Subgroups — #15681824

Order 2944.

556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625078	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625072	C₂	1	1	yes	maximal, characteristic
3	#13092885	C₂	1	1	yes	maximal, characteristic
4	#13092374	C₂	1	2	yes	maximal, complemented
5	#13092373	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092371	C₂	1	2	yes	maximal, complemented
7	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#67262	C₂	1	2	yes	maximal, complemented
9	C₂×#10624997	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂×#67267	C₂²	1	1	yes	characteristic
18	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	#10625078	C₄	1	8	yes	complemented
20	#10625072	C₄	1	8	yes	complemented
21	C₂×#67262	C₂²	1	2	yes	—
22	C₂×#67262	C₂²	1	2	yes	—
23	C₂×C₈×D₂₃	C₂²	1	2	yes	—
24	C₂×C₈×D₂₃	C₂²	1	2	yes	—
25	#10625064	C₄	1	2	yes	—
26	#10625058	C₄	1	2	yes	—
27	(C₂²⋊D₂₃)⋊C₄	C₄	1	4	yes	—
28	#10625043	C₂²	1	2	yes	—
29	C₄²×D₂₃	C₂²	1	2	yes	—
30	C₂₃×#97	C₂²	1	2	yes	—
31	C₄×C₈×C₂₃	C₂²	1	2	yes	—
32	#10624997	C₂²	1	1	yes	characteristic
33	#10624997	C₂²	1	2	yes	—
34	#10624997	C₂²	1	1	yes	characteristic
35	#10624997	C₂²	1	1	yes	characteristic
36	#10624997	C₂²	1	1	yes	characteristic
37	C₈×Dic₂₃	C₂²	1	2	yes	—
38	#10624985	C₂²	1	2	yes	—
39	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
42	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
43	#67288	C₂×C₄	1	4	yes	—
44	#67288	C₂×C₄	1	4	yes	—
45	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₂₃	C₂³	1	2	yes	—
48	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
49	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
56	C₄²×C₂₃	C₂³	1	2	yes	—
57	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
58	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
60	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
62	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
63	#67267	C₂×C₄	1	4	yes	complemented
64	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
65	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
67	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
68	#67262	C₂×C₄	1	8	yes	complemented
69	#67262	C₂×C₄	1	8	yes	complemented
70	C₈×D₂₃	C₂×C₄	1	8	yes	complemented
71	C₈×D₂₃	C₂×C₄	1	8	yes	complemented
72	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊D₂₃	C₄²	1	8	yes	—
80	C₂×Dic₂₃	—	4	2	no	—
81	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
84	D₉₂	C₄²	1	4	yes	—
85	C₄×D₂₃	—	4	4	no	—
86	C₄×D₂₃	C₂²×C₄	1	4	yes	—
87	C₄×D₂₃	C₂²×C₄	1	4	yes	—
88	Dic₄₆	C₄²	1	4	yes	—
89	C₈×C₂₃	C₂²×C₄	1	2	yes	—
90	C₈×C₂₃	C₂²×C₄	1	4	yes	—
91	C₈×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
95	#2875	—	23	1	no	maximal, Sylow
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	C₂×C₄²	1	2	yes	—
98	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₂₃	—	2	4	no	—
100	C₂×D₂₃	C₂×C₄²	1	4	yes	—
101	C₄×C₂₃	—	2	4	no	—
102	C₄×C₂₃	C₂×C₄²	1	2	yes	—
103	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
104	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
105	Dic₂₃	—	2	2	no	—
106	Dic₂₃	—	2	2	no	—
107	Dic₂₃	C₂×C₄²	1	4	yes	—
108	C₂×#131	—	23	1	no	—
109	C₂×#131	—	23	1	no	—
110	C₄×#41	—	23	1	no	—
111	#405	—	23	1	no	—
112	#405	—	23	2	no	—
113	#405	—	23	2	no	—
114	#405	D₂₃	1	1	yes	characteristic, complemented
115	C₄×M₁₆	—	23	1	no	—
116	C₄×M₁₆	—	23	2	no	—
117	C₂×#97	—	23	1	no	—
118	C₂×#97	—	23	2	no	—
119	C₂×C₂₃	—	2	2	no	—
120	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
121	C₂×C₂₃	#568	1	2	yes	—
122	D₂₃	—	2	4	no	—
123	C₂×#41	—	23	1	no	—
124	#131	—	23	8	no	—
125	#131	—	23	8	no	—
126	C₂×M₁₆	—	23	1	no	—
127	C₂×M₁₆	—	23	2	no	—
