Subgroups — C₂₃×#33740 #19181366

Order 5888.

532 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3176	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2244	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2218	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2215	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2177	C₂	1	2	yes	maximal, complemented
8	C₄×C₂₃×D₁₆	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#540	—	2	2	no	—
11	C₂×C₂₃×D₁₆	C₂²	1	2	yes	—
12	C₂×C₂₃×D₁₆	C₂²	1	2	yes	—
13	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#499	C₂²	1	2	yes	—
16	C₂₃×#493	C₂²	1	2	yes	—
17	C₂₃×#490	—	2	2	no	—
18	C₂₃×#469	—	2	2	no	—
19	C₂₃×#466	C₂²	1	4	yes	—
20	C₂₃×#442	—	2	2	no	—
21	C₄×D₈×C₂₃	C₂²	1	4	yes	—
22	C₂₃×#405	C₂²	1	1	yes	characteristic
23	C₂₃×#367	C₂²	1	2	yes	—
24	C₂₃×#366	C₂²	1	2	yes	—
25	C₂₃×#357	C₂²	1	4	yes	—
26	C₂₃×#357	C₂²	1	4	yes	—
27	C₂₃×#346	C₂²	1	1	yes	characteristic
28	C₂₃×#346	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×D₈×C₂₃	—	2	2	no	—
32	C₂×D₈×C₂₃	C₂³	1	4	yes	—
33	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#127	—	2	2	no	—
36	C₂₃×#125	—	2	2	no	—
37	C₂₃×#123	—	4	4	no	—
38	C₂₃×#121	—	2	4	no	—
39	C₂₃×#121	—	4	4	no	—
40	C₄×D₄×C₂₃	—	2	4	no	—
41	C₂₃×#117	D₄	1	1	yes	characteristic
42	C₂₃×D₁₆	—	2	4	no	—
43	C₂₃×D₁₆	—	2	4	no	—
44	C₂₃×M₃₂	—	2	2	no	—
45	C₂×C₁₆×C₂₃	—	2	2	no	—
46	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#107	C₂³	1	4	yes	—
51	C₂₃×#106	—	2	2	no	—
52	C₂₃×#102	—	2	4	no	—
53	C₂₃×#102	—	2	4	no	—
54	C₂₃×#97	D₄	1	1	yes	characteristic
55	C₂₃×#97	D₄	1	1	yes	characteristic
56	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₂₃	—	2	4	no	—
59	C₂×D₄×C₂₃	—	4	4	no	—
60	C₂×D₄×C₂₃	—	4	4	no	—
61	C₂²×C₄×C₂₃	—	4	4	no	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	D₈×C₂₃	—	2	4	no	—
64	D₈×C₂₃	—	2	4	no	—
65	D₈×C₂₃	—	4	4	no	—
66	M₁₆×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	1	no	—
68	C₂×C₈×C₂₃	—	2	1	no	—
69	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	4	4	no	—
75	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
76	C₂²⋊C₄×C₂₃	—	4	4	no	—
77	C₂²⋊C₄×C₂₃	—	4	4	no	—
78	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
79	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₂₃	—	2	1	no	—
82	C₁₆×C₂₃	—	2	1	no	—
83	C₁₆×C₂₃	#41	1	2	yes	—
84	C₁₆×C₂₃	#41	1	2	yes	—
85	#33740	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	4	4	no	—
87	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
88	D₄×C₂₃	—	4	4	no	—
89	D₄×C₂₃	—	4	4	no	—
90	D₄×C₂₃	—	16	4	no	—
91	D₄×C₂₃	—	16	4	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	16	4	no	—
96	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₂₃	—	2	1	no	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	#135	1	2	yes	—
103	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
104	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
105	#3176	C₂×C₂₃	1	2	yes	complemented
106	#2253	C₂×C₂₃	1	1	yes	characteristic, complemented
107	#2246	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#2244	C₂×C₂₃	1	2	yes	complemented
109	#2218	C₂×C₂₃	1	2	yes	complemented
110	#2215	C₂×C₂₃	1	2	yes	complemented
111	#2177	C₂×C₂₃	1	2	yes	complemented
112	C₄×D₁₆	C₂×C₂₃	1	2	yes	complemented
113	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	8	4	no	—
117	C₂²×C₂₃	—	8	4	no	—
118	C₂²×C₂₃	—	8	4	no	—
119	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	8	4	no	—
124	C₄×C₂₃	#532	1	1	yes	characteristic
125	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
126	#540	—	2	2	no	—
127	C₂×D₁₆	C₂²×C₂₃	1	2	yes	—
