Subgroups — C₂₃×#23798 #19171424

Order 5888.

1322 conjugacy classes of subgroups.

The subgroup lattice has 214 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	4	yes	maximal, complemented
2	C₄×C₂₃×#127	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2093	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2011	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₂₃	C₂²	1	4	yes	—
7	C₂×C₂₃×#127	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₂₃	—	2	4	no	—
9	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
10	C₄×D₈×C₂₃	—	2	16	no	—
11	C₂×C₂₃×#107	C₂²	1	4	yes	—
12	C₂×C₂₃×#102	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#390	—	2	4	no	—
15	C₂₃×#390	C₂²	1	4	yes	—
16	C₂₃×#384	C₂²	1	2	yes	—
17	C₄³×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#336	C₂²	1	4	yes	—
19	C₂²×D₄×C₂₃	—	2	4	no	—
20	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
21	C₂³×C₄×C₂₃	—	2	4	no	—
22	C₂×D₈×C₂₃	—	2	16	no	—
23	C₂×D₈×C₂₃	—	2	8	no	—
24	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
25	C₂₃×#127	—	2	8	no	—
26	C₄×D₄×C₂₃	—	2	16	no	—
27	C₄×D₄×C₂₃	—	4	16	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
30	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
31	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
32	C₂×C₄²×C₂₃	D₄	1	2	yes	—
33	C₂×C₄²×C₂₃	D₄	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×#107	—	2	8	no	—
37	C₂₃×#102	—	2	16	no	—
38	C₄×C₈×C₂₃	D₄	1	8	yes	complemented
39	C₂⁴×C₂₃	—	2	4	no	—
40	C₂×D₄×C₂₃	—	2	16	no	—
41	C₂×D₄×C₂₃	—	2	8	no	—
42	C₂×D₄×C₂₃	—	2	8	no	—
43	C₂×D₄×C₂₃	—	2	4	no	—
44	C₂×D₄×C₂₃	—	4	8	no	—
45	C₂×D₄×C₂₃	—	4	8	no	—
46	C₂×D₄×C₂₃	#41	1	8	yes	—
47	C₂²×C₄×C₂₃	—	2	4	no	—
48	C₂²×C₄×C₂₃	—	4	8	no	—
49	C₂²×C₄×C₂₃	—	4	16	no	—
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
54	D₈×C₂₃	—	2	32	no	—
55	C₂×C₈×C₂₃	—	2	8	no	—
56	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
57	C₄⋊C₄×C₂₃	—	2	8	no	—
58	C₄⋊C₄×C₂₃	—	4	8	no	—
59	C₂²⋊C₄×C₂₃	—	4	8	no	—
60	C₂²⋊C₄×C₂₃	—	4	16	no	—
61	C₄²×C₂₃	—	2	4	no	—
62	C₄²×C₂₃	—	4	2	no	—
63	C₄²×C₂₃	D₈	1	8	yes	complemented
64	C₄²×C₂₃	C₂×D₄	1	4	yes	—
65	#23798	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	2	8	no	—
67	C₂³×C₂₃	—	4	4	no	—
68	C₂³×C₂₃	—	4	4	no	—
69	C₂³×C₂₃	—	4	8	no	—
70	C₂³×C₂₃	—	4	8	no	—
71	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
72	D₄×C₂₃	—	2	16	no	—
73	D₄×C₂₃	—	4	32	no	—
74	D₄×C₂₃	—	4	8	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	—	4	8	no	—
80	C₂×C₄×C₂₃	—	8	16	no	—
81	C₂×C₄×C₂₃	#127	1	2	yes	—
82	C₂×C₄×C₂₃	#127	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×D₈	1	8	yes	—
84	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
85	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
87	C₈×C₂₃	—	2	8	no	—
88	C₂×C₄×D₈	C₂×C₂₃	1	4	yes	complemented
89	C₄×#127	C₂×C₂₃	1	2	yes	complemented
90	#2093	C₂×C₂₃	1	4	yes	complemented
91	#2011	C₂×C₂₃	1	4	yes	complemented
92	#1744	C₂×C₂₃	1	1	yes	characteristic, complemented
93	C₂²×C₂₃	—	4	16	no	—
94	C₂²×C₂₃	—	4	8	no	—
95	C₂²×C₂₃	—	8	8	no	—
96	C₂²×C₂₃	—	8	4	no	—
97	C₂²×C₂₃	—	8	4	no	—
98	C₂²×C₂₃	#525	1	2	yes	—
99	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
100	C₂²×C₂₃	C₂²×D₈	1	2	yes	—
101	C₂²×C₂₃	#575	1	2	yes	—
102	C₄×C₂₃	—	2	8	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	8	4	no	—
105	C₄×C₂₃	#493	1	4	yes	—
106	C₄×C₂₃	#525	1	4	yes	—
107	C₂²×D₈	C₂²×C₂₃	1	4	yes	—
108	C₂×#127	C₂²×C₂₃	1	2	yes	—
109	C₂×C₄×D₄	—	2	4	no	—
110	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
111	C₄×D₈	—	2	16	no	—
112	C₂×#107	C₂²×C₂₃	1	4	yes	—
113	C₂×#102	C₂²×C₂₃	1	8	yes	—
