Subgroups — #15671627

Order 2944.

656 conjugacy classes of subgroups.

The subgroup lattice has 164 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	C₂×C₂₃⋊#95	C₂	1	4	yes	maximal
3	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₂₃	C₂²	1	4	yes	—
7	C₂×C₄²×C₂₃	—	2	4	no	—
8	C₂×C₄²×C₂₃	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₃	—	2	4	no	—
10	C₂×C₄⋊Dic₂₃	—	2	4	no	—
11	C₂×C₄⋊Dic₂₃	C₄	1	8	yes	complemented
12	C₂₃⋊#95	—	2	8	no	—
13	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₂₃	—	2	4	no	—
15	C₂²×C₄×C₂₃	—	2	4	no	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₃	D₄	1	4	yes	—
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	D₄	1	4	yes	—
22	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
23	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
24	C₂²×Dic₂₃	—	2	4	no	—
25	C₂²×Dic₂₃	—	2	4	no	—
26	C₂²×Dic₂₃	—	2	8	no	—
27	C₂²×Dic₂₃	—	2	8	no	—
28	C₂²×Dic₂₃	C₂×C₄	1	8	yes	—
29	C₄²×C₂₃	—	2	8	no	—
30	C₄⋊Dic₂₃	—	2	16	no	—
31	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
32	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂³×C₂₃	C₂×D₄	1	2	yes	—
35	C₂³×C₂₃	C₂×D₄	1	2	yes	—
36	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
38	C₂³×C₂₃	#41	1	4	yes	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	8	no	—
42	C₂×C₄×C₂₃	—	2	8	no	—
43	C₂×C₄×C₂₃	C₄²	1	4	yes	—
44	C₂×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
45	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
47	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₂₃	C₄⋊C₄	1	8	yes	complemented
49	C₂×Dic₂₃	—	2	8	no	—
50	C₂×Dic₂₃	—	2	16	no	—
51	C₂×Dic₂₃	—	4	8	no	—
52	C₂×Dic₂₃	—	4	4	no	—
53	C₂×Dic₂₃	—	4	4	no	—
54	C₄⋊#95	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
56	C₂²×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
57	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
59	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₂²×C₂₃	#117	1	2	yes	—
62	C₂²×C₂₃	C₄×D₄	1	4	yes	—
63	C₂²×C₂₃	C₄×Q₈	1	4	yes	—
64	C₂²×C₂₃	#121	1	4	yes	—
65	C₂²×C₂₃	C₂²⋊Q₈	1	4	yes	—
66	C₂²×C₂₃	#124	1	2	yes	—
67	C₂²×C₂₃	#125	1	2	yes	—
68	C₂²×C₂₃	#127	1	1	yes	characteristic
69	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
70	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₄×C₂₃	—	2	8	no	—
72	C₄×C₂₃	#95	1	8	yes	complemented
73	Dic₂₃	—	4	8	no	—
74	C₂²×C₄⋊C₄	—	23	2	no	—
75	C₂²×C₄²	D₂₃	1	1	yes	characteristic
76	C₂×#95	—	23	4	no	—
77	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
78	C₂×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
79	C₂×C₂₃	#380	1	2	yes	—
80	C₂×C₂₃	#382	1	1	yes	characteristic
81	C₂×C₂₃	#384	1	1	yes	characteristic
82	C₂×C₂₃	#389	1	4	yes	—
83	C₂×C₂₃	#390	1	2	yes	—
84	C₂×C₂₃	#391	1	2	yes	—
85	C₂³×C₄	—	23	4	no	—
86	C₂³×C₄	C₂×D₂₃	1	2	yes	—
87	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	23	8	no	—
89	C₂×C₄⋊C₄	—	46	4	no	—
90	C₂×C₄⋊C₄	—	46	4	no	—
91	C₂×C₄²	—	2	4	no	—
