Subgroups — C₂₃×#33748 #19181374

Order 5888.

592 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3169	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2218	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2177	C₂	1	4	yes	maximal, complemented
7	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#532	—	2	2	no	—
9	C₂×C₂₃×D₁₆	C₂²	1	4	yes	—
10	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
11	C₂₃×#500	C₂²	1	2	yes	—
12	C₂₃×#493	C₂²	1	2	yes	—
13	C₂₃×#486	—	2	2	no	—
14	C₂₃×#466	—	2	2	no	—
15	C₂₃×#466	C₂²	1	4	yes	—
16	C₄×D₈×C₂₃	—	2	2	no	—
17	C₄×D₈×C₂₃	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#367	C₂²	1	4	yes	—
20	C₂₃×#357	C₂²	1	8	yes	—
21	C₂₃×#346	C₂²	1	2	yes	—
22	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×#41×C₂₃	—	4	2	no	—
25	C₂×D₈×C₂₃	—	2	2	no	—
26	C₂×D₈×C₂₃	C₂³	1	4	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×#127	—	2	2	no	—
31	C₂₃×#124	—	4	2	no	—
32	C₂₃×#121	—	2	4	no	—
33	C₂₃×#121	—	4	2	no	—
34	C₄×D₄×C₂₃	—	2	4	no	—
35	C₄×D₄×C₂₃	—	4	2	no	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂₃×D₁₆	—	2	8	no	—
38	C₂₃×M₃₂	—	2	4	no	—
39	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
40	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
41	C₂₃×#107	—	2	2	no	—
42	C₂₃×#107	C₂³	1	4	yes	—
43	C₂₃×#102	—	2	4	no	—
44	C₂₃×#102	—	2	4	no	—
45	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	#41×C₂₃	—	8	8	no	—
50	C₂×Q₈×C₂₃	—	4	2	no	—
51	C₂×D₄×C₂₃	—	2	4	no	—
52	C₂×D₄×C₂₃	—	4	4	no	—
53	C₂×D₄×C₂₃	—	4	2	no	—
54	C₂×D₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	4	4	no	—
56	C₂²×C₄×C₂₃	D₈	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	D₈×C₂₃	—	2	8	no	—
59	D₈×C₂₃	—	4	4	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	4	2	no	—
68	C₂²⋊C₄×C₂₃	—	4	4	no	—
69	C₂²⋊C₄×C₂₃	—	4	4	no	—
70	C₄²×C₂₃	D₈	1	2	yes	—
71	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₂₃	—	2	2	no	—
74	C₁₆×C₂₃	#41	1	4	yes	complemented
75	#33748	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	4	4	no	—
77	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₂₃	—	8	4	no	—
79	D₄×C₂₃	—	4	4	no	—
80	D₄×C₂₃	—	4	4	no	—
81	D₄×C₂₃	—	8	8	no	—
82	D₄×C₂₃	—	8	4	no	—
83	D₄×C₂₃	—	8	4	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	4	no	—
87	C₂×C₄×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	—	8	8	no	—
89	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	2	1	no	—
95	C₈×C₂₃	#135	1	2	yes	—
96	C₈×C₂₃	C₂×#41	1	2	yes	—
97	#3169	C₂×C₂₃	1	2	yes	complemented
98	#2254	C₂×C₂₃	1	1	yes	characteristic, complemented
99	#2246	C₂×C₂₃	1	1	yes	characteristic, complemented
100	#2240	C₂×C₂₃	1	2	yes	complemented
101	#2218	C₂×C₂₃	1	4	yes	complemented
102	#2177	C₂×C₂₃	1	4	yes	complemented
103	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	1	no	—
106	C₂²×C₂₃	—	8	4	no	—
107	C₂²×C₂₃	—	8	4	no	—
108	C₂²×C₂₃	—	8	4	no	—
109	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	8	4	no	—
113	C₄×C₂₃	#509	1	4	yes	—
114	C₄×C₂₃	#532	1	1	yes	characteristic
115	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
116	#532	—	2	2	no	—
117	C₂×D₁₆	C₂²×C₂₃	1	4	yes	—
118	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
119	#500	C₂²×C₂₃	1	2	yes	—
120	#493	C₂²×C₂₃	1	2	yes	—
121	#486	—	2	2	no	—
122	#466	—	2	2	no	—
123	#466	C₂²×C₂₃	1	4	yes	—
124	C₄×D₈	—	2	2	no	—
125	C₄×D₈	C₂²×C₂₃	1	4	yes	—
