Subgroups — C₂₃×#11322 #19158948

Order 5888.

958 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊C₈	C₂	1	3	yes	maximal
2	C₂×C₂₃×#336	C₂	1	3	yes	maximal
3	C₄×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₂₃	C₂²	1	3	yes	—
5	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
7	C₂×C₂₃×C₄⋊C₈	—	2	12	no	—
8	C₂×C₂₃×C₄⋊C₈	—	2	6	no	—
9	C₂×C₂₃×#95	C₄	1	3	yes	—
10	C₂×C₂₃×#95	C₄	1	1	yes	characteristic
11	C₂₃×#336	—	2	6	no	—
12	C₂³×C₄×C₂₃	C₂×C₄	1	3	yes	—
13	C₂³×C₄×C₂₃	C₂×C₄	1	3	yes	—
14	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	—	2	3	no	—
16	C₂²×C₈×C₂₃	—	2	3	no	—
17	C₂²×C₈×C₂₃	—	2	6	no	—
18	C₂²×C₈×C₂₃	—	2	6	no	—
19	C₂²×C₈×C₂₃	—	2	3	no	—
20	C₂²×C₈×C₂₃	—	2	3	no	—
21	C₂²×C₈×C₂₃	D₄	1	6	yes	—
22	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₂₃	—	2	6	no	—
24	C₂×C₄²×C₂₃	—	2	6	no	—
25	C₂×C₄²×C₂₃	D₄	1	6	yes	—
26	C₂×C₄²×C₂₃	Q₈	1	6	yes	—
27	C₂₃×C₄⋊C₈	—	2	24	no	—
28	C₂₃×#95	—	2	6	no	—
29	C₂₃×#95	C₈	1	4	yes	complemented
30	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	—	2	6	no	—
32	C₂²×C₄×C₂₃	—	2	6	no	—
33	C₂²×C₄×C₂₃	—	2	6	no	—
34	C₂²×C₄×C₂₃	—	2	6	no	—
35	C₂²×C₄×C₂₃	—	2	3	no	—
36	C₂²×C₄×C₂₃	—	2	3	no	—
37	C₂²×C₄×C₂₃	C₄⋊C₄	1	6	yes	—
38	C₂²×C₄×C₂₃	C₄⋊C₄	1	6	yes	—
39	C₂²×C₄×C₂₃	C₂×C₈	1	6	yes	—
40	C₂²×C₄×C₂₃	M₁₆	1	6	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	#41	1	3	yes	—
47	C₂×C₈×C₂₃	—	2	12	no	—
48	C₂×C₈×C₂₃	—	2	6	no	—
49	C₂×C₈×C₂₃	—	2	6	no	—
50	C₂×C₈×C₂₃	—	4	3	no	—
51	C₂×C₈×C₂₃	—	4	6	no	—
52	C₂×C₈×C₂₃	—	4	3	no	—
53	C₂×C₈×C₂₃	—	4	3	no	—
54	C₂×C₈×C₂₃	—	4	1	no	—
55	C₄²×C₂₃	—	2	6	no	—
56	C₄²×C₂₃	—	2	12	no	—
57	C₄²×C₂₃	—	2	6	no	—
58	#11322	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
60	C₂³×C₂₃	C₄×D₄	1	3	yes	—
61	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
62	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
63	C₂³×C₂₃	C₂×M₁₆	1	3	yes	—
64	C₂³×C₂₃	#131	1	3	yes	—
65	C₂³×C₂₃	#131	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	6	no	—
67	C₂×C₄×C₂₃	—	2	6	no	—
68	C₂×C₄×C₂₃	—	2	12	no	—
69	C₂×C₄×C₂₃	—	2	12	no	—
70	C₂×C₄×C₂₃	—	2	6	no	—
71	C₂×C₄×C₂₃	—	2	6	no	—
72	C₂×C₄×C₂₃	—	2	6	no	—
73	C₂×C₄×C₂₃	—	4	6	no	—
74	C₂×C₄×C₂₃	—	4	6	no	—
75	C₂×C₄×C₂₃	C₄⋊C₈	1	12	yes	complemented
76	C₂×C₄×C₂₃	#120	1	3	yes	—
77	C₂×C₄×C₂₃	#121	1	3	yes	—
78	C₂×C₄×C₂₃	C₂²⋊Q₈	1	6	yes	—
79	C₂×C₄×C₂₃	C₂²⋊Q₈	1	3	yes	—
80	C₂×C₄×C₂₃	#123	1	3	yes	—
81	C₂×C₄×C₂₃	#125	1	3	yes	—
