Subgroups — C₂₃×#9611 #19157237

Order 5888.

382 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1947	C₂	1	2	yes	maximal
2	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1296	C₂	1	2	yes	maximal
6	C₂×C₂₃×#107	—	2	2	no	—
7	C₂×C₂₃×#106	—	2	1	no	—
8	C₂×C₂₃×#106	—	2	1	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×#334	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	2	yes	—
26	C₂×C₄²×C₂₃	—	2	1	no	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×#107	—	4	4	no	—
29	C₂₃×#106	—	4	2	no	—
30	C₂₃×#106	—	4	2	no	—
31	C₂₃×C₄⋊C₈	—	2	2	no	—
32	C₂₃×C₄⋊C₈	—	2	1	no	—
33	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
34	C₂₃×C₄⋊C₈	Q₈	1	2	yes	—
35	C₄×C₈×C₂₃	—	4	2	no	—
36	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₂₃	Q₈	1	2	yes	—
38	C₂₃×#95	—	2	1	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₂₃	—	2	1	no	—
45	C₂×C₈×C₂₃	—	2	1	no	—
46	C₂×C₈×C₂₃	—	2	4	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	4	1	no	—
49	C₂×C₈×C₂₃	—	4	2	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	—	4	1	no	—
52	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₂₃	D₈	1	2	yes	—
55	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	4	2	no	—
58	C₄⋊C₄×C₂₃	—	4	2	no	—
59	C₄⋊C₄×C₂₃	—	4	2	no	—
60	C₄⋊C₄×C₂₃	—	4	2	no	—
61	C₄⋊C₄×C₂₃	—	8	2	no	—
62	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	—
63	C₄²×C₂₃	—	4	1	no	—
64	C₄²×C₂₃	—	4	1	no	—
65	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₂₃	#41	1	1	yes	characteristic
69	#9611	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	—	8	2	no	—
76	C₂×C₄×C₂₃	—	8	2	no	—
77	C₂×C₄×C₂₃	—	8	2	no	—
78	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#135	1	2	yes	—
88	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
90	C₈×C₂₃	—	2	2	no	—
91	C₈×C₂₃	—	4	4	no	—
92	C₈×C₂₃	—	4	2	no	—
93	C₈×C₂₃	#107	1	4	yes	complemented
94	#1947	C₂×C₂₃	1	2	yes	—
95	#1844	C₂×C₂₃	1	1	yes	characteristic
96	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
97	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
98	#1296	C₂×C₂₃	1	2	yes	—
99	C₂²×C₂₃	#389	1	1	yes	characteristic
100	C₂²×C₂₃	#459	1	1	yes	characteristic
101	C₂²×C₂₃	#462	1	1	yes	characteristic
102	C₂²×C₂₃	#465	1	2	yes	—
103	C₂²×C₂₃	#471	1	2	yes	—
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	4	4	no	—
107	C₄×C₂₃	—	16	2	no	—
108	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
109	C₄×C₂₃	#428	1	1	yes	characteristic
110	C₄×C₂₃	C₄×SD₁₆	1	2	yes	—
111	C₄×C₂₃	#474	1	1	yes	characteristic
112	C₄×C₂₃	#477	1	1	yes	characteristic
113	C₄×C₂₃	#498	1	1	yes	characteristic
114	C₄×C₂₃	#499	1	1	yes	characteristic
115	C₂×#107	—	2	2	no	—
116	C₂×#106	—	2	1	no	—
117	C₂×#106	—	2	1	no	—
118	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×C₄×C₈	—	2	1	no	—
121	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
122	#384	C₂²×C₂₃	1	2	yes	—
123	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
124	#340	—	2	2	no	—
125	#336	—	2	1	no	—
126	#334	C₄×C₂₃	1	4	yes	complemented
127	C₂×C₂₃	#1665	1	2	yes	—
128	C₂×C₂₃	#1672	1	2	yes	—
129	C₂×C₂₃	#1941	1	1	yes	characteristic
130	C₂×C₂₃	#1947	1	1	yes	characteristic
131	C₂×C₂₃	#1951	1	1	yes	characteristic
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
135	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
142	#107	—	4	4	no	—
143	#106	—	4	2	no	—
144	#106	—	4	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
148	C₄⋊C₈	Q₈×C₂₃	1	2	yes	—
149	C₄×C₈	—	4	2	no	—
150	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
151	C₄×C₈	Q₈×C₂₃	1	2	yes	—
152	#95	—	2	1	no	—
153	C₂₃	#9611	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
168	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
169	C₂×C₈	D₈×C₂₃	1	2	yes	—
170	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	8	2	no	—
177	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
181	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
182	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
183	C₄²	#41×C₂₃	1	1	yes	characteristic
184	C₂³	C₂₃×#121	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
196	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
197	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
198	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
199	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
200	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂₃×#135	1	2	yes	—
202	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	4	2	no	—
207	C₈	C₂₃×#107	1	4	yes	complemented
208	C₂²	C₂₃×#389	1	1	yes	characteristic
209	C₂²	C₂₃×#459	1	1	yes	characteristic
210	C₂²	C₂₃×#462	1	1	yes	characteristic
211	C₂²	C₂₃×#465	1	2	yes	—
212	C₂²	C₂₃×#471	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	16	2	no	—
217	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
218	C₄	C₂₃×#428	1	1	yes	characteristic
219	C₄	C₄×SD₁₆×C₂₃	1	2	yes	—
220	C₄	C₂₃×#474	1	1	yes	characteristic
221	C₄	C₂₃×#477	1	1	yes	characteristic
222	C₄	C₂₃×#498	1	1	yes	characteristic
223	C₄	C₂₃×#499	1	1	yes	characteristic
224	C₂	C₂₃×#1665	1	2	yes	—
225	C₂	C₂₃×#1672	1	2	yes	—
226	C₂	C₂₃×#1941	1	1	yes	characteristic
227	C₂	C₂₃×#1947	1	1	yes	characteristic
228	C₂	C₂₃×#1951	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×#334	C₄	semidirect
69. #9611	C₂₃	direct (N × H)
93. C₈×C₂₃	#107	semidirect
126. #334	C₄×C₂₃	semidirect
153. C₂₃	#9611	direct (N × H)
207. C₈	C₂₃×#107	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1947	2
C₂₃×#1844	2
C₈×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#334	2
C₂₃×#1296	2
#9611	23

Subgroups — C23×#9611 #19157237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9611 #19157237