Subgroups — C₂₃×#7555 #19155181

Order 5888.

4566 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	C₂³×C₂²⋊C₄×C₂₃	C₂	1	3	yes	maximal
2	C₂²×C₂₃×#95	C₂	1	4	yes	maximal, complemented
3	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
4	C₂⁴×C₄×C₂₃	C₂²	1	6	yes	—
5	C₂²×C₂²⋊C₄×C₂₃	—	2	12	no	—
6	C₂²×C₂²⋊C₄×C₂₃	—	2	24	no	—
7	C₂²×C₂²⋊C₄×C₂₃	C₄	1	12	yes	complemented
8	C₂×C₂₃×#95	—	2	24	no	—
9	C₂⁵×C₂₃	—	2	12	no	—
10	C₂⁵×C₂₃	—	2	12	no	—
11	C₂⁵×C₂₃	C₂×C₄	1	6	yes	—
12	C₂⁵×C₂₃	D₄	1	6	yes	—
13	C₂⁵×C₂₃	Q₈	1	2	yes	—
14	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
15	C₂³×C₄×C₂₃	—	2	24	no	—
16	C₂³×C₄×C₂₃	—	2	12	no	—
17	C₂³×C₄×C₂₃	—	2	12	no	—
18	C₂³×C₄×C₂₃	—	2	24	no	—
19	C₂³×C₄×C₂₃	C₂×C₄	1	12	yes	—
20	C₂³×C₄×C₂₃	D₄	1	24	yes	—
21	C₂×C₂²⋊C₄×C₂₃	—	2	96	no	—
22	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
23	C₂×C₂²⋊C₄×C₂₃	—	2	48	no	—
24	C₂₃×#95	—	2	32	no	—
25	C₂⁴×C₂₃	—	2	12	no	—
26	C₂⁴×C₂₃	—	2	12	no	—
27	C₂⁴×C₂₃	—	2	48	no	—
28	C₂⁴×C₂₃	—	2	24	no	—
29	C₂⁴×C₂₃	—	2	24	no	—
30	C₂⁴×C₂₃	—	2	24	no	—
31	C₂⁴×C₂₃	—	2	24	no	—
32	C₂⁴×C₂₃	—	4	4	no	—
33	C₂⁴×C₂₃	—	4	24	no	—
34	C₂⁴×C₂₃	—	4	12	no	—
35	C₂⁴×C₂₃	—	4	12	no	—
36	C₂⁴×C₂₃	—	4	8	no	—
37	C₂⁴×C₂₃	—	4	4	no	—
38	C₂⁴×C₂₃	C₄²	1	4	yes	—
39	C₂⁴×C₂₃	C₂²⋊C₄	1	12	yes	—
40	C₂⁴×C₂₃	C₄⋊C₄	1	12	yes	—
41	C₂⁴×C₂₃	C₂²×C₄	1	3	yes	—
42	C₂⁴×C₂₃	C₂×D₄	1	3	yes	—
43	C₂⁴×C₂₃	C₂×D₄	1	12	yes	—
44	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₂⁴×C₂₃	#41	1	12	yes	—
46	C₂²×C₄×C₂₃	—	2	12	no	—
47	C₂²×C₄×C₂₃	—	2	12	no	—
48	C₂²×C₄×C₂₃	—	2	48	no	—
49	C₂²×C₄×C₂₃	—	2	48	no	—
50	C₂²×C₄×C₂₃	—	2	24	no	—
51	C₂²×C₄×C₂₃	—	2	48	no	—
52	C₂²×C₄×C₂₃	—	2	24	no	—
53	C₂²×C₄×C₂₃	—	4	12	no	—
54	C₂²×C₄×C₂₃	—	4	24	no	—
55	C₂²×C₄×C₂₃	—	4	24	no	—
56	C₂²×C₄×C₂₃	—	4	12	no	—
57	C₂²×C₄×C₂₃	—	4	12	no	—
58	C₂²×C₄×C₂₃	—	4	12	no	—
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	24	yes	complemented
60	C₂²⋊C₄×C₂₃	—	2	96	no	—
61	#7555	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	—	2	24	no	—
63	C₂³×C₂₃	—	2	12	no	—
64	C₂³×C₂₃	—	2	24	no	—