114	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
115	#390	—	2	4	no	—
116	#390	C₂²×C₂₃	1	4	yes	—
117	#384	C₂²×C₂₃	1	2	yes	—
118	C₄³	C₂²×C₂₃	1	1	yes	characteristic
119	#336	C₂²×C₂₃	1	4	yes	—
120	C₂×C₂₃	—	8	8	no	—
121	C₂×C₂₃	#2397	1	1	yes	characteristic
122	C₂×C₂₃	#3100	1	4	yes	—
123	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
124	C₂×C₂₃	#3150	1	1	yes	characteristic
125	C₂²×D₄	—	2	4	no	—
126	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
127	C₂³×C₄	—	2	4	no	—
128	C₂×D₈	—	2	16	no	—
129	C₂×D₈	—	2	8	no	—
130	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
131	#127	—	2	8	no	—
132	C₄×D₄	—	2	16	no	—
133	C₄×D₄	—	4	16	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
136	C₂×C₂²⋊C₄	—	2	8	no	—
137	C₂×C₂²⋊C₄	—	2	4	no	—
138	C₂×C₄²	D₄×C₂₃	1	2	yes	—
139	C₂×C₄²	D₄×C₂₃	1	2	yes	—
140	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
141	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
142	#107	—	2	8	no	—
143	#102	—	2	16	no	—
144	C₄×C₈	D₄×C₂₃	1	8	yes	complemented
145	C₂₃	#23798	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	—	2	4	no	—
147	C₂×D₄	—	2	16	no	—
148	C₂×D₄	—	2	8	no	—
149	C₂×D₄	—	2	8	no	—
150	C₂×D₄	—	2	4	no	—
151	C₂×D₄	—	4	8	no	—
152	C₂×D₄	—	4	8	no	—
153	C₂×D₄	#41×C₂₃	1	8	yes	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	8	no	—
156	C₂²×C₄	—	4	16	no	—
157	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
159	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
161	D₈	—	2	32	no	—
162	C₂×C₈	—	2	8	no	—
163	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	4	8	no	—
166	C₂²⋊C₄	—	4	8	no	—
167	C₂²⋊C₄	—	4	16	no	—
168	C₄²	—	2	4	no	—
169	C₄²	—	4	2	no	—
170	C₄²	D₈×C₂₃	1	8	yes	complemented
171	C₄²	C₂×D₄×C₂₃	1	4	yes	—
172	C₂³	—	2	8	no	—
173	C₂³	—	4	8	no	—
174	C₂³	—	4	8	no	—
175	C₂³	—	4	4	no	—
176	C₂³	—	4	4	no	—
177	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
178	D₄	—	2	16	no	—
179	D₄	—	4	32	no	—
180	D₄	—	4	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	16	no	—
187	C₂×C₄	C₂₃×#127	1	2	yes	—
188	C₂×C₄	C₂₃×#127	1	2	yes	—
189	C₂×C₄	C₂×D₈×C₂₃	1	8	yes	—
190	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
191	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
192	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
193	C₈	—	2	8	no	—
194	C₂²	—	4	8	no	—
195	C₂²	—	4	16	no	—
196	C₂²	—	8	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	4	no	—
199	C₂²	C₂₃×#525	1	2	yes	—
200	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
201	C₂²	C₂²×D₈×C₂₃	1	2	yes	—
202	C₂²	C₂₃×#575	1	2	yes	—
203	C₄	—	2	8	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	C₂₃×#493	1	4	yes	—
207	C₄	C₂₃×#525	1	4	yes	—
208	C₂	—	8	8	no	—
209	C₂	C₂₃×#2397	1	1	yes	characteristic
210	C₂	C₂₃×#3100	1	4	yes	—
211	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
212	C₂	C₂₃×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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
1. C₂×C₄×D₈×C₂₃	C₂	semidirect
2. C₄×C₂₃×#127	C₂	semidirect
3. C₂₃×#2093	C₂	semidirect
4. C₂₃×#2011	C₂	semidirect
5. C₂₃×#1744	C₂	semidirect
38. C₄×C₈×C₂₃	D₄	semidirect
63. C₄²×C₂₃	D₈	semidirect
65. #23798	C₂₃	direct (N × H)
88. C₂×C₄×D₈	C₂×C₂₃	semidirect
89. C₄×#127	C₂×C₂₃	semidirect
90. #2093	C₂×C₂₃	semidirect
91. #2011	C₂×C₂₃	semidirect
92. #1744	C₂×C₂₃	semidirect
144. C₄×C₈	D₄×C₂₃	semidirect
145. C₂₃	#23798	direct (N × H)
170. C₄²	D₈×C₂₃	semidirect

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×#127	2
C₂₃×#2093	2
C₂₃×#2011	2
C₂₃×#1744	2
#23798	23

Subgroups — C23×#23798 #19171424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23798 #19171424