Subgroups — C₂₃×#36640 #19184266

Order 5888.

1990 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2864	C₂	1	4	yes	maximal, complemented
2	C₂₃×#2859	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2851	C₂	1	8	yes	maximal
4	C₄×C₂₃×#125	C₂	1	8	yes	maximal, complemented
5	C₂₃×#2296	C₂	1	4	yes	maximal
6	C₂₃×#2285	C₂	1	2	yes	maximal
7	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×#125	C₂²	1	8	yes	—
9	C₂×C₂₃×#117	C₂²	1	4	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
12	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#389	C₂²	1	16	yes	—
14	C₂₃×#389	C₂²	1	16	yes	—
15	C₂₃×#387	C₂²	1	32	yes	—
16	C₂₃×#385	C₂²	1	16	yes	—
17	C₂₃×#384	C₂²	1	4	yes	—
18	C₂₃×#382	C₂²	1	4	yes	—
19	C₂₃×#382	C₂²	1	8	yes	—
20	C₂₃×#380	C₂²	1	8	yes	—
21	C₄×C₄⋊C₄×C₂₃	C₂²	1	8	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	16	yes	—
23	C₄³×C₂₃	C₂²	1	4	yes	—
24	C₄³×C₂₃	C₂²	1	2	yes	—
25	C₄³×C₂₃	C₂²	1	2	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×#125	—	2	32	no	—
30	C₂₃×#117	—	2	16	no	—
31	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
32	C₂×C₄⋊C₄×C₂₃	C₂³	1	32	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
34	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
35	C₂×C₄²×C₂₃	—	2	8	no	—
36	C₂×C₄²×C₂₃	—	2	16	no	—
37	C₂×C₄²×C₂₃	Q₈	1	8	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
47	C₂₃×#95	C₂³	1	16	yes	—
48	C₂₃×#95	C₂³	1	32	yes	—
49	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	8	no	—
51	C₂²×C₄×C₂₃	—	2	16	no	—
52	C₂²×C₄×C₂₃	—	2	8	no	—
53	C₂²×C₄×C₂₃	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₂₃	#41	1	8	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	16	yes	—
57	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	—	2	16	no	—
63	C₄⋊C₄×C₂₃	—	2	64	no	—
64	C₄⋊C₄×C₂₃	—	2	32	no	—
65	C₂²⋊C₄×C₂₃	—	2	16	no	—
66	C₄²×C₂₃	—	2	16	no	—
67	C₄²×C₂₃	—	2	8	no	—
68	C₄²×C₂₃	—	2	8	no	—
69	C₄²×C₂₃	—	2	8	no	—
70	C₄²×C₂₃	—	2	16	no	—
71	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
72	C₄²×C₂₃	C₂×Q₈	1	8	yes	—
73	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
74	C₄²×C₂₃	#41	1	16	yes	—
75	#36640	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	2	4	no	—
77	C₂³×C₂₃	C₂²×Q₈	1	2	yes	—
78	C₂³×C₂₃	C₂×#41	1	4	yes	—
79	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
80	C₂×C₄×C₂₃	—	2	32	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	8	no	—
83	C₂×C₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	2	16	no	—
85	C₂×C₄×C₂₃	—	2	16	no	—
86	C₂×C₄×C₂₃	—	2	32	no	—
87	C₂×C₄×C₂₃	#125	1	16	yes	complemented
88	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
89	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
90	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
92	C₂×C₄×C₂₃	C₂×#41	1	16	yes	—
93	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
94	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
95	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
96	#2864	C₂×C₂₃	1	4	yes	complemented
97	#2859	C₂×C₂₃	1	4	yes	complemented
98	#2851	C₂×C₂₃	1	8	yes	—
99	C₄×#125	C₂×C₂₃	1	8	yes	complemented
100	#2296	C₂×C₂₃	1	4	yes	—
101	#2285	C₂×C₂₃	1	2	yes	—
102	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic
103	C₂²×C₂₃	—	2	4	no	—
104	C₂²×C₂₃	—	2	8	no	—
105	C₂²×C₂₃	C₂×#125	1	4	yes	—
106	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
107	C₂²×C₂₃	C₂²×#41	1	2	yes	—
108	C₂²×C₂₃	C₂²×#41	1	4	yes	—
109	C₄×C₂₃	—	2	4	no	—
110	C₄×C₂₃	—	2	16	no	—
111	C₄×C₂₃	—	4	16	no	—
112	C₄×C₂₃	C₂×#125	1	4	yes	—
113	C₄×C₂₃	C₂×#125	1	4	yes	—
114	C₄×C₂₃	#533	1	8	yes	—
115	C₂×#125	C₂²×C₂₃	1	8	yes	—
116	C₂×#117	C₂²×C₂₃	1	4	yes	—
117	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
118	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
119	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
120	#389	C₂²×C₂₃	1	16	yes	—
121	#389	C₂²×C₂₃	1	16	yes	—
122	#387	C₂²×C₂₃	1	32	yes	—
