Subgroups — C₂₃×#14680 #19162306

Order 5888.

6494 conjugacy classes of subgroups.

The subgroup lattice has 252 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×#380	C₂	1	6	yes	maximal, complemented
3	C₂×C₂₃×#379	C₂	1	2	yes	maximal
4	C₂₃×C₂²⋊#95	C₂	1	6	yes	maximal, complemented
5	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₂₃	—	2	24	no	—
7	C₂⁴×C₄×C₂₃	C₄	1	8	yes	complemented
8	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
11	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	—
12	C₂₃×#380	—	2	24	no	—
13	C₂₃×#379	—	2	8	no	—
14	C₂×C₂₃×#95	C₂²	1	12	yes	complemented
15	C₂⁵×C₂₃	—	2	12	no	—
16	C₂⁵×C₂₃	—	2	12	no	—
17	C₂⁵×C₂₃	C₂×C₄	1	4	yes	—
18	C₂⁵×C₂₃	D₄	1	4	yes	—
19	C₂⁵×C₂₃	Q₈	1	4	yes	—
20	C₂⁵×C₂₃	C₂³	1	3	yes	—
21	C₂³×C₄×C₂₃	—	2	128	no	—
22	C₂³×C₄×C₂₃	—	2	12	no	—
23	C₂³×C₄×C₂₃	—	2	96	no	—
24	C₂³×C₄×C₂₃	—	2	24	no	—
25	C₂³×C₄×C₂₃	—	2	24	no	—
26	C₂³×C₄×C₂₃	C₂×C₄	1	12	yes	—
27	C₂³×C₄×C₂₃	C₂×C₄	1	12	yes	—
28	C₂³×C₄×C₂₃	D₄	1	24	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
31	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
33	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
34	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
35	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
36	C₂₃×#95	—	2	24	no	—
37	C₂⁴×C₂₃	—	2	32	no	—
38	C₂⁴×C₂₃	—	2	96	no	—
39	C₂⁴×C₂₃	—	2	24	no	—
40	C₂⁴×C₂₃	—	2	24	no	—
41	C₂⁴×C₂₃	—	2	48	no	—
42	C₂⁴×C₂₃	—	2	48	no	—
43	C₂⁴×C₂₃	—	2	12	no	—
44	C₂⁴×C₂₃	—	2	12	no	—
45	C₂⁴×C₂₃	C₄⋊C₄	1	16	yes	complemented
46	C₂⁴×C₂₃	C₂²×C₄	1	6	yes	—
47	C₂⁴×C₂₃	C₂×D₄	1	12	yes	—
48	C₂⁴×C₂₃	C₂×D₄	1	6	yes	—
49	C₂⁴×C₂₃	C₂×Q₈	1	6	yes	—
50	C₂⁴×C₂₃	#41	1	12	yes	—
51	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	—	2	96	no	—
53	C₂²×C₄×C₂₃	—	2	256	no	—
54	C₂²×C₄×C₂₃	—	2	24	no	—
55	C₂²×C₄×C₂₃	—	2	12	no	—
56	C₂²×C₄×C₂₃	—	2	48	no	—
57	C₂²×C₄×C₂₃	—	2	48	no	—
58	C₂²×C₄×C₂₃	—	2	96	no	—
59	C₂²×C₄×C₂₃	—	2	8	no	—
60	C₂²×C₄×C₂₃	—	2	24	no	—
61	C₂²×C₄×C₂₃	—	2	24	no	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	48	yes	complemented
63	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
67	C₄⋊C₄×C₂₃	—	4	16	no	—
68	C₂²⋊C₄×C₂₃	—	4	48	no	—
69	#14680	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	—	2	32	no	—
71	C₂³×C₂₃	—	2	128	no	—
72	C₂³×C₂₃	—	2	128	no	—
73	C₂³×C₂₃	—	2	96	no	—
74	C₂³×C₂₃	—	2	12	no	—
75	C₂³×C₂₃	—	2	12	no	—
76	C₂³×C₂₃	—	2	24	no	—
77	C₂³×C₂₃	—	2	24	no	—
78	C₂³×C₂₃	—	2	24	no	—
79	C₂³×C₂₃	—	2	24	no	—
80	C₂³×C₂₃	—	2	48	no	—
81	C₂³×C₂₃	—	2	96	no	—
82	C₂³×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
83	C₂³×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
84	C₂³×C₂₃	#117	1	12	yes	—
85	C₂³×C₂₃	#121	1	24	yes	—
86	C₂³×C₂₃	C₂²⋊Q₈	1	24	yes	—
87	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
88	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
89	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂³×C₂₃	C₂×#41	1	6	yes	—
92	C₂×C₄×C₂₃	—	2	96	no	—