65	C₂³×C₂₃	—	2	12	no	—
66	C₂³×C₂₃	—	2	48	no	—
67	C₂³×C₂₃	—	2	48	no	—
68	C₂³×C₂₃	—	4	24	no	—
69	C₂³×C₂₃	—	4	48	no	—
70	C₂³×C₂₃	—	4	24	no	—
71	C₂³×C₂₃	—	4	24	no	—
72	C₂³×C₂₃	—	4	8	no	—
73	C₂³×C₂₃	—	4	48	no	—
74	C₂³×C₂₃	—	4	16	no	—
75	C₂³×C₂₃	—	4	24	no	—
76	C₂³×C₂₃	—	4	8	no	—
77	C₂³×C₂₃	#95	1	8	yes	complemented
78	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
79	C₂³×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
80	C₂³×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
81	C₂³×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
82	C₂³×C₂₃	#117	1	6	yes	—
83	C₂³×C₂₃	C₄×D₄	1	24	yes	—
84	C₂³×C₂₃	#120	1	12	yes	—
85	C₂³×C₂₃	#120	1	4	yes	—
86	C₂³×C₂₃	#121	1	12	yes	—
87	C₂³×C₂₃	#121	1	24	yes	—
88	C₂³×C₂₃	C₂²⋊Q₈	1	12	yes	—
89	C₂³×C₂₃	#123	1	12	yes	—
90	C₂³×C₂₃	#123	1	12	yes	—
91	C₂³×C₂₃	#124	1	12	yes	—
92	C₂³×C₂₃	#126	1	8	yes	—
93	C₂³×C₂₃	#127	1	4	yes	—
94	C₂×C₄×C₂₃	—	2	24	no	—
95	C₂×C₄×C₂₃	—	2	48	no	—
96	C₂×C₄×C₂₃	—	4	24	no	—
97	C₂×C₄×C₂₃	—	4	24	no	—
98	C₂×C₄×C₂₃	—	4	48	no	—
99	C₂×C₄×C₂₃	—	4	48	no	—
100	C₂³×C₂²⋊C₄	C₂×C₂₃	1	3	yes	—
101	C₂²×#95	C₂×C₂₃	1	4	yes	complemented
102	C₂²×C₂₃	—	2	24	no	—
103	C₂²×C₂₃	—	4	16	no	—
104	C₂²×C₂₃	—	4	24	no	—
105	C₂²×C₂₃	—	4	48	no	—
106	C₂²×C₂₃	—	4	24	no	—
107	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
108	C₂²×C₂₃	C₄×C₂²⋊C₄	1	6	yes	—
109	C₂²×C₂₃	#379	1	3	yes	—
110	C₂²×C₂₃	#380	1	6	yes	—
111	C₂²×C₂₃	#381	1	3	yes	—
112	C₂²×C₂₃	#385	1	12	yes	—
113	C₂²×C₂₃	#386	1	12	yes	—
114	C₂²×C₂₃	#386	1	12	yes	—
115	C₂²×C₂₃	#388	1	24	yes	—
116	C₂²×C₂₃	#390	1	12	yes	—
117	C₂²×C₂₃	#392	1	12	yes	—
118	C₂²×C₂₃	#393	1	4	yes	—
119	C₂²×C₂₃	#394	1	24	yes	—
120	C₂²×C₂₃	#396	1	8	yes	—
121	C₂²×C₂₃	#397	1	12	yes	—
122	C₂²×C₂₃	#399	1	4	yes	—
123	C₄×C₂₃	—	4	24	no	—
124	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
125	C₂⁴×C₄	C₂²×C₂₃	1	6	yes	—
126	C₂²×C₂²⋊C₄	—	2	12	no	—
127	C₂²×C₂²⋊C₄	—	2	24	no	—
128	C₂²×C₂²⋊C₄	C₄×C₂₃	1	12	yes	complemented
129	C₂×#95	—	2	24	no	—
130	C₂×C₂₃	—	4	8	no	—
131	C₂×C₂₃	C₂²⋊#95	1	3	yes	—
