Subgroups — C₂₃×#2572 #15670191

Order 2944.

476 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#394	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#393	C₂	1	2	yes	maximal, complemented
4	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#388	C₂	1	2	yes	maximal, complemented
6	C₂₃×#386	C₂	1	2	yes	maximal
7	C₂₃×#383	C₂	1	2	yes	maximal, complemented
8	C₂₃×#381	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#95	C₂²	1	2	yes	—
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂₃×#95	C₂²	1	1	yes	characteristic
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	1	yes	characteristic
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂⁴×C₂₃	C₂³	1	2	yes	—
32	C₂×Q₈×C₂₃	—	2	2	no	—
33	C₂×Q₈×C₂₃	—	4	1	no	—
34	C₂×Q₈×C₂₃	—	4	1	no	—
35	C₂×D₄×C₂₃	—	2	4	no	—
36	C₂×D₄×C₂₃	—	4	1	no	—
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₂₃	—	4	1	no	—
49	C₄⋊C₄×C₂₃	—	4	1	no	—
50	C₂²⋊C₄×C₂₃	—	2	4	no	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	—	2	4	no	—
53	C₂²⋊C₄×C₂₃	—	4	2	no	—
54	C₂²⋊C₄×C₂₃	—	4	2	no	—
55	C₄²×C₂₃	—	2	4	no	—
56	C₄²×C₂₃	—	4	1	no	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	#41	1	4	yes	—
64	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
65	Q₈×C₂₃	—	4	4	no	—
66	D₄×C₂₃	—	4	4	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	4	8	no	—
88	C₂×C₄×C₂₃	#41	1	4	yes	—
89	C₂×C₄×C₂₃	#41	1	2	yes	—
90	#2572	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₂₃	—	2	4	no	—
92	C₂²×C₂₃	—	2	4	no	—
93	C₂²×C₂₃	—	2	4	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
99	C₂²×C₂₃	C₂×#41	1	2	yes	—
100	C₂²×C₂₃	C₂×#41	1	2	yes	—
101	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	4	2	no	—
106	C₄×C₂₃	—	4	2	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	—	4	2	no	—
111	#400	C₂×C₂₃	1	1	yes	characteristic, complemented
112	#394	C₂×C₂₃	1	1	yes	characteristic
113	#393	C₂×C₂₃	1	2	yes	complemented
114	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
115	#388	C₂×C₂₃	1	2	yes	complemented
116	#386	C₂×C₂₃	1	2	yes	—
117	#383	C₂×C₂₃	1	2	yes	complemented
118	#381	C₂×C₂₃	1	2	yes	complemented
119	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
120	C₂×C₂₃	—	4	4	no	—
121	C₂×C₂₃	#529	1	2	yes	—
122	C₂×C₂₃	#538	1	1	yes	characteristic
123	C₂×C₂₃	#551	1	2	yes	—
124	C₂×C₂₃	#551	1	1	yes	characteristic
125	C₂×C₂₃	#555	1	1	yes	characteristic
126	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
136	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
137	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
138	#95	C₂²×C₂₃	1	2	yes	—
139	#95	C₂²×C₂₃	1	2	yes	—
140	#95	C₂²×C₂₃	1	1	yes	characteristic
141	#95	C₂²×C₂₃	1	2	yes	—
142	#95	C₂²×C₂₃	1	2	yes	—
143	#95	C₂²×C₂₃	1	1	yes	characteristic
144	#95	C₂²×C₂₃	1	2	yes	—
145	#95	C₂²×C₂₃	1	2	yes	—
146	#95	C₂²×C₂₃	1	2	yes	—
147	C₂₃	#2572	1	1	yes	characteristic, direct product factor, Sylow
148	C₂⁴	C₂³×C₂₃	1	2	yes	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	4	1	no	—
151	C₂×Q₈	—	4	1	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₄⋊C₄	—	4	1	no	—
166	C₄⋊C₄	—	4	1	no	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₄²	—	2	4	no	—
173	C₄²	—	4	1	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	#41×C₂₃	1	4	yes	—
181	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
182	Q₈	—	4	4	no	—
183	D₄	—	4	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	#41×C₂₃	1	4	yes	—
206	C₂×C₄	#41×C₂₃	1	2	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₂×#41×C₂₃	1	2	yes	—
215	C₂²	C₂×#41×C₂₃	1	2	yes	—
216	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
217	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
218	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂₃×#529	1	2	yes	—
229	C₂	C₂₃×#538	1	1	yes	characteristic
230	C₂	C₂₃×#551	1	2	yes	—
231	C₂	C₂₃×#551	1	1	yes	characteristic
232	C₂	C₂₃×#555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#400	C₂	semidirect
3. C₂₃×#393	C₂	semidirect
4. C₂₃×#391	C₂	semidirect
5. C₂₃×#388	C₂	semidirect
7. C₂₃×#383	C₂	semidirect
8. C₂₃×#381	C₂	semidirect
9. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
90. #2572	C₂₃	direct (N × H)
111. #400	C₂×C₂₃	semidirect
113. #393	C₂×C₂₃	semidirect
114. #391	C₂×C₂₃	semidirect
115. #388	C₂×C₂₃	semidirect
117. #383	C₂×C₂₃	semidirect
118. #381	C₂×C₂₃	semidirect
119. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
147. C₂₃	#2572	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#400	2
C₂₃×#394	2
C₂₃×#393	2
C₂₃×#391	2
C₂₃×#388	2
C₂₃×#386	2
C₂₃×#383	2
C₂₃×#381	2
C₄×C₂²⋊C₄×C₂₃	2
#2572	23

Subgroups — C23×#2572 #15670191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2572 #15670191