Subgroups — C₂₃×#2617 #15670236

Order 2944.

824 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, complemented
2	C₂×C₂₃×#120	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#397	C₂	1	2	yes	maximal, complemented
5	C₂₃×#394	C₂	1	4	yes	maximal, complemented
6	C₂₃×#392	C₂	1	2	yes	maximal, complemented
7	C₂₃×#386	C₂	1	2	yes	maximal, complemented
8	C₂₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₃	C₂²	1	4	yes	—
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#121	—	2	4	no	—
16	C₂₃×#120	—	2	8	no	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
19	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
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	4	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
26	C₂₃×#95	C₂²	1	2	yes	complemented
27	C₂₃×#95	C₂²	1	4	yes	complemented
28	C₂⁴×C₂₃	—	2	2	no	—
29	C₂⁴×C₂₃	—	2	2	no	—
30	C₂⁴×C₂₃	—	4	2	no	—
31	C₂⁴×C₂₃	—	4	2	no	—
32	C₂⁴×C₂₃	D₄	1	4	yes	complemented
33	C₂⁴×C₂₃	C₂³	1	2	yes	—
34	C₂⁴×C₂₃	C₂³	1	2	yes	—
35	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
36	C₂×D₄×C₂₃	—	2	8	no	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	—	2	4	no	—
39	C₂×D₄×C₂₃	—	2	4	no	—
40	C₂×D₄×C₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	—	4	4	no	—
42	C₂×D₄×C₂₃	—	4	4	no	—
43	C₂×D₄×C₂₃	—	4	2	no	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	4	2	no	—
46	C₂²×C₄×C₂₃	—	4	2	no	—
47	C₂²×C₄×C₂₃	D₄	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	4	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₂²⋊C₄×C₂₃	—	2	8	no	—
56	C₂²⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	—	2	4	no	—
58	C₂²⋊C₄×C₂₃	—	2	4	no	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	4	4	no	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	2	8	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	—	2	2	no	—
71	C₂³×C₂₃	—	2	4	no	—
72	C₂³×C₂₃	—	2	4	no	—
73	C₂³×C₂₃	—	4	8	no	—
74	C₂³×C₂₃	—	4	4	no	—
75	C₂³×C₂₃	—	4	4	no	—
76	C₂³×C₂₃	—	4	8	no	—
77	C₂³×C₂₃	C₂×D₄	1	4	yes	—
78	C₂³×C₂₃	C₂×D₄	1	2	yes	—
79	C₂³×C₂₃	C₂×D₄	1	2	yes	—
80	C₂³×C₂₃	#41	1	2	yes	—
81	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
82	D₄×C₂₃	—	4	4	no	—
83	D₄×C₂₃	—	4	16	no	—
84	D₄×C₂₃	—	4	8	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	4	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
99	#2617	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	4	no	—
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	2	4	no	—
104	C₂²×C₂₃	—	2	4	no	—
105	C₂²×C₂₃	—	2	8	no	—
106	C₂²×C₂₃	—	4	16	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	—	4	2	no	—
109	C₂²×C₂₃	—	4	4	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	4	no	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	—	4	4	no	—
115	C₂²×C₂₃	#121	1	4	yes	—
116	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
120	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	4	2	no	—
124	C₄×C₂₃	—	4	4	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₂×#121	C₂×C₂₃	1	1	yes	characteristic, complemented
127	C₂×#120	C₂×C₂₃	1	2	yes	complemented
128	C₂²×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#397	C₂×C₂₃	1	2	yes	complemented
130	#394	C₂×C₂₃	1	4	yes	complemented
131	#392	C₂×C₂₃	1	2	yes	complemented
132	#386	C₂×C₂₃	1	2	yes	complemented
133	#380	C₂×C₂₃	1	1	yes	characteristic, complemented
134	C₂×C₂₃	—	2	4	no	—
135	C₂×C₂₃	—	4	4	no	—
136	C₂×C₂₃	—	8	2	no	—
137	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
138	C₂×C₂₃	#534	1	1	yes	characteristic
139	C₂×C₂₃	#534	1	2	yes	—
140	C₂×C₂₃	#535	1	1	yes	characteristic
141	C₂×C₂₃	#538	1	2	yes	—
142	C₂⁵	C₂²×C₂₃	1	1	yes	characteristic
143	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
144	C₂²×D₄	C₂²×C₂₃	1	4	yes	—
145	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
