Subgroups — C₂₃×#7855 #19155481

Order 5888.

706 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×#2920	C₂	1	2	yes	maximal
2	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1274	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₂₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#432	—	2	2	no	—
7	C₂₃×#432	—	2	2	no	—
8	C₂₃×#431	C₄	1	4	yes	complemented
9	C₂₃×#431	C₂²	1	4	yes	—
10	C₂₃×#423	—	2	2	no	—
11	C₂₃×#407	—	2	2	no	—
12	C₄×M₁₆×C₂₃	C₄	1	4	yes	complemented
13	C₂₃×#376	—	2	2	no	—
14	C₂₃×#376	C₄	1	2	yes	—
15	C₂₃×#376	C₂²	1	2	yes	—
16	C₂×C₂₃×#95	—	2	2	no	—
17	C₂³×C₄×C₂₃	—	2	2	no	—
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₃	D₄	1	2	yes	—
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂×C₄²×C₂₃	—	2	2	no	—
22	C₂×C₄²×C₂₃	—	2	8	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	—	2	4	no	—
32	C₂₃×C₄⋊C₈	—	2	4	no	—
33	C₂₃×C₂²⋊C₈	—	2	4	no	—
34	C₂₃×C₂²⋊C₈	—	2	4	no	—
35	C₂₃×#97	—	2	4	no	—
36	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
37	C₂₃×#95	—	2	4	no	—
38	C₂₃×#95	—	2	4	no	—
39	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	2	8	no	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	4	4	no	—
44	C₂²×C₄×C₂₃	—	4	8	no	—
45	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	M₁₆×C₂₃	—	4	8	no	—
56	C₂×C₈×C₂₃	—	2	8	no	—
57	C₄²×C₂₃	—	2	8	no	—
58	C₄²×C₂₃	—	2	4	no	—
59	C₄²×C₂₃	C₄²	1	1	yes	characteristic
60	C₄²×C₂₃	C₄²	1	1	yes	characteristic
61	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₂₃	M₁₆	1	8	yes	complemented
67	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
70	#7855	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₂₃	—	2	4	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	#95	1	2	yes	—
74	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	8	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	—	4	16	no	—
86	C₂×C₄×C₂₃	—	4	8	no	—
87	C₂×C₄×C₂₃	#95	1	2	yes	—
88	C₂×C₄×C₂₃	#95	1	2	yes	—
89	C₂×C₄×C₂₃	#95	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
95	C₈×C₂₃	—	4	8	no	—
96	#2920	C₂×C₂₃	1	2	yes	—
97	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
98	#1274	C₂×C₂₃	1	4	yes	complemented
99	C₂²×C₂₃	—	2	8	no	—
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	H(3,ℤ/4)	1	4	yes	—
104	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
105	C₂²×C₂₃	#407	1	2	yes	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	4	8	no	—
112	C₄×C₂₃	C₄×M₁₆	1	2	yes	—
113	C₄×C₂₃	#423	1	2	yes	—
114	C₂²×C₄²	C₄×C₂₃	1	2	yes	—
115	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
116	#432	—	2	2	no	—
117	#432	—	2	2	no	—
118	#431	C₄×C₂₃	1	4	yes	complemented
119	#431	C₂²×C₂₃	1	4	yes	—
120	#423	—	2	2	no	—
121	#407	—	2	2	no	—
122	C₄×M₁₆	C₄×C₂₃	1	4	yes	complemented
123	#376	—	2	2	no	—
124	#376	C₄×C₂₃	1	2	yes	—
125	#376	C₂²×C₂₃	1	2	yes	—
126	C₂×#95	—	2	2	no	—
127	C₂×C₂₃	—	2	2	no	—
128	C₂×C₂₃	—	2	4	no	—
129	C₂×C₂₃	#1728	1	1	yes	characteristic
130	C₂×C₂₃	C₂×H(3,ℤ/4)	1	2	yes	—
131	C₂³×C₄	—	2	2	no	—
132	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂³×C₄	D₄×C₂₃	1	2	yes	—
134	C₂×M₁₆	—	2	4	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	—	2	8	no	—
137	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
138	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
141	C₂×C₄²	D₄×C₂₃	1	2	yes	—
142	C₂×C₄²	D₄×C₂₃	1	2	yes	—
143	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
144	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₂²⋊C₈	—	2	4	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	#97	—	2	4	no	—
150	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
151	#95	—	2	4	no	—
152	#95	—	2	4	no	—
153	C₂₃	#7855	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
162	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
165	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
166	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
167	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
170	M₁₆	—	4	8	no	—
171	C₂×C₈	—	2	8	no	—
172	C₄²	—	2	8	no	—
173	C₄²	—	2	4	no	—
174	C₄²	C₄²×C₂₃	1	1	yes	characteristic
175	C₄²	C₄²×C₂₃	1	1	yes	characteristic
176	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
177	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
178	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
179	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
180	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
181	C₄²	M₁₆×C₂₃	1	8	yes	complemented
182	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
183	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
185	C₂³	—	2	4	no	—
186	C₂³	—	2	2	no	—
187	C₂³	C₂₃×#95	1	2	yes	—
188	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	16	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	C₂₃×#95	1	2	yes	—
202	C₂×C₄	C₂₃×#95	1	2	yes	—
203	C₂×C₄	C₂₃×#95	1	2	yes	—
204	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
209	C₈	—	4	8	no	—
210	C₂²	—	2	8	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	C₂₃×H(3,ℤ/4)	1	4	yes	—
215	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
216	C₂²	C₂₃×#407	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	8	no	—
223	C₄	C₄×M₁₆×C₂₃	1	2	yes	—
224	C₄	C₂₃×#423	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	—	2	4	no	—
227	C₂	C₂₃×#1728	1	1	yes	characteristic
228	C₂	C₂×C₂₃×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#95
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
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
3. C₂₃×#1274	C₂	semidirect
8. C₂₃×#431	C₄	semidirect
12. C₄×M₁₆×C₂₃	C₄	semidirect
36. C₄×C₈×C₂₃	C₂×C₄	semidirect
66. C₄²×C₂₃	M₁₆	semidirect
70. #7855	C₂₃	direct (N × H)
98. #1274	C₂×C₂₃	semidirect
118. #431	C₄×C₂₃	semidirect
122. C₄×M₁₆	C₄×C₂₃	semidirect
150. C₄×C₈	C₂×C₄×C₂₃	semidirect
153. C₂₃	#7855	direct (N × H)
181. C₄²	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2920	2
C₂×C₂₃×#376	2
C₂₃×#1274	2
#7855	23

Subgroups — C23×#7855 #19155481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7855 #19155481