Subgroups — #15681958

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67262	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic
3	#117×D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092445	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092444	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092443	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092442	C₂	1	4	yes	maximal, complemented
8	#13092439	C₂	1	4	yes	maximal, complemented
9	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#117	C₂²	1	1	yes	characteristic
14	#10625097	C₂²	1	1	yes	characteristic
15	C₂×#67267	C₂²	1	1	yes	characteristic
16	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
17	M₁₆×D₂₃	—	2	4	no	—
18	C₂×#67262	—	2	4	no	—
19	C₂×#67262	C₂²	1	2	yes	—
20	C₂×#67262	C₂²	1	2	yes	—
21	C₂×#67262	C₂²	1	2	yes	—
22	C₂×C₈×D₂₃	C₂²	1	2	yes	—
23	#10625063	C₄	1	2	yes	—
24	C₄⋊C₄×D₂₃	C₄	1	2	yes	—
25	C₂²⋊C₄×D₂₃	C₄	1	2	yes	—
26	#10625048	C₄	1	2	yes	—
27	#10625043	C₂²	1	2	yes	—
28	C₄²×D₂₃	C₂²	1	2	yes	—
29	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
30	Dic₂₃⋊C₈	C₂²	1	4	yes	complemented
31	Dic₂₃⋊C₈	C₂²	1	4	yes	complemented
32	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
33	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₂₃	D₄	1	4	yes	—
40	C₂×C₄×D₂₃	Q₈	1	4	yes	—
41	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
43	M₁₆×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂×C₈×C₂₃	C₂³	1	2	yes	—
47	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
49	C₄²×C₂₃	C₂³	1	2	yes	—
50	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
51	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
53	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
55	C₄×Dic₂₃	C₂³	1	2	yes	—
56	#67267	—	2	2	no	—
57	C₂×C₂₃⋊C₈	—	2	2	no	—
58	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
59	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
60	#67262	—	2	4	no	—
61	#67262	—	2	4	no	—
62	#67262	—	2	4	no	—
63	C₈×D₂₃	—	2	4	no	—
64	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₂₃	—	4	2	no	—
66	C₂²×D₂₃	C₄⋊C₄	1	4	yes	—
67	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
75	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
76	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₂₃	—	4	4	no	—
79	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
80	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
81	C₄×D₂₃	C₂×D₄	1	2	yes	—
82	C₄×D₂₃	C₂×D₄	1	2	yes	—
83	C₄×D₂₃	C₂×Q₈	1	2	yes	—
84	C₄×D₂₃	C₂×Q₈	1	2	yes	—
85	C₈×C₂₃	—	2	2	no	—
86	C₈×C₂₃	—	2	2	no	—
87	C₂₃⋊C₈	—	2	2	no	—
88	C₂₃⋊C₈	—	2	2	no	—
89	#2905	—	23	1	no	maximal, Sylow
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₂₃	—	2	2	no	—
94	C₂×D₂₃	—	2	2	no	—
95	C₂×D₂₃	—	4	4	no	—
96	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₄×C₂₃	—	2	4	no	—
99	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
102	Dic₂₃	—	2	4	no	—
103	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
104	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂²×M₁₆	—	23	1	no	—
106	C₂²×M₁₆	—	23	1	no	—
107	C₂×#117	—	23	1	no	—
108	#424	—	23	1	no	—
109	#424	—	23	1	no	—
110	#424	—	23	1	no	—
111	#424	—	23	4	no	—
112	#424	D₂₃	1	1	yes	characteristic, complemented
113	#423	—	23	4	no	—
114	C₂×C₂₃	—	2	2	no	—
115	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₂₃	#568	1	2	yes	—
117	D₂₃	—	2	2	no	—
118	D₂₃	—	2	2	no	—
119	C₂³×C₄	—	23	1	no	—
120	C₂×M₁₆	—	23	1	no	—
121	C₂×M₁₆	—	23	2	no	—
