Subgroups — C₂×C₄×C₄⋊Dic₂₃ #15675732

Order 2944.

928 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₃	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₂₃	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₂₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃⋊#95	C₂	1	2	yes	maximal
5	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×Dic₂₃	C₂²	1	4	yes	—
10	C₂×C₄²×C₂₃	C₄	1	8	yes	complemented
11	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₂₃	C₄	1	16	yes	direct product factor
13	C₂×C₄⋊Dic₂₃	C₂²	1	8	yes	—
14	C₂×C₄×Dic₂₃	—	2	8	no	—
15	C₂×C₄×Dic₂₃	C₂²	1	8	yes	—
16	C₂₃⋊#95	C₂²	1	8	yes	—
17	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	—	2	4	no	—
19	C₂²×C₄×C₂₃	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₂₃	D₄	1	4	yes	—
24	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂²×Dic₂₃	—	2	8	no	—
29	C₂²×Dic₂₃	—	2	8	no	—
30	C₂²×Dic₂₃	C₂×C₄	1	16	yes	—
31	C₂²×Dic₂₃	C₂³	1	8	yes	—
32	C₄²×C₂₃	C₂×C₄	1	16	yes	complemented
33	C₄⋊Dic₂₃	C₂×C₄	1	32	yes	direct product factor
34	C₄×Dic₂₃	—	2	16	no	—
35	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
36	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
37	C₂³×C₂₃	C₂×D₄	1	2	yes	—
38	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
39	C₂³×C₂₃	#41	1	4	yes	—
40	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₂₃	—	2	8	no	—
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	C₄²	1	16	yes	complemented
44	C₂×C₄×C₂₃	C₄⋊C₄	1	16	yes	complemented
45	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
51	C₂×Dic₂₃	—	2	32	no	—
52	C₂×Dic₂₃	—	2	8	no	—
53	C₂×Dic₂₃	—	2	8	no	—
54	C₂×Dic₂₃	C₂²×C₄	1	16	yes	—
55	C₂×C₄×C₄⋊C₄	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	C₂×C₄²	1	4	yes	—
57	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
58	C₂²×C₂₃	#117	1	4	yes	—
59	C₂²×C₂₃	C₄×D₄	1	8	yes	—
60	C₂²×C₂₃	C₄×Q₈	1	8	yes	—
61	C₂²×C₂₃	C₂³×C₄	1	2	yes	—
62	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
63	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂²×C₂₃	C₂×#41	1	2	yes	—
66	C₄×C₂₃	—	2	4	no	—
67	C₄×C₂₃	C₂×C₄²	1	8	yes	complemented
68	C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	complemented
69	Dic₂₃	—	2	16	no	—
70	C₂²×C₄⋊C₄	—	23	2	no	—
71	C₂²×C₄²	—	23	2	no	—
72	C₂²×C₄²	D₂₃	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	23	8	no	—
74	C₂×#95	—	23	2	no	—
75	C₂×C₂₃	C₄×C₄⋊C₄	1	8	yes	complemented
76	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
77	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
79	C₂×C₂₃	C₂×C₄×D₄	1	2	yes	—
80	C₂×C₂₃	C₂×C₄×Q₈	1	2	yes	—
81	C₂³×C₄	—	23	4	no	—
82	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
83	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
85	C₂×C₄⋊C₄	—	23	16	no	—
86	C₂×C₄⋊C₄	—	23	8	no	—
87	C₂×C₄²	—	23	8	no	—
88	C₂×C₄²	—	46	8	no	—
89	C₂×C₄²	Dic₂₃	1	8	yes	complemented
90	C₂×C₄²	C₂×D₂₃	1	4	yes	—
91	#95	—	23	8	no	—