146	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
147	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
148	#121	—	2	4	no	—
149	#120	—	2	8	no	—
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
152	C₂×C₂²⋊C₄	—	2	4	no	—
153	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
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	4	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	complemented
158	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
159	#95	C₂²×C₂₃	1	2	yes	complemented
160	#95	C₂²×C₂₃	1	4	yes	complemented
161	C₂₃	#2617	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	—	2	2	no	—
163	C₂⁴	—	2	2	no	—
164	C₂⁴	—	4	2	no	—
165	C₂⁴	—	4	2	no	—
166	C₂⁴	D₄×C₂₃	1	4	yes	complemented
167	C₂⁴	C₂³×C₂₃	1	2	yes	—
168	C₂⁴	C₂³×C₂₃	1	2	yes	—
169	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
170	C₂×D₄	—	2	8	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₄×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	4	yes	—
185	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	2	8	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	8	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	4	8	no	—
208	C₂³	—	4	4	no	—
209	C₂³	—	4	4	no	—
210	C₂³	—	4	8	no	—
211	C₂³	C₂×D₄×C₂₃	1	4	yes	—
212	C₂³	C₂×D₄×C₂₃	1	2	yes	—
213	C₂³	C₂×D₄×C₂₃	1	2	yes	—
214	C₂³	#41×C₂₃	1	2	yes	—
215	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	D₄	—	4	16	no	—
218	D₄	—	4	8	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	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
232	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	4	16	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	4	no	—
248	C₂²	C₂₃×#121	1	4	yes	—
249	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
250	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
251	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
252	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
253	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	2	no	—
259	C₂	—	2	4	no	—
260	C₂	—	4	4	no	—
261	C₂	—	8	2	no	—
262	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
263	C₂	C₂₃×#534	1	1	yes	characteristic
264	C₂	C₂₃×#534	1	2	yes	—
265	C₂	C₂₃×#535	1	1	yes	characteristic
266	C₂	C₂₃×#538	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#121	C₂	semidirect
2. C₂×C₂₃×#120	C₂	semidirect
3. C₂²×C₂²⋊C₄×C₂₃	C₂	semidirect
4. C₂₃×#397	C₂	semidirect
5. C₂₃×#394	C₂	semidirect
6. C₂₃×#392	C₂	semidirect
7. C₂₃×#386	C₂	semidirect
8. C₂₃×#380	C₂	semidirect
12. C₂²×D₄×C₂₃	C₂²	semidirect
17. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
18. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
24. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
25. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
26. C₂₃×#95	C₂²	semidirect
27. C₂₃×#95	C₂²	semidirect
32. C₂⁴×C₂₃	D₄	semidirect
47. C₂²×C₄×C₂₃	D₄	semidirect
99. #2617	C₂₃	direct (N × H)
126. C₂×#121	C₂×C₂₃	semidirect
127. C₂×#120	C₂×C₂₃	semidirect
128. C₂²×C₂²⋊C₄	C₂×C₂₃	semidirect
129. #397	C₂×C₂₃	semidirect
130. #394	C₂×C₂₃	semidirect
131. #392	C₂×C₂₃	semidirect
132. #386	C₂×C₂₃	semidirect
133. #380	C₂×C₂₃	semidirect
145. C₂²×D₄	C₂²×C₂₃	semidirect
150. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
151. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
157. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
158. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
159. #95	C₂²×C₂₃	semidirect
160. #95	C₂²×C₂₃	semidirect
161. C₂₃	#2617	direct (N × H)
166. C₂⁴	D₄×C₂₃	semidirect
181. 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×C₂₃×#120	2
C₂²×C₂²⋊C₄×C₂₃	2
C₂₃×#397	2
C₂₃×#394	2
C₂₃×#392	2
C₂₃×#386	2
C₂₃×#380	2
#2617	23

Subgroups — C23×#2617 #15670236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2617 #15670236