122	C₂×M₁₆	—	23	2	no	—
123	C₂×M₁₆	—	23	2	no	—
124	C₂×M₁₆	—	46	4	no	—
125	C₂×M₁₆	—	46	4	no	—
126	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
127	C₂²×C₈	—	23	1	no	—
128	C₂²×C₈	—	23	2	no	—
129	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
130	#117	—	23	1	no	—
131	#117	—	23	2	no	—
132	#117	—	23	2	no	—
133	#117	—	23	2	no	—
134	#117	C₂×D₂₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	23	2	no	—
136	C₂×C₂²⋊C₄	—	23	2	no	—
137	C₂×C₄²	—	23	2	no	—
138	C₄⋊C₈	—	23	4	no	—
139	C₄⋊C₈	—	23	4	no	—
140	C₄⋊C₈	—	23	4	no	—
141	C₄⋊C₈	C₂×D₂₃	1	4	yes	complemented
142	C₂₃	#2905	1	1	yes	characteristic, complemented, Sylow
143	C₂⁴	—	23	1	no	—
144	C₂²×C₄	—	23	1	no	—
145	C₂²×C₄	—	23	1	no	—
146	C₂²×C₄	—	23	1	no	—
147	C₂²×C₄	—	23	4	no	—
148	C₂²×C₄	—	23	4	no	—
149	C₂²×C₄	—	23	1	no	—
150	C₂²×C₄	—	23	1	no	—
151	C₂²×C₄	—	23	4	no	—
152	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	46	4	no	—
155	M₁₆	—	46	4	no	—
156	M₁₆	—	46	4	no	—
157	M₁₆	—	46	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	23	2	no	—
160	C₂×C₈	—	23	2	no	—
161	C₂×C₈	—	46	4	no	—
162	C₂×C₈	—	46	2	no	—
163	C₂×C₈	C₂²×D₂₃	1	2	yes	—
164	C₂×C₈	C₂²×D₂₃	1	2	yes	—
165	C₄⋊C₄	—	23	4	no	—
166	C₄⋊C₄	—	23	2	no	—
167	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
168	C₂²⋊C₄	—	23	2	no	—
169	C₂²⋊C₄	—	23	4	no	—
170	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
171	C₄²	—	23	4	no	—
172	C₄²	—	23	2	no	—
173	C₄²	C₂²×D₂₃	1	2	yes	—
174	C₂³	—	23	1	no	—
175	C₂³	—	23	1	no	—
176	C₂³	—	23	4	no	—
177	C₂³	—	92	2	no	—
178	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
179	C₂×C₄	—	23	4	no	—
180	C₂×C₄	—	23	1	no	—
181	C₂×C₄	—	23	1	no	—
182	C₂×C₄	—	23	4	no	—
183	C₂×C₄	—	23	4	no	—
184	C₂×C₄	—	23	2	no	—
185	C₂×C₄	—	23	2	no	—
186	C₂×C₄	—	23	2	no	—
187	C₂×C₄	—	23	2	no	—
188	C₂×C₄	—	23	4	no	—
189	C₂×C₄	—	92	4	no	—
190	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
191	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
192	C₂×C₄	D₄×D₂₃	1	2	yes	—
193	C₂×C₄	Q₈×D₂₃	1	2	yes	—
194	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	46	2	no	—
198	C₈	—	46	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	23	2	no	—
201	C₂²	—	23	2	no	—
202	C₂²	—	46	2	no	—
203	C₂²	—	46	2	no	—
204	C₂²	—	92	4	no	—
205	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
206	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	23	2	no	—
209	C₄	—	23	2	no	—
210	C₄	—	46	4	no	—
211	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
212	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
213	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	46	2	no	—
216	C₂	—	46	2	no	—
217	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
218	C₂	#13093182	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#424	C₂
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#424	D₂₃
hypocenter	C₂₃	#2905
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #13092445	C₂	semidirect
5. #13092444	C₂	semidirect
6. #13092443	C₂	semidirect
7. #13092442	C₂	semidirect
8. #13092439	C₂	semidirect
9. C₂₃×#424	C₂	semidirect
29. C₂₃×C₄⋊C₈	C₂²	semidirect
30. Dic₂₃⋊C₈	C₂²	semidirect
31. Dic₂₃⋊C₈	C₂²	semidirect
32. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
112. #424	D₂₃	semidirect
141. C₄⋊C₈	C₂×D₂₃	semidirect
142. C₂₃	#2905	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67262	2
C₂×M₁₆×D₂₃	2
#117×D₂₃	2
#13092445	2
#13092444	2
#13092443	2
#13092442	2
#13092439	2
C₂₃×#424	2
#2905	23