Subgroups — C₂₃×#14681 #19162307

Order 5888.

6350 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#381	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₃×#379	C₂	1	2	yes	maximal
4	C₂₃×C₂²⋊#95	C₂	1	8	yes	maximal, complemented
5	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₂₃	—	2	8	no	—
7	C₂⁴×C₄×C₂₃	—	2	16	no	—
8	C₂⁴×C₄×C₂₃	C₄	1	8	yes	complemented
9	C₂⁴×C₄×C₂₃	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
12	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
13	C₂₃×#381	—	2	16	no	—
14	C₂₃×#379	—	2	8	no	—
15	C₂×C₂₃×#95	C₂²	1	16	yes	complemented
16	C₂⁵×C₂₃	—	2	8	no	—
17	C₂⁵×C₂₃	—	2	16	no	—
18	C₂⁵×C₂₃	C₂×C₄	1	4	yes	—
19	C₂⁵×C₂₃	D₄	1	8	yes	—
20	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
21	C₂⁵×C₂₃	C₂³	1	2	yes	—
22	C₂³×C₄×C₂₃	—	2	128	no	—
23	C₂³×C₄×C₂₃	—	2	32	no	—
24	C₂³×C₄×C₂₃	—	2	16	no	—
25	C₂³×C₄×C₂₃	—	2	8	no	—
26	C₂³×C₄×C₂₃	—	2	8	no	—
27	C₂³×C₄×C₂₃	—	2	64	no	—
28	C₂³×C₄×C₂₃	—	2	32	no	—
29	C₂³×C₄×C₂₃	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₂₃	D₄	1	16	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
35	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
36	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
37	C₂×C₂²⋊C₄×C₂₃	—	2	32	no	—
38	C₂₃×#95	—	2	32	no	—
39	C₂⁴×C₂₃	—	2	128	no	—
40	C₂⁴×C₂₃	—	2	16	no	—
41	C₂⁴×C₂₃	—	2	32	no	—
42	C₂⁴×C₂₃	—	2	32	no	—
43	C₂⁴×C₂₃	—	2	64	no	—
44	C₂⁴×C₂₃	—	2	8	no	—
45	C₂⁴×C₂₃	—	2	8	no	—
46	C₂⁴×C₂₃	—	2	8	no	—
47	C₂⁴×C₂₃	C₂²⋊C₄	1	16	yes	—
48	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
50	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
51	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
52	C₂⁴×C₂₃	C₂×D₄	1	8	yes	—
53	C₂⁴×C₂₃	#41	1	16	yes	—
54	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	—	2	128	no	—
56	C₂²×C₄×C₂₃	—	2	256	no	—
57	C₂²×C₄×C₂₃	—	2	32	no	—
58	C₂²×C₄×C₂₃	—	2	16	no	—
59	C₂²×C₄×C₂₃	—	2	16	no	—
60	C₂²×C₄×C₂₃	—	2	64	no	—
61	C₂²×C₄×C₂₃	—	2	8	no	—
62	C₂²×C₄×C₂₃	—	2	8	no	—
63	C₂²×C₄×C₂₃	—	2	64	no	—
64	C₂²×C₄×C₂₃	—	2	8	no	—
65	C₂²×C₄×C₂₃	—	2	16	no	—
66	C₂²×C₄×C₂₃	—	2	32	no	—
67	C₂²×C₄×C₂₃	C₂²⋊C₄	1	32	yes	complemented
68	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	16	yes	—
70	C₂²⋊C₄×C₂₃	—	4	16	no	—
71	C₂²⋊C₄×C₂₃	—	4	32	no	—
72	#14681	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	—	2	256	no	—
74	C₂³×C₂₃	—	2	128	no	—
75	C₂³×C₂₃	—	2	8	no	—
76	C₂³×C₂₃	—	2	8	no	—
77	C₂³×C₂₃	—	2	8	no	—
78	C₂³×C₂₃	—	2	16	no	—
79	C₂³×C₂₃	—	2	16	no	—
80	C₂³×C₂₃	—	2	16	no	—
81	C₂³×C₂₃	—	2	32	no	—
82	C₂³×C₂₃	—	2	64	no	—
83	C₂³×C₂₃	—	2	32	no	—
84	C₂³×C₂₃	—	2	64	no	—
85	C₂³×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
86	C₂³×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
87	C₂³×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
88	C₂³×C₂₃	#117	1	16	yes	—
89	C₂³×C₂₃	#120	1	16	yes	—
90	C₂³×C₂₃	#123	1	32	yes	—
91	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
92	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
