Subgroups — #15682335

Order 2944.

348 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092970	C₂	1	1	yes	maximal, characteristic
2	#13092547	C₂	1	4	yes	maximal, complemented
3	#13092545	C₂	1	4	yes	maximal, complemented
4	#13092445	C₂	1	1	yes	maximal, characteristic
5	#13092444	C₂	1	1	yes	maximal, characteristic
6	#13092440	C₂	1	1	yes	maximal, characteristic
7	#13092372	C₂	1	1	yes	maximal, characteristic
8	C₂×#10624997	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#117	C₂²	1	1	yes	characteristic
13	#10625097	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×#67267	C₂²	1	1	yes	characteristic
16	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
17	#10625061	C₄	1	2	yes	—
18	#10625059	C₄	1	2	yes	—
19	#10625049	C₄	1	4	yes	—
20	C₄×Dic₄₆	C₂²	1	4	yes	—
21	C₂₃×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	#10624997	C₂²	1	2	yes	—
24	#10624997	C₂²	1	2	yes	—
25	#10624997	C₂²	1	2	yes	—
26	Dic₂₃⋊C₈	C₂²	1	4	yes	—
27	Dic₂₃⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₂₃	C₂²	1	2	yes	—
29	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
32	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
33	M₁₆×C₂₃	—	2	2	no	—
34	C₂×C₈×C₂₃	Q₈	1	4	yes	complemented
35	C₂×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
40	C₄²×C₂₃	C₂³	1	2	yes	—
41	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
44	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
45	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
46	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₂₃	C₂³	1	2	yes	—
49	C₄×Dic₂₃	C₂³	1	2	yes	—
50	#67267	—	2	2	no	—
51	C₂×C₂₃⋊C₈	—	2	2	no	—
52	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
53	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
54	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₂₃	#41	1	2	yes	—
59	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₂₃	—	4	2	no	—
61	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
62	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
63	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
64	Dic₄₆	—	4	4	no	—
65	C₈×C₂₃	—	2	2	no	—
66	C₈×C₂₃	C₂×Q₈	1	2	yes	—
67	C₈×C₂₃	C₂×Q₈	1	2	yes	—
68	C₂₃⋊C₈	—	2	2	no	—
69	C₂₃⋊C₈	—	2	2	no	—
70	#2961	—	23	1	no	maximal, Sylow
71	C₂²×C₂₃	—	2	2	no	—
72	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
73	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₂₃	—	2	4	no	—
75	C₄×C₂₃	C₄×Q₈	1	2	yes	—
76	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
78	Dic₂₃	—	2	2	no	—
79	Dic₂₃	—	2	4	no	—
80	Dic₂₃	—	2	2	no	—
81	#533	—	23	1	no	—
82	#446	—	23	4	no	—
83	#446	—	23	4	no	—
84	#424	—	23	1	no	—
85	#424	—	23	1	no	—
86	#423	—	23	1	no	—
87	#405	—	23	1	no	—
88	#405	D₂₃	1	1	yes	characteristic
89	C₂×#97	—	23	1	no	—
90	C₂×C₂₃	—	2	2	no	—
91	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₂₃	#568	1	2	yes	—
93	C₂×M₁₆	—	23	1	no	—
94	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
95	C₂²×C₈	—	23	1	no	—
96	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
97	C₄⋊Q₈	—	23	2	no	—
98	#125	—	23	2	no	—
99	C₂²⋊Q₈	—	23	4	no	—
100	C₄×Q₈	—	23	4	no	—
101	#117	—	23	1	no	—
102	#117	C₂×D₂₃	1	1	yes	characteristic
103	C₂×C₄²	—	23	1	no	—
104	C₄⋊C₈	—	23	4	no	—
105	C₄⋊C₈	—	23	4	no	—
106	C₄⋊C₈	—	23	4	no	—
107	#97	—	23	2	no	—
108	#97	—	23	2	no	—
109	#97	—	23	2	no	—
110	#97	C₂×D₂₃	1	2	yes	—
111	C₄×C₈	—	23	2	no	—
112	C₄×C₈	C₂×D₂₃	1	2	yes	—
113	C₂₃	#2961	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	23	4	no	—
115	C₂²×C₄	—	23	2	no	—
116	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	46	2	no	—
119	C₂×C₈	—	23	2	no	—
120	C₂×C₈	—	23	2	no	—
121	C₂×C₈	—	46	2	no	—
122	C₂×C₈	Dic₄₆	1	4	yes	complemented
123	C₂×C₈	C₂²×D₂₃	1	2	yes	—
124	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
126	C₄⋊C₄	—	23	2	no	—
127	C₄⋊C₄	—	23	4	no	—
128	C₄⋊C₄	—	23	4	no	—
129	C₄⋊C₄	—	23	4	no	—
130	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
131	C₂²⋊C₄	—	23	2	no	—
132	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
133	C₄²	—	23	1	no	—
134	C₄²	—	23	1	no	—
135	C₄²	—	23	2	no	—
136	C₄²	—	23	2	no	—
137	C₄²	C₂²×D₂₃	1	2	yes	—
138	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
139	Q₈	—	92	4	no	—
140	C₂×C₄	—	23	2	no	—
141	C₂×C₄	—	23	2	no	—
142	C₂×C₄	—	23	4	no	—
143	C₂×C₄	—	92	2	no	—
144	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
147	C₂×C₄	#67288	1	2	yes	—
148	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	46	2	no	—
151	C₈	—	46	2	no	—
152	C₈	C₂×Dic₄₆	1	2	yes	—
153	C₈	C₂×Dic₄₆	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₄₆	1	2	yes	—
156	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	46	2	no	—
159	C₄	—	46	2	no	—
160	C₄	—	46	4	no	—
161	C₄	C₄×Dic₄₆	1	2	yes	—
162	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
163	C₄	C₂×#67288	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₄₆	1	1	yes	characteristic
166	C₂	#13093182	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#405	C₂
prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#405	D₂₃
hypocenter	C₂₃	#2961
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #13092547	C₂	semidirect
3. #13092545	C₂	semidirect
34. C₂×C₈×C₂₃	Q₈	semidirect
113. C₂₃	#2961	semidirect
122. C₂×C₈	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
#13092970	2
#13092547	2
#13092545	2
#13092445	2
#13092444	2
#13092440	2
#13092372	2
C₂×#10624997	2
C₂₃×#405	2
#2961	23