Subgroups — #15682345

Order 2944.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092971	C₂	1	1	yes	maximal, characteristic
2	#13092545	C₂	1	2	yes	maximal, complemented
3	#13092544	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092544	C₂	1	2	yes	maximal, complemented
5	Q₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092442	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
9	#13092374	C₂	1	2	yes	maximal
10	C₄×#67262	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	#10625117	C₄	1	1	yes	characteristic
14	#10625115	C₄	1	1	yes	characteristic
15	C₂×#67262	C₂²	1	1	yes	characteristic
16	C₂×#67262	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₃	C₂²	1	2	yes	—
18	#10625068	C₄	1	1	yes	characteristic
19	#10625067	C₄	1	2	yes	—
20	#10625060	C₄	1	2	yes	—
21	#10625059	C₄	1	1	yes	characteristic
22	#10625043	C₂²	1	2	yes	—
23	C₄²×D₂₃	C₂²	1	1	yes	characteristic
24	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₄₆	C₂²	1	2	yes	—
26	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
28	C₂₃×#97	C₂²	1	2	yes	—
29	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
30	#10624997	C₂²	1	1	yes	characteristic
31	#10624997	C₂²	1	1	yes	characteristic
32	#10624997	C₂²	1	2	yes	—
33	Dic₂₃⋊C₈	C₂²	1	2	yes	—
34	Dic₂₃⋊C₈	C₂²	1	2	yes	—
35	Dic₂₃⋊C₈	C₂²	1	2	yes	—
36	C₈×Dic₂₃	C₂²	1	2	yes	—
37	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
39	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₄²×C₂₃	C₂³	1	2	yes	—
54	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
58	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
66	C₄×Dic₂₃	C₂³	1	2	yes	—
67	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
68	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
70	#67262	—	2	2	no	—
71	#67262	Q₈	1	4	yes	complemented
72	C₈×D₂₃	—	2	4	no	—
73	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₂₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₂₃	—	2	2	no	—
86	C₄×D₂₃	—	4	1	no	—
87	C₄×D₂₃	C₂×Q₈	1	2	yes	—
88	C₄×D₂₃	#41	1	2	yes	—
89	Dic₄₆	—	2	2	no	—
90	Dic₄₆	—	4	2	no	—
91	C₈×C₂₃	—	2	2	no	—
92	C₈×C₂₃	—	2	1	no	—
93	C₈×C₂₃	C₂×Q₈	1	2	yes	—
94	C₂₃⋊C₈	—	2	2	no	—
95	C₂₃⋊C₈	—	2	1	no	—
96	C₂₃⋊C₈	C₂×Q₈	1	2	yes	—
97	#2962	—	23	1	no	maximal, Sylow
98	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₂₃	—	2	1	no	—
100	C₂×D₂₃	—	2	1	no	—
101	C₂×D₂₃	C₄×Q₈	1	2	yes	—
102	C₄×C₂₃	—	2	2	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	#131	1	2	yes	—
106	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
107	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
108	Dic₂₃	—	2	2	no	—
109	Dic₂₃	—	2	1	no	—
110	Dic₂₃	—	2	1	no	—
111	Dic₂₃	—	2	2	no	—
112	Dic₂₃	C₄×Q₈	1	2	yes	—
113	#533	—	23	1	no	—
114	#446	—	23	1	no	—
115	#446	—	23	2	no	—
116	#446	—	23	2	no	—
117	#446	D₂₃	1	1	yes	characteristic, complemented
118	C₈×Q₈	—	23	1	no	—
119	C₈×Q₈	—	23	1	no	—
120	#424	—	23	2	no	—
121	C₂×C₄⋊C₈	—	23	1	no	—
122	#405	—	23	2	no	—
123	C₄×M₁₆	—	23	1	no	—
124	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
126	C₂×C₂₃	#568	1	1	yes	characteristic
127	D₂₃	—	2	2	no	—
128	C₂×M₁₆	—	23	1	no	—
129	C₂×M₁₆	—	23	1	no	—
130	C₂²×C₈	—	23	2	no	—
131	C₄⋊Q₈	—	23	1	no	—
132	C₄⋊Q₈	—	23	1	no	—