93	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
94	C₂³×C₂₃	C₂×#41	1	8	yes	—
95	C₂×C₄×C₂₃	—	2	128	no	—
96	C₂×C₄×C₂₃	—	2	8	no	—
97	C₂×C₄×C₂₃	—	2	8	no	—
98	C₂×C₄×C₂₃	—	2	32	no	—
99	C₂×C₄×C₂₃	—	2	64	no	—
100	C₂×C₄×C₂₃	—	4	8	no	—
101	C₂×C₄×C₂₃	—	4	16	no	—
102	C₂×C₄×C₂₃	—	4	32	no	—
103	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	16	yes	—
104	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic
105	C₂×#381	C₂×C₂₃	1	4	yes	complemented
106	C₂×#379	C₂×C₂₃	1	2	yes	—
107	C₂²⋊#95	C₂×C₂₃	1	8	yes	complemented
108	C₂²×C₂₃	—	2	128	no	—
109	C₂²×C₂₃	—	2	32	no	—
110	C₂²×C₂₃	—	2	64	no	—
111	C₂²×C₂₃	—	2	8	no	—
112	C₂²×C₂₃	—	2	8	no	—
113	C₂²×C₂₃	—	2	8	no	—
114	C₂²×C₂₃	—	2	16	no	—
115	C₂²×C₂₃	—	2	32	no	—
116	C₂²×C₂₃	#379	1	8	yes	—
117	C₂²×C₂₃	#381	1	16	yes	complemented
118	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	2	yes	—
120	C₂²×C₂₃	C₂×#117	1	4	yes	—
121	C₂²×C₂₃	C₂×#120	1	4	yes	—
122	C₂²×C₂₃	C₂×#123	1	8	yes	—
123	C₂²×C₂₃	#525	1	16	yes	—
124	C₄×C₂₃	—	2	16	no	—
125	C₄×C₂₃	—	8	8	no	—
126	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
127	C₂⁴×C₄	—	2	8	no	—
128	C₂⁴×C₄	—	2	16	no	—
129	C₂⁴×C₄	C₄×C₂₃	1	8	yes	complemented
130	C₂⁴×C₄	C₂²×C₂₃	1	4	yes	—
131	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
132	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
133	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
134	#381	—	2	16	no	—
135	#379	—	2	8	no	—
136	C₂×#95	C₂²×C₂₃	1	16	yes	complemented
137	C₂×C₂₃	—	2	8	no	—
138	C₂×C₂₃	—	2	16	no	—
139	C₂×C₂₃	C₂×#379	1	1	yes	characteristic
140	C₂×C₂₃	C₂×#381	1	2	yes	—
141	C₂×C₂₃	#2280	1	4	yes	—
142	C₂×C₂₃	#2849	1	8	yes	—
143	C₂⁵	—	2	8	no	—
144	C₂⁵	—	2	16	no	—
145	C₂⁵	C₂×C₄×C₂₃	1	4	yes	—
146	C₂⁵	D₄×C₂₃	1	8	yes	—
147	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
148	C₂⁵	C₂³×C₂₃	1	2	yes	—
149	C₂³×C₄	—	2	128	no	—
150	C₂³×C₄	—	2	32	no	—
151	C₂³×C₄	—	2	16	no	—
152	C₂³×C₄	—	2	8	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	64	no	—
155	C₂³×C₄	—	2	32	no	—
156	C₂³×C₄	C₂×C₄×C₂₃	1	16	yes	—
157	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
158	C₂³×C₄	D₄×C₂₃	1	16	yes	—
159	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
160	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
161	C₂×C₂²⋊C₄	—	2	8	no	—
162	C₂×C₂²⋊C₄	—	2	16	no	—
163	C₂×C₂²⋊C₄	—	2	16	no	—
164	C₂×C₂²⋊C₄	—	2	32	no	—
165	#95	—	2	32	no	—
166	C₂₃	#14681	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	—	2	128	no	—
168	C₂⁴	—	2	16	no	—
169	C₂⁴	—	2	32	no	—
170	C₂⁴	—	2	32	no	—
171	C₂⁴	—	2	64	no	—
172	C₂⁴	—	2	8	no	—
173	C₂⁴	—	2	8	no	—
174	C₂⁴	—	2	8	no	—
175	C₂⁴	C₂²⋊C₄×C₂₃	1	16	yes	—
176	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
177	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
178	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
179	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
180	C₂⁴	C₂×D₄×C₂₃	1	8	yes	—
181	C₂⁴	#41×C₂₃	1	16	yes	—
