Subgroups — C₂₃×#7274 #19154900

Order 5888.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₂₃	—	2	1	no	—
5	C₂×C₂₃×#107	—	2	1	no	—
6	C₂×C₂₃×#107	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#384	—	2	1	no	—
10	C₂₃×C₄⋊C₁₆	—	2	1	no	—
11	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
12	C₂₃×#335	—	2	1	no	—
13	C₂₃×#335	C₄	1	2	yes	complemented
14	C₂²×C₈×C₂₃	—	4	1	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
19	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
20	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂×C₁₆×C₂₃	—	4	1	no	—
23	C₂×C₁₆×C₂₃	—	4	1	no	—
24	C₂₃×#107	—	4	1	no	—
25	C₂₃×#107	—	4	1	no	—
26	C₂₃×#107	—	4	2	no	—
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	—	4	1	no	—
29	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	—	4	2	no	—
34	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	4	2	no	—
40	C₂×C₈×C₂₃	—	8	1	no	—
41	C₂×C₈×C₂₃	—	8	1	no	—
42	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₄⋊C₄×C₂₃	—	4	2	no	—
52	C₄⋊C₄×C₂₃	—	8	2	no	—
53	C₄⋊C₄×C₂₃	—	8	1	no	—
54	C₄⋊C₄×C₂₃	—	8	1	no	—
55	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
59	C₁₆×C₂₃	—	4	2	no	—
60	#7274	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	2	1	no	—
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	8	2	no	—
65	C₂×C₄×C₂₃	—	8	2	no	—
66	C₂×C₄×C₂₃	—	8	2	no	—
67	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
68	C₂×C₄×C₂₃	#100	1	1	yes	characteristic
69	C₂×C₄×C₂₃	#101	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
71	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
72	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
73	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
75	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
77	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
78	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
79	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
80	C₂×C₄×C₂₃	Q₃₂	1	1	yes	characteristic
81	C₈×C₂₃	—	2	1	no	—
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	8	2	no	—
84	C₈×C₂₃	#102	1	1	yes	characteristic
85	C₈×C₂₃	#103	1	1	yes	characteristic
86	C₈×C₂₃	#106	1	1	yes	characteristic
87	C₈×C₂₃	#107	1	1	yes	characteristic
88	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
89	#1845	C₂×C₂₃	1	1	yes	characteristic
90	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
91	C₂²×C₂₃	#331	1	1	yes	characteristic
92	C₂²×C₂₃	#332	1	1	yes	characteristic
93	C₂²×C₂₃	#340	1	1	yes	characteristic
94	C₂²×C₂₃	#357	1	1	yes	characteristic
95	C₂²×C₂₃	#358	1	1	yes	characteristic
96	C₂²×C₂₃	#361	1	1	yes	characteristic
97	C₂²×C₂₃	#362	1	1	yes	characteristic
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	1	no	—
100	C₄×C₂₃	—	16	2	no	—
101	C₄×C₂₃	#332	1	1	yes	characteristic
102	C₄×C₂₃	#333	1	1	yes	characteristic
103	C₄×C₂₃	#340	1	1	yes	characteristic
104	C₄×C₂₃	#343	1	1	yes	characteristic
105	C₄×C₂₃	#357	1	1	yes	characteristic
106	C₄×C₂₃	#358	1	1	yes	characteristic
107	C₄×C₂₃	#366	1	1	yes	characteristic
108	C₄×C₂₃	#367	1	1	yes	characteristic
109	C₂²×C₁₆	—	2	1	no	—
110	C₂×#107	—	2	1	no	—
111	C₂×#107	C₄×C₂₃	1	2	yes	—
112	C₂×C₄⋊C₈	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
114	#384	—	2	1	no	—
115	C₄⋊C₁₆	—	2	1	no	—
116	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	complemented
117	#335	—	2	1	no	—
118	#335	C₄×C₂₃	1	2	yes	complemented
119	C₂×C₂₃	#1296	1	1	yes	characteristic
120	C₂×C₂₃	#1361	1	1	yes	characteristic
121	C₂×C₂₃	#1362	1	1	yes	characteristic
122	C₂×C₂₃	#1364	1	1	yes	characteristic
123	C₂×C₂₃	#1365	1	1	yes	characteristic
124	C₂×C₂₃	#1380	1	1	yes	characteristic
125	C₂×C₂₃	#1387	1	1	yes	characteristic
126	C₂²×C₈	—	4	1	no	—
127	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	4	1	no	—
135	C₂×C₁₆	—	4	1	no	—
136	#107	—	4	1	no	—
137	#107	—	4	1	no	—
138	#107	—	4	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	4	1	no	—
141	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
142	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
143	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
144	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
145	C₂₃	#7274	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
160	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
161	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
162	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	8	2	no	—
166	C₄⋊C₄	—	8	1	no	—
167	C₄⋊C₄	—	8	1	no	—
168	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
169	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
170	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
171	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
172	C₁₆	—	4	2	no	—
173	C₂³	C₂₃×#102	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
180	C₂×C₄	C₂₃×#100	1	1	yes	characteristic
181	C₂×C₄	C₂₃×#101	1	1	yes	characteristic
182	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
183	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
184	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
185	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
186	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
187	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
188	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
189	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
190	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
191	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
192	C₂×C₄	C₂₃×Q₃₂	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	8	2	no	—
196	C₈	C₂₃×#102	1	1	yes	characteristic
197	C₈	C₂₃×#103	1	1	yes	characteristic
198	C₈	C₂₃×#106	1	1	yes	characteristic
199	C₈	C₂₃×#107	1	1	yes	characteristic
200	C₂²	C₂₃×#331	1	1	yes	characteristic
201	C₂²	C₂₃×#332	1	1	yes	characteristic
202	C₂²	C₂₃×#340	1	1	yes	characteristic
203	C₂²	C₂₃×#357	1	1	yes	characteristic
204	C₂²	C₂₃×#358	1	1	yes	characteristic
205	C₂²	C₂₃×#361	1	1	yes	characteristic
206	C₂²	C₂₃×#362	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	16	2	no	—
210	C₄	C₂₃×#332	1	1	yes	characteristic
211	C₄	C₂₃×#333	1	1	yes	characteristic
212	C₄	C₂₃×#340	1	1	yes	characteristic
213	C₄	C₂₃×#343	1	1	yes	characteristic
214	C₄	C₂₃×#357	1	1	yes	characteristic
215	C₄	C₂₃×#358	1	1	yes	characteristic
216	C₄	C₂₃×#366	1	1	yes	characteristic
217	C₄	C₂₃×#367	1	1	yes	characteristic
218	C₂	C₂₃×#1296	1	1	yes	characteristic
219	C₂	C₂₃×#1361	1	1	yes	characteristic
220	C₂	C₂₃×#1362	1	1	yes	characteristic
221	C₂	C₂₃×#1364	1	1	yes	characteristic
222	C₂	C₂₃×#1365	1	1	yes	characteristic
223	C₂	C₂₃×#1380	1	1	yes	characteristic
224	C₂	C₂₃×#1387	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#102
commutator	C₂×C₈	C₄²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×C₄⋊C₁₆	C₄	semidirect
13. C₂₃×#335	C₄	semidirect
60. #7274	C₂₃	direct (N × H)
116. C₄⋊C₁₆	C₄×C₂₃	semidirect
118. #335	C₄×C₂₃	semidirect
145. C₂₃	#7274	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₄⋊C₁₆	2
C₂₃×#1845	2
C₂×C₂₃×#335	2
#7274	23

Subgroups — C23×#7274 #19154900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7274 #19154900