Subgroups — C₂₃×#2364 #15669983

Order 2944.

716 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	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂₃×#388	C₂	1	2	yes	maximal, complemented
4	C₂₃×#387	C₂	1	2	yes	maximal, complemented
5	C₂₃×#386	C₂	1	2	yes	maximal, complemented
6	C₂₃×#386	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#121	C₄	1	8	yes	complemented
16	C₄×D₄×C₂₃	—	2	8	no	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	1	yes	characteristic
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂⁴×C₂₃	C₂³	1	2	yes	—
34	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×D₄×C₂₃	—	2	4	no	—
36	C₂×D₄×C₂₃	—	2	4	no	—
37	C₂×D₄×C₂₃	—	4	2	no	—
38	C₂×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
39	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	—	4	2	no	—
46	C₂²×C₄×C₂₃	—	4	2	no	—
47	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	complemented
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₃	—	2	4	no	—
56	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	4	2	no	—
61	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
62	C₂²⋊C₄×C₂₃	D₄	1	4	yes	complemented
63	C₄²×C₂₃	—	2	4	no	—
64	C₄²×C₂₃	—	4	2	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	—	2	2	no	—
71	C₂³×C₂₃	—	2	2	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
74	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₂₃	C₂×D₄	1	2	yes	—
76	C₂³×C₂₃	#41	1	2	yes	—
77	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
78	D₄×C₂₃	—	2	16	no	—
79	D₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	4	8	no	—
96	C₂×C₄×C₂₃	—	4	8	no	—
97	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
99	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₂₃	#41	1	4	yes	—
102	#2364	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	4	no	—
106	C₂²×C₂₃	—	2	4	no	—
107	C₂²×C₂₃	—	2	4	no	—
108	C₂²×C₂₃	—	2	4	no	—
109	C₂²×C₂₃	—	2	4	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	2	no	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	C₄×D₄	1	4	yes	—
115	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×#41	1	2	yes	—
119	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
120	C₄×C₂₃	—	2	4	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	4	no	—
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	—	4	2	no	—
127	C₂×#121	C₂×C₂₃	1	1	yes	characteristic
128	C₂×C₄×D₄	C₂×C₂₃	1	2	yes	complemented
129	#388	C₂×C₂₃	1	2	yes	complemented
130	#387	C₂×C₂₃	1	2	yes	complemented
131	#386	C₂×C₂₃	1	2	yes	complemented
132	#386	C₂×C₂₃	1	2	yes	complemented
133	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
134	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
135	C₂×#95	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂×C₂₃	—	2	4	no	—
137	C₂×C₂₃	—	4	4	no	—
138	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
140	C₂×C₂₃	#518	1	1	yes	characteristic
141	C₂×C₂₃	#546	1	2	yes	—
142	C₂×C₂₃	#553	1	2	yes	—
143	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
144	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
146	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
147	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
148	#121	C₄×C₂₃	1	8	yes	complemented
149	C₄×D₄	—	2	8	no	—
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
159	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
160	#95	C₂²×C₂₃	1	1	yes	characteristic
161	#95	C₂²×C₂₃	1	2	yes	—
162	#95	C₂²×C₂₃	1	1	yes	characteristic
163	#95	C₂²×C₂₃	1	2	yes	—
164	#95	C₂²×C₂₃	1	2	yes	—
165	#95	C₂²×C₂₃	1	2	yes	—
166	C₂₃	#2364	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	C₂³×C₂₃	1	2	yes	—
168	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
173	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	complemented
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	complemented
182	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
183	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
187	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
188	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	complemented
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	complemented
196	C₂²⋊C₄	D₄×C₂₃	1	4	yes	complemented
197	C₄²	—	2	4	no	—
198	C₄²	—	4	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
208	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
209	C₂³	C₂×D₄×C₂₃	1	2	yes	—
210	C₂³	#41×C₂₃	1	2	yes	—
211	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
212	D₄	—	2	16	no	—
213	D₄	—	4	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
232	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
233	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
234	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
235	C₂×C₄	#41×C₂₃	1	4	yes	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₄×D₄×C₂₃	1	4	yes	—
248	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
249	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
250	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
251	C₂²	C₂×#41×C₂₃	1	2	yes	—
252	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	4	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₂	—	2	4	no	—
261	C₂	—	4	4	no	—
262	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
263	C₂	C₄×#41×C₂₃	1	1	yes	characteristic
264	C₂	C₂₃×#518	1	1	yes	characteristic
265	C₂	C₂₃×#546	1	2	yes	—
266	C₂	C₂₃×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄×C₂₃	C₂	semidirect
3. C₂₃×#388	C₂	semidirect
4. C₂₃×#387	C₂	semidirect
5. C₂₃×#386	C₂	semidirect
6. C₂₃×#386	C₂	semidirect
7. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
8. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
9. C₂×C₂₃×#95	C₂	semidirect
15. C₂₃×#121	C₄	semidirect
38. C₂×D₄×C₂₃	C₂×C₄	semidirect
39. C₂×D₄×C₂₃	C₂×C₄	semidirect
47. C₂²×C₄×C₂₃	C₂×C₄	semidirect
56. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
61. C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
62. C₂²⋊C₄×C₂₃	D₄	semidirect
102. #2364	C₂₃	direct (N × H)
128. C₂×C₄×D₄	C₂×C₂₃	semidirect
129. #388	C₂×C₂₃	semidirect
130. #387	C₂×C₂₃	semidirect
131. #386	C₂×C₂₃	semidirect
132. #386	C₂×C₂₃	semidirect
133. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
134. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
135. C₂×#95	C₂×C₂₃	semidirect
148. #121	C₄×C₂₃	semidirect
166. C₂₃	#2364	direct (N × H)
172. C₂×D₄	C₂×C₄×C₂₃	semidirect
173. C₂×D₄	C₂×C₄×C₂₃	semidirect
181. C₂²×C₄	C₂×C₄×C₂₃	semidirect
190. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
195. C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
196. C₂²⋊C₄	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#121	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#388	2
C₂₃×#387	2
C₂₃×#386	2
C₄×C₂²⋊C₄×C₂₃	2
C₂×C₂₃×#95	2
#2364	23

Subgroups — C23×#2364 #15669983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2364 #15669983