82	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₄⋊Q₈	1	3	yes	—
84	C₂×C₄×C₂₃	C₄⋊Q₈	1	3	yes	—
85	C₈×C₂₃	—	4	6	no	—
86	C₈×C₂₃	—	4	2	no	—
87	C₂²×C₄⋊C₈	C₂×C₂₃	1	3	yes	—
88	C₂×#336	C₂×C₂₃	1	3	yes	—
89	C₄×#95	C₂×C₂₃	1	1	yes	characteristic
90	C₂²×C₂₃	#384	1	1	yes	characteristic
91	C₂²×C₂₃	#385	1	3	yes	—
92	C₂²×C₂₃	#389	1	3	yes	—
93	C₂²×C₂₃	C₂×C₄⋊C₈	1	3	yes	—
94	C₂²×C₂₃	#423	1	3	yes	—
95	C₂²×C₂₃	#424	1	3	yes	—
96	C₂²×C₂₃	#424	1	3	yes	—
97	C₂²×C₂₃	C₈×D₄	1	3	yes	—
98	C₂²×C₂₃	#435	1	6	yes	—
99	C₂²×C₂₃	#436	1	3	yes	—
100	C₂²×C₂₃	C₈×Q₈	1	1	yes	characteristic
101	C₂²×C₂₃	#446	1	3	yes	—
102	C₄×C₂₃	—	2	12	no	—
103	C₄×C₂₃	—	4	6	no	—
104	C₄×C₂₃	#392	1	1	yes	characteristic
105	C₄×C₂₃	#395	1	3	yes	—
106	C₄×C₂₃	#398	1	3	yes	—
107	C₄×C₂₃	#399	1	1	yes	characteristic
108	C₂³×C₈	C₂²×C₂₃	1	3	yes	—
109	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
110	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
111	C₂×C₄⋊C₈	—	2	12	no	—
112	C₂×C₄⋊C₈	—	2	6	no	—
113	C₂×#95	C₄×C₂₃	1	3	yes	—
114	C₂×#95	C₄×C₂₃	1	1	yes	characteristic
115	#336	—	2	6	no	—
116	C₂×C₂₃	#1444	1	1	yes	characteristic
117	C₂×C₂₃	#1842	1	1	yes	characteristic
118	C₂×C₂₃	#1843	1	1	yes	characteristic
119	C₂×C₂₃	#1850	1	3	yes	—
120	C₂×C₂₃	#1851	1	3	yes	—
121	C₂×C₂₃	#1939	1	3	yes	—
122	C₂×C₂₃	#1940	1	3	yes	—
123	C₂³×C₄	C₂×C₄×C₂₃	1	3	yes	—
124	C₂³×C₄	C₂×C₄×C₂₃	1	3	yes	—
125	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	C₂²×C₈	—	2	3	no	—
127	C₂²×C₈	—	2	3	no	—
128	C₂²×C₈	—	2	6	no	—
129	C₂²×C₈	—	2	6	no	—
130	C₂²×C₈	—	2	3	no	—
131	C₂²×C₈	—	2	3	no	—
132	C₂²×C₈	D₄×C₂₃	1	6	yes	—
133	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
134	C₂×C₄²	—	2	6	no	—
135	C₂×C₄²	—	2	6	no	—
136	C₂×C₄²	D₄×C₂₃	1	6	yes	—
137	C₂×C₄²	Q₈×C₂₃	1	6	yes	—
138	C₄⋊C₈	—	2	24	no	—
139	#95	—	2	6	no	—
140	#95	C₈×C₂₃	1	4	yes	complemented
141	C₂₃	#11322	1	1	yes	characteristic, direct product factor, Sylow
142	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	6	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	3	no	—
148	C₂²×C₄	—	2	3	no	—
149	C₂²×C₄	C₄⋊C₄×C₂₃	1	6	yes	—
150	C₂²×C₄	C₄⋊C₄×C₂₃	1	6	yes	—
151	C₂²×C₄	C₂×C₈×C₂₃	1	6	yes	—
152	C₂²×C₄	M₁₆×C₂₃	1	6	yes	—
153	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
154	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
155	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Q₈×C₂₃	1	3	yes	—
