Subgroups — C₂×C₄×C₂²⋊Dic₂₃ #15675727

Order 2944.

1360 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Dic₂₃	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₂₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	—	2	8	no	—
8	C₂³×C₄×C₂₃	C₄	1	8	yes	complemented
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	4	yes	—
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×Dic₂₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₂₃	C₄	1	16	yes	direct product factor
14	C₂×C₂²⋊Dic₂₃	C₂²	1	8	yes	—
15	C₂×C₄×Dic₂₃	—	2	8	no	—
16	C₂×C₄×Dic₂₃	C₂²	1	8	yes	complemented
17	C₂₃⋊#95	C₂²	1	8	yes	complemented
18	C₂⁴×C₂₃	—	2	4	no	—
19	C₂⁴×C₂₃	—	2	4	no	—
20	C₂⁴×C₂₃	C₂×C₄	1	8	yes	—
21	C₂⁴×C₂₃	C₂×C₄	1	4	yes	—
22	C₂⁴×C₂₃	C₂³	1	2	yes	—
23	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	—	2	4	no	—
25	C₂²×C₄×C₂₃	—	2	32	no	—
26	C₂²×C₄×C₂₃	—	2	16	no	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	16	yes	complemented
30	C₂²×C₄×C₂₃	D₄	1	8	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×Dic₂₃	—	2	8	no	—
34	C₂²×Dic₂₃	—	2	8	no	—
35	C₂²×Dic₂₃	C₂×C₄	1	16	yes	complemented
36	C₂²×Dic₂₃	C₂³	1	8	yes	—
37	C₂²⋊Dic₂₃	C₂×C₄	1	32	yes	direct product factor
38	C₄×Dic₂₃	—	2	16	no	—
39	C₂³×C₂₃	—	2	8	no	—
40	C₂³×C₂₃	—	2	8	no	—
41	C₂³×C₂₃	—	2	32	no	—
42	C₂³×C₂₃	—	2	4	no	—
43	C₂³×C₂₃	—	2	4	no	—
44	C₂³×C₂₃	C₄²	1	16	yes	complemented
45	C₂³×C₂₃	C₂²×C₄	1	8	yes	—
46	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
47	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
48	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂³×C₂₃	C₂×D₄	1	4	yes	—
50	C₂³×C₂₃	#41	1	4	yes	—
51	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₃	—	2	8	no	—
53	C₂×C₄×C₂₃	—	2	4	no	—
54	C₂×C₄×C₂₃	—	2	32	no	—
55	C₂×C₄×C₂₃	C₂²⋊C₄	1	16	yes	complemented
56	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₂₃	C₂×D₄	1	8	yes	—
59	C₂×Dic₂₃	—	2	32	no	—
60	C₂×Dic₂₃	—	2	8	no	—
61	C₂×Dic₂₃	—	2	8	no	—
62	C₂×Dic₂₃	C₂²×C₄	1	16	yes	complemented
63	C₂×C₄×C₂²⋊C₄	—	23	1	no	maximal, Sylow
64	C₂²×C₂₃	—	2	4	no	—
65	C₂²×C₂₃	—	2	4	no	—
66	C₂²×C₂₃	—	2	16	no	—
67	C₂²×C₂₃	—	2	32	no	—
68	C₂²×C₂₃	C₂×C₄²	1	4	yes	—
69	C₂²×C₂₃	C₂×C₄²	1	8	yes	complemented
70	C₂²×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
71	C₂²×C₂₃	#117	1	4	yes	—
72	C₂²×C₂₃	C₄×D₄	1	16	yes	—
73	C₂²×C₂₃	C₂³×C₄	1	2	yes	—
74	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
76	C₂²×C₂₃	C₂×#41	1	2	yes	—
77	C₄×C₂₃	—	2	4	no	—
78	C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	complemented
79	Dic₂₃	—	2	16	no	—
80	C₂⁴×C₄	D₂₃	1	1	yes	characteristic
81	C₂²×C₂²⋊C₄	—	23	2	no	—
82	C₂²×C₄²	—	23	2	no	—
83	C₄×C₂²⋊C₄	—	23	8	no	—
84	C₂×#95	—	23	2	no	—
85	C₂×C₂₃	—	2	8	no	—
86	C₂×C₂₃	C₄×C₂²⋊C₄	1	8	yes	complemented
87	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
88	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
90	C₂×C₂₃	C₂×C₄×D₄	1	4	yes	—
91	C₂⁵	C₂×D₂₃	1	1	yes	characteristic
92	C₂³×C₄	—	2	8	no	—
93	C₂³×C₄	—	23	4	no	—
94	C₂³×C₄	Dic₂₃	1	8	yes	complemented
95	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
96	C₂³×C₄	C₂×D₂₃	1	4	yes	—
97	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
98	C₂×C₂²⋊C₄	—	23	16	no	—
99	C₂×C₂²⋊C₄	—	23	8	no	—
100	C₂×C₄²	—	23	8	no	—
101	C₂×C₄²	—	46	8	no	—
102	#95	—	23	8	no	—
103	C₂₃	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
104	C₂⁴	—	2	4	no	—
105	C₂⁴	—	2	4	no	—
