Subgroups — C₂×#368×C₂₂₃ #43251175

Order 28544.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₂₂₃	C₂	1	2	yes	maximal
3	#368×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₁₆×C₂₂₃	—	2	2	no	—
5	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₂₂₃	C₄	1	2	yes	—
7	C₂×C₁₆×C₂₂₃	C₄	1	2	yes	—
8	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
9	#108×C₂₂₃	—	2	2	no	—
10	#108×C₂₂₃	C₂²	1	4	yes	—
11	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
12	M₁₆×C₂₂₃	—	2	4	no	—
13	M₁₆×C₂₂₃	—	4	2	no	—
14	C₂×C₈×C₂₂₃	—	4	2	no	—
15	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
19	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₁₆×C₂₂₃	C₂×C₄	1	4	yes	—
22	C₂³×C₂₂₃	Q₁₆	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₈×C₂₂₃	—	4	4	no	—
30	C₈×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₈×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₈×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₂₂₃	—	2	2	no	—
35	C₂²×C₂₂₃	#107	1	1	yes	characteristic
36	C₂²×C₂₂₃	#107	1	1	yes	characteristic
37	C₂²×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
38	C₄×C₂₂₃	#107	1	1	yes	characteristic
39	C₄×C₂₂₃	#107	1	1	yes	characteristic
40	C₄×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₂₂₃	C₂×D₈	1	1	yes	characteristic
42	C₂×C₂₂₃	—	2	1	no	—
43	C₂×C₂₂₃	—	2	1	no	—
44	C₂×C₂₂₃	#368	1	2	yes	direct product factor
45	C₂×C₂₂₃	C₂×#107	1	1	yes	characteristic
46	C₂₂₃	C₂×#368	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×#368	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
49	C₂×#108	C₂×C₂₂₃	1	2	yes	—
50	#368	C₂×C₂₂₃	1	4	yes	direct product factor
51	C₂×M₁₆	—	2	2	no	—
52	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
53	C₂×C₁₆	C₄×C₂₂₃	1	2	yes	—
54	C₂×C₁₆	C₄×C₂₂₃	1	2	yes	—
55	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
56	#108	—	2	2	no	—
57	#108	C₂²×C₂₂₃	1	4	yes	—
58	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
59	M₁₆	—	2	4	no	—
60	M₁₆	—	4	2	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
63	C₂×C₈	C₂×C₄×C₂₂₃	1	1	yes	characteristic
64	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
65	C₂×C₈	Q₈×C₂₂₃	1	1	yes	characteristic
66	C₂×C₈	Q₈×C₂₂₃	1	1	yes	characteristic
67	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
68	C₁₆	C₂×C₄×C₂₂₃	1	4	yes	—
69	C₂³	Q₁₆×C₂₂₃	1	1	yes	characteristic
70	C₂×C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
71	C₂×C₄	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
72	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
73	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
74	C₂×C₄	Q₁₆×C₂₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
76	C₈	—	4	4	no	—
77	C₈	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
78	C₈	C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
79	C₈	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
80	C₈	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
81	C₂²	—	2	2	no	—
82	C₂²	#107×C₂₂₃	1	1	yes	characteristic
83	C₂²	#107×C₂₂₃	1	1	yes	characteristic
84	C₂²	C₂×Q₁₆×C₂₂₃	1	1	yes	characteristic
85	C₄	#107×C₂₂₃	1	1	yes	characteristic
86	C₄	#107×C₂₂₃	1	1	yes	characteristic
87	C₄	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
88	C₄	C₂×D₈×C₂₂₃	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2	1	no	—
91	C₂	#368×C₂₂₃	1	2	yes	direct product factor
92	C₂	C₂×#107×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	D₈
commutator	C₈	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
socle	C₂²×C₂₂₃	#107
prime radical	C₂³×C₂₂₃	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ C₂²×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. #368×C₂₂₃	C₂	direct (N × H)
44. C₂×C₂₂₃	#368	direct (N × H)
46. C₂₂₃	C₂×#368	direct (N × H)
47. C₂×#368	C₂₂₃	direct (N × H)
50. #368	C₂×C₂₂₃	direct (N × H)
91. C₂	#368×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×C₂₂₃	2
C₂×#108×C₂₂₃	2
#368×C₂₂₃	2
C₂×#368	223

Subgroups — C2×#368×C223 #43251175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#368×C₂₂₃ #43251175