123	#385	C₂²×C₂₃	1	16	yes	—
124	#384	C₂²×C₂₃	1	4	yes	—
125	#382	C₂²×C₂₃	1	8	yes	—
126	#382	C₂²×C₂₃	1	4	yes	—
127	#380	C₂²×C₂₃	1	8	yes	—
128	C₄×C₄⋊C₄	C₂²×C₂₃	1	8	yes	—
129	C₄×C₄⋊C₄	C₂²×C₂₃	1	16	yes	—
130	C₄³	C₂²×C₂₃	1	4	yes	—
131	C₄³	C₂²×C₂₃	1	2	yes	—
132	C₄³	C₂²×C₂₃	1	2	yes	—
133	C₂×C₂₃	—	2	4	no	—
134	C₂×C₂₃	C₂²×#125	1	1	yes	characteristic
135	C₂×C₂₃	C₂×#533	1	2	yes	—
136	C₂×C₂₃	#3444	1	4	yes	—
137	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
138	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
139	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
140	#125	—	2	32	no	—
141	#117	—	2	16	no	—
142	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
143	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
144	C₂×C₄⋊C₄	C₂³×C₂₃	1	32	yes	—
145	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
146	C₂×C₄²	—	2	8	no	—
147	C₂×C₄²	—	2	16	no	—
148	C₂×C₄²	Q₈×C₂₃	1	8	yes	—
149	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
150	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
151	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
152	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
153	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
154	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
158	#95	C₂³×C₂₃	1	16	yes	—
159	#95	C₂³×C₂₃	1	32	yes	—
160	C₂₃	#36640	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	16	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	C₂×Q₈×C₂₃	1	8	yes	—
166	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
167	C₂²×C₄	#41×C₂₃	1	8	yes	—
168	C₂²×C₄	C₂⁴×C₂₃	1	16	yes	—
169	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
171	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
174	C₄⋊C₄	—	2	16	no	—
175	C₄⋊C₄	—	2	64	no	—
176	C₄⋊C₄	—	2	32	no	—
177	C₂²⋊C₄	—	2	16	no	—
178	C₄²	—	2	16	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	2	8	no	—
182	C₄²	—	2	16	no	—
183	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
184	C₄²	C₂×Q₈×C₂₃	1	8	yes	—
185	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
186	C₄²	#41×C₂₃	1	16	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂²×Q₈×C₂₃	1	2	yes	—
189	C₂³	C₂×#41×C₂₃	1	4	yes	—
190	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
191	C₂×C₄	—	2	32	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	2	32	no	—
198	C₂×C₄	C₂₃×#125	1	16	yes	complemented
199	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
200	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
201	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
202	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
203	C₂×C₄	C₂×#41×C₂₃	1	16	yes	—
204	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
205	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
206	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	C₂×C₂₃×#125	1	4	yes	—
210	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
211	C₂²	C₂²×#41×C₂₃	1	2	yes	—
212	C₂²	C₂²×#41×C₂₃	1	4	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	16	no	—
215	C₄	—	4	16	no	—
216	C₄	C₂×C₂₃×#125	1	4	yes	—
217	C₄	C₂×C₂₃×#125	1	4	yes	—
218	C₄	C₂₃×#533	1	8	yes	—
219	C₂	—	2	4	no	—
220	C₂	C₂²×C₂₃×#125	1	1	yes	characteristic
221	C₂	C₂×C₂₃×#533	1	2	yes	—
222	C₂	C₂₃×#3444	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂⁴×C₂₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2864	C₂	semidirect
2. C₂₃×#2859	C₂	semidirect
4. C₄×C₂₃×#125	C₂	semidirect
75. #36640	C₂₃	direct (N × H)
87. C₂×C₄×C₂₃	#125	semidirect
96. #2864	C₂×C₂₃	semidirect
97. #2859	C₂×C₂₃	semidirect
99. C₄×#125	C₂×C₂₃	semidirect
160. C₂₃	#36640	direct (N × H)
198. C₂×C₄	C₂₃×#125	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2864	2
C₂₃×#2859	2
C₂₃×#2851	2
C₄×C₂₃×#125	2
C₂₃×#2296	2
C₂₃×#2285	2
C₂×C₄³×C₂₃	2
#36640	23

Subgroups — C23×#36640 #19184266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36640 #19184266