128	C₂×M₁₆	—	23	2	no	—
129	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
130	C₂²×C₈	—	23	1	no	—
131	C₂²×C₈	—	23	2	no	—
132	C₂²×C₈	—	23	2	no	—
133	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
134	C₄×Q₈	—	23	2	no	—
135	C₄×D₄	—	23	2	no	—
136	C₄×D₄	—	23	4	no	—
137	#117	—	23	2	no	—
138	#117	C₂×D₂₃	1	1	yes	characteristic
139	C₂×C₄²	—	23	1	no	—
140	C₂×C₄²	—	23	2	no	—
141	#97	—	23	1	no	—
142	#97	—	23	2	no	—
143	#97	—	23	1	no	—
144	#97	—	23	1	no	—
145	#97	—	23	1	no	—
146	#97	—	23	2	no	—
147	#97	C₂×D₂₃	1	2	yes	—
148	C₄×C₈	—	23	2	no	—
149	C₄×C₈	—	23	2	no	—
150	C₄×C₈	C₂×D₂₃	1	2	yes	—
151	C₂₃	#2875	1	1	yes	characteristic, complemented, Sylow
152	#41	—	23	4	no	—
153	#41	—	23	4	no	—
154	C₂×Q₈	—	23	1	no	—
155	C₂×D₄	—	23	2	no	—
156	C₂×D₄	—	23	1	no	—
157	C₂²×C₄	—	23	2	no	—
158	C₂²×C₄	—	23	2	no	—
159	C₂²×C₄	—	23	4	no	—
160	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
161	M₁₆	—	23	4	no	—
162	M₁₆	—	23	8	no	—
163	M₁₆	—	23	8	no	—
164	M₁₆	C₄×D₂₃	1	4	yes	complemented
165	C₂×C₈	—	23	4	no	—
166	C₂×C₈	—	23	2	no	—
167	C₂×C₈	—	23	1	no	—
168	C₂×C₈	—	23	1	no	—
169	C₂×C₈	—	23	8	no	—
170	C₂×C₈	—	23	8	no	—
171	C₂×C₈	C₄×D₂₃	1	4	yes	complemented
172	C₂×C₈	C₂²×D₂₃	1	2	yes	—
173	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
175	C₄⋊C₄	—	23	4	no	—
176	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
177	C₂²⋊C₄	—	23	4	no	—
178	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
179	C₄²	—	23	4	no	—
180	C₄²	—	23	2	no	—
181	C₄²	—	23	1	no	—
182	C₄²	—	23	1	no	—
183	C₄²	C₂²×D₂₃	1	2	yes	—
184	C₂³	—	23	2	no	—
185	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
186	Q₈	—	23	4	no	—
187	D₄	—	23	8	no	—
188	D₄	—	23	4	no	—
189	C₂×C₄	—	23	2	no	—
190	C₂×C₄	—	23	4	no	—
191	C₂×C₄	—	23	4	no	—
192	C₂×C₄	—	23	2	no	—
193	C₂×C₄	—	23	2	no	—
194	C₂×C₄	—	92	4	no	—
195	C₂×C₄	—	92	2	no	—
196	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
197	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
200	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
201	C₈	—	23	2	no	—
202	C₈	—	23	2	no	—
203	C₈	—	23	4	no	—
204	C₈	C₂×C₄×D₂₃	1	2	yes	—
205	C₈	C₂×C₄×D₂₃	1	4	yes	—
206	C₈	C₂×C₄×D₂₃	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	23	4	no	—
209	C₂²	—	46	4	no	—
210	C₂²	C₄²×D₂₃	1	2	yes	—
211	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	23	4	no	—
214	C₄	—	46	2	no	—
215	C₄	—	46	2	no	—
216	C₄	C₄²×D₂₃	1	2	yes	—
217	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
218	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	46	4	no	—
221	C₂	C₂×C₄²×D₂₃	1	1	yes	characteristic
222	C₂	#13093182	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Fitting	C₂₃×#405	C₂
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#405	D₂₃
hypocenter	C₂₃	#2875
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #13092374	C₂	semidirect
5. #13092373	C₂	semidirect
6. #13092371	C₂	semidirect
7. M₁₆×Dic₂₃	C₂	semidirect
8. C₄×#67262	C₂	semidirect
9. C₂×#10624997	C₂	semidirect
10. #97×D₂₃	C₂	semidirect
11. C₂₃×#405	C₂	semidirect
19. #10625078	C₄	semidirect
20. #10625072	C₄	semidirect
49. M₁₆×C₂₃	C₂×C₄	semidirect
50. C₂×C₈×C₂₃	C₂×C₄	semidirect
63. #67267	C₂×C₄	semidirect
64. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
68. #67262	C₂×C₄	semidirect
69. #67262	C₂×C₄	semidirect
70. C₈×D₂₃	C₂×C₄	semidirect
71. C₈×D₂₃	C₂×C₄	semidirect
114. #405	D₂₃	semidirect
151. C₂₃	#2875	semidirect
164. M₁₆	C₄×D₂₃	semidirect
171. C₂×C₈	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625078	2
C₂×#10625072	2
#13092885	2
#13092374	2
#13092373	2
#13092371	2
M₁₆×Dic₂₃	2
C₄×#67262	2
C₂×#10624997	2
#97×D₂₃	2
C₂₃×#405	2
#2875	23