93	C₂×C₄×C₂₃	—	2	12	no	—
94	C₂×C₄×C₂₃	—	2	24	no	—
95	C₂×C₄×C₂₃	—	2	96	no	—
96	C₂×C₄×C₂₃	—	4	8	no	—
97	C₂×C₄×C₂₃	—	4	24	no	—
98	C₂×C₄×C₂₃	—	4	24	no	—
99	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
100	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
101	C₂×C₄×C₂₃	#120	1	16	yes	—
102	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic
103	C₂×#380	C₂×C₂₃	1	6	yes	complemented
104	C₂×#379	C₂×C₂₃	1	2	yes	—
105	C₂²⋊#95	C₂×C₂₃	1	6	yes	complemented
106	C₂²×C₂₃	—	2	128	no	—
107	C₂²×C₂₃	—	2	48	no	—
108	C₂²×C₂₃	—	2	48	no	—
109	C₂²×C₂₃	—	2	12	no	—
110	C₂²×C₂₃	—	2	24	no	—
111	C₂²×C₂₃	—	2	12	no	—
112	C₂²×C₂₃	—	2	24	no	—
113	C₂²×C₂₃	#380	1	24	yes	complemented
114	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	3	yes	—
115	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×#117	1	3	yes	—
117	C₂²×C₂₃	C₂×#120	1	4	yes	—
118	C₂²×C₂₃	C₂×#121	1	6	yes	—
119	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	6	yes	—
120	C₂²×C₂₃	#525	1	12	yes	—
121	C₄×C₂₃	—	2	12	no	—
122	C₄×C₂₃	—	8	8	no	—
123	C₄×C₂₃	#379	1	8	yes	complemented
124	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
125	C₂⁴×C₄	—	2	24	no	—
126	C₂⁴×C₄	C₄×C₂₃	1	8	yes	complemented
127	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
128	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
130	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	—
131	#380	—	2	24	no	—
132	#379	—	2	8	no	—
133	C₂×#95	C₂²×C₂₃	1	12	yes	complemented
134	C₂×C₂₃	—	2	24	no	—
135	C₂×C₂₃	C₂×#379	1	1	yes	characteristic
136	C₂×C₂₃	C₂×#380	1	3	yes	—
137	C₂×C₂₃	#2280	1	3	yes	—
138	C₂×C₂₃	#2847	1	4	yes	—
139	C₂×C₂₃	#2848	1	4	yes	—
140	C₂⁵	—	2	12	no	—
141	C₂⁵	—	2	12	no	—
142	C₂⁵	C₂×C₄×C₂₃	1	4	yes	—
143	C₂⁵	D₄×C₂₃	1	4	yes	—
144	C₂⁵	Q₈×C₂₃	1	4	yes	—
145	C₂⁵	C₂³×C₂₃	1	3	yes	—
146	C₂³×C₄	—	2	128	no	—
147	C₂³×C₄	—	2	12	no	—
148	C₂³×C₄	—	2	96	no	—
149	C₂³×C₄	—	2	24	no	—
150	C₂³×C₄	—	2	24	no	—
151	C₂³×C₄	C₂×C₄×C₂₃	1	12	yes	—
152	C₂³×C₄	C₂×C₄×C₂₃	1	12	yes	—
153	C₂³×C₄	D₄×C₂₃	1	24	yes	—
154	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
155	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
156	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	2	24	no	—
158	C₂×C₂²⋊C₄	—	2	24	no	—
159	C₂×C₂²⋊C₄	—	2	24	no	—
160	C₂×C₂²⋊C₄	—	2	24	no	—
161	#95	—	2	24	no	—
162	C₂₃	#14680	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	—	2	32	no	—
164	C₂⁴	—	2	96	no	—
165	C₂⁴	—	2	24	no	—
166	C₂⁴	—	2	24	no	—
167	C₂⁴	—	2	48	no	—
168	C₂⁴	—	2	48	no	—
169	C₂⁴	—	2	12	no	—
170	C₂⁴	—	2	12	no	—
171	C₂⁴	C₄⋊C₄×C₂₃	1	16	yes	complemented
172	C₂⁴	C₂²×C₄×C₂₃	1	6	yes	—
173	C₂⁴	C₂×D₄×C₂₃	1	12	yes	—
174	C₂⁴	C₂×D₄×C₂₃	1	6	yes	—
175	C₂⁴	C₂×Q₈×C₂₃	1	6	yes	—
176	C₂⁴	#41×C₂₃	1	12	yes	—
177	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	—	2	96	no	—
179	C₂²×C₄	—	2	256	no	—
180	C₂²×C₄	—	2	24	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	48	no	—
183	C₂²×C₄	—	2	48	no	—