92	C₂×C₄²	Dic₂₃	1	4	yes	complemented
93	#95	—	46	8	no	—
94	C₂₃	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
95	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
96	C₂²×C₄	—	2	4	no	—
97	C₂²×C₄	—	2	4	no	—
98	C₂²×C₄	—	23	8	no	—
99	C₂²×C₄	—	46	8	no	—
100	C₂²×C₄	—	46	4	no	—
101	C₂²×C₄	—	46	8	no	—
102	C₂²×C₄	—	46	4	no	—
103	C₂²×C₄	Dic₄₆	1	4	yes	—
104	C₂²×C₄	Dic₄₆	1	2	yes	—
105	C₂²×C₄	C₄×D₂₃	1	4	yes	—
106	C₂²×C₄	D₉₂	1	4	yes	—
107	C₂²×C₄	D₉₂	1	2	yes	—
108	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
109	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
110	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
111	C₄⋊C₄	—	46	16	no	—
112	C₄²	—	2	8	no	—
113	C₂³	C₂×Dic₄₆	1	2	yes	—
114	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
115	C₂³	C₂×C₄×D₂₃	1	2	yes	—
116	C₂³	C₂×D₉₂	1	2	yes	—
117	C₂³	C₂×D₉₂	1	1	yes	characteristic
118	C₂³	#67288	1	4	yes	—
119	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
120	C₂³	C₂×C₂²⋊D₂₃	1	2	yes	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	46	16	no	—
126	C₂×C₄	—	46	8	no	—
127	C₂×C₄	—	92	4	no	—
128	C₂×C₄	—	92	4	no	—
129	C₂×C₄	—	92	8	no	—
130	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
131	C₂×C₄	Dic₂₃⋊C₄	1	8	yes	—
132	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
133	C₂×C₄	C₄⋊Dic₂₃	1	8	yes	complemented
134	C₂×C₄	(C₂×D₂₃)⋊C₄	1	8	yes	—
135	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
136	C₂²	C₄×Dic₄₆	1	4	yes	—
137	C₂²	#10625040	1	2	yes	—
138	C₂²	#10625040	1	1	yes	characteristic
139	C₂²	#10625041	1	2	yes	—
140	C₂²	C₄×D₉₂	1	4	yes	—
141	C₂²	#10625045	1	1	yes	characteristic
142	C₂²	#10625046	1	2	yes	—
143	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
144	C₂²	C₂×Dic₂₃⋊C₄	1	2	yes	—
145	C₂²	#10625095	1	4	yes	—
146	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
147	C₂²	C₂×C₄⋊Dic₂₃	1	2	yes	—
148	C₂²	#10625097	1	2	yes	—
149	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	2	yes	—
150	C₂²	#10625101	1	4	yes	—
151	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	92	8	no	—
154	C₄	C₂₃⋊#95	1	8	yes	complemented
155	C₂	C₂×C₂₃⋊#95	1	1	yes	characteristic
156	C₂	C₄×C₄⋊Dic₂₃	1	2	yes	—
157	C₂	#13092277	1	2	yes	—
158	C₂	#13092283	1	1	yes	characteristic
159	C₂	#13092287	1	1	yes	characteristic
160	C₂	#13092315	1	4	yes	—
161	C₂	#13092319	1	2	yes	—
162	C₂	#13092323	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Fitting	C₂²×C₄²×C₂₃	C₂
socle, prime radical, cosocle	C₂⁴×C₂₃	C₂³
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	C₄⋊#95
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
8. C₂×C₄²×C₂₃	C₄	semidirect
11. C₂×C₄⋊Dic₂₃	C₄	semidirect
48. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
72. C₄×C₂₃	#95	semidirect
92. C₂×C₄²	Dic₂₃	semidirect
94. C₂₃	C₄⋊#95	semidirect
133. C₂×C₄	C₄⋊Dic₂₃	semidirect
154. C₄	C₂₃⋊#95	semidirect

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
C₂×C₂₃⋊#95	2
C₂²×C₄²×C₂₃	2
C₄⋊#95	23