Subgroups — C₂₃×#11599 #19159225

Order 5888.

450 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1333	C₂	1	4	yes	maximal, complemented
5	C₂²×D₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₂₃	—	2	1	no	—
7	C₂×C₂₃×M₃₂	—	2	1	no	—
8	C₄×D₈×C₂₃	—	2	1	no	—
9	C₄×D₈×C₂₃	—	2	1	no	—
10	C₄×D₈×C₂₃	C₄	1	2	yes	—
11	C₄×D₈×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#431	—	2	1	no	—
13	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
14	C₂×C₂₃×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#346	C₂²	1	2	yes	complemented
17	C₄×C₁₆×C₂₃	—	2	1	no	—
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	—	4	1	no	—
22	C₂×D₈×C₂₃	—	2	1	no	—
23	C₂×D₈×C₂₃	—	2	2	no	—
24	C₂×D₈×C₂₃	C₈	1	4	yes	complemented
25	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₄×D₄×C₂₃	—	2	2	no	—
29	C₄×D₄×C₂₃	—	2	2	no	—
30	C₄×D₄×C₂₃	—	4	1	no	—
31	C₄×D₄×C₂₃	—	4	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂₃×M₃₂	—	2	4	no	—
36	C₂×C₁₆×C₂₃	—	2	2	no	—
37	C₂₃×#107	—	2	1	no	—
38	C₂₃×#107	C₂×C₄	1	2	yes	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×#102	—	2	2	no	—
41	C₂₃×#102	—	2	2	no	—
42	C₂₃×C₂²⋊C₈	—	4	1	no	—
43	C₄×C₈×C₂₃	—	4	1	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₂₃	—	4	1	no	—
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	—	2	2	no	—
51	C₂×D₄×C₂₃	—	4	1	no	—
52	C₂×D₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	—	4	1	no	—
54	C₂²×C₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	—	8	4	no	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
60	D₈×C₂₃	—	2	4	no	—
61	D₈×C₂₃	—	2	2	no	—
62	D₈×C₂₃	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₄⋊C₄×C₂₃	—	4	1	no	—
73	C₂²⋊C₄×C₂₃	—	4	2	no	—
74	C₂²⋊C₄×C₂₃	—	4	2	no	—
75	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₂₃	D₈	1	1	yes	characteristic
77	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
78	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₁₆×C₂₃	—	2	4	no	—
80	#11599	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	—	4	4	no	—
82	C₂³×C₂₃	—	4	2	no	—
83	C₂³×C₂₃	—	8	2	no	—
84	C₂³×C₂₃	—	8	2	no	—
85	C₂³×C₂₃	#102	1	1	yes	characteristic
86	D₄×C₂₃	—	2	4	no	—
87	D₄×C₂₃	—	4	4	no	—
88	D₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	4	1	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	8	4	no	—
93	C₂×C₄×C₂₃	—	8	2	no	—
94	C₂×C₄×C₂₃	—	8	4	no	—
95	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#104	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₂₃	—	2	1	no	—
104	C₈×C₂₃	—	2	1	no	—
105	C₈×C₂₃	—	8	2	no	—
106	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
107	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
108	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
110	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
111	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
112	#1333	C₂×C₂₃	1	4	yes	complemented
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	4	4	no	—
116	C₂²×C₂₃	—	8	8	no	—
117	C₂²×C₂₃	—	8	2	no	—
118	C₂²×C₂₃	—	8	2	no	—
119	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
120	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
121	C₄×C₂₃	—	2	1	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	8	2	no	—
125	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
126	C₄×C₂₃	#361	1	2	yes	—
127	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
129	C₂²×D₈	C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₄×D₄	—	2	1	no	—
131	C₂×M₃₂	—	2	1	no	—
132	C₄×D₈	—	2	1	no	—
133	C₄×D₈	—	2	1	no	—
134	C₄×D₈	C₄×C₂₃	1	2	yes	—
135	C₄×D₈	C₂²×C₂₃	1	2	yes	—
136	#431	—	2	1	no	—
137	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
138	C₂×#102	—	2	1	no	—
139	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
140	#346	C₂²×C₂₃	1	2	yes	complemented
141	C₄×C₁₆	—	2	1	no	—
142	#335	—	2	1	no	—
143	#334	C₂²×C₂₃	1	2	yes	complemented
144	C₂×C₂₃	—	2	2	no	—
145	C₂×C₂₃	—	8	4	no	—
146	C₂×C₂₃	C₂×D₄⋊C₈	1	1	yes	characteristic
147	C₂×C₂₃	#2145	1	1	yes	characteristic
148	C₂×C₂₃	C₂×#361	1	1	yes	characteristic
149	C₂²×D₄	—	2	1	no	—
150	C₂³×C₄	—	4	1	no	—
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	C₈×C₂₃	1	4	yes	complemented
154	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
155	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	4	1	no	—
160	C₄×D₄	—	4	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₂²⋊C₄	—	4	1	no	—
163	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
164	M₃₂	—	2	4	no	—
165	C₂×C₁₆	—	2	2	no	—
166	#107	—	2	1	no	—
167	#107	C₂×C₄×C₂₃	1	2	yes	—
168	C₄⋊C₈	—	2	2	no	—
169	#102	—	2	2	no	—
170	#102	—	2	2	no	—
171	C₂²⋊C₈	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
175	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
176	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₂₃	#11599	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	—	4	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	8	4	no	—
187	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
190	D₈	—	2	2	no	—
191	D₈	—	2	4	no	—
192	D₈	C₂×C₈×C₂₃	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
198	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
199	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₄²	D₈×C₂₃	1	1	yes	characteristic
207	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₁₆	—	2	4	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	—	8	2	no	—
214	C₂³	C₂₃×#102	1	1	yes	characteristic
215	D₄	—	2	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
225	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
226	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
227	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
228	C₂×C₄	C₂₃×#104	1	2	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
231	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	8	2	no	—
235	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
236	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
237	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	8	8	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	2	no	—
244	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
245	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	8	2	no	—
250	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
251	C₄	C₂₃×#361	1	2	yes	—
252	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
253	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	8	4	no	—
256	C₂	C₂×C₂₃×D₄⋊C₈	1	1	yes	characteristic
257	C₂	C₂₃×#2145	1	1	yes	characteristic
258	C₂	C₂×C₂₃×#361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#102
prime radical	C₂²×D₈×C₂₃	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×M₃₂	C₂	semidirect
3. C₂₃×#1564	C₂	semidirect
4. C₂₃×#1333	C₂	semidirect
16. C₂₃×#346	C₂²	semidirect
19. C₂₃×#334	C₂²	semidirect
24. C₂×D₈×C₂₃	C₈	semidirect
80. #11599	C₂₃	direct (N × H)
110. C₄×M₃₂	C₂×C₂₃	semidirect
111. #1564	C₂×C₂₃	semidirect
112. #1333	C₂×C₂₃	semidirect
140. #346	C₂²×C₂₃	semidirect
143. #334	C₂²×C₂₃	semidirect
153. C₂×D₈	C₈×C₂₃	semidirect
177. C₂₃	#11599	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₂₃	2
C₄×C₂₃×M₃₂	2
C₂₃×#1564	2
C₂₃×#1333	2
#11599	23

Subgroups — C23×#11599 #19159225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11599 #19159225