132	C₂×C₂₃	#1437	1	4	yes	—
133	C₂×C₂₃	#1438	1	6	yes	—
134	C₂×C₂₃	#1439	1	12	yes	—
135	C₂×C₂₃	#1440	1	6	yes	—
136	C₂⁵	—	2	12	no	—
137	C₂⁵	—	2	12	no	—
138	C₂⁵	C₂×C₄×C₂₃	1	6	yes	—
139	C₂⁵	D₄×C₂₃	1	6	yes	—
140	C₂⁵	Q₈×C₂₃	1	2	yes	—
141	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
142	C₂³×C₄	—	2	12	no	—
143	C₂³×C₄	—	2	12	no	—
144	C₂³×C₄	—	2	24	no	—
145	C₂³×C₄	—	2	24	no	—
146	C₂³×C₄	C₂×C₄×C₂₃	1	12	yes	—
147	C₂³×C₄	D₄×C₂₃	1	24	yes	—
148	C₂×C₂²⋊C₄	—	2	96	no	—
149	C₂×C₂²⋊C₄	—	2	24	no	—
150	C₂×C₂²⋊C₄	—	2	48	no	—
151	#95	—	2	32	no	—
152	C₂₃	#7555	1	1	yes	characteristic, direct product factor, Sylow
153	C₂⁴	—	2	12	no	—
154	C₂⁴	—	2	12	no	—
155	C₂⁴	—	2	48	no	—
156	C₂⁴	—	2	24	no	—
157	C₂⁴	—	2	24	no	—
158	C₂⁴	—	2	24	no	—
159	C₂⁴	—	2	24	no	—
160	C₂⁴	—	4	4	no	—
161	C₂⁴	—	4	24	no	—
162	C₂⁴	—	4	12	no	—
163	C₂⁴	—	4	12	no	—
164	C₂⁴	—	4	8	no	—
165	C₂⁴	—	4	4	no	—
166	C₂⁴	C₄²×C₂₃	1	4	yes	—
167	C₂⁴	C₂²⋊C₄×C₂₃	1	12	yes	—
168	C₂⁴	C₄⋊C₄×C₂₃	1	12	yes	—
169	C₂⁴	C₂²×C₄×C₂₃	1	3	yes	—
170	C₂⁴	C₂×D₄×C₂₃	1	3	yes	—
171	C₂⁴	C₂×D₄×C₂₃	1	12	yes	—
172	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
173	C₂⁴	#41×C₂₃	1	12	yes	—
174	C₂²×C₄	—	2	12	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	2	48	no	—
177	C₂²×C₄	—	2	24	no	—
178	C₂²×C₄	—	2	48	no	—
179	C₂²×C₄	—	2	48	no	—
180	C₂²×C₄	—	2	24	no	—
181	C₂²×C₄	—	4	12	no	—
182	C₂²×C₄	—	4	24	no	—
183	C₂²×C₄	—	4	24	no	—
184	C₂²×C₄	—	4	12	no	—
185	C₂²×C₄	—	4	12	no	—
186	C₂²×C₄	—	4	12	no	—
187	C₂²×C₄	C₂²⋊C₄×C₂₃	1	24	yes	complemented
188	C₂²⋊C₄	—	2	96	no	—
189	C₂³	—	2	24	no	—
190	C₂³	—	2	12	no	—
191	C₂³	—	2	24	no	—
192	C₂³	—	2	12	no	—
193	C₂³	—	2	48	no	—
194	C₂³	—	2	48	no	—
195	C₂³	—	4	24	no	—
196	C₂³	—	4	48	no	—
197	C₂³	—	4	24	no	—
198	C₂³	—	4	24	no	—
199	C₂³	—	4	8	no	—
200	C₂³	—	4	48	no	—
201	C₂³	—	4	16	no	—
202	C₂³	—	4	24	no	—
203	C₂³	—	4	8	no	—
204	C₂³	C₂₃×#95	1	8	yes	complemented
205	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
206	C₂³	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
