Subgroups — C₂₃×#7569 #19155195

Order 5888.

684 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	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1270	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#127	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄×C₂₃	—	2	2	no	—
6	C₂×C₄×D₄×C₂₃	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₄×C₈×C₂₃	—	2	2	no	—
9	C₂₃×#390	—	2	2	no	—
10	C₂₃×#390	C₂²	1	2	yes	—
11	C₂₃×#384	C₄	1	1	yes	characteristic
12	C₄³×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#336	C₂²	1	4	yes	—
14	C₂²×D₄×C₂₃	—	2	2	no	—
15	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂₃×#127	—	2	2	no	—
19	C₂₃×#127	C₈	1	4	yes	complemented
20	C₄×D₄×C₂₃	—	4	4	no	—
21	C₄×D₄×C₂₃	—	4	4	no	—
22	C₄×D₄×C₂₃	—	4	4	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
26	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	D₄	1	2	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×C₄⋊C₈	—	2	8	no	—
33	C₄×C₈×C₂₃	—	4	4	no	—
34	C₂⁴×C₂₃	—	2	2	no	—
35	C₂×D₄×C₂₃	—	2	8	no	—
36	C₂×D₄×C₂₃	—	2	2	no	—
37	C₂×D₄×C₂₃	—	4	2	no	—
38	C₂×D₄×C₂₃	—	4	2	no	—
39	C₂×D₄×C₂₃	—	4	4	no	—
40	C₂×D₄×C₂₃	—	4	4	no	—
41	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	—	4	4	no	—
44	C₂²×C₄×C₂₃	—	4	8	no	—
45	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₂₃	—	2	8	no	—
51	C₂×C₈×C₂₃	—	4	4	no	—
52	C₂×C₈×C₂₃	—	4	4	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	—	4	4	no	—
55	C₂²⋊C₄×C₂₃	—	4	4	no	—
56	C₂²⋊C₄×C₂₃	—	4	4	no	—
57	C₂²⋊C₄×C₂₃	—	4	4	no	—
58	C₄²×C₂₃	—	2	2	no	—
59	C₄²×C₂₃	—	2	4	no	—
60	C₄²×C₂₃	—	4	1	no	—
61	C₄²×C₂₃	C₂×C₈	1	2	yes	—
62	C₄²×C₂₃	M₁₆	1	2	yes	—
63	C₄²×C₂₃	D₈	1	4	yes	—
64	C₄²×C₂₃	SD₁₆	1	4	yes	—
65	#7569	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	—	4	2	no	—
68	C₂³×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	—	4	4	no	—
70	C₂³×C₂₃	—	4	4	no	—
71	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	D₄×C₂₃	—	4	4	no	—
73	D₄×C₂₃	—	4	4	no	—
74	D₄×C₂₃	—	4	16	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	—	8	8	no	—
84	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
85	C₂×C₄×C₂₃	#100	1	2	yes	—
86	C₂×C₄×C₂₃	#102	1	4	yes	—
87	C₂×C₄×C₂₃	#102	1	4	yes	—
88	C₂×C₄×C₂₃	#104	1	4	yes	—
89	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
93	C₈×C₂₃	—	4	8	no	—
94	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
95	C₄×C₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
96	#1270	C₂×C₂₃	1	4	yes	complemented
97	C₂²×C₂₃	—	4	4	no	—
98	C₂²×C₂₃	—	4	8	no	—
99	C₂²×C₂₃	—	8	2	no	—
100	C₂²×C₂₃	—	8	4	no	—
101	C₂²×C₂₃	—	8	2	no	—
102	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₂₃	#410	1	1	yes	characteristic
104	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
105	C₂²×C₂₃	C₂×#102	1	2	yes	—
106	C₂²×C₂₃	C₂×#104	1	2	yes	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	—	8	2	no	—
113	C₄×C₂₃	D₄⋊C₈	1	8	yes	—
114	C₄×C₂₃	#331	1	4	yes	—
115	C₂×#127	C₄×C₂₃	1	1	yes	characteristic
116	C₂×C₄×D₄	—	2	2	no	—
117	C₂×C₄×D₄	C₄×C₂₃	1	2	yes	—
118	C₂×C₄⋊C₈	—	2	2	no	—
119	C₂×C₄×C₈	—	2	2	no	—
120	#390	—	2	2	no	—
121	#390	C₂²×C₂₃	1	2	yes	—
122	#384	C₄×C₂₃	1	1	yes	characteristic
123	C₄³	C₂²×C₂₃	1	1	yes	characteristic
124	#336	C₂²×C₂₃	1	4	yes	—
