Subgroups — #15682344

Order 2944.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092974	C₂	1	1	yes	maximal, characteristic
2	#13092547	C₂	1	2	yes	maximal, complemented
3	#13092545	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₄₆	C₂	1	2	yes	maximal, complemented
5	#13092445	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#13092375	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	#10625097	C₂²	1	2	yes	—
13	#10625095	C₄	1	2	yes	—
14	#10625095	C₄	1	2	yes	—
15	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
16	#10625041	C₄	1	2	yes	—
17	#10625040	C₄	1	1	yes	characteristic
18	#10625040	C₄	1	1	yes	characteristic
19	C₄×Dic₄₆	C₂²	1	2	yes	—
20	C₄×Dic₄₆	C₂²	1	2	yes	—
21	C₂₃×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	#10624997	C₂²	1	4	yes	—
24	Dic₂₃⋊C₈	C₂²	1	4	yes	—
25	Dic₂₃⋊C₈	C₂²	1	4	yes	—
26	C₈×Dic₂₃	C₂²	1	4	yes	—
27	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
29	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
35	M₁₆×C₂₃	—	2	2	no	—
36	M₁₆×C₂₃	Q₈	1	4	yes	complemented
37	C₂×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂×C₈×C₂₃	C₂³	1	2	yes	—
39	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
48	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
49	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
50	C₄×Dic₂₃	C₂³	1	4	yes	—
51	C₂×C₂₃⋊C₈	—	2	4	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	—	2	2	no	—
56	C₂×C₄×C₂₃	—	4	1	no	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₂₃	#41	1	2	yes	—
64	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
66	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
67	Dic₄₆	—	2	4	no	—
68	Dic₄₆	—	4	2	no	—
69	C₈×C₂₃	—	2	2	no	—
70	C₈×C₂₃	C₂×Q₈	1	4	yes	—
71	C₂₃⋊C₈	—	2	2	no	—
72	C₂₃⋊C₈	—	2	2	no	—
73	#2962	—	23	1	no	maximal, Sylow
74	C₂²×C₂₃	—	2	1	no	—
75	C₂²×C₂₃	—	2	1	no	—
76	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
77	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₂₃	—	2	1	no	—
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	—	2	1	no	—
81	C₄×C₂₃	C₄×Q₈	1	2	yes	—
82	C₄×C₂₃	#131	1	2	yes	—
83	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
84	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
85	Dic₂₃	—	2	4	no	—
86	Dic₂₃	—	2	4	no	—
87	#533	—	23	1	no	—
88	#446	—	23	4	no	—
89	#446	—	23	2	no	—
90	C₈×Q₈	—	23	2	no	—
91	#424	—	23	2	no	—
92	C₂×C₄⋊C₈	—	23	1	no	—
93	#405	—	23	2	no	—
94	C₄×M₁₆	D₂₃	1	1	yes	characteristic
95	C₂×C₂₃	—	2	2	no	—
96	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
98	C₂×C₂₃	#568	1	1	yes	characteristic
99	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
101	C₂²×C₈	—	23	2	no	—
102	C₄⋊Q₈	—	23	1	no	—
103	C₄⋊Q₈	—	23	1	no	—
104	#125	—	23	2	no	—
105	C₂²⋊Q₈	—	23	2	no	—
106	C₂²⋊Q₈	—	23	2	no	—
107	C₄×Q₈	—	23	2	no	—
108	C₄×Q₈	—	23	2	no	—
109	#117	—	23	2	no	—
110	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
111	C₄⋊C₈	—	23	4	no	—
112	C₄⋊C₈	—	23	4	no	—
113	C₄⋊C₈	—	23	1	no	—
114	C₄⋊C₈	—	23	1	no	—
115	C₄⋊C₈	—	23	2	no	—
116	#97	—	23	4	no	—
117	#97	C₂×D₂₃	1	2	yes	—
118	C₄×C₈	—	23	4	no	—
119	C₄×C₈	C₂×D₂₃	1	2	yes	—
120	C₂₃	#2962	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	—	23	2	no	—
122	C₂×Q₈	—	23	2	no	—
123	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₄₆	1	4	yes	complemented
128	C₂×C₈	—	23	2	no	—
129	C₂×C₈	—	23	2	no	—
130	C₂×C₈	—	46	4	no	—
131	C₂×C₈	C₂²×D₂₃	1	2	yes	—
132	C₂×C₈	C₂²×D₂₃	1	2	yes	—
133	C₄⋊C₄	—	23	2	no	—
134	C₄⋊C₄	—	23	2	no	—
135	C₄⋊C₄	—	23	4	no	—
136	C₄⋊C₄	—	23	4	no	—
137	C₄⋊C₄	—	23	2	no	—
138	C₄⋊C₄	—	23	2	no	—
139	C₂²⋊C₄	—	23	4	no	—
140	C₄²	—	23	4	no	—
141	C₄²	C₄×D₂₃	1	1	yes	characteristic
142	C₄²	C₄×D₂₃	1	1	yes	characteristic
143	C₄²	C₂²×D₂₃	1	1	yes	characteristic
144	C₄²	C₂²×D₂₃	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
146	Q₈	—	46	4	no	—
147	Q₈	—	92	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	23	4	no	—
151	C₂×C₄	—	23	4	no	—
152	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
153	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
158	C₂×C₄	#67288	1	2	yes	—
159	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	46	2	no	—
162	C₈	—	46	2	no	—
163	C₈	C₂×Dic₄₆	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₄₆	1	2	yes	—
167	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	46	4	no	—
172	C₄	—	46	4	no	—
173	C₄	C₄×Dic₄₆	1	2	yes	—
174	C₄	#10625078	1	2	yes	—
175	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
176	C₄	C₂×#67288	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₄₆	1	1	yes	characteristic
179	C₂	C₂×#10625078	1	1	yes	characteristic
180	C₂	#13093183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₄×M₁₆×C₂₃	C₂
prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	#2962
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13092547	C₂	semidirect
3. #13092545	C₂	semidirect
4. C₈×Dic₄₆	C₂	semidirect
36. M₁₆×C₂₃	Q₈	semidirect
120. C₂₃	#2962	semidirect
127. M₁₆	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
#13092974	2
#13092547	2
#13092545	2
C₈×Dic₄₆	2
#13092445	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092375	2
C₄×M₁₆×C₂₃	2
#2962	23