Subgroups — C₂₃×#12982 #19160608

Order 5888.

474 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2117	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1376	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
8	C₂₃×#532	C₄	1	1	yes	characteristic
9	C₂₃×#504	—	2	4	no	—
10	C₂×C₂₃×M₃₂	—	2	2	no	—
11	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
13	C₂₃×#436	—	2	2	no	—
14	C₈×D₄×C₂₃	—	2	2	no	—
15	C₂₃×#423	C₄	1	1	yes	characteristic
16	C₂₃×#408	C₄	1	1	yes	characteristic
17	C₂₃×#408	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#350	—	2	4	no	—
20	C₂₃×#349	—	2	2	no	—
21	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×#131	—	2	2	no	—
24	C₂₃×#131	—	2	2	no	—
25	C₂₃×#131	—	2	2	no	—
26	C₂₃×#131	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	2	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
36	C₂₃×#127	C₈	1	1	yes	characteristic
37	C₂₃×#124	C₈	1	1	yes	characteristic
38	C₂₃×#124	C₈	1	1	yes	characteristic
39	C₂₃×#121	—	2	2	no	—
40	C₄×D₄×C₂₃	—	2	2	no	—
41	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂₃×M₃₂	—	2	4	no	—
43	C₂₃×M₃₂	—	2	2	no	—
44	C₂₃×M₃₂	—	2	2	no	—
45	C₂₃×M₃₂	—	2	2	no	—
46	C₂₃×M₃₂	D₄	1	4	yes	complemented
47	C₂×C₁₆×C₂₃	—	2	4	no	—
48	C₂×C₁₆×C₂₃	—	2	2	no	—
49	C₂×C₁₆×C₂₃	—	2	2	no	—
50	C₂₃×C₄⋊C₈	—	2	2	no	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×C₂²⋊C₈	—	2	2	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₄×C₈×C₂₃	—	2	1	no	—
55	#41×C₂₃	—	2	2	no	—
56	#41×C₂₃	—	2	2	no	—
57	#41×C₂₃	—	2	2	no	—
58	#41×C₂₃	—	2	2	no	—
59	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
60	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
64	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₂₃	M₁₆	1	2	yes	—
66	C₂²×C₄×C₂₃	—	2	1	no	—
67	C₂²×C₄×C₂₃	—	2	1	no	—
68	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₂₃	—	2	2	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	2	2	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	1	no	—
77	C₂×C₈×C₂₃	—	2	1	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	—	4	1	no	—
81	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	#41	1	2	yes	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂²⋊C₄×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	—	2	1	no	—
90	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
92	C₁₆×C₂₃	—	2	4	no	—
93	C₁₆×C₂₃	—	4	2	no	—
94	#12982	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	2	1	no	—
96	C₂³×C₂₃	C₂²⋊C₈	1	2	yes	—
97	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
98	Q₈×C₂₃	—	2	2	no	—
99	Q₈×C₂₃	—	2	2	no	—
100	D₄×C₂₃	—	2	4	no	—
101	D₄×C₂₃	—	2	2	no	—
102	D₄×C₂₃	—	2	2	no	—
103	D₄×C₂₃	—	4	4	no	—
104	D₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₂₃	#131	1	2	yes	—
118	C₈×C₂₃	—	2	2	no	—
119	C₈×C₂₃	—	4	1	no	—
120	C₈×C₂₃	—	4	1	no	—
121	C₈×C₂₃	#120	1	1	yes	characteristic
122	C₈×C₂₃	#121	1	2	yes	—
123	C₈×C₂₃	#123	1	1	yes	characteristic
124	#3407	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#2931	C₂×C₂₃	1	1	yes	characteristic
126	#2117	C₂×C₂₃	1	2	yes	complemented
127	#2115	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#1376	C₂×C₂₃	1	2	yes	complemented
129	C₂²×C₂₃	—	2	2	no	—
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	—	4	2	no	—
132	C₂²×C₂₃	—	4	2	no	—
133	C₂²×C₂₃	—	4	1	no	—
134	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₂₃	#435	1	2	yes	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	#386	1	1	yes	characteristic
141	C₄×C₂₃	#408	1	1	yes	characteristic
142	C₄×C₂₃	C₈×D₄	1	2	yes	—
143	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
145	#532	C₄×C₂₃	1	1	yes	characteristic
146	#504	—	2	4	no	—
147	C₂×M₃₂	—	2	2	no	—
148	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
149	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
150	#436	—	2	2	no	—
151	C₈×D₄	—	2	2	no	—
152	#423	C₄×C₂₃	1	1	yes	characteristic
153	#408	C₄×C₂₃	1	1	yes	characteristic
154	#408	C₄×C₂₃	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	#350	—	2	4	no	—
157	#349	—	2	2	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	—	2	2	no	—
160	C₂×C₂₃	—	4	2	no	—
161	C₂×C₂₃	—	8	1	no	—
162	C₂×C₂₃	#1869	1	1	yes	characteristic
163	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
165	#131	—	2	2	no	—
166	#131	—	2	2	no	—
167	#131	—	2	2	no	—
168	#131	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₈	D₄×C₂₃	1	2	yes	—
176	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
178	#127	C₈×C₂₃	1	1	yes	characteristic
179	#124	C₈×C₂₃	1	1	yes	characteristic
180	#124	C₈×C₂₃	1	1	yes	characteristic
181	#121	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
184	M₃₂	—	2	4	no	—
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	M₃₂	—	2	2	no	—
188	M₃₂	D₄×C₂₃	1	4	yes	complemented
189	C₂×C₁₆	—	2	4	no	—
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₂₃	#12982	1	1	yes	characteristic, direct product factor, Sylow
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	#41	—	2	2	no	—
202	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
203	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
207	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
208	C₂×D₄	M₁₆×C₂₃	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	#41×C₂₃	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
234	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
235	C₁₆	—	2	4	no	—
236	C₁₆	—	4	2	no	—
237	C₂³	—	2	1	no	—
238	C₂³	C₂₃×C₂²⋊C₈	1	2	yes	—
239	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
258	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂₃×#131	1	2	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	C₂₃×#120	1	1	yes	characteristic
264	C₈	C₂₃×#121	1	2	yes	—
265	C₈	C₂₃×#123	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
272	C₂²	C₂₃×#435	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	C₂₃×#386	1	1	yes	characteristic
278	C₄	C₂₃×#408	1	1	yes	characteristic
279	C₄	C₈×D₄×C₂₃	1	2	yes	—
280	C₂	—	2	1	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂₃×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#120
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1869
prime radical	C₂₃×#532	C₄
LCS	G ▷ C₂×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
1. C₂₃×#3407	C₂	semidirect
3. C₂₃×#2117	C₂	semidirect
4. C₂₃×#2115	C₂	semidirect
5. C₂₃×#1376	C₂	semidirect
46. C₂₃×M₃₂	D₄	semidirect
94. #12982	C₂₃	direct (N × H)
124. #3407	C₂×C₂₃	semidirect
126. #2117	C₂×C₂₃	semidirect
127. #2115	C₂×C₂₃	semidirect
128. #1376	C₂×C₂₃	semidirect
188. M₃₂	D₄×C₂₃	semidirect
197. C₂₃	#12982	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×#3407	2
C₂₃×#2931	2
C₂₃×#2117	2
C₂₃×#2115	2
C₂₃×#1376	2
#12982	23

Subgroups — C23×#12982 #19160608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12982 #19160608