Subgroups — C₂₃×#9600 #19157226

Order 5888.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1939	C₂	1	2	yes	maximal
2	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₂₃×#106	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#340	C₄	1	1	yes	characteristic
14	C₂₃×#340	C₄	1	1	yes	characteristic
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×#335	—	2	2	no	—
17	C₂₃×#322	—	2	2	no	—
18	C₂²×C₈×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	—	2	2	no	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×#106	—	2	4	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	2	2	no	—
31	C₂₃×C₄⋊C₈	—	4	4	no	—
32	C₄×C₈×C₂₃	—	2	1	no	—
33	C₄×C₈×C₂₃	—	2	1	no	—
34	C₄×C₈×C₂₃	—	2	1	no	—
35	C₄×C₈×C₂₃	D₄	1	2	yes	—
36	C₄×C₈×C₂₃	Q₈	1	2	yes	—
37	C₄×C₈×C₂₃	Q₈	1	2	yes	—
38	C₂₃×#95	—	2	2	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	C₂²×C₄	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	2	no	—
45	C₂×C₈×C₂₃	—	2	2	no	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	—	2	1	no	—
50	C₂×C₈×C₂₃	—	2	1	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	4	2	no	—
54	C₂×C₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	—	4	2	no	—
56	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	M₁₆	1	4	yes	—
62	C₄²×C₂₃	—	4	2	no	—
63	C₄²×C₂₃	—	4	2	no	—
64	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₂₃	#41	1	1	yes	characteristic
68	#9600	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	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	—	4	2	no	—
76	C₂×C₄×C₂₃	—	4	2	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#121	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×M₁₆	1	2	yes	—
85	C₂×C₄×C₂₃	#131	1	2	yes	—
86	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
87	C₂×C₄×C₂₃	#137	1	2	yes	—
88	C₈×C₂₃	—	2	4	no	—
89	C₈×C₂₃	—	2	2	no	—
90	C₈×C₂₃	—	2	2	no	—
91	C₈×C₂₃	—	4	1	no	—
92	C₈×C₂₃	—	4	1	no	—
93	C₈×C₂₃	—	8	2	no	—
94	#1939	C₂×C₂₃	1	2	yes	—
95	C₄×#106	C₂×C₂₃	1	1	yes	characteristic
96	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
97	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
98	#1279	C₂×C₂₃	1	2	yes	—
99	C₂²×C₂₃	#389	1	1	yes	characteristic
100	C₂²×C₂₃	#435	1	2	yes	—
101	C₂²×C₂₃	#440	1	2	yes	—
102	C₂²×C₂₃	#443	1	1	yes	characteristic
103	C₂²×C₂₃	#443	1	1	yes	characteristic
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	4	4	no	—
107	C₄×C₂₃	—	8	2	no	—
108	C₄×C₂₃	#423	1	1	yes	characteristic
109	C₄×C₂₃	#424	1	1	yes	characteristic
110	C₄×C₂₃	#446	1	2	yes	—
111	C₄×C₂₃	#469	1	1	yes	characteristic
112	C₄×C₂₃	#470	1	1	yes	characteristic
113	C₄×C₂₃	#501	1	2	yes	—
114	C₂×#106	C₄×C₂₃	1	2	yes	—
115	C₂×C₄⋊C₈	—	2	2	no	—
116	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
121	#340	C₄×C₂₃	1	1	yes	characteristic
122	#340	C₄×C₂₃	1	1	yes	characteristic
123	#336	—	2	2	no	—
124	#335	—	2	2	no	—
125	#322	—	2	2	no	—
126	C₂×C₂₃	#1585	1	2	yes	—
127	C₂×C₂₃	#1588	1	2	yes	—
128	C₂×C₂₃	#1940	1	1	yes	characteristic
129	C₂×C₂₃	#1944	1	1	yes	characteristic
130	C₂×C₂₃	#1945	1	1	yes	characteristic
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
135	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
138	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
141	#106	—	2	4	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	4	4	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	D₄×C₂₃	1	2	yes	—
149	C₄×C₈	Q₈×C₂₃	1	2	yes	—
150	C₄×C₈	Q₈×C₂₃	1	2	yes	—
151	#95	—	2	2	no	—
152	C₂₃	#9600	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
171	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	M₁₆×C₂₃	1	4	yes	—
176	C₄²	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
180	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
181	C₄²	#41×C₂₃	1	1	yes	characteristic
182	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
193	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
194	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
195	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
196	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
198	C₂×C₄	C₂₃×#131	1	2	yes	—
199	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
200	C₂×C₄	C₂₃×#137	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	—	8	2	no	—
207	C₂²	C₂₃×#389	1	1	yes	characteristic
208	C₂²	C₂₃×#435	1	2	yes	—
209	C₂²	C₂₃×#440	1	2	yes	—
210	C₂²	C₂₃×#443	1	1	yes	characteristic
211	C₂²	C₂₃×#443	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	8	2	no	—
216	C₄	C₂₃×#423	1	1	yes	characteristic
217	C₄	C₂₃×#424	1	1	yes	characteristic
218	C₄	C₂₃×#446	1	2	yes	—
219	C₄	C₂₃×#469	1	1	yes	characteristic
220	C₄	C₂₃×#470	1	1	yes	characteristic
221	C₄	C₂₃×#501	1	2	yes	—
222	C₂	C₂₃×#1585	1	2	yes	—
223	C₂	C₂₃×#1588	1	2	yes	—
224	C₂	C₂₃×#1940	1	1	yes	characteristic
225	C₂	C₂₃×#1944	1	1	yes	characteristic
226	C₂	C₂₃×#1945	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	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
68. #9600	C₂₃	direct (N × H)
152. C₂₃	#9600	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×#1939	2
C₄×C₂₃×#106	2
C₂×C₂₃×#335	2
C₂×C₂₃×#322	2
C₂₃×#1279	2
#9600	23

Subgroups — C23×#9600 #19157226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9600 #19157226