Subgroups — #15674116

Order 2944.

260 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	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
3	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#13092172	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂₃×#95	C₄	1	2	yes	—
14	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
15	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
16	C₄×C₂₃⋊C₈	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	C₈	1	4	yes	complemented
24	C₄²×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	D₄	1	2	yes	—
27	C₄²×C₂₃	Q₈	1	2	yes	—
28	C₂×C₂₃⋊C₈	—	2	1	no	—
29	C₂×C₂₃⋊C₈	—	2	1	no	—
30	C₂×C₂₃⋊C₈	—	2	1	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	2	no	—
35	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
36	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
37	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	1	no	—
40	C₂×C₄×C₂₃	—	2	1	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
48	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
56	C₂₃⋊C₈	—	2	2	no	—
57	C₂₃⋊C₈	—	2	2	no	—
58	C₂₃⋊C₈	—	4	2	no	—
59	#1939	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
63	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
64	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂²×C₂₃	#131	1	1	yes	characteristic
66	C₂²×C₂₃	#131	1	1	yes	characteristic
67	C₄×C₂₃	—	2	2	no	—
68	C₄×C₂₃	—	2	4	no	—
69	C₄×C₂₃	—	4	2	no	—
70	C₄×C₂₃	C₄⋊C₈	1	4	yes	complemented
71	C₄×C₂₃	#121	1	1	yes	characteristic
72	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₄×C₂₃	#125	1	1	yes	characteristic
74	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄⋊C₈	—	23	2	no	—
76	C₂×C₄⋊C₈	—	23	1	no	—
77	C₂×C₄×C₈	—	23	1	no	—
78	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
79	#336	—	23	2	no	—
80	C₂×C₂₃	#389	1	1	yes	characteristic
81	C₂×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₂₃	#424	1	1	yes	characteristic
83	C₂×C₂₃	C₈×D₄	1	1	yes	characteristic
84	C₂×C₂₃	#436	1	1	yes	characteristic
85	C₂×C₂₃	C₈×Q₈	1	1	yes	characteristic
86	C₂×C₂₃	#446	1	1	yes	characteristic
87	C₂²×C₈	—	23	1	no	—
88	C₂²×C₈	—	23	1	no	—
89	C₂²×C₈	—	23	2	no	—
90	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
91	C₂×C₄⋊C₄	Dic₂₃	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₂₃	1	2	yes	—
93	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
94	C₄⋊C₈	—	46	4	no	—
95	C₄⋊C₈	—	46	2	no	—
96	C₄×C₈	—	46	2	no	—
97	#95	Dic₂₃	1	2	yes	—
98	C₂₃	#1939	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
100	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
101	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
102	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
104	C₂×C₈	—	23	2	no	—
105	C₂×C₈	—	23	2	no	—
106	C₂×C₈	—	46	1	no	—
107	C₂×C₈	—	46	1	no	—
108	C₂×C₈	—	46	1	no	—
109	C₂×C₈	—	46	1	no	—
110	C₂×C₈	—	46	2	no	—
111	C₂×C₈	—	46	2	no	—
112	C₂×C₈	—	46	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₂₃⋊C₈	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	Dic₄₆	1	2	yes	—
118	C₄²	D₉₂	1	2	yes	—
119	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₂×C₂₃⋊C₈	1	4	yes	—
128	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
129	C₂×C₄	#67267	1	2	yes	—
130	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
131	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
132	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
134	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
135	C₂×C₄	#67290	1	1	yes	characteristic
136	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
137	C₂×C₄	#67292	1	1	yes	characteristic
138	C₈	—	46	2	no	—
139	C₈	—	46	2	no	—
140	C₈	—	92	2	no	—
141	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
142	C₂²	C₂×#67267	1	1	yes	characteristic
143	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
144	C₂²	D₄×Dic₂₃	1	1	yes	characteristic
145	C₂²	Q₈×Dic₂₃	1	1	yes	characteristic
146	C₂²	#10625119	1	1	yes	characteristic
147	C₂²	#10625119	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊(C₂₃⋊C₈)	1	4	yes	complemented
152	C₄	#10625059	1	1	yes	characteristic
153	C₄	#10625061	1	1	yes	characteristic
154	C₄	#10625066	1	1	yes	characteristic
155	C₄	#10625068	1	1	yes	characteristic
156	C₂	#13092316	1	1	yes	characteristic
157	C₂	C₂×C₄⋊(C₂₃⋊C₈)	1	1	yes	characteristic
158	C₂	#13092444	1	1	yes	characteristic
159	C₂	D₄×C₂₃⋊C₈	1	1	yes	characteristic
160	C₂	#13092495	1	1	yes	characteristic
161	C₂	Q₈×C₂₃⋊C₈	1	1	yes	characteristic
162	C₂	#13092548	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#1939
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. C₄⋊C₄×C₂₃	C₈	semidirect
70. C₄×C₂₃	C₄⋊C₈	semidirect
98. C₂₃	#1939	semidirect
114. C₄⋊C₄	C₂₃⋊C₈	semidirect
151. C₄	C₄⋊(C₂₃⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×C₄×C₂₃⋊C₈	2
#13092172	2
C₄×C₄⋊C₄×C₂₃	2
#1939	23