Subgroups — C₂₃×#23825 #19171451

Order 5888.

714 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2100	C₂	1	2	yes	maximal
4	C₂₃×#2074	C₂	1	2	yes	maximal
5	C₂₃×#2043	C₂	1	2	yes	maximal
6	C₂₃×#2026	C₂	1	2	yes	maximal
7	C₂₃×#1763	C₂	1	4	yes	maximal
8	C₂₃×#1745	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×#103	C₂²	1	4	yes	—
13	C₂×C₂₃×#103	C₂²	1	4	yes	—
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂₃×#391	C₂²	1	2	yes	—
16	C₂₃×#389	—	2	2	no	—
17	C₂₃×#389	C₂²	1	2	yes	—
18	C₂₃×#387	—	2	2	no	—
19	C₂₃×#384	C₂²	1	1	yes	characteristic
20	C₂₃×#382	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₄³×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#340	C₂²	1	4	yes	—
26	C₂₃×#340	C₂²	1	4	yes	—
27	C₂₃×#336	C₂²	1	2	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
31	C₂₃×C₄⋊Q₈	—	2	4	no	—
32	C₂₃×#125	—	2	4	no	—
33	C₄×Q₈×C₂₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₂₃	—	2	2	no	—
43	C₂×C₄²×C₂₃	—	2	2	no	—
44	C₂×C₄²×C₂₃	D₄	1	2	yes	—
45	C₂×C₄²×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×#103	—	2	8	no	—
50	C₂₃×#103	—	2	8	no	—
51	C₂₃×#97	—	2	4	no	—
52	C₂₃×#95	—	2	2	no	—
53	C₂₃×#95	—	2	2	no	—
54	C₂₃×#95	—	2	4	no	—
55	C₂₃×#95	—	2	2	no	—
56	C₂×Q₈×C₂₃	—	2	2	no	—
57	C₂×Q₈×C₂₃	—	2	4	no	—
58	C₂×Q₈×C₂₃	—	2	4	no	—
59	C₂×Q₈×C₂₃	#41	1	4	yes	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₄⋊C₄×C₂₃	—	2	8	no	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	4	4	no	—
78	C₄⋊C₄×C₂₃	—	4	4	no	—
79	C₄⋊C₄×C₂₃	—	4	4	no	—
80	C₄⋊C₄×C₂₃	#41	1	4	yes	—
81	C₄²×C₂₃	—	2	4	no	—
82	C₄²×C₂₃	—	2	2	no	—
83	C₄²×C₂₃	—	2	2	no	—
84	C₄²×C₂₃	—	4	2	no	—
85	C₄²×C₂₃	—	4	4	no	—
86	C₄²×C₂₃	—	4	4	no	—
87	C₄²×C₂₃	#41	1	4	yes	—
88	#23825	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₂₃	—	2	8	no	—
91	Q₈×C₂₃	—	4	4	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	#124	1	2	yes	—
111	C₂×C₄×C₂₃	#124	1	2	yes	—
112	C₂×C₄×C₂₃	#137	1	4	yes	—
113	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
114	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
115	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
116	C₈×C₂₃	—	4	4	no	—
117	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
118	C₄×#125	C₂×C₂₃	1	1	yes	characteristic
119	#2100	C₂×C₂₃	1	2	yes	—
120	#2074	C₂×C₂₃	1	2	yes	—
121	#2043	C₂×C₂₃	1	2	yes	—
122	#2026	C₂×C₂₃	1	2	yes	—
123	#1763	C₂×C₂₃	1	4	yes	—
124	#1745	C₂×C₂₃	1	1	yes	characteristic
125	C₂²×C₂₃	#525	1	1	yes	characteristic
126	C₂²×C₂₃	#525	1	1	yes	characteristic
127	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
128	C₂²×C₂₃	C₂×#137	1	2	yes	—
129	C₂²×C₂₃	#575	1	2	yes	—
130	C₄×C₂₃	—	2	4	no	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	4	2	no	—
134	C₄×C₂₃	—	4	4	no	—
135	C₄×C₂₃	—	4	4	no	—
136	C₄×C₂₃	—	8	2	no	—
137	C₄×C₂₃	—	8	2	no	—
138	C₄×C₂₃	#529	1	4	yes	—
139	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
142	C₂×#103	C₂²×C₂₃	1	4	yes	—
143	C₂×#103	C₂²×C₂₃	1	4	yes	—
144	C₂×#97	C₂²×C₂₃	1	2	yes	—
145	#391	C₂²×C₂₃	1	2	yes	—
146	#389	—	2	2	no	—
147	#389	C₂²×C₂₃	1	2	yes	—
148	#387	—	2	2	no	—
149	#384	C₂²×C₂₃	1	1	yes	characteristic
150	#382	C₂²×C₂₃	1	1	yes	characteristic
151	C₄×C₄⋊C₄	—	2	2	no	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
154	C₄³	C₂²×C₂₃	1	1	yes	characteristic
155	#340	C₂²×C₂₃	1	4	yes	—
156	#340	C₂²×C₂₃	1	4	yes	—
157	#336	C₂²×C₂₃	1	2	yes	—
158	#336	C₂²×C₂₃	1	2	yes	—
159	C₂×C₂₃	#2398	1	1	yes	characteristic
160	C₂×C₂₃	#3116	1	2	yes	—
161	C₂×C₂₃	#3117	1	2	yes	—
162	C₂×C₂₃	#3139	1	1	yes	characteristic
163	C₂×C₂₃	#3142	1	1	yes	characteristic
164	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
165	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
166	C₄⋊Q₈	—	2	4	no	—
167	#125	—	2	4	no	—
168	C₄×Q₈	—	2	8	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	D₄×C₂₃	1	2	yes	—
180	C₂×C₄²	D₄×C₂₃	1	2	yes	—
181	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
184	#103	—	2	8	no	—
185	#103	—	2	8	no	—
186	#97	—	2	4	no	—
187	#95	—	2	2	no	—
188	#95	—	2	2	no	—
189	#95	—	2	4	no	—
190	#95	—	2	2	no	—
191	C₂₃	#23825	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	#41×C₂₃	1	4	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	#41×C₂₃	1	4	yes	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	#41×C₂₃	1	4	yes	—
224	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	Q₈	—	4	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₂₃×#124	1	2	yes	—
246	C₂×C₄	C₂₃×#124	1	2	yes	—
247	C₂×C₄	C₂₃×#137	1	4	yes	—
248	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
249	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
250	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
251	C₈	—	4	4	no	—
252	C₂²	C₂₃×#525	1	1	yes	characteristic
253	C₂²	C₂₃×#525	1	1	yes	characteristic
254	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
255	C₂²	C₂×C₂₃×#137	1	2	yes	—
256	C₂²	C₂₃×#575	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₄	—	8	2	no	—
265	C₄	C₂₃×#529	1	4	yes	—
266	C₂	C₂₃×#2398	1	1	yes	characteristic
267	C₂	C₂₃×#3116	1	2	yes	—
268	C₂	C₂₃×#3117	1	2	yes	—
269	C₂	C₂₃×#3139	1	1	yes	characteristic
270	C₂	C₂₃×#3142	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
88. #23825	C₂₃	direct (N × H)
191. C₂₃	#23825	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×C₄⋊Q₈	2
C₄×C₂₃×#125	2
C₂₃×#2100	2
C₂₃×#2074	2
C₂₃×#2043	2
C₂₃×#2026	2
C₂₃×#1763	2
C₂₃×#1745	2
#23825	23

Subgroups — C23×#23825 #19171451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23825 #19171451