Subgroups — #15682054

Order 2944.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092477	C₂	1	2	yes	maximal, complemented
3	#13092477	C₂	1	2	yes	maximal, complemented
4	#13092473	C₂	1	2	yes	maximal, complemented
5	#13092469	C₂	1	2	yes	maximal, complemented
6	#13092444	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092380	C₂	1	1	yes	maximal, characteristic
8	C₂×(C₂×D₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
9	#13092375	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092372	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₃	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	#10625097	C₂²	1	1	yes	characteristic
17	C₂×#67267	C₂²	1	1	yes	characteristic
18	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	#10625063	C₄	1	1	yes	characteristic
20	#10625063	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₂₃	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₂₃	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₂₃	C₄	1	2	yes	—
24	#10625048	C₄	1	2	yes	—
25	#10625043	C₂²	1	2	yes	—
26	C₄²×D₂₃	C₂²	1	2	yes	—
27	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
29	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
33	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
34	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
35	#10624997	C₂²	1	2	yes	complemented
36	#10624997	C₂²	1	2	yes	complemented
37	C₈×Dic₂₃	C₂²	1	2	yes	complemented
38	C₈×Dic₂₃	C₂²	1	2	yes	complemented
39	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
41	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₂₃	—	2	2	no	—
45	C₂×C₄×D₂₃	—	2	2	no	—
46	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
52	M₁₆×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
60	C₄²×C₂₃	C₂³	1	2	yes	—
61	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
62	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
63	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
65	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
66	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
67	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
69	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
70	C₄×Dic₂₃	C₂³	1	2	yes	—
71	#67267	—	2	2	no	—
72	C₂×C₂₃⋊C₈	—	2	2	no	—
73	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
76	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₂₃	—	2	2	no	—
78	C₂²×D₂₃	—	2	2	no	—
79	C₂²×D₂₃	—	4	1	no	—
80	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#41	1	2	yes	—
86	C₂×C₄×C₂₃	#41	1	2	yes	—
87	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₂₃	—	2	2	no	—
89	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
91	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
92	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₂₃	—	2	4	no	—
94	C₄×D₂₃	—	2	2	no	—
95	C₄×D₂₃	—	2	2	no	—
96	C₄×D₂₃	—	2	1	no	—
97	C₄×D₂₃	—	2	1	no	—
98	C₄×D₂₃	—	2	1	no	—
99	C₄×D₂₃	—	2	1	no	—
100	C₄×D₂₃	—	4	2	no	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	—	2	1	no	—
103	C₈×C₂₃	—	2	2	no	—
104	C₂₃⋊C₈	—	2	1	no	—
105	C₂₃⋊C₈	—	2	1	no	—
106	C₂₃⋊C₈	—	2	2	no	—
107	#2924	—	23	1	no	maximal, Sylow
108	C₂²×C₂₃	—	2	1	no	—
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	#117	1	2	yes	—
111	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₂₃	—	2	1	no	—
113	C₂×D₂₃	—	2	1	no	—
114	C₂×D₂₃	—	2	1	no	—
115	C₂×D₂₃	—	2	1	no	—
116	C₂×D₂₃	—	2	1	no	—
117	C₂×D₂₃	—	4	4	no	—
118	C₂×D₂₃	#131	1	2	yes	—
119	C₄×C₂₃	—	2	2	no	—
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	#117	1	2	yes	—
122	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
123	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
124	Dic₂₃	—	2	1	no	—
125	Dic₂₃	—	2	1	no	—
126	Dic₂₃	—	2	2	no	—
127	Dic₂₃	—	2	2	no	—
128	C₂×#117	—	23	1	no	—
129	#433	—	23	2	no	—
130	#433	—	23	2	no	—
131	#432	—	23	2	no	—
132	#431	—	23	2	no	—
133	#424	—	23	1	no	—
134	#424	D₂₃	1	1	yes	characteristic, complemented
135	#407	—	23	1	no	—
136	C₂×C₂²⋊C₈	—	23	1	no	—
137	#405	—	23	1	no	—
138	#405	—	23	1	no	—
139	C₂×C₂₃	—	2	2	no	—
140	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
141	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
142	C₂×C₂₃	#568	1	1	yes	characteristic
