Subgroups — C₂₃×#8599 #19156225

Order 5888.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1797	C₂	1	2	yes	maximal
3	C₂×C₂₃×#335	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1277	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#431	—	2	4	no	—
7	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#336	—	2	2	no	—
14	C₂₃×#335	—	2	4	no	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	—	2	4	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₂₃	—	2	2	no	—
22	C₂×C₄²×C₂₃	—	2	2	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	C₂₃×C₄⋊C₈	—	2	4	no	—
28	C₂₃×C₄⋊C₈	—	2	4	no	—
29	C₂₃×C₄⋊C₈	—	2	4	no	—
30	C₂₃×C₂²⋊C₈	—	2	4	no	—
31	C₂₃×C₂²⋊C₈	—	4	4	no	—
32	C₄×C₈×C₂₃	—	2	2	no	—
33	C₄×C₈×C₂₃	D₄	1	4	yes	—
34	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	1	no	—
38	C₂²×C₄×C₂₃	—	2	1	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	4	2	no	—
42	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	D₈	1	2	yes	—
46	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	4	no	—
53	C₂×C₈×C₂₃	—	4	4	no	—
54	C₂×C₈×C₂₃	—	4	4	no	—
55	C₂×C₈×C₂₃	—	4	4	no	—
56	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₂₃	—	2	1	no	—
58	C₄²×C₂₃	—	2	1	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	—	2	2	no	—
61	C₄²×C₂₃	C₂×D₄	1	2	yes	—
62	C₄²×C₂₃	#41	1	2	yes	—
63	#8599	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	—	2	1	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	1	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	#107	1	2	yes	—
69	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	#107	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#121	1	2	yes	—
86	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
87	C₂×C₄×C₂₃	#131	1	4	yes	—
88	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₂₃	#135	1	2	yes	—
91	C₈×C₂₃	—	2	4	no	—
92	C₈×C₂₃	—	4	2	no	—
93	C₈×C₂₃	—	8	4	no	—
94	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
95	#1797	C₂×C₂₃	1	2	yes	—
96	C₂×#335	C₂×C₂₃	1	2	yes	complemented
97	#1277	C₂×C₂₃	1	2	yes	complemented
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	4	no	—
100	C₂²×C₂₃	—	4	1	no	—
101	C₂²×C₂₃	—	4	1	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	#380	1	1	yes	characteristic
104	C₂²×C₂₃	#424	1	2	yes	—
105	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
106	C₂²×C₂₃	#427	1	1	yes	characteristic
107	C₂²×C₂₃	#430	1	2	yes	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	#408	1	2	yes	—
113	C₄×C₂₃	#433	1	2	yes	—
114	C₄×C₂₃	#466	1	2	yes	—
115	C₄×C₂₃	#467	1	2	yes	—
116	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
117	#431	—	2	4	no	—
118	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
120	C₂×C₂²⋊C₈	—	2	2	no	—
121	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	#336	—	2	2	no	—
125	#335	—	2	4	no	—
126	C₂×C₂₃	—	4	2	no	—
127	C₂×C₂₃	#1571	1	2	yes	—
128	C₂×C₂₃	#1572	1	2	yes	—
129	C₂×C₂₃	#1798	1	1	yes	characteristic
130	C₂×C₂₃	#1808	1	1	yes	characteristic
131	C₂×C₂₃	#1811	1	1	yes	characteristic
132	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
134	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
135	C₂²×C₈	—	2	4	no	—
136	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
137	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
142	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
144	C₄⋊C₈	—	2	4	no	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	2	4	no	—
147	C₂²⋊C₈	—	2	4	no	—
148	C₂²⋊C₈	—	4	4	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	D₄×C₂₃	1	4	yes	—
151	C₂₃	#8599	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	D₈×C₂₃	1	2	yes	—
164	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₂×D₄×C₂₃	1	2	yes	—
180	C₄²	#41×C₂₃	1	2	yes	—
181	C₂³	—	2	1	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	1	no	—
184	C₂³	—	2	2	no	—
185	C₂³	C₂₃×#107	1	2	yes	—
186	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂₃×#107	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
202	C₂×C₄	C₂₃×#121	1	2	yes	—
203	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
204	C₂×C₄	C₂₃×#131	1	4	yes	—
205	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂₃×#135	1	2	yes	—
208	C₈	—	2	4	no	—
209	C₈	—	4	2	no	—
210	C₈	—	8	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂₃×#380	1	1	yes	characteristic
217	C₂²	C₂₃×#424	1	2	yes	—
218	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
219	C₂²	C₂₃×#427	1	1	yes	characteristic
220	C₂²	C₂₃×#430	1	2	yes	—
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	C₂₃×#408	1	2	yes	—
226	C₄	C₂₃×#433	1	2	yes	—
227	C₄	C₂₃×#466	1	2	yes	—
228	C₄	C₂₃×#467	1	2	yes	—
229	C₂	—	4	2	no	—
230	C₂	C₂₃×#1571	1	2	yes	—
231	C₂	C₂₃×#1572	1	2	yes	—
232	C₂	C₂₃×#1798	1	1	yes	characteristic
233	C₂	C₂₃×#1808	1	1	yes	characteristic
234	C₂	C₂₃×#1811	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#335	C₂	semidirect
4. C₂₃×#1277	C₂	semidirect
63. #8599	C₂₃	direct (N × H)
96. C₂×#335	C₂×C₂₃	semidirect
97. #1277	C₂×C₂₃	semidirect
151. C₂₃	#8599	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×C₂₃×#431	2
C₂₃×#1797	2
C₂×C₂₃×#335	2
C₂₃×#1277	2
#8599	23

Subgroups — C23×#8599 #19156225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8599 #19156225