92	C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
93	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	23	16	no	—
96	C₂²×C₄	—	23	8	no	—
97	C₂²×C₄	—	46	8	no	—
98	C₂²×C₄	—	46	8	no	—
99	C₂²×C₄	Dic₄₆	1	4	yes	—
100	C₂²×C₄	C₄×D₂₃	1	8	yes	—
101	C₂²×C₄	D₉₂	1	4	yes	—
102	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
103	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
104	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
105	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
108	C₄⋊C₄	—	23	32	no	—
109	C₄²	—	46	16	no	—
110	C₄²	C₂×Dic₂₃	1	16	yes	complemented
111	C₂³	C₂×Dic₄₆	1	2	yes	—
112	C₂³	C₂×C₄×D₂₃	1	4	yes	—
113	C₂³	C₂×D₉₂	1	2	yes	—
114	C₂³	#67288	1	4	yes	—
115	C₂³	C₂²×Dic₂₃	1	2	yes	—
116	C₂³	C₂³×D₂₃	1	1	yes	characteristic
117	C₂×C₄	—	2	8	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	23	16	no	—
120	C₂×C₄	—	46	32	no	—
121	C₂×C₄	—	46	8	no	—
122	C₂×C₄	—	46	8	no	—
123	C₂×C₄	C₄×Dic₂₃	1	16	yes	complemented
124	C₂×C₄	C₄⋊Dic₂₃	1	16	yes	direct product factor
125	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₂₃	1	8	yes	—
127	C₂×C₄	C₂×D₉₂	1	4	yes	—
128	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
129	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
130	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
131	C₂²	C₄×Dic₄₆	1	8	yes	—
132	C₂²	C₄×D₉₂	1	8	yes	—
133	C₂²	C₂×C₄×Dic₂₃	1	4	yes	—
134	C₂²	C₂×C₄⋊Dic₂₃	1	4	yes	—
135	C₂²	#10625097	1	4	yes	—
136	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
137	C₂²	C₂²×C₄×D₂₃	1	2	yes	—
138	C₂²	C₂²×D₉₂	1	1	yes	characteristic
139	C₂²	C₂×#67288	1	2	yes	—
140	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	46	16	no	—
143	C₄	C₂×C₄×Dic₂₃	1	8	yes	complemented
144	C₄	C₂×C₄⋊Dic₂₃	1	8	yes	direct product factor
145	C₂	C₄×C₄⋊Dic₂₃	1	8	yes	direct product factor
146	C₂	C₂²×C₄×Dic₂₃	1	1	yes	characteristic
147	C₂	C₂²×C₄⋊Dic₂₃	1	1	yes	characteristic
148	C₂	C₂×#10625097	1	1	yes	characteristic
149	C₂	C₂×C₄×D₉₂	1	2	yes	—
150	C₂	C₂×C₄×Dic₄₆	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊Dic₂₃	C₂	direct (N × H)
10. C₂×C₄²×C₂₃	C₄	semidirect
12. C₂×C₄⋊Dic₂₃	C₄	direct (N × H)
32. C₄²×C₂₃	C₂×C₄	semidirect
33. C₄⋊Dic₂₃	C₂×C₄	direct (N × H)
43. C₂×C₄×C₂₃	C₄²	semidirect
44. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
67. C₄×C₂₃	C₂×C₄²	semidirect
68. C₄×C₂₃	C₂×C₄⋊C₄	semidirect
75. C₂×C₂₃	C₄×C₄⋊C₄	semidirect
89. C₂×C₄²	Dic₂₃	semidirect
92. C₂₃	C₂×C₄×C₄⋊C₄	semidirect
110. C₄²	C₂×Dic₂₃	semidirect
123. C₂×C₄	C₄×Dic₂₃	semidirect
124. C₂×C₄	C₄⋊Dic₂₃	direct (N × H)
143. C₄	C₂×C₄×Dic₂₃	semidirect
144. C₄	C₂×C₄⋊Dic₂₃	direct (N × H)
145. C₂	C₄×C₄⋊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×C₄⋊Dic₂₃	2
C₂²×C₄×Dic₂₃	2
C₄×C₄⋊Dic₂₃	2
C₂×C₂₃⋊#95	2
C₂²×C₄²×C₂₃	2
C₂×C₄×C₄⋊C₄	23

Subgroups — C2×C4×C4⋊Dic23 #15675732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊Dic₂₃ #15675732