184	C₂²×C₄	—	2	96	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	48	yes	complemented
189	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
190	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
192	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
193	C₄⋊C₄	—	4	16	no	—
194	C₂²⋊C₄	—	4	48	no	—
195	C₂³	—	2	32	no	—
196	C₂³	—	2	96	no	—
197	C₂³	—	2	128	no	—
198	C₂³	—	2	128	no	—
199	C₂³	—	2	96	no	—
200	C₂³	—	2	12	no	—
201	C₂³	—	2	12	no	—
202	C₂³	—	2	24	no	—
203	C₂³	—	2	24	no	—
204	C₂³	—	2	24	no	—
205	C₂³	—	2	24	no	—
206	C₂³	—	2	48	no	—
207	C₂³	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
208	C₂³	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
209	C₂³	C₂₃×#117	1	12	yes	—
210	C₂³	C₂₃×#121	1	24	yes	—
211	C₂³	C₂₃×C₂²⋊Q₈	1	24	yes	—
212	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
213	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
214	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
215	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
216	C₂³	C₂×#41×C₂₃	1	6	yes	—
217	C₂×C₄	—	2	96	no	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	24	no	—
220	C₂×C₄	—	2	96	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	24	no	—
223	C₂×C₄	—	4	24	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
226	C₂×C₄	C₂₃×#120	1	16	yes	—
227	C₂²	—	2	128	no	—
228	C₂²	—	2	12	no	—
229	C₂²	—	2	24	no	—
230	C₂²	—	2	12	no	—
231	C₂²	—	2	24	no	—
232	C₂²	—	2	48	no	—
233	C₂²	—	2	48	no	—
234	C₂²	C₂₃×#380	1	24	yes	complemented
235	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	3	yes	—
236	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
237	C₂²	C₂×C₂₃×#117	1	3	yes	—
238	C₂²	C₂×C₂₃×#120	1	4	yes	—
239	C₂²	C₂×C₂₃×#121	1	6	yes	—
240	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	6	yes	—
241	C₂²	C₂₃×#525	1	12	yes	—
242	C₄	—	2	12	no	—
243	C₄	—	8	8	no	—
244	C₄	C₂₃×#379	1	8	yes	complemented
245	C₂	—	2	24	no	—
246	C₂	C₂×C₂₃×#379	1	1	yes	characteristic
247	C₂	C₂×C₂₃×#380	1	3	yes	—
248	C₂	C₂₃×#2280	1	3	yes	—
249	C₂	C₂₃×#2847	1	4	yes	—
250	C₂	C₂₃×#2848	1	4	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×#380	C₂	semidirect
4. C₂₃×C₂²⋊#95	C₂	semidirect
7. C₂⁴×C₄×C₂₃	C₄	semidirect
10. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
14. C₂×C₂₃×#95	C₂²	semidirect
45. C₂⁴×C₂₃	C₄⋊C₄	semidirect
62. C₂²×C₄×C₂₃	C₂²⋊C₄	semidirect
69. #14680	C₂₃	direct (N × H)
103. C₂×#380	C₂×C₂₃	semidirect
105. C₂²⋊#95	C₂×C₂₃	semidirect
113. C₂²×C₂₃	#380	semidirect
123. C₄×C₂₃	#379	semidirect
126. C₂⁴×C₄	C₄×C₂₃	semidirect
129. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
133. C₂×#95	C₂²×C₂₃	semidirect
162. C₂₃	#14680	direct (N × H)
171. C₂⁴	C₄⋊C₄×C₂₃	semidirect
188. C₂²×C₄	C₂²⋊C₄×C₂₃	semidirect
234. C₂²	C₂₃×#380	semidirect
244. C₄	C₂₃×#379	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×#380	2
C₂×C₂₃×#379	2
C₂₃×C₂²⋊#95	2
#14680	23

Subgroups — C23×#14680 #19162306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#14680 #19162306