	C₂×C₂₂₃	1	4	yes	complemented
40	#121	C₂×C₂₂₃	1	4	yes	complemented
41	C₄×D₄	C₂×C₂₂₃	1	2	yes	complemented
42	#117	C₂×C₂₂₃	1	4	yes	complemented
43	C₂×Q₈	C₂²×C₂₂₃	1	2	yes	—
44	C₂×D₄	C₂²×C₂₂₃	1	2	yes	—
45	C₂×D₄	C₂²×C₂₂₃	1	4	yes	—
46	C₂²×C₄	C₂²×C₂₂₃	1	4	yes	—
47	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
48	C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
49	C₂²⋊C₄	C₂²×C₂₂₃	1	8	yes	—
50	C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	—
51	C₄²	C₂²×C₂₂₃	1	4	yes	—
52	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
53	C₂³	C₂³×C₂₂₃	1	4	yes	—
54	Q₈	—	4	2	no	—
55	D₄	—	2	4	no	—
56	D₄	—	4	4	no	—
57	C₂×C₄	—	2	8	no	—
58	C₂×C₄	C₂³×C₂₂₃	1	4	yes	—
59	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂³×C₂₂₃	1	4	yes	—
61	C₂×C₄	C₂³×C₂₂₃	1	2	yes	—
62	C₂²	—	2	4	no	—
63	C₂²	—	2	4	no	—
64	C₂²	—	2	4	no	—
65	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	4	no	—
68	C₄	—	2	1	no	—
69	C₄	#41×C₂₂₃	1	4	yes	—
70	C₂	—	4	4	no	—
71	C₂	C₂×#41×C₂₂₃	1	2	yes	—
72	C₂	H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₂₂₃	C₂	semidirect
2. #124×C₂₂₃	C₂	semidirect
3. #121×C₂₂₃	C₂	semidirect
4. C₄×D₄×C₂₂₃	C₂	semidirect
5. #117×C₂₂₃	C₂	semidirect
36. C₂₂₃	#555	direct (N × H)
37. #555	C₂₂₃	direct (N × H)
38. C₄⋊Q₈	C₂×C₂₂₃	semidirect
39. #124	C₂×C₂₂₃	semidirect
40. #121	C₂×C₂₂₃	semidirect
41. C₄×D₄	C₂×C₂₂₃	semidirect
42. #117	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₂₂₃	2
#124×C₂₂₃	2
#121×C₂₂₃	2
C₄×D₄×C₂₂₃	2
#117×C₂₂₃	2
#555	223

Subgroups — #555×C223 #28383267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #555×C₂₂₃ #28383267