Subgroups — C₂₃×#9615 #19157241

Order 5888.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1948	C₂	1	2	yes	maximal
2	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1296	C₂	1	2	yes	maximal
6	C₂×C₂₃×#107	—	2	1	no	—
7	C₂×C₂₃×#107	—	2	1	no	—
8	C₂×C₂₃×#106	—	2	2	no	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#384	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#340	—	2	2	no	—
16	C₂₃×#336	—	2	1	no	—
17	C₂₃×#335	C₄	1	4	yes	complemented
18	C₂²×C₈×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	—	2	1	no	—
28	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×#107	—	4	2	no	—
30	C₂₃×#107	—	4	2	no	—
31	C₂₃×#106	—	4	4	no	—
32	C₂₃×C₄⋊C₈	—	2	2	no	—
33	C₂₃×C₄⋊C₈	—	2	1	no	—
34	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
35	C₂₃×C₄⋊C₈	Q₈	1	2	yes	—
36	C₄×C₈×C₂₃	—	4	2	no	—
37	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₂₃	Q₈	1	2	yes	—
39	C₂₃×#95	—	2	1	no	—
40	C₂²×C₄×C₂₃	—	2	1	no	—
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	—	4	2	no	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₂₃	—	2	1	no	—
47	C₂×C₈×C₂₃	—	2	1	no	—
48	C₂×C₈×C₂₃	—	2	4	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	—	4	1	no	—
52	C₂×C₈×C₂₃	—	4	2	no	—
53	C₂×C₈×C₂₃	—	4	1	no	—
54	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	4	2	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	—	4	2	no	—
63	C₄⋊C₄×C₂₃	—	4	2	no	—
64	C₄⋊C₄×C₂₃	—	8	2	no	—
65	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
66	C₄⋊C₄×C₂₃	Q₁₆	1	2	yes	—
67	C₄²×C₂₃	—	4	1	no	—
68	C₄²×C₂₃	—	4	1	no	—
69	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₂₃	#41	1	1	yes	characteristic
73	#9615	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	1	no	—
80	C₂×C₄×C₂₃	—	4	1	no	—
81	C₂×C₄×C₂₃	—	4	1	no	—
82	C₂×C₄×C₂₃	—	4	1	no	—
83	C₂×C₄×C₂₃	—	8	2	no	—
84	C₂×C₄×C₂₃	—	8	2	no	—
85	C₂×C₄×C₂₃	—	8	2	no	—
86	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
98	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
100	C₈×C₂₃	—	2	2	no	—
101	C₈×C₂₃	—	4	4	no	—
102	C₈×C₂₃	—	4	2	no	—
103	C₈×C₂₃	#106	1	4	yes	complemented
104	#1948	C₂×C₂₃	1	2	yes	—
105	#1845	C₂×C₂₃	1	1	yes	characteristic
106	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
108	#1296	C₂×C₂₃	1	2	yes	—
109	C₂²×C₂₃	#389	1	1	yes	characteristic
110	C₂²×C₂₃	#460	1	1	yes	characteristic
111	C₂²×C₂₃	#461	1	1	yes	characteristic
112	C₂²×C₂₃	#466	1	1	yes	characteristic
113	C₂²×C₂₃	#467	1	1	yes	characteristic
114	C₂²×C₂₃	#472	1	1	yes	characteristic
115	C₂²×C₂₃	#473	1	1	yes	characteristic
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	16	2	no	—
121	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
122	C₄×C₂₃	#428	1	1	yes	characteristic
123	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
124	C₄×C₂₃	C₄×Q₁₆	1	1	yes	characteristic
125	C₄×C₂₃	#475	1	1	yes	characteristic
126	C₄×C₂₃	#476	1	1	yes	characteristic
127	C₄×C₂₃	#499	1	1	yes	characteristic
128	C₄×C₂₃	#500	1	1	yes	characteristic
129	C₂×#107	—	2	1	no	—
130	C₂×#107	—	2	1	no	—
131	C₂×#106	—	2	2	no	—
132	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
133	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×C₄×C₈	—	2	1	no	—
135	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
136	#384	C₂²×C₂₃	1	2	yes	—
137	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
138	#340	—	2	2	no	—
139	#336	—	2	1	no	—
140	#335	C₄×C₂₃	1	4	yes	complemented
141	C₂×C₂₃	#1668	1	1	yes	characteristic
142	C₂×C₂₃	#1670	1	1	yes	characteristic
143	C₂×C₂₃	#1673	1	1	yes	characteristic
144	C₂×C₂₃	#1675	1	1	yes	characteristic
145	C₂×C₂₃	#1942	1	1	yes	characteristic
146	C₂×C₂₃	#1948	1	1	yes	characteristic
147	C₂×C₂₃	#1952	1	1	yes	characteristic
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
159	#107	—	4	2	no	—
160	#107	—	4	2	no	—
161	#106	—	4	4	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
165	C₄⋊C₈	Q₈×C₂₃	1	2	yes	—
166	C₄×C₈	—	4	2	no	—
167	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
168	C₄×C₈	Q₈×C₂₃	1	2	yes	—
169	#95	—	2	1	no	—
170	C₂₃	#9615	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
186	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
187	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
188	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	8	2	no	—
196	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
197	C₄⋊C₄	Q₁₆×C₂₃	1	2	yes	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
201	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
202	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
203	C₄²	#41×C₂₃	1	1	yes	characteristic
204	C₂³	C₂₃×#121	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
219	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
221	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
222	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
223	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
226	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
227	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
228	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	4	2	no	—
233	C₈	C₂₃×#106	1	4	yes	complemented
234	C₂²	C₂₃×#389	1	1	yes	characteristic
235	C₂²	C₂₃×#460	1	1	yes	characteristic
236	C₂²	C₂₃×#461	1	1	yes	characteristic
237	C₂²	C₂₃×#466	1	1	yes	characteristic
238	C₂²	C₂₃×#467	1	1	yes	characteristic
239	C₂²	C₂₃×#472	1	1	yes	characteristic
240	C₂²	C₂₃×#473	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	16	2	no	—
246	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
247	C₄	C₂₃×#428	1	1	yes	characteristic
248	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
249	C₄	C₄×Q₁₆×C₂₃	1	1	yes	characteristic
250	C₄	C₂₃×#475	1	1	yes	characteristic
251	C₄	C₂₃×#476	1	1	yes	characteristic
252	C₄	C₂₃×#499	1	1	yes	characteristic
253	C₄	C₂₃×#500	1	1	yes	characteristic
254	C₂	C₂₃×#1668	1	1	yes	characteristic
255	C₂	C₂₃×#1670	1	1	yes	characteristic
256	C₂	C₂₃×#1673	1	1	yes	characteristic
257	C₂	C₂₃×#1675	1	1	yes	characteristic
258	C₂	C₂₃×#1942	1	1	yes	characteristic
259	C₂	C₂₃×#1948	1	1	yes	characteristic
260	C₂	C₂₃×#1952	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#121
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₂₃×#335	C₄	semidirect
73. #9615	C₂₃	direct (N × H)
103. C₈×C₂₃	#106	semidirect
140. #335	C₄×C₂₃	semidirect
170. C₂₃	#9615	direct (N × H)
233. C₈	C₂₃×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1948	2
C₂₃×#1845	2
C₈×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#335	2
C₂₃×#1296	2
#9615	23

Subgroups — C23×#9615 #19157241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9615 #19157241