Subgroups — C₂₃×#8958 #19156584

Order 5888.

534 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2982	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1479	C₂	1	2	yes	maximal
4	C₂₃×#1461	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1276	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₂₃	—	2	4	no	—
10	C₂₃×#431	—	2	2	no	—
11	C₂₃×#431	C₂²	1	2	yes	—
12	C₂₃×#431	C₂²	1	2	yes	—
13	C₂₃×#391	C₄	1	1	yes	characteristic
14	C₂₃×#391	C₄	1	1	yes	characteristic
15	C₂₃×#389	—	2	2	no	—
16	C₂₃×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₄³×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×Q₈⋊C₈	—	2	4	no	—
20	C₂₃×#324	—	2	2	no	—
21	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×C₄⋊Q₈	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	C₈	1	4	yes	complemented
25	C₄×Q₈×C₂₃	—	2	2	no	—
26	C₄×Q₈×C₂₃	—	2	2	no	—
27	C₄×Q₈×C₂₃	—	2	2	no	—
28	C₄×Q₈×C₂₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	—	2	1	no	—
36	C₂×C₄²×C₂₃	—	2	1	no	—
37	C₂×C₄²×C₂₃	—	2	1	no	—
38	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	D₄	1	2	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	—	2	4	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
45	C₂₃×C₂²⋊C₈	—	2	2	no	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₈×C₂₃	—	2	4	no	—
48	C₄×C₈×C₂₃	—	2	2	no	—
49	C₄×C₈×C₂₃	—	2	2	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂×Q₈×C₂₃	—	2	2	no	—
53	C₂×Q₈×C₂₃	—	2	1	no	—
54	C₂×Q₈×C₂₃	—	2	2	no	—
55	C₂×Q₈×C₂₃	—	2	2	no	—
56	C₂×Q₈×C₂₃	—	2	2	no	—
57	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
58	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
59	C₂×Q₈×C₂₃	M₁₆	1	2	yes	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	2	8	no	—
72	C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	4	4	no	—
76	C₄⋊C₄×C₂₃	—	4	2	no	—
77	C₄²×C₂₃	—	2	4	no	—
78	C₄²×C₂₃	—	2	2	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	1	no	—
81	C₄²×C₂₃	—	2	2	no	—
82	C₄²×C₂₃	—	4	2	no	—
83	C₄²×C₂₃	—	4	1	no	—
84	C₄²×C₂₃	—	4	1	no	—
85	C₄²×C₂₃	C₂×C₈	1	2	yes	—
86	C₄²×C₂₃	C₂×D₄	1	2	yes	—
87	C₄²×C₂₃	#41	1	2	yes	—
88	#8958	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₂₃	—	2	4	no	—
91	Q₈×C₂₃	—	2	4	no	—
92	Q₈×C₂₃	—	2	4	no	—
93	Q₈×C₂₃	—	4	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	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	4	1	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	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
112	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#121	1	2	yes	—
115	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#131	1	2	yes	—
119	C₈×C₂₃	—	4	4	no	—
120	C₈×C₂₃	—	8	2	no	—
121	#2982	C₂×C₂₃	1	1	yes	characteristic
122	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
123	#1479	C₂×C₂₃	1	2	yes	—
124	#1461	C₂×C₂₃	1	1	yes	characteristic
125	#1276	C₂×C₂₃	1	2	yes	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	2	1	no	—
128	C₂²×C₂₃	#386	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₂₃	#408	1	1	yes	characteristic
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	4	2	no	—
136	C₄×C₂₃	—	4	2	no	—
137	C₄×C₂₃	—	4	2	no	—
138	C₄×C₂₃	—	8	1	no	—
139	C₄×C₂₃	C₈×D₄	1	2	yes	—
140	C₄×C₂₃	#435	1	2	yes	—
141	C₄×C₂₃	#456	1	2	yes	—
142	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₈×Q₈	—	2	4	no	—
146	#431	—	2	2	no	—
147	#431	C₂²×C₂₃	1	2	yes	—
148	#431	C₂²×C₂₃	1	2	yes	—
149	#391	C₄×C₂₃	1	1	yes	characteristic
150	#391	C₄×C₂₃	1	1	yes	characteristic
151	#389	—	2	2	no	—
152	#384	C₄×C₂₃	1	1	yes	characteristic
153	C₄×C₄⋊C₄	—	2	2	no	—
154	C₄³	C₂²×C₂₃	1	1	yes	characteristic
155	Q₈⋊C₈	—	2	4	no	—
156	#324	—	2	2	no	—
157	C₂×C₂₃	—	2	1	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	#1869	1	1	yes	characteristic
160	C₂×C₂₃	#1884	1	1	yes	characteristic
161	C₂×C₂₃	#1886	1	1	yes	characteristic
162	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	C₈×C₂₃	1	4	yes	complemented
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₂₃	1	2	yes	—
182	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₂₃	#8958	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
200	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
201	C₂×Q₈	M₁₆×C₂₃	1	2	yes	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₄²	C₂×C₈×C₂₃	1	2	yes	—
228	C₄²	C₂×D₄×C₂₃	1	2	yes	—
229	C₄²	#41×C₂₃	1	2	yes	—
230	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	2	4	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
253	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
254	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#121	1	2	yes	—
256	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂₃×#131	1	2	yes	—
260	C₈	—	4	4	no	—
261	C₈	—	8	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	C₂₃×#386	1	1	yes	characteristic
265	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
266	C₂²	C₂₃×#408	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	4	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	1	no	—
275	C₄	C₈×D₄×C₂₃	1	2	yes	—
276	C₄	C₂₃×#435	1	2	yes	—
277	C₄	C₂₃×#456	1	2	yes	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	C₂₃×#1869	1	1	yes	characteristic
281	C₂	C₂₃×#1884	1	1	yes	characteristic
282	C₂	C₂₃×#1886	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#386
prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₂₃×C₄⋊Q₈	C₈	semidirect
88. #8958	C₂₃	direct (N × H)
165. C₄⋊Q₈	C₈×C₂₃	semidirect
193. C₂₃	#8958	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×#2982	2
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#1479	2
C₂₃×#1461	2
C₂₃×#1276	2
#8958	23

Subgroups — C23×#8958 #19156584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8958 #19156584