Subgroups — C₂₃×#11485 #19159111

Order 5888.

372 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2115	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1327	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1321	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂₃×#532	C₄	1	1	yes	characteristic
8	C₂×C₂₃×M₃₂	—	2	2	no	—
9	C₂₃×#436	—	2	2	no	—
10	C₈×D₄×C₂₃	—	2	2	no	—
11	C₂₃×#423	C₂²	1	1	yes	characteristic
12	C₂₃×#408	C₄	1	1	yes	characteristic
13	C₂₃×#408	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
15	C₂₃×#349	—	2	2	no	—
16	C₂₃×#349	C₂²	1	4	yes	—
17	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂₃×#131	—	4	1	no	—
20	C₂₃×#131	—	4	1	no	—
21	C₂₃×#131	—	4	1	no	—
22	C₂₃×#131	—	4	1	no	—
23	C₂×M₁₆×C₂₃	—	2	1	no	—
24	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
26	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	—	2	1	no	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	D₄	1	2	yes	—
31	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
32	C₂₃×#127	C₈	1	1	yes	characteristic
33	C₂₃×#124	C₈	1	1	yes	characteristic
34	C₂₃×#124	C₈	1	1	yes	characteristic
35	C₂₃×#121	—	2	2	no	—
36	C₄×D₄×C₂₃	—	2	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂₃×M₃₂	—	2	4	no	—
39	C₂₃×M₃₂	—	4	2	no	—
40	C₂×C₁₆×C₂₃	—	4	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₂²⋊C₈	—	2	2	no	—
43	C₂₃×C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈×C₂₃	—	2	1	no	—
45	C₄×C₈×C₂₃	—	2	1	no	—
46	#41×C₂₃	—	4	1	no	—
47	#41×C₂₃	—	4	1	no	—
48	#41×C₂₃	—	4	1	no	—
49	#41×C₂₃	—	4	1	no	—
50	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
51	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
52	C₂×D₄×C₂₃	—	2	2	no	—
53	C₂×D₄×C₂₃	—	2	1	no	—
54	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
55	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
56	C₂×D₄×C₂₃	M₁₆	1	2	yes	—
57	C₂²×C₄×C₂₃	—	2	1	no	—
58	C₂²×C₄×C₂₃	—	2	1	no	—
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₂₃	—	4	2	no	—
62	M₁₆×C₂₃	—	4	1	no	—
63	M₁₆×C₂₃	—	4	2	no	—
64	M₁₆×C₂₃	—	4	1	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	1	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	1	no	—
69	C₂×C₈×C₂₃	—	2	1	no	—
70	C₂×C₈×C₂₃	—	2	1	no	—
71	C₂×C₈×C₂₃	—	4	1	no	—
72	C₂×C₈×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₂²⋊C₄×C₂₃	—	2	2	no	—
79	C₂²⋊C₄×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	1	no	—
81	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
83	C₁₆×C₂₃	—	4	4	no	—
84	#11485	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₂₃	—	2	1	no	—
86	C₂³×C₂₃	C₂²⋊C₈	1	2	yes	—
87	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
88	Q₈×C₂₃	—	4	1	no	—
89	Q₈×C₂₃	—	4	1	no	—
90	D₄×C₂₃	—	4	1	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	1	no	—
93	D₄×C₂₃	—	4	4	no	—
94	D₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
108	C₈×C₂₃	—	2	2	no	—
109	C₈×C₂₃	—	2	1	no	—
110	C₈×C₂₃	—	4	1	no	—
111	C₈×C₂₃	—	4	1	no	—
112	C₈×C₂₃	#99	1	1	yes	characteristic
113	C₈×C₂₃	#99	1	1	yes	characteristic
114	#2931	C₂×C₂₃	1	1	yes	characteristic
115	#2115	C₂×C₂₃	1	2	yes	complemented
116	#1327	C₂×C₂₃	1	2	yes	complemented
117	#1321	C₂×C₂₃	1	2	yes	complemented
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	4	1	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	—	4	1	no	—
130	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
131	C₄×C₂₃	#413	1	1	yes	characteristic
132	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
133	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
134	#532	C₄×C₂₃	1	1	yes	characteristic
135	C₂×M₃₂	—	2	2	no	—
136	#436	—	2	2	no	—
137	C₈×D₄	—	2	2	no	—
138	#423	C₂²×C₂₃	1	1	yes	characteristic
139	#408	C₄×C₂₃	1	1	yes	characteristic
140	#408	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
142	#349	—	2	2	no	—
143	#349	C₂²×C₂₃	1	4	yes	—
144	C₂×C₂₃	—	2	1	no	—
145	C₂×C₂₃	—	4	1	no	—
146	C₂×C₂₃	—	4	2	no	—
147	C₂×C₂₃	—	8	1	no	—
148	C₂×C₂₃	#1459	1	1	yes	characteristic
149	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
150	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
151	#131	—	4	1	no	—
152	#131	—	4	1	no	—
153	#131	—	4	1	no	—
154	#131	—	4	1	no	—
155	C₂×M₁₆	—	2	1	no	—
156	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
157	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
158	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	D₄×C₂₃	1	2	yes	—
163	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
164	#127	C₈×C₂₃	1	1	yes	characteristic
165	#124	C₈×C₂₃	1	1	yes	characteristic
166	#124	C₈×C₂₃	1	1	yes	characteristic
167	#121	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
170	M₃₂	—	2	4	no	—
171	M₃₂	—	4	2	no	—
172	C₂×C₁₆	—	4	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	2	1	no	—
178	C₂₃	#11485	1	1	yes	characteristic, direct product factor, Sylow
179	#41	—	4	1	no	—
180	#41	—	4	1	no	—
181	#41	—	4	1	no	—
182	#41	—	4	1	no	—
183	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
184	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
188	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
189	C₂×D₄	M₁₆×C₂₃	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	1	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
215	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
216	C₁₆	—	4	4	no	—
217	C₂³	—	2	1	no	—
218	C₂³	C₂₃×C₂²⋊C₈	1	2	yes	—
219	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
220	Q₈	—	4	1	no	—
221	Q₈	—	4	1	no	—
222	D₄	—	4	1	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	1	no	—
243	C₈	—	4	1	no	—
244	C₈	C₂₃×#99	1	1	yes	characteristic
245	C₈	C₂₃×#99	1	1	yes	characteristic
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
259	C₄	C₂₃×#413	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₂₃×#1459	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#99
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×M₁₆	C₂³×C₂₃
socle	C₂×C₂₃	#1459
prime radical	C₂₃×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2115	C₂	semidirect
3. C₂₃×#1327	C₂	semidirect
4. C₂₃×#1321	C₂	semidirect
84. #11485	C₂₃	direct (N × H)
115. #2115	C₂×C₂₃	semidirect
116. #1327	C₂×C₂₃	semidirect
117. #1321	C₂×C₂₃	semidirect
178. C₂₃	#11485	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×#2931	2
C₂₃×#2115	2
C₂₃×#1327	2
C₂₃×#1321	2
#11485	23

Subgroups — C23×#11485 #19159111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11485 #19159111