157	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	#41×C₂₃	1	3	yes	—
159	C₂×C₈	—	2	12	no	—
160	C₂×C₈	—	2	6	no	—
161	C₂×C₈	—	2	6	no	—
162	C₂×C₈	—	4	3	no	—
163	C₂×C₈	—	4	6	no	—
164	C₂×C₈	—	4	3	no	—
165	C₂×C₈	—	4	3	no	—
166	C₂×C₈	—	4	1	no	—
167	C₄²	—	2	6	no	—
168	C₄²	—	2	12	no	—
169	C₄²	—	2	6	no	—
170	C₂³	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
171	C₂³	C₄×D₄×C₂₃	1	3	yes	—
172	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
173	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
174	C₂³	C₂×M₁₆×C₂₃	1	3	yes	—
175	C₂³	C₂₃×#131	1	3	yes	—
176	C₂³	C₂₃×#131	1	1	yes	characteristic
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	4	6	no	—
185	C₂×C₄	—	4	6	no	—
186	C₂×C₄	C₂₃×C₄⋊C₈	1	12	yes	complemented
187	C₂×C₄	C₂₃×#120	1	3	yes	—
188	C₂×C₄	C₂₃×#121	1	3	yes	—
189	C₂×C₄	C₂₃×C₂²⋊Q₈	1	6	yes	—
190	C₂×C₄	C₂₃×C₂²⋊Q₈	1	3	yes	—
191	C₂×C₄	C₂₃×#123	1	3	yes	—
192	C₂×C₄	C₂₃×#125	1	3	yes	—
193	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
194	C₂×C₄	C₂₃×C₄⋊Q₈	1	3	yes	—
195	C₂×C₄	C₂₃×C₄⋊Q₈	1	3	yes	—
196	C₈	—	4	6	no	—
197	C₈	—	4	2	no	—
198	C₂²	C₂₃×#384	1	1	yes	characteristic
199	C₂²	C₂₃×#385	1	3	yes	—
200	C₂²	C₂₃×#389	1	3	yes	—
201	C₂²	C₂×C₂₃×C₄⋊C₈	1	3	yes	—
202	C₂²	C₂₃×#423	1	3	yes	—
203	C₂²	C₂₃×#424	1	3	yes	—
204	C₂²	C₂₃×#424	1	3	yes	—
205	C₂²	C₈×D₄×C₂₃	1	3	yes	—
206	C₂²	C₂₃×#435	1	6	yes	—
207	C₂²	C₂₃×#436	1	3	yes	—
208	C₂²	C₈×Q₈×C₂₃	1	1	yes	characteristic
209	C₂²	C₂₃×#446	1	3	yes	—
210	C₄	—	2	12	no	—
211	C₄	—	4	6	no	—
212	C₄	C₂₃×#392	1	1	yes	characteristic
213	C₄	C₂₃×#395	1	3	yes	—
214	C₄	C₂₃×#398	1	3	yes	—
215	C₄	C₂₃×#399	1	1	yes	characteristic
216	C₂	C₂₃×#1444	1	1	yes	characteristic
217	C₂	C₂₃×#1842	1	1	yes	characteristic
218	C₂	C₂₃×#1843	1	1	yes	characteristic
219	C₂	C₂₃×#1850	1	3	yes	—
220	C₂	C₂₃×#1851	1	3	yes	—
221	C₂	C₂₃×#1939	1	3	yes	—
222	C₂	C₂₃×#1940	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator	C₂³	C₂²×C₈×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle, prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
29. C₂₃×#95	C₈	semidirect
58. #11322	C₂₃	direct (N × H)
75. C₂×C₄×C₂₃	C₄⋊C₈	semidirect
140. #95	C₈×C₂₃	semidirect
141. C₂₃	#11322	direct (N × H)
186. C₂×C₄	C₂₃×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×C₄⋊C₈	2
C₂×C₂₃×#336	2
C₄×C₂₃×#95	2
#11322	23

Subgroups — C23×#11322 #19158948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11322 #19158948