126	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
127	#367	C₂²×C₂₃	1	4	yes	—
128	#357	C₂²×C₂₃	1	8	yes	—
129	#346	C₂²×C₂₃	1	2	yes	—
130	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
131	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×C₂₃	—	4	1	no	—
133	C₂×C₂₃	—	16	4	no	—
134	C₂×C₂₃	#3169	1	1	yes	characteristic
135	C₂×C₂₃	C₂×#509	1	2	yes	—
136	C₂×#41	—	4	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	C₂³×C₂₃	1	4	yes	—
139	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	#127	—	2	2	no	—
143	#124	—	4	2	no	—
144	#121	—	2	4	no	—
145	#121	—	4	2	no	—
146	C₄×D₄	—	2	4	no	—
147	C₄×D₄	—	4	2	no	—
148	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
149	D₁₆	—	2	8	no	—
150	M₃₂	—	2	4	no	—
151	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
152	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
153	#107	—	2	2	no	—
154	#107	C₂³×C₂₃	1	4	yes	—
155	#102	—	2	4	no	—
156	#102	—	2	4	no	—
157	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
158	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
159	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
161	C₂₃	#33748	1	1	yes	characteristic, direct product factor, Sylow
162	#41	—	8	8	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	4	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	D₈×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
171	D₈	—	2	8	no	—
172	D₈	—	4	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
178	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	4	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₄²	D₈×C₂₃	1	2	yes	—
184	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
186	C₁₆	—	2	2	no	—
187	C₁₆	#41×C₂₃	1	4	yes	complemented
188	C₂³	—	4	4	no	—
189	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
190	Q₈	—	8	4	no	—
191	D₄	—	4	4	no	—
192	D₄	—	4	4	no	—
193	D₄	—	8	8	no	—
194	D₄	—	8	4	no	—
195	D₄	—	8	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
203	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
204	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	C₂₃×#135	1	2	yes	—
208	C₈	C₂×#41×C₂₃	1	2	yes	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	8	4	no	—
212	C₂²	—	8	4	no	—
213	C₂²	—	8	4	no	—
214	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	8	4	no	—
218	C₄	C₂₃×#509	1	4	yes	—
219	C₄	C₂₃×#532	1	1	yes	characteristic
220	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	16	4	no	—
223	C₂	C₂₃×#3169	1	1	yes	characteristic
224	C₂	C₂×C₂₃×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
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
1. C₂₃×#3169	C₂	semidirect
2. C₂₃×#2254	C₂	semidirect
3. C₂₃×#2246	C₂	semidirect
4. C₂₃×#2240	C₂	semidirect
5. C₂₃×#2218	C₂	semidirect
6. C₂₃×#2177	C₂	semidirect
7. C₄×C₂₃×M₃₂	C₂	semidirect
74. C₁₆×C₂₃	#41	semidirect
75. #33748	C₂₃	direct (N × H)
97. #3169	C₂×C₂₃	semidirect
98. #2254	C₂×C₂₃	semidirect
99. #2246	C₂×C₂₃	semidirect
100. #2240	C₂×C₂₃	semidirect
101. #2218	C₂×C₂₃	semidirect
102. #2177	C₂×C₂₃	semidirect
103. C₄×M₃₂	C₂×C₂₃	semidirect
161. C₂₃	#33748	direct (N × H)
187. C₁₆	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3169	2
C₂₃×#2254	2
C₂₃×#2246	2
C₂₃×#2240	2
C₂₃×#2218	2
C₂₃×#2177	2
C₄×C₂₃×M₃₂	2
#33748	23

Subgroups — C23×#33748 #19181374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33748 #19181374