Subgroups — C₂₃×#9604 #19157230

Order 5888.

388 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	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#335	C₂	1	2	yes	maximal
5	C₂₃×#1277	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#97	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
14	C₄³×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#336	—	2	1	no	—
16	C₂₃×#336	C₄	1	1	yes	characteristic
17	C₂₃×#336	C₄	1	1	yes	characteristic
18	C₂₃×#335	—	2	4	no	—
19	C₂²×C₈×C₂₃	—	2	1	no	—
20	C₂²×C₈×C₂₃	—	2	1	no	—
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	—	2	4	no	—
32	C₂₃×C₄⋊C₈	—	2	2	no	—
33	C₂₃×C₄⋊C₈	—	2	1	no	—
34	C₂₃×C₄⋊C₈	—	2	1	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	4	2	no	—
37	C₂₃×C₄⋊C₈	—	4	2	no	—
38	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
39	C₂₃×C₄⋊C₈	Q₈	1	2	yes	—
40	C₂₃×#97	—	4	2	no	—
41	C₄×C₈×C₂₃	—	2	2	no	—
42	C₄×C₈×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	4	1	no	—
58	C₂×C₈×C₂₃	—	4	1	no	—
59	C₂×C₈×C₂₃	—	4	1	no	—
60	C₂×C₈×C₂₃	—	4	1	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₄²×C₂₃	—	2	1	no	—
69	C₄²×C₂₃	—	2	1	no	—
70	C₄²×C₂₃	—	4	1	no	—
71	C₄²×C₂₃	D₈	1	2	yes	—
72	C₄²×C₂₃	Q₁₆	1	2	yes	—
73	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₂₃	#41	1	1	yes	characteristic
76	C₄²×C₂₃	#41	1	1	yes	characteristic
77	#9604	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	#107	1	2	yes	—
91	C₂×C₄×C₂₃	#107	1	2	yes	—
92	C₂×C₄×C₂₃	#108	1	2	yes	—
93	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₂₃	#131	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
105	C₈×C₂₃	—	2	4	no	—
106	C₈×C₂₃	—	4	2	no	—
107	C₈×C₂₃	—	4	2	no	—
108	C₈×C₂₃	—	4	2	no	—
109	C₈×C₂₃	—	8	2	no	—
110	#1842	C₂×C₂₃	1	1	yes	characteristic
111	#1767	C₂×C₂₃	1	1	yes	characteristic
112	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
113	C₂×#335	C₂×C₂₃	1	2	yes	—
114	#1277	C₂×C₂₃	1	2	yes	—
115	C₂²×C₂₃	#389	1	1	yes	characteristic
116	C₂²×C₂₃	#423	1	1	yes	characteristic
117	C₂²×C₂₃	#424	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
119	C₂²×C₂₃	#428	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×#108	1	1	yes	characteristic
121	C₂²×C₂₃	#430	1	1	yes	characteristic
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	#435	1	2	yes	—
129	C₄×C₂₃	#446	1	2	yes	—
130	C₄×C₂₃	#459	1	1	yes	characteristic
131	C₄×C₂₃	#462	1	1	yes	characteristic
132	C₄×C₂₃	#474	1	1	yes	characteristic
133	C₄×C₂₃	#477	1	1	yes	characteristic
134	C₂×C₄⋊C₈	—	2	1	no	—
135	C₂×C₄⋊C₈	—	2	1	no	—
136	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
139	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×#97	—	2	1	no	—
141	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
142	C₄³	C₂²×C₂₃	1	1	yes	characteristic
143	#336	—	2	1	no	—
144	#336	C₄×C₂₃	1	1	yes	characteristic
145	#336	C₄×C₂₃	1	1	yes	characteristic
146	#335	—	2	4	no	—
147	C₂×C₂₃	#1567	1	1	yes	characteristic
148	C₂×C₂₃	#1569	1	1	yes	characteristic
149	C₂×C₂₃	#1571	1	1	yes	characteristic
150	C₂×C₂₃	#1573	1	1	yes	characteristic
151	C₂×C₂₃	#1940	1	1	yes	characteristic
152	C₂×C₂₃	#1947	1	1	yes	characteristic
153	C₂×C₂₃	#1950	1	1	yes	characteristic
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
162	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
164	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	4	2	no	—
172	C₄⋊C₈	—	4	2	no	—
173	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
174	C₄⋊C₈	Q₈×C₂₃	1	2	yes	—
175	#97	—	4	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₂₃	#9604	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	4	1	no	—
207	C₄²	D₈×C₂₃	1	2	yes	—
208	C₄²	Q₁₆×C₂₃	1	2	yes	—
209	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
211	C₄²	#41×C₂₃	1	1	yes	characteristic
212	C₄²	#41×C₂₃	1	1	yes	characteristic
213	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	C₂₃×#107	1	2	yes	—
226	C₂×C₄	C₂₃×#107	1	2	yes	—
227	C₂×C₄	C₂₃×#108	1	2	yes	—
228	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
229	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
230	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
231	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
233	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
234	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
235	C₂×C₄	C₂₃×#131	1	2	yes	—
236	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	8	2	no	—
245	C₂²	C₂₃×#389	1	1	yes	characteristic
246	C₂²	C₂₃×#423	1	1	yes	characteristic
247	C₂²	C₂₃×#424	1	1	yes	characteristic
248	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
249	C₂²	C₂₃×#428	1	1	yes	characteristic
250	C₂²	C₂×C₂₃×#108	1	1	yes	characteristic
251	C₂²	C₂₃×#430	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	C₂₃×#435	1	2	yes	—
259	C₄	C₂₃×#446	1	2	yes	—
260	C₄	C₂₃×#459	1	1	yes	characteristic
261	C₄	C₂₃×#462	1	1	yes	characteristic
262	C₄	C₂₃×#474	1	1	yes	characteristic
263	C₄	C₂₃×#477	1	1	yes	characteristic
264	C₂	C₂₃×#1567	1	1	yes	characteristic
265	C₂	C₂₃×#1569	1	1	yes	characteristic
266	C₂	C₂₃×#1571	1	1	yes	characteristic
267	C₂	C₂₃×#1573	1	1	yes	characteristic
268	C₂	C₂₃×#1940	1	1	yes	characteristic
269	C₂	C₂₃×#1947	1	1	yes	characteristic
270	C₂	C₂₃×#1950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
77. #9604	C₂₃	direct (N × H)
178. C₂₃	#9604	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×#1842	2
C₂₃×#1767	2
C₄×C₂₃×C₄⋊C₈	2
C₂×C₂₃×#335	2
C₂₃×#1277	2
#9604	23

Subgroups — C23×#9604 #19157230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9604 #19157230