128	C₂×D₁₆	C₂²×C₂₃	1	2	yes	—
129	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
130	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
131	#499	C₂²×C₂₃	1	2	yes	—
132	#493	C₂²×C₂₃	1	2	yes	—
133	#490	—	2	2	no	—
134	#469	—	2	2	no	—
135	#466	C₂²×C₂₃	1	4	yes	—
136	#442	—	2	2	no	—
137	C₄×D₈	C₂²×C₂₃	1	4	yes	—
138	#405	C₂²×C₂₃	1	1	yes	characteristic
139	#367	C₂²×C₂₃	1	2	yes	—
140	#366	C₂²×C₂₃	1	2	yes	—
141	#357	C₂²×C₂₃	1	4	yes	—
142	#357	C₂²×C₂₃	1	4	yes	—
143	#346	C₂²×C₂₃	1	1	yes	characteristic
144	#346	C₂²×C₂₃	1	1	yes	characteristic
145	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
146	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₂₃	—	4	1	no	—
148	C₂×C₂₃	—	16	4	no	—
149	C₂×C₂₃	#3169	1	1	yes	characteristic
150	C₂×C₂₃	#3415	1	2	yes	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	C₂³×C₂₃	1	4	yes	—
153	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
155	#127	—	2	2	no	—
156	#125	—	2	2	no	—
157	#123	—	4	4	no	—
158	#121	—	2	4	no	—
159	#121	—	4	4	no	—
160	C₄×D₄	—	2	4	no	—
161	#117	D₄×C₂₃	1	1	yes	characteristic
162	D₁₆	—	2	4	no	—
163	D₁₆	—	2	4	no	—
164	M₃₂	—	2	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
167	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
168	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
170	#107	C₂³×C₂₃	1	4	yes	—
171	#106	—	2	2	no	—
172	#102	—	2	4	no	—
173	#102	—	2	4	no	—
174	#97	D₄×C₂₃	1	1	yes	characteristic
175	#97	D₄×C₂₃	1	1	yes	characteristic
176	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
178	C₂₃	#33740	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
184	D₈	—	2	4	no	—
185	D₈	—	2	4	no	—
186	D₈	—	4	4	no	—
187	M₁₆	—	2	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₂²⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
200	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₁₆	—	2	1	no	—
203	C₁₆	—	2	1	no	—
204	C₁₆	#41×C₂₃	1	2	yes	—
205	C₁₆	#41×C₂₃	1	2	yes	—
206	C₂³	—	4	4	no	—
207	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	16	4	no	—
211	D₄	—	16	4	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	16	4	no	—
216	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
217	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
218	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	C₂₃×#135	1	2	yes	—
223	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
224	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	8	4	no	—
235	C₄	C₂₃×#532	1	1	yes	characteristic
236	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
237	C₂	—	4	1	no	—
238	C₂	—	16	4	no	—
239	C₂	C₂₃×#3169	1	1	yes	characteristic
240	C₂	C₂₃×#3415	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 ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3176	C₂	semidirect
2. C₂₃×#2253	C₂	semidirect
3. C₂₃×#2246	C₂	semidirect
4. C₂₃×#2244	C₂	semidirect
5. C₂₃×#2218	C₂	semidirect
6. C₂₃×#2215	C₂	semidirect
7. C₂₃×#2177	C₂	semidirect
8. C₄×C₂₃×D₁₆	C₂	semidirect
9. C₂₃×#2108	C₂	semidirect
85. #33740	C₂₃	direct (N × H)
105. #3176	C₂×C₂₃	semidirect
106. #2253	C₂×C₂₃	semidirect
107. #2246	C₂×C₂₃	semidirect
108. #2244	C₂×C₂₃	semidirect
109. #2218	C₂×C₂₃	semidirect
110. #2215	C₂×C₂₃	semidirect
111. #2177	C₂×C₂₃	semidirect
112. C₄×D₁₆	C₂×C₂₃	semidirect
113. #2108	C₂×C₂₃	semidirect
178. C₂₃	#33740	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×#3176	2
C₂₃×#2253	2
C₂₃×#2246	2
C₂₃×#2244	2
C₂₃×#2218	2
C₂₃×#2215	2
C₂₃×#2177	2
C₄×C₂₃×D₁₆	2
C₂₃×#2108	2
#33740	23

Subgroups — C23×#33740 #19181366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33740 #19181366