182	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	—	2	256	no	—
184	C₂²×C₄	—	2	128	no	—
185	C₂²×C₄	—	2	32	no	—
186	C₂²×C₄	—	2	16	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	2	64	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	64	no	—
192	C₂²×C₄	—	2	16	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	32	no	—
195	C₂²×C₄	C₂²⋊C₄×C₂₃	1	32	yes	complemented
196	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
197	C₂²×C₄	C₂×D₄×C₂₃	1	16	yes	—
198	C₂²⋊C₄	—	4	16	no	—
199	C₂²⋊C₄	—	4	32	no	—
200	C₂³	—	2	128	no	—
201	C₂³	—	2	256	no	—
202	C₂³	—	2	32	no	—
203	C₂³	—	2	64	no	—
204	C₂³	—	2	8	no	—
205	C₂³	—	2	8	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	16	no	—
208	C₂³	—	2	16	no	—
209	C₂³	—	2	16	no	—
210	C₂³	—	2	32	no	—
211	C₂³	—	2	64	no	—
212	C₂³	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
213	C₂³	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
214	C₂³	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
215	C₂³	C₂₃×#117	1	16	yes	—
216	C₂³	C₂₃×#120	1	16	yes	—
217	C₂³	C₂₃×#123	1	32	yes	—
218	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
219	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
220	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
221	C₂³	C₂×#41×C₂₃	1	8	yes	—
222	C₂×C₄	—	2	128	no	—
223	C₂×C₄	—	2	64	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	32	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	16	no	—
229	C₂×C₄	—	4	32	no	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	16	yes	—
231	C₂²	—	2	128	no	—
232	C₂²	—	2	8	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	16	no	—
236	C₂²	—	2	32	no	—
237	C₂²	—	2	32	no	—
238	C₂²	—	2	64	no	—
239	C₂²	C₂₃×#379	1	8	yes	—
240	C₂²	C₂₃×#381	1	16	yes	complemented
241	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
242	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	2	yes	—
243	C₂²	C₂×C₂₃×#117	1	4	yes	—
244	C₂²	C₂×C₂₃×#120	1	4	yes	—
245	C₂²	C₂×C₂₃×#123	1	8	yes	—
246	C₂²	C₂₃×#525	1	16	yes	—
247	C₄	—	2	16	no	—
248	C₄	—	8	8	no	—
249	C₂	—	2	8	no	—
250	C₂	—	2	16	no	—
251	C₂	C₂×C₂₃×#379	1	1	yes	characteristic
252	C₂	C₂×C₂₃×#381	1	2	yes	—
253	C₂	C₂₃×#2280	1	4	yes	—
254	C₂	C₂₃×#2849	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
prime radical	C₂⁶×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#381	C₂	semidirect
4. C₂₃×C₂²⋊#95	C₂	semidirect
8. C₂⁴×C₄×C₂₃	C₄	semidirect
15. C₂×C₂₃×#95	C₂²	semidirect
67. C₂²×C₄×C₂₃	C₂²⋊C₄	semidirect
72. #14681	C₂₃	direct (N × H)
105. C₂×#381	C₂×C₂₃	semidirect
107. C₂²⋊#95	C₂×C₂₃	semidirect
117. C₂²×C₂₃	#381	semidirect
129. C₂⁴×C₄	C₄×C₂₃	semidirect
136. C₂×#95	C₂²×C₂₃	semidirect
166. C₂₃	#14681	direct (N × H)
195. C₂²×C₄	C₂²⋊C₄×C₂₃	semidirect
240. C₂²	C₂₃×#381	semidirect

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₂₃	2
C₂×C₂₃×#381	2
C₂×C₂₃×#379	2
C₂₃×C₂²⋊#95	2
#14681	23

Subgroups — C23×#14681 #19162307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#14681 #19162307