Subgroups — C₂×#424×C₂₂₃ #43251919

Order 28544.

512 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#424×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊C₈×C₂₂₃	C₂	1	4	yes	maximal, complemented
6	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₂₃	—	2	4	no	—
8	C₂×M₁₆×C₂₂₃	C₂²	1	4	yes	—
9	C₂²×C₈×C₂₂₃	—	2	4	no	—
10	C₂²×C₈×C₂₂₃	C₂²	1	4	yes	—
11	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
13	#117×C₂₂₃	C₄	1	4	yes	—
14	#117×C₂₂₃	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₂₂₃	C₄	1	2	yes	—
16	C₂×C₂²⋊C₄×C₂₂₃	C₄	1	2	yes	—
17	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
18	C₄⋊C₈×C₂₂₃	C₂²	1	16	yes	complemented
19	C₂⁴×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₂₃	D₄	1	4	yes	—
24	C₂²×C₄×C₂₂₃	Q₈	1	4	yes	—
25	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₂₃	—	2	8	no	—
28	C₂×C₈×C₂₂₃	—	2	4	no	—
29	C₂×C₈×C₂₂₃	—	2	8	no	—
30	C₂×C₈×C₂₂₃	—	2	4	no	—
31	C₂×C₈×C₂₂₃	C₂³	1	4	yes	—
32	C₂×C₈×C₂₂₃	C₂³	1	4	yes	—
33	C₄⋊C₄×C₂₂₃	C₂×C₄	1	8	yes	—
34	C₂²⋊C₄×C₂₂₃	C₂×C₄	1	8	yes	—
35	C₄²×C₂₂₃	C₂×C₄	1	4	yes	—
36	C₄²×C₂₂₃	C₂³	1	4	yes	—
37	C₂³×C₂₂₃	—	2	4	no	—
38	C₂³×C₂₂₃	C₄⋊C₄	1	4	yes	—
39	C₂³×C₂₂₃	C₂²×C₄	1	2	yes	—
40	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₂₃	—	2	8	no	—
42	C₂×C₄×C₂₂₃	C₄⋊C₄	1	8	yes	—
43	C₂×C₄×C₂₂₃	C₄⋊C₄	1	4	yes	—
44	C₂×C₄×C₂₂₃	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₂₂₃	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
52	C₈×C₂₂₃	—	2	4	no	—
53	C₈×C₂₂₃	—	2	4	no	—
54	C₂²×C₂₂₃	—	2	4	no	—
55	C₂²×C₂₂₃	—	2	8	no	—
56	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₂₂₃	#131	1	4	yes	—
59	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₂₂₃	—	2	8	no	—
61	C₄×C₂₂₃	C₂×C₄⋊C₄	1	2	yes	—
62	C₄×C₂₂₃	C₂×C₄⋊C₄	1	4	yes	—
63	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₂₂₃	—	2	4	no	—
66	C₂×C₂₂₃	#424	1	4	yes	direct product factor
67	C₂×C₂₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₂₂₃	C₂×#131	1	2	yes	—
69	C₂₂₃	C₂×#424	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×#424	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×M₁₆	C₂×C₂₂₃	1	1	yes	characteristic
72	C₂³×C₈	C₂×C₂₂₃	1	1	yes	characteristic
73	C₂×#117	C₂×C₂₂₃	1	1	yes	characteristic
74	#424	C₂×C₂₂₃	1	8	yes	direct product factor
75	C₂×C₄⋊C₈	C₂×C₂₂₃	1	4	yes	complemented
76	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
77	C₂×M₁₆	—	2	4	no	—
78	C₂×M₁₆	C₂²×C₂₂₃	1	4	yes	—
79	C₂²×C₈	—	2	4	no	—
80	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
81	C₂²×C₈	C₂²×C₂₂₃	1	4	yes	—
82	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
83	#117	C₄×C₂₂₃	1	4	yes	—
84	#117	C₂²×C₂₂₃	1	4	yes	—
85	C₂×C₄⋊C₄	C₄×C₂₂₃	1	2	yes	—
86	C₂×C₂²⋊C₄	C₄×C₂₂₃	1	2	yes	—
87	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
88	C₄⋊C₈	C₂²×C₂₂₃	1	16	yes	complemented
89	C₂⁴	C₂×C₄×C₂₂₃	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₄×C₂₂₃	1	4	yes	—
91	C₂²×C₄	C₂×C₄×C₂₂₃	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
93	C₂²×C₄	D₄×C₂₂₃	1	4	yes	—
94	C₂²×C₄	Q₈×C₂₂₃	1	4	yes	—
95	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
96	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
97	M₁₆	—	2	8	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂³×C₂₂₃	1	4	yes	—
102	C₂×C₈	C₂³×C₂₂₃	1	4	yes	—
103	C₄⋊C₄	C₂×C₄×C₂₂₃	1	8	yes	—
104	C₂²⋊C₄	C₂×C₄×C₂₂₃	1	8	yes	—
105	C₄²	C₂×C₄×C₂₂₃	1	4	yes	—
106	C₄²	C₂³×C₂₂₃	1	4	yes	—
107	C₂³	—	2	4	no	—
108	C₂³	C₄⋊C₄×C₂₂₃	1	4	yes	—
109	C₂³	C₂²×C₄×C₂₂₃	1	2	yes	—
110	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	C₄⋊C₄×C₂₂₃	1	8	yes	—
113	C₂×C₄	C₄⋊C₄×C₂₂₃	1	4	yes	—
114	C₂×C₄	C₂²×C₄×C₂₂₃	1	8	yes	—
115	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
117	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
118	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
119	C₂×C₄	C₂×Q₈×C₂₂₃	1	4	yes	—
120	C₂×C₄	C₂×Q₈×C₂₂₃	1	2	yes	—
121	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	2	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	8	no	—
126	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	4	yes	—
127	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	2	yes	—
128	C₂²	#131×C₂₂₃	1	4	yes	—
129	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
130	C₄	—	2	8	no	—
131	C₄	C₂×C₄⋊C₄×C₂₂₃	1	2	yes	—
132	C₄	C₂×C₄⋊C₄×C₂₂₃	1	4	yes	—
133	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
134	C₄	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#424×C₂₂₃	1	4	yes	direct product factor
137	C₂	C₂²×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
138	C₂	C₂×#131×C₂₂₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle	C₂³×C₂₂₃	C₂²×C₄
prime radical	C₂⁴×C₂₂₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. #424×C₂₂₃	C₂	direct (N × H)
5. C₂×C₄⋊C₈×C₂₂₃	C₂	semidirect
18. C₄⋊C₈×C₂₂₃	C₂²	semidirect
66. C₂×C₂₂₃	#424	direct (N × H)
69. C₂₂₃	C₂×#424	direct (N × H)
70. C₂×#424	C₂₂₃	direct (N × H)
74. #424	C₂×C₂₂₃	direct (N × H)
75. C₂×C₄⋊C₈	C₂×C₂₂₃	semidirect
88. C₄⋊C₈	C₂²×C₂₂₃	semidirect
136. C₂	#424×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×M₁₆×C₂₂₃	2
C₂³×C₈×C₂₂₃	2
C₂×#117×C₂₂₃	2
#424×C₂₂₃	2
C₂×C₄⋊C₈×C₂₂₃	2
C₂×#424	223

Subgroups — C2×#424×C223 #43251919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#424×C₂₂₃ #43251919