Subgroups — C₂₃×#13200 #19160826

Order 5888.

438 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2117	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2112	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#525	C₄	1	1	yes	characteristic
9	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
11	C₂₃×#435	—	2	2	no	—
12	C₈×D₄×C₂₃	—	2	2	no	—
13	C₂₃×#424	C₄	1	1	yes	characteristic
14	C₂₃×#408	C₄	1	1	yes	characteristic
15	C₂₃×#407	C₄	1	1	yes	characteristic
16	C₂₃×#405	C₂²	1	1	yes	characteristic
17	C₂₃×#364	—	2	2	no	—
18	C₂₃×#350	—	2	4	no	—
19	C₂₃×C₂²⋊C₁₆	—	2	4	no	—
20	C₂₃×#347	—	2	2	no	—
21	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×#131	—	2	2	no	—
24	C₂₃×#131	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	—	2	2	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#123	—	2	1	no	—
36	C₂₃×C₂²⋊Q₈	—	2	1	no	—
37	C₂₃×#121	—	2	1	no	—
38	C₂₃×#120	—	2	1	no	—
39	C₄×D₄×C₂₃	—	2	2	no	—
40	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
41	C₂₃×M₃₂	—	2	2	no	—
42	C₂₃×M₃₂	—	2	2	no	—
43	C₂₃×M₃₂	—	2	2	no	—
44	C₂₃×M₃₂	—	2	2	no	—
45	C₂×C₁₆×C₂₃	—	2	2	no	—
46	C₂×C₁₆×C₂₃	D₄	1	4	yes	complemented
47	C₂₃×C₄⋊C₈	—	2	2	no	—
48	C₂₃×C₂²⋊C₈	—	2	2	no	—
49	C₂₃×C₂²⋊C₈	—	2	2	no	—
50	C₂₃×#97	D₄	1	2	yes	—
51	C₄×C₈×C₂₃	D₄	1	2	yes	—
52	C₂⁴×C₂₃	M₁₆	1	1	yes	characteristic
53	#41×C₂₃	—	2	2	no	—
54	#41×C₂₃	—	2	2	no	—
55	C₂×Q₈×C₂₃	M₁₆	1	1	yes	characteristic
56	C₂×D₄×C₂₃	—	2	2	no	—
57	C₂×D₄×C₂₃	—	2	1	no	—
58	C₂×D₄×C₂₃	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	M₁₆×C₂₃	—	2	2	no	—
67	M₁₆×C₂₃	—	2	2	no	—
68	M₁₆×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	#41	1	2	yes	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂²⋊C₄×C₂₃	—	2	2	no	—
85	C₂²⋊C₄×C₂₃	—	2	2	no	—
86	C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
88	C₁₆×C₂₃	—	2	4	no	—
89	C₁₆×C₂₃	—	4	2	no	—
90	#13200	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	1	no	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	—	4	2	no	—
95	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	2	no	—
97	D₄×C₂₃	—	2	2	no	—
98	D₄×C₂₃	—	4	4	no	—
99	D₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#131	1	2	yes	—
113	C₈×C₂₃	—	2	2	no	—
114	C₈×C₂₃	—	4	1	no	—
115	C₈×C₂₃	—	4	1	no	—
116	C₈×C₂₃	#121	1	2	yes	—
117	C₈×C₂₃	#124	1	1	yes	characteristic
118	C₈×C₂₃	#127	1	1	yes	characteristic
119	#2929	C₂×C₂₃	1	1	yes	characteristic
120	#2153	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#2117	C₂×C₂₃	1	2	yes	complemented
122	#2112	C₂×C₂₃	1	2	yes	complemented
123	#2110	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	2	4	no	—
126	C₂²×C₂₃	—	4	1	no	—
127	C₂²×C₂₃	—	4	1	no	—
128	C₂²×C₂₃	—	4	2	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	#407	1	1	yes	characteristic
133	C₂²×C₂₃	#435	1	2	yes	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	4	1	no	—
136	C₄×C₂₃	—	4	1	no	—
137	C₄×C₂₃	#390	1	1	yes	characteristic
138	C₄×C₂₃	#408	1	1	yes	characteristic
139	C₄×C₂₃	#435	1	2	yes	—
140	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
141	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	#525	C₄×C₂₃	1	1	yes	characteristic
143	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
144	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
145	#435	—	2	2	no	—
146	C₈×D₄	—	2	2	no	—
147	#424	C₄×C₂₃	1	1	yes	characteristic
148	#408	C₄×C₂₃	1	1	yes	characteristic
149	#407	C₄×C₂₃	1	1	yes	characteristic
150	#405	C₂²×C₂₃	1	1	yes	characteristic
151	#364	—	2	2	no	—
152	#350	—	2	4	no	—
153	C₂²⋊C₁₆	—	2	4	no	—
154	#347	—	2	2	no	—
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	—	2	2	no	—
157	C₂×C₂₃	—	4	2	no	—
158	C₂×C₂₃	—	8	1	no	—
159	C₂×C₂₃	#1955	1	1	yes	characteristic
160	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
174	#123	—	2	1	no	—
175	C₂²⋊Q₈	—	2	1	no	—
176	#121	—	2	1	no	—
177	#120	—	2	1	no	—
178	C₄×D₄	—	2	2	no	—
179	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
180	M₃₂	—	2	2	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	—	2	2	no	—
184	C₂×C₁₆	—	2	2	no	—
185	C₂×C₁₆	D₄×C₂₃	1	4	yes	complemented
186	C₄⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	#97	D₄×C₂₃	1	2	yes	—
190	C₄×C₈	D₄×C₂₃	1	2	yes	—
191	C₂₃	#13200	1	1	yes	characteristic, direct product factor, Sylow
192	C₂⁴	M₁₆×C₂₃	1	1	yes	characteristic
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	M₁₆×C₂₃	1	1	yes	characteristic
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	M₁₆×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	4	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂×C₈	#41×C₂₃	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
228	C₁₆	—	2	4	no	—
229	C₁₆	—	4	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	2	no	—
234	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	D₄	—	2	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
250	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#131	1	2	yes	—
252	C₈	—	2	2	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	C₂₃×#121	1	2	yes	—
256	C₈	C₂₃×#124	1	1	yes	characteristic
257	C₈	C₂₃×#127	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	C₂₃×#407	1	1	yes	characteristic
267	C₂²	C₂₃×#435	1	2	yes	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	C₂₃×#390	1	1	yes	characteristic
272	C₄	C₂₃×#408	1	1	yes	characteristic
273	C₄	C₂₃×#435	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	2	no	—
276	C₂	—	4	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₂₃×#1955	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#127
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1955
prime radical	C₂₃×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2153	C₂	semidirect
3. C₂₃×#2117	C₂	semidirect
4. C₂₃×#2112	C₂	semidirect
5. C₂₃×#2110	C₂	semidirect
46. C₂×C₁₆×C₂₃	D₄	semidirect
90. #13200	C₂₃	direct (N × H)
120. #2153	C₂×C₂₃	semidirect
121. #2117	C₂×C₂₃	semidirect
122. #2112	C₂×C₂₃	semidirect
123. #2110	C₂×C₂₃	semidirect
185. C₂×C₁₆	D₄×C₂₃	semidirect
191. C₂₃	#13200	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2929	2
C₂₃×#2153	2
C₂₃×#2117	2
C₂₃×#2112	2
C₂₃×#2110	2
#13200	23

Subgroups — C23×#13200 #19160826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13200 #19160826