125	C₂×C₂₃	—	8	4	no	—
126	C₂×C₂₃	#1460	1	1	yes	characteristic
127	C₂×C₂₃	C₂×D₄⋊C₈	1	2	yes	—
128	C₂×C₂₃	#1502	1	2	yes	—
129	C₂×C₂₃	#1525	1	1	yes	characteristic
130	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
131	C₂²×D₄	—	2	2	no	—
132	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
133	C₂³×C₄	—	2	2	no	—
134	C₂²×C₈	—	2	4	no	—
135	#127	—	2	2	no	—
136	#127	C₈×C₂₃	1	4	yes	complemented
137	C₄×D₄	—	4	4	no	—
138	C₄×D₄	—	4	4	no	—
139	C₄×D₄	—	4	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
142	C₂×C₂²⋊C₄	—	2	2	no	—
143	C₂×C₂²⋊C₄	—	2	4	no	—
144	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
146	C₂×C₄²	D₄×C₂₃	1	2	yes	—
147	C₂×C₄²	D₄×C₂₃	1	2	yes	—
148	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
149	C₄⋊C₈	—	2	8	no	—
150	C₄×C₈	—	4	4	no	—
151	C₂₃	#7569	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	8	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	4	no	—
158	C₂×D₄	—	4	4	no	—
159	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	—	4	8	no	—
163	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
166	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	—	4	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₂²⋊C₄	—	4	4	no	—
174	C₂²⋊C₄	—	4	4	no	—
175	C₂²⋊C₄	—	4	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	4	1	no	—
179	C₄²	C₂×C₈×C₂₃	1	2	yes	—
180	C₄²	M₁₆×C₂₃	1	2	yes	—
181	C₄²	D₈×C₂₃	1	4	yes	—
182	C₄²	SD₁₆×C₂₃	1	4	yes	—
183	C₂³	—	2	4	no	—
184	C₂³	—	4	2	no	—
185	C₂³	—	4	2	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	4	4	no	—
188	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	D₄	—	4	4	no	—
190	D₄	—	4	4	no	—
191	D₄	—	4	16	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
202	C₂×C₄	C₂₃×#100	1	2	yes	—
203	C₂×C₄	C₂₃×#102	1	4	yes	—
204	C₂×C₄	C₂₃×#102	1	4	yes	—
205	C₂×C₄	C₂₃×#104	1	4	yes	—
206	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
209	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
210	C₈	—	4	8	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	4	8	no	—
213	C₂²	—	8	4	no	—
214	C₂²	—	8	2	no	—
215	C₂²	—	8	2	no	—
216	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
217	C₂²	C₂₃×#410	1	1	yes	characteristic
218	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
219	C₂²	C₂×C₂₃×#102	1	2	yes	—
220	C₂²	C₂×C₂₃×#104	1	2	yes	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	1	no	—
226	C₄	—	8	2	no	—
227	C₄	C₂₃×D₄⋊C₈	1	8	yes	—
228	C₄	C₂₃×#331	1	4	yes	—
229	C₂	—	8	4	no	—
230	C₂	C₂₃×#1460	1	1	yes	characteristic
231	C₂	C₂×C₂₃×D₄⋊C₈	1	2	yes	—
232	C₂	C₂₃×#1502	1	2	yes	—
233	C₂	C₂₃×#1525	1	1	yes	characteristic
234	C₂	C₂×C₂₃×#331	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂×C₂₃×#127	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
3. C₂₃×#1270	C₂	semidirect
19. C₂₃×#127	C₈	semidirect
65. #7569	C₂₃	direct (N × H)
95. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
96. #1270	C₂×C₂₃	semidirect
136. #127	C₈×C₂₃	semidirect
151. C₂₃	#7569	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#127	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1270	2
#7569	23

Subgroups — C23×#7569 #19155195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7569 #19155195