143	D₂₃	—	2	2	no	—
144	D₂₃	—	4	1	no	—
145	C₂³×C₄	—	23	1	no	—
146	C₂×M₁₆	—	23	1	no	—
147	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
148	C₂²×C₈	—	23	1	no	—
149	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
150	#117	—	23	1	no	—
151	#117	—	23	1	no	—
152	#117	—	23	1	no	—
153	#117	—	23	2	no	—
154	#117	—	23	2	no	—
155	#117	C₂×D₂₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	23	1	no	—
157	C₂×C₄⋊C₄	—	23	1	no	—
158	C₂×C₂²⋊C₄	—	23	2	no	—
159	C₂×C₄²	—	23	2	no	—
160	C₄⋊C₈	—	23	2	no	—
161	C₄⋊C₈	—	23	2	no	—
162	C₄⋊C₈	C₂×D₂₃	1	2	yes	complemented
163	C₄⋊C₈	C₂×D₂₃	1	2	yes	complemented
164	C₂²⋊C₈	—	23	1	no	—
165	C₂²⋊C₈	—	23	1	no	—
166	C₂²⋊C₈	—	23	1	no	—
167	C₂²⋊C₈	—	23	1	no	—
168	C₂²⋊C₈	—	23	2	no	—
169	C₂²⋊C₈	—	23	2	no	—
170	#97	—	23	2	no	—
171	#97	—	23	2	no	—
172	C₄×C₈	—	23	2	no	—
173	C₄×C₈	—	23	2	no	—
174	C₂₃	#2924	1	1	yes	characteristic, complemented, Sylow
175	C₂⁴	—	23	1	no	—
176	C₂²×C₄	—	23	1	no	—
177	C₂²×C₄	—	23	2	no	—
178	C₂²×C₄	—	23	1	no	—
179	C₂²×C₄	—	23	1	no	—
180	C₂²×C₄	—	23	1	no	—
181	C₂²×C₄	—	23	1	no	—
182	C₂²×C₄	—	23	2	no	—
183	C₂²×C₄	—	46	2	no	—
184	C₂²×C₄	—	46	2	no	—
185	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	46	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	23	1	no	—
190	C₂×C₈	—	23	2	no	—
191	C₂×C₈	—	23	1	no	—
192	C₂×C₈	—	46	2	no	—
193	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
194	C₂×C₈	C₂²×D₂₃	1	2	yes	—
195	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
196	C₄⋊C₄	—	23	2	no	—
197	C₄⋊C₄	—	23	2	no	—
198	C₄⋊C₄	—	23	1	no	—
199	C₄⋊C₄	—	23	1	no	—
200	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
201	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	23	2	no	—
203	C₂²⋊C₄	—	23	2	no	—
204	C₂²⋊C₄	—	23	2	no	—
205	C₂²⋊C₄	C₄×D₂₃	1	2	yes	—
206	C₄²	—	23	2	no	—
207	C₄²	—	23	2	no	—
208	C₄²	—	23	2	no	—
209	C₄²	C₂²×D₂₃	1	2	yes	—
210	C₂³	—	23	1	no	—
211	C₂³	—	23	1	no	—
212	C₂³	—	46	2	no	—
213	C₂³	—	46	2	no	—
214	C₂³	—	92	1	no	—
215	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
216	C₂×C₄	—	23	1	no	—
217	C₂×C₄	—	23	2	no	—
218	C₂×C₄	—	23	2	no	—
219	C₂×C₄	—	23	1	no	—
220	C₂×C₄	—	46	4	no	—
221	C₂×C₄	—	46	2	no	—
222	C₂×C₄	—	46	2	no	—
223	C₂×C₄	—	46	2	no	—
224	C₂×C₄	—	46	1	no	—
225	C₂×C₄	—	46	1	no	—
226	C₂×C₄	—	46	1	no	—
227	C₂×C₄	—	46	1	no	—
228	C₂×C₄	—	92	2	no	—
229	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
230	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
232	C₂×C₄	#67290	1	2	yes	—
233	C₂×C₄	#67292	1	2	yes	—
234	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	46	1	no	—
239	C₈	—	46	1	no	—
240	C₈	—	46	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	23	2	no	—
244	C₂²	—	46	1	no	—
245	C₂²	—	46	1	no	—
246	C₂²	—	46	1	no	—
247	C₂²	—	46	1	no	—
248	C₂²	—	46	1	no	—
249	C₂²	—	92	4	no	—
250	C₂²	#10625063	1	2	yes	—
251	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	46	1	no	—
255	C₄	—	46	1	no	—
256	C₄	—	46	2	no	—
257	C₄	—	46	2	no	—
258	C₄	#10625063	1	2	yes	—
259	C₄	C₂×#67290	1	1	yes	characteristic
260	C₄	C₂×#67292	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	46	2	no	—
263	C₂	—	92	1	no	—
264	C₂	C₂×#10625063	1	1	yes	characteristic
265	C₂	#131×D₂₃	1	1	yes	characteristic
266	C₂	#13093182	1	1	yes	characteristic

	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	#2924
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
2. #13092477	C₂	semidirect
3. #13092477	C₂	semidirect
4. #13092473	C₂	semidirect
5. #13092469	C₂	semidirect
6. #13092444	C₂	semidirect
9. #13092375	C₂	semidirect
10. #13092372	C₂	semidirect
11. C₂₃×#424	C₂	semidirect
27. C₂₃×C₄⋊C₈	C₂²	semidirect
28. C₂₃×C₄⋊C₈	C₂²	semidirect
35. #10624997	C₂²	semidirect
36. #10624997	C₂²	semidirect
37. C₈×Dic₂₃	C₂²	semidirect
38. C₈×Dic₂₃	C₂²	semidirect
39. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
40. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
134. #424	D₂₃	semidirect
162. C₄⋊C₈	C₂×D₂₃	semidirect
163. C₄⋊C₈	C₂×D₂₃	semidirect
174. C₂₃	#2924	semidirect

Maximal subgroups

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

subgroup	index
#117×D₂₃	2
#13092477	2
#13092473	2
#13092469	2
#13092444	2
#13092380	2
C₂×(C₂×D₂₃)⋊C₈	2
#13092375	2
#13092372	2
C₂₃×#424	2
#2924	23