133	#125	—	23	2	no	—
134	C₂²⋊Q₈	—	23	1	no	—
135	C₂²⋊Q₈	—	23	2	no	—
136	C₂²⋊Q₈	—	23	1	no	—
137	C₄×Q₈	—	23	1	no	—
138	C₄×Q₈	—	23	2	no	—
139	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
140	#117	—	23	2	no	—
141	C₂×C₄²	—	23	1	no	—
142	C₄⋊C₈	—	23	2	no	—
143	C₄⋊C₈	—	23	2	no	—
144	C₄⋊C₈	—	23	2	no	—
145	C₄⋊C₈	—	23	1	no	—
146	C₄⋊C₈	—	23	2	no	—
147	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
149	#97	—	23	1	no	—
150	#97	—	23	1	no	—
151	#97	—	23	2	no	—
152	#97	C₂×D₂₃	1	2	yes	—
153	C₄×C₈	—	23	2	no	—
154	C₄×C₈	—	23	1	no	—
155	C₄×C₈	—	23	2	no	—
156	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
157	C₂₃	#2962	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	23	1	no	—
159	C₂×Q₈	—	23	2	no	—
160	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
161	C₂²×C₄	—	23	1	no	—
162	C₂²×C₄	—	23	1	no	—
163	C₂²×C₄	—	23	1	no	—
164	M₁₆	—	23	4	no	—
165	M₁₆	—	46	2	no	—
166	C₂×C₈	—	23	1	no	—
167	C₂×C₈	—	23	2	no	—
168	C₂×C₈	—	23	1	no	—
169	C₂×C₈	—	46	4	no	—
170	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₂₃	1	2	yes	—
172	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
173	C₄⋊C₄	—	23	2	no	—
174	C₄⋊C₄	—	23	2	no	—
175	C₄⋊C₄	—	23	2	no	—
176	C₄⋊C₄	—	23	2	no	—
177	C₄⋊C₄	—	23	2	no	—
178	C₄⋊C₄	—	23	2	no	—
179	C₄⋊C₄	—	23	1	no	—
180	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
181	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
182	C₂²⋊C₄	—	23	2	no	—
183	C₂²⋊C₄	—	23	2	no	—
184	C₄²	—	23	1	no	—
185	C₄²	—	23	1	no	—
186	C₄²	—	23	2	no	—
187	C₄²	—	23	1	no	—
188	C₄²	C₂²×D₂₃	1	1	yes	characteristic
189	C₄²	C₂²×D₂₃	1	2	yes	—
190	C₂³	—	23	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	46	2	no	—
193	Q₈	—	92	2	no	—
194	C₂×C₄	—	23	1	no	—
195	C₂×C₄	—	23	2	no	—
196	C₂×C₄	—	23	2	no	—
197	C₂×C₄	—	23	1	no	—
198	C₂×C₄	—	23	1	no	—
199	C₂×C₄	—	23	2	no	—
200	C₂×C₄	—	23	2	no	—
201	C₂×C₄	—	46	2	no	—
202	C₂×C₄	—	92	1	no	—
203	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	23	2	no	—
211	C₈	—	46	2	no	—
212	C₈	—	46	1	no	—
213	C₈	Q₈×D₂₃	1	2	yes	—
214	C₂²	—	23	2	no	—
215	C₂²	—	46	1	no	—
216	C₂²	—	46	1	no	—
217	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	23	2	no	—
222	C₄	—	46	1	no	—
223	C₄	—	46	2	no	—
224	C₄	—	46	1	no	—
225	C₄	—	46	2	no	—
226	C₄	#10625078	1	2	yes	—
227	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
228	C₄	#41×D₂₃	1	1	yes	characteristic
229	C₂	—	46	2	no	—
230	C₂	C₄×Q₈×D₂₃	1	1	yes	characteristic
231	C₂	C₂×#10625078	1	1	yes	characteristic
232	C₂	#13093182	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#446	C₂
prime radical	C₂²×D₂₃	C₂²×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#446	D₂₃
hypocenter	C₂₃	#2962
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #13092545	C₂	semidirect
3. #13092544	C₂	semidirect
4. #13092544	C₂	semidirect
5. Q₈×C₂₃⋊C₈	C₂	semidirect
6. C₈×Dic₄₆	C₂	semidirect
11. C₂₃×#446	C₂	semidirect
71. #67262	Q₈	semidirect
117. #446	D₂₃	semidirect
157. C₂₃	#2962	semidirect

Maximal subgroups

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

subgroup	index
#13092971	2
#13092545	2
#13092544	2
Q₈×C₂₃⋊C₈	2
C₈×Dic₄₆	2
#13092442	2
C₄⋊C₈×D₂₃	2
#13092374	2
C₄×#67262	2
C₂₃×#446	2
#2962	23