Subgroups — C₂₃×#33732 #19181358

Order 5888.

482 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	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2255	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2253	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2159	C₂	1	4	yes	maximal
6	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#533	—	2	1	no	—
8	C₂×C₂₃×#125	—	2	1	no	—
9	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
10	C₂₃×#500	C₂²	1	2	yes	—
11	C₂₃×#499	—	2	1	no	—
12	C₂₃×#499	—	2	1	no	—
13	C₂₃×#499	—	2	1	no	—
14	C₂₃×#499	C₂²	1	2	yes	—
15	C₂₃×#499	C₂²	1	2	yes	—
16	C₂₃×#499	C₂²	1	2	yes	—
17	C₂₃×#498	—	2	1	no	—
18	C₂₃×#427	—	2	1	no	—
19	C₂₃×#427	C₂²	1	2	yes	—
20	C₂×C₂₃×#107	C₂²	1	2	yes	—
21	C₂×C₂₃×#106	—	2	1	no	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#367	C₂²	1	8	yes	—
24	C₂₃×#366	C₂²	1	8	yes	—
25	C₂₃×#346	C₂²	1	2	yes	—
26	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×C₄⋊Q₈	—	2	2	no	—
30	C₂₃×#125	—	2	2	no	—
31	C₂₃×#125	—	2	2	no	—
32	C₂₃×#125	—	2	2	no	—
33	C₂₃×#125	—	4	2	no	—
34	C₂₃×C₂²⋊Q₈	—	4	2	no	—
35	C₄×Q₈×C₂₃	—	4	2	no	—
36	C₂₃×#117	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂₃×M₃₂	Q₈	1	8	yes	complemented
41	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
42	C₂₃×#107	—	2	2	no	—
43	C₂₃×#107	—	2	2	no	—
44	C₂₃×#107	C₂³	1	4	yes	—
45	C₂₃×#107	C₂³	1	4	yes	—
46	C₂₃×#106	—	2	2	no	—
47	C₂₃×#106	—	2	2	no	—
48	C₂₃×#106	—	2	2	no	—
49	C₂₃×#106	—	2	2	no	—
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	D₈	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
59	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	—	2	4	no	—
65	C₄⋊C₄×C₂₃	—	2	4	no	—
66	C₄⋊C₄×C₂₃	—	4	2	no	—
67	C₄⋊C₄×C₂₃	—	4	4	no	—
68	C₄⋊C₄×C₂₃	—	4	4	no	—
69	C₄⋊C₄×C₂₃	—	4	4	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₂²⋊C₄×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	D₈	1	2	yes	—
74	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₂₃	C₂×Q₈	1	8	yes	—
77	#33732	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₂₃	—	8	4	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	4	4	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	—	8	4	no	—
84	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₈×C₂₃	C₄⋊Q₈	1	2	yes	—
91	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
92	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
93	C₈×C₂₃	C₂²×Q₈	1	2	yes	—
94	#3160	C₂×C₂₃	1	1	yes	characteristic
95	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
96	#2255	C₂×C₂₃	1	4	yes	complemented
97	#2253	C₂×C₂₃	1	4	yes	complemented
98	#2159	C₂×C₂₃	1	4	yes	—
99	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	#500	1	2	yes	—
102	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	8	4	no	—
106	C₄×C₂₃	—	8	4	no	—
107	C₄×C₂₃	#500	1	2	yes	—
108	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
110	#533	—	2	1	no	—
111	C₂×#125	—	2	1	no	—
112	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
113	#500	C₂²×C₂₃	1	2	yes	—
114	#499	—	2	1	no	—
115	#499	—	2	1	no	—
116	#499	—	2	1	no	—
117	#499	C₂²×C₂₃	1	2	yes	—
118	#499	C₂²×C₂₃	1	2	yes	—
119	#499	C₂²×C₂₃	1	2	yes	—
120	#498	—	2	1	no	—
121	#427	—	2	1	no	—
122	#427	C₂²×C₂₃	1	2	yes	—
123	C₂×#107	C₂²×C₂₃	1	2	yes	—
124	C₂×#106	—	2	1	no	—
125	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
126	#367	C₂²×C₂₃	1	8	yes	—
127	#366	C₂²×C₂₃	1	8	yes	—
128	#346	C₂²×C₂₃	1	2	yes	—
129	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
130	C₂×C₂₃	—	2	2	no	—
131	C₂×C₂₃	C₂×#500	1	1	yes	characteristic
132	C₂×C₂₃	#3414	1	2	yes	—
133	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	2	no	—
136	#125	—	2	2	no	—
137	#125	—	2	2	no	—
138	#125	—	2	2	no	—
139	#125	—	4	2	no	—
140	C₂²⋊Q₈	—	4	2	no	—
141	C₄×Q₈	—	4	2	no	—
142	#117	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	4	2	no	—
145	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
146	M₃₂	Q₈×C₂₃	1	8	yes	complemented
147	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#107	C₂³×C₂₃	1	4	yes	—
151	#107	C₂³×C₂₃	1	4	yes	—
152	#106	—	2	2	no	—
153	#106	—	2	2	no	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
157	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₂₃	#33732	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	D₈×C₂₃	1	2	yes	—
164	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
165	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
166	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
168	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
170	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	D₈×C₂₃	1	2	yes	—
181	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₁₆	C₂×Q₈×C₂₃	1	8	yes	—
184	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
185	Q₈	—	8	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
191	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
192	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
193	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
195	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
196	C₈	C₂₃×C₄⋊Q₈	1	2	yes	—
197	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
198	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
199	C₈	C₂²×Q₈×C₂₃	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	C₂₃×#500	1	2	yes	—
202	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	8	4	no	—
207	C₄	C₂₃×#500	1	2	yes	—
208	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
209	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×C₂₃×#500	1	1	yes	characteristic
212	C₂	C₂₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
3. C₂₃×#2255	C₂	semidirect
4. C₂₃×#2253	C₂	semidirect
40. C₂₃×M₃₂	Q₈	semidirect
77. #33732	C₂₃	direct (N × H)
96. #2255	C₂×C₂₃	semidirect
97. #2253	C₂×C₂₃	semidirect
146. M₃₂	Q₈×C₂₃	semidirect
160. C₂₃	#33732	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3160	2
C₂×C₂₃×#499	2
C₂₃×#2255	2
C₂₃×#2253	2
C₂₃×#2159	2
C₄×C₂₃×M₃₂	2
#33732	23

Subgroups — C23×#33732 #19181358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33732 #19181358