Subgroups — C₂₃×#12666 #19160292

Order 5888.

318 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	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#347	C₂	1	2	yes	maximal
3	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1375	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₂₃×#97	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
13	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
14	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
15	C₂₃×#347	—	2	4	no	—
16	C₂₃×#346	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₂₃	C₄	1	2	yes	complemented
18	C₂₃×#336	C₄	1	2	yes	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃×M₃₂	—	2	2	no	—
26	C₂₃×M₃₂	—	2	1	no	—
27	C₂₃×M₃₂	—	2	1	no	—
28	C₂₃×M₃₂	—	2	2	no	—
29	C₂₃×M₃₂	D₄	1	2	yes	—
30	C₂₃×M₃₂	Q₈	1	2	yes	—
31	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
34	C₂₃×C₄⋊C₈	—	2	1	no	—
35	C₂₃×C₄⋊C₈	—	2	1	no	—
36	C₂₃×#97	—	2	2	no	—
37	C₂₃×#97	—	2	2	no	—
38	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂₃×#95	—	2	1	no	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	—	2	1	no	—
48	C₂×C₈×C₂₃	—	2	1	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	—	4	2	no	—
61	C₄²×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	C₄²	1	1	yes	characteristic
64	C₄²×C₂₃	C₄²	1	1	yes	characteristic
65	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
66	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
67	C₁₆×C₂₃	—	2	2	no	—
68	C₁₆×C₂₃	—	2	4	no	—
69	C₁₆×C₂₃	C₄⋊C₄	1	4	yes	complemented
70	#12666	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	4	1	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	#97	1	1	yes	characteristic
78	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
79	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
87	C₈×C₂₃	—	2	1	no	—
88	C₈×C₂₃	—	2	1	no	—
89	C₈×C₂₃	—	4	2	no	—
90	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₈×C₂₃	#117	1	1	yes	characteristic
92	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
93	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
94	#2150	C₂×C₂₃	1	1	yes	characteristic
95	C₂×#347	C₂×C₂₃	1	2	yes	—
96	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
97	#1770	C₂×C₂₃	1	1	yes	characteristic
98	#1375	C₂×C₂₃	1	2	yes	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
102	C₂²×C₂₃	#436	1	1	yes	characteristic
103	C₂²×C₂₃	#446	1	1	yes	characteristic
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	8	2	no	—
108	C₄×C₂₃	#347	1	2	yes	complemented
109	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₂₃	#405	1	1	yes	characteristic
111	C₄×C₂₃	#436	1	1	yes	characteristic
112	C₄×C₂₃	#446	1	1	yes	characteristic
113	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
115	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
116	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
117	C₂×#97	C₂²×C₂₃	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
119	C₄×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
120	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	complemented
121	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	complemented
122	#347	—	2	4	no	—
123	#346	C₄×C₂₃	1	2	yes	complemented
124	C₄×C₁₆	C₄×C₂₃	1	2	yes	complemented
125	#336	C₄×C₂₃	1	2	yes	—
126	C₂×C₂₃	—	2	1	no	—
127	C₂×C₂₃	—	2	1	no	—
128	C₂×C₂₃	#1770	1	1	yes	characteristic
129	C₂×C₂₃	C₂×#347	1	1	yes	characteristic
130	C₂×C₂₃	#2110	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
132	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
136	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	1	no	—
139	M₃₂	—	2	1	no	—
140	M₃₂	—	2	2	no	—
141	M₃₂	D₄×C₂₃	1	2	yes	—
142	M₃₂	Q₈×C₂₃	1	2	yes	—
143	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
144	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
145	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
146	C₄⋊C₈	—	2	1	no	—
147	C₄⋊C₈	—	2	1	no	—
148	#97	—	2	2	no	—
149	#97	—	2	2	no	—
150	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
151	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
152	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
153	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
154	#95	—	2	1	no	—
155	C₂₃	#12666	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
166	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
167	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
170	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
171	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
172	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₄²×C₂₃	1	1	yes	characteristic
177	C₄²	C₄²×C₂₃	1	1	yes	characteristic
178	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
179	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	2	4	no	—
182	C₁₆	C₄⋊C₄×C₂₃	1	4	yes	complemented
183	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
190	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
191	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
192	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
196	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
197	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
198	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
203	C₈	C₂₃×#117	1	1	yes	characteristic
204	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
205	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
209	C₂²	C₂₃×#436	1	1	yes	characteristic
210	C₂²	C₂₃×#446	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	C₂₃×#347	1	2	yes	complemented
216	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
217	C₄	C₂₃×#405	1	1	yes	characteristic
218	C₄	C₂₃×#436	1	1	yes	characteristic
219	C₄	C₂₃×#446	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₂₃×#1770	1	1	yes	characteristic
223	C₂	C₂×C₂₃×#347	1	1	yes	characteristic
224	C₂	C₂₃×#2110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#97
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×C₄⋊C₁₆	C₄	semidirect
14. C₂₃×C₄⋊C₁₆	C₄	semidirect
16. C₂₃×#346	C₄	semidirect
17. C₄×C₁₆×C₂₃	C₄	semidirect
69. C₁₆×C₂₃	C₄⋊C₄	semidirect
70. #12666	C₂₃	direct (N × H)
108. C₄×C₂₃	#347	semidirect
120. C₄⋊C₁₆	C₄×C₂₃	semidirect
121. C₄⋊C₁₆	C₄×C₂₃	semidirect
123. #346	C₄×C₂₃	semidirect
124. C₄×C₁₆	C₄×C₂₃	semidirect
155. C₂₃	#12666	direct (N × H)
182. C₁₆	C₄⋊C₄×C₂₃	semidirect
215. C₄	C₂₃×#347	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2150	2
C₂×C₂₃×#347	2
C₄×C₂₃×M₃₂	2
C₂₃×#1770	2
C₂₃×#1375	2
#12666	23

Subgroups — C23×#12666 #19160292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12666 #19160292