Subgroups — C₂₃×#7867 #19155493

Order 5888.

922 conjugacy classes of subgroups.

The subgroup lattice has 236 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×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1276	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₂₃	—	2	2	no	—
5	C₂²×C₄²×C₂₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#432	—	2	2	no	—
8	C₂₃×#432	C₄	1	4	yes	complemented
9	C₂₃×#431	—	2	2	no	—
10	C₂₃×#431	C₂²	1	4	yes	—
11	C₂₃×#423	C₄	1	4	yes	complemented
12	C₂₃×#407	—	2	2	no	—
13	C₄×M₁₆×C₂₃	—	2	2	no	—
14	C₄³×C₂₃	—	2	2	no	—
15	C₄³×C₂₃	C₄	1	2	yes	—
16	C₄³×C₂₃	C₂²	1	2	yes	—
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	16	no	—
22	C₂×C₄²×C₂₃	—	2	2	no	—
23	C₂×C₄²×C₂₃	—	2	4	no	—
24	C₂×C₄²×C₂₃	—	2	4	no	—
25	C₂×C₄²×C₂₃	—	2	8	no	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₂₃	D₄	1	2	yes	—
31	C₂×C₄²×C₂₃	D₄	1	2	yes	—
32	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	—	2	4	no	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
36	C₂₃×C₂²⋊C₈	—	2	4	no	—
37	C₂₃×C₂²⋊C₈	—	2	4	no	—
38	C₂₃×#97	—	2	4	no	—
39	C₄×C₈×C₂₃	—	2	4	no	—
40	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	—	2	16	no	—
42	C₂²×C₄×C₂₃	—	2	8	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	2	8	no	—
45	C₂²×C₄×C₂₃	—	2	4	no	—
46	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂²⋊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	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	M₁₆×C₂₃	—	4	8	no	—
57	C₂×C₈×C₂₃	—	2	8	no	—
58	C₄²×C₂₃	—	2	16	no	—
59	C₄²×C₂₃	—	2	16	no	—
60	C₄²×C₂₃	—	2	8	no	—
61	C₄²×C₂₃	—	2	4	no	—
62	C₄²×C₂₃	C₄²	1	1	yes	characteristic
63	C₄²×C₂₃	C₄²	1	1	yes	characteristic
64	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₂₃	M₁₆	1	8	yes	complemented
70	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
73	#7867	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	—	2	4	no	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	#95	1	2	yes	—
77	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	—	2	16	no	—
79	C₂×C₄×C₂₃	—	2	32	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	8	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	2	4	no	—
86	C₂×C₄×C₂₃	—	2	4	no	—
87	C₂×C₄×C₂₃	—	2	4	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	#95	1	2	yes	—
90	C₂×C₄×C₂₃	#95	1	2	yes	—
91	C₂×C₄×C₂₃	#95	1	2	yes	—
92	C₂×C₄×C₂₃	#104	1	8	yes	—
93	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
98	C₈×C₂₃	—	4	8	no	—
99	#2920	C₂×C₂₃	1	2	yes	—
100	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic
101	#1276	C₂×C₂₃	1	4	yes	complemented
102	C₂²×C₂₃	—	2	8	no	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	#339	1	4	yes	—
107	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
108	C₂²×C₂₃	#407	1	2	yes	—
109	C₄×C₂₃	—	2	16	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	#339	1	4	yes	—
114	C₄×C₂₃	C₄×M₁₆	1	2	yes	—
115	C₄×C₂₃	C₂×#104	1	4	yes	—
116	C₄×C₂₃	#423	1	2	yes	—
117	C₂²×C₄²	—	2	2	no	—
118	C₂²×C₄²	C₄×C₂₃	1	2	yes	—
119	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
120	#432	—	2	2	no	—
121	#432	C₄×C₂₃	1	4	yes	complemented
122	#431	—	2	2	no	—
123	#431	C₂²×C₂₃	1	4	yes	—
124	#423	C₄×C₂₃	1	4	yes	complemented
125	#407	—	2	2	no	—
126	C₄×M₁₆	—	2	2	no	—
127	C₄³	—	2	2	no	—
128	C₄³	C₄×C₂₃	1	2	yes	—
129	C₄³	C₂²×C₂₃	1	2	yes	—
130	C₂×C₂₃	—	2	2	no	—
131	C₂×C₂₃	—	2	4	no	—
132	C₂×C₂₃	#1728	1	1	yes	characteristic
133	C₂×C₂₃	C₂×#339	1	2	yes	—
134	C₂³×C₄	—	2	2	no	—
135	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₂³×C₄	D₄×C₂₃	1	2	yes	—
137	C₂×M₁₆	—	2	4	no	—
138	C₂×C₄²	—	2	16	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	2	4	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	8	no	—
143	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
144	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
147	C₂×C₄²	D₄×C₂₃	1	2	yes	—
148	C₂×C₄²	D₄×C₂₃	1	2	yes	—
149	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
150	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	C₂₃	#7867	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	16	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
167	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
170	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
171	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
174	M₁₆	—	4	8	no	—
175	C₂×C₈	—	2	8	no	—
176	C₄²	—	2	16	no	—
177	C₄²	—	2	16	no	—
178	C₄²	—	2	8	no	—
179	C₄²	—	2	4	no	—
180	C₄²	C₄²×C₂₃	1	1	yes	characteristic
181	C₄²	C₄²×C₂₃	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
183	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
184	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
185	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
186	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
187	C₄²	M₁₆×C₂₃	1	8	yes	complemented
188	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
189	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
190	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
191	C₂³	—	2	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	C₂₃×#95	1	2	yes	—
194	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	32	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	C₂₃×#95	1	2	yes	—
207	C₂×C₄	C₂₃×#95	1	2	yes	—
208	C₂×C₄	C₂₃×#95	1	2	yes	—
209	C₂×C₄	C₂₃×#104	1	8	yes	—
210	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
212	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
215	C₈	—	4	8	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	C₂₃×#339	1	4	yes	—
221	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
222	C₂²	C₂₃×#407	1	2	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	C₂₃×#339	1	4	yes	—
228	C₄	C₄×M₁₆×C₂₃	1	2	yes	—
229	C₄	C₂×C₂₃×#104	1	4	yes	—
230	C₄	C₂₃×#423	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	C₂₃×#1728	1	1	yes	characteristic
234	C₂	C₂×C₂₃×#339	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₂×D₄
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1276	C₂	semidirect
8. C₂₃×#432	C₄	semidirect
11. C₂₃×#423	C₄	semidirect
35. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
69. C₄²×C₂₃	M₁₆	semidirect
73. #7867	C₂₃	direct (N × H)
101. #1276	C₂×C₂₃	semidirect
121. #432	C₄×C₂₃	semidirect
124. #423	C₄×C₂₃	semidirect
152. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
157. C₂₃	#7867	direct (N × H)
187. 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×C₄³×C₂₃	2
C₂₃×#1276	2
#7867	23

Subgroups — C23×#7867 #19155493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7867 #19155493