207	C₂³	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
208	C₂³	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
209	C₂³	C₂₃×#117	1	6	yes	—
210	C₂³	C₄×D₄×C₂₃	1	24	yes	—
211	C₂³	C₂₃×#120	1	12	yes	—
212	C₂³	C₂₃×#120	1	4	yes	—
213	C₂³	C₂₃×#121	1	12	yes	—
214	C₂³	C₂₃×#121	1	24	yes	—
215	C₂³	C₂₃×C₂²⋊Q₈	1	12	yes	—
216	C₂³	C₂₃×#123	1	12	yes	—
217	C₂³	C₂₃×#123	1	12	yes	—
218	C₂³	C₂₃×#124	1	12	yes	—
219	C₂³	C₂₃×#126	1	8	yes	—
220	C₂³	C₂₃×#127	1	4	yes	—
221	C₂×C₄	—	2	24	no	—
222	C₂×C₄	—	2	48	no	—
223	C₂×C₄	—	4	24	no	—
224	C₂×C₄	—	4	24	no	—
225	C₂×C₄	—	4	48	no	—
226	C₂×C₄	—	4	48	no	—
227	C₂²	—	2	24	no	—
228	C₂²	—	4	16	no	—
229	C₂²	—	4	24	no	—
230	C₂²	—	4	48	no	—
231	C₂²	—	4	24	no	—
232	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
233	C₂²	C₄×C₂²⋊C₄×C₂₃	1	6	yes	—
234	C₂²	C₂₃×#379	1	3	yes	—
235	C₂²	C₂₃×#380	1	6	yes	—
236	C₂²	C₂₃×#381	1	3	yes	—
237	C₂²	C₂₃×#385	1	12	yes	—
238	C₂²	C₂₃×#386	1	12	yes	—
239	C₂²	C₂₃×#386	1	12	yes	—
240	C₂²	C₂₃×#388	1	24	yes	—
241	C₂²	C₂₃×#390	1	12	yes	—
242	C₂²	C₂₃×#392	1	12	yes	—
243	C₂²	C₂₃×#393	1	4	yes	—
244	C₂²	C₂₃×#394	1	24	yes	—
245	C₂²	C₂₃×#396	1	8	yes	—
246	C₂²	C₂₃×#397	1	12	yes	—
247	C₂²	C₂₃×#399	1	4	yes	—
248	C₄	—	4	24	no	—
249	C₂	—	4	8	no	—
250	C₂	C₂₃×C₂²⋊#95	1	3	yes	—
251	C₂	C₂₃×#1437	1	4	yes	—
252	C₂	C₂₃×#1438	1	6	yes	—
253	C₂	C₂₃×#1439	1	12	yes	—
254	C₂	C₂₃×#1440	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₂₃	C₂³
commutator	C₂³	C₂×C₄²×C₂₃
Frattini, cosocle	C₂⁵	C₂³×C₂₃
prime radical	C₂⁶×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×#95	C₂	semidirect
7. C₂²×C₂²⋊C₄×C₂₃	C₄	semidirect
59. C₂²×C₄×C₂₃	C₂²⋊C₄	semidirect
61. #7555	C₂₃	direct (N × H)
77. C₂³×C₂₃	#95	semidirect
101. C₂²×#95	C₂×C₂₃	semidirect
128. C₂²×C₂²⋊C₄	C₄×C₂₃	semidirect
152. C₂₃	#7555	direct (N × H)
187. C₂²×C₄	C₂²⋊C₄×C₂₃	semidirect
204. C₂³	C₂₃×#95	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊C₄×C₂₃	2
C₂²×C₂₃×#95	2
#7555	23

Subgroups — C23×#7555 #19155181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7555 #19155181