106	C₂⁴	C₄×D₂₃	1	8	yes	—
107	C₂⁴	C₂×Dic₂₃	1	4	yes	—
108	C₂⁴	C₂²×D₂₃	1	2	yes	—
109	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	32	no	—
112	C₂²×C₄	—	2	16	no	—
113	C₂²×C₄	—	23	16	no	—
114	C₂²×C₄	—	23	8	no	—
115	C₂²×C₄	—	46	8	no	—
116	C₂²×C₄	—	46	8	no	—
117	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
118	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
119	C₂²×C₄	C₂×Dic₂₃	1	16	yes	complemented
120	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	—
121	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
122	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
123	C₂²⋊C₄	—	23	32	no	—
124	C₄²	—	46	16	no	—
125	C₂³	—	2	8	no	—
126	C₂³	—	2	32	no	—
127	C₂³	—	2	8	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	C₄×Dic₂₃	1	16	yes	complemented
131	C₂³	C₂×C₄×D₂₃	1	8	yes	—
132	C₂³	C₂×C₄×D₂₃	1	4	yes	—
133	C₂³	#67288	1	4	yes	—
134	C₂³	C₂²×Dic₂₃	1	4	yes	—
135	C₂³	C₂²×Dic₂₃	1	2	yes	—
136	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
137	C₂³	C₂³×D₂₃	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	32	no	—
141	C₂×C₄	—	23	16	no	—
142	C₂×C₄	—	46	32	no	—
143	C₂×C₄	—	46	8	no	—
144	C₂×C₄	—	46	8	no	—
145	C₂×C₄	C₂²⋊Dic₂₃	1	16	yes	direct product factor
146	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
147	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
148	C₂×C₄	C₂×C₂²⋊D₂₃	1	8	yes	—
149	C₂²	—	2	4	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	16	no	—
152	C₂²	—	2	32	no	—
153	C₂²	C₂×C₄×Dic₂₃	1	4	yes	—
154	C₂²	C₂×C₄×Dic₂₃	1	8	yes	complemented
155	C₂²	#10625097	1	4	yes	—
156	C₂²	C₄×C₂²⋊D₂₃	1	16	yes	—
157	C₂²	C₂×C₂²⋊Dic₂₃	1	4	yes	—
158	C₂²	C₂²×C₄×D₂₃	1	2	yes	—
159	C₂²	C₂×#67288	1	2	yes	—
160	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
161	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
162	C₄	—	2	4	no	—
163	C₄	—	46	16	no	—
164	C₄	C₂×C₂²⋊Dic₂₃	1	8	yes	direct product factor
165	C₂	—	2	8	no	—
166	C₂	C₄×C₂²⋊Dic₂₃	1	8	yes	direct product factor
167	C₂	C₂²×C₄×Dic₂₃	1	1	yes	characteristic
168	C₂	C₂²×C₂²⋊Dic₂₃	1	1	yes	characteristic
169	C₂	C₂×#10625097	1	1	yes	characteristic
170	C₂	C₂×C₄×C₂²⋊D₂₃	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂⁴×C₄×C₂₃	C₂
socle	C₂⁴×C₂₃	C₂³
prime radical	C₂⁵×C₂₃	C₂²
cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂⁴×C₄	D₂₃
hypocenter	C₂₃	C₂×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₂₃	C₂	semidirect
3. C₄×C₂²⋊Dic₂₃	C₂	direct (N × H)
4. C₂×C₂₃⋊#95	C₂	semidirect
8. C₂³×C₄×C₂₃	C₄	semidirect
13. C₂×C₂²⋊Dic₂₃	C₄	direct (N × H)
16. C₂×C₄×Dic₂₃	C₂²	semidirect
17. C₂₃⋊#95	C₂²	semidirect
29. C₂²×C₄×C₂₃	C₂×C₄	semidirect
35. C₂²×Dic₂₃	C₂×C₄	semidirect
37. C₂²⋊Dic₂₃	C₂×C₄	direct (N × H)
44. C₂³×C₂₃	C₄²	semidirect
55. C₂×C₄×C₂₃	C₂²⋊C₄	semidirect
62. C₂×Dic₂₃	C₂²×C₄	semidirect
69. C₂²×C₂₃	C₂×C₄²	semidirect
78. C₄×C₂₃	C₂×C₂²⋊C₄	semidirect
86. C₂×C₂₃	C₄×C₂²⋊C₄	semidirect
94. C₂³×C₄	Dic₂₃	semidirect
103. C₂₃	C₂×C₄×C₂²⋊C₄	semidirect
119. C₂²×C₄	C₂×Dic₂₃	semidirect
130. C₂³	C₄×Dic₂₃	semidirect
145. C₂×C₄	C₂²⋊Dic₂₃	direct (N × H)
154. C₂²	C₂×C₄×Dic₂₃	semidirect
164. C₄	C₂×C₂²⋊Dic₂₃	direct (N × H)
166. C₂	C₄×C₂²⋊Dic₂₃	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×C₂²⋊Dic₂₃	2
C₂²×C₄×Dic₂₃	2
C₄×C₂²⋊Dic₂₃	2
C₂×C₂₃⋊#95	2
C₂⁴×C₄×C₂₃	2
C₂×C₄×C₂²⋊C₄	23

Subgroups — C2×C4×C22⋊Dic23 #15675727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂²⋊Dic₂₃ #15675727