Subgroups — C₂₃×#33460 #19181086

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2951	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2177	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2176	C₂	1	2	yes	maximal, complemented
5	C₄×C₂₃×SD₃₂	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2139	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₂²	1	2	yes	—
10	C₂₃×#520	—	2	2	no	—
11	C₂₃×#508	C₄	1	8	yes	complemented
12	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#440	—	2	4	no	—
18	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	—
19	C₄×D₈×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂₃×#107	C₂²	1	2	yes	—
21	C₂₃×#417	—	2	2	no	—
22	C₂₃×#405	C₂²	1	1	yes	characteristic
23	C₂₃×#367	C₂²	1	2	yes	—
24	C₂₃×#367	C₂²	1	2	yes	—
25	C₂₃×#358	C₂²	1	4	yes	—
26	C₂₃×#357	C₂²	1	4	yes	—
27	C₂₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
29	C₂×#41×C₂₃	—	2	2	no	—
30	C₂₃×#135	—	2	4	no	—
31	C₂₃×#135	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₂₃	—	2	2	no	—
34	C₂×D₈×C₂₃	C₂³	1	2	yes	—
35	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₂₃	—	2	4	no	—
38	C₄×D₄×C₂₃	—	2	4	no	—
39	C₄×D₄×C₂₃	—	4	4	no	—
40	C₂₃×#117	—	4	2	no	—
41	C₂₃×#117	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
44	C₂₃×Q₃₂	C₂×C₄	1	4	yes	complemented
45	C₂₃×SD₃₂	C₂×C₄	1	8	yes	complemented
46	C₂₃×D₁₆	C₂×C₄	1	4	yes	complemented
47	C₂₃×M₃₂	—	2	2	no	—
48	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#107	—	2	2	no	—
53	C₂₃×#107	C₂³	1	4	yes	—
54	C₂₃×#103	—	2	4	no	—
55	C₂₃×#102	—	2	4	no	—
56	C₂₃×#97	D₄	1	2	yes	—
57	C₄×C₈×C₂₃	C₂³	1	2	yes	—
58	#41×C₂₃	—	2	8	no	—
59	#41×C₂₃	—	4	4	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	—	4	4	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	2	2	no	—
68	Q₁₆×C₂₃	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₂₃	—	2	8	no	—
70	D₈×C₂₃	—	2	2	no	—
71	D₈×C₂₃	C₂²×C₄	1	4	yes	—
72	M₁₆×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	#41	1	2	yes	—
77	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
82	C₂²⋊C₄×C₂₃	—	4	4	no	—
83	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
84	C₄²×C₂₃	—	4	4	no	—
85	C₄²×C₂₃	C₂×D₄	1	2	yes	—
86	C₁₆×C₂₃	—	2	2	no	—
87	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
89	#33460	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	4	2	no	—
91	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₂₃	—	2	4	no	—
93	Q₈×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	2	4	no	—
95	D₄×C₂₃	—	4	8	no	—
96	D₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	8	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	—	16	4	no	—
101	C₂×C₄×C₂₃	—	16	2	no	—
102	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
104	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#135	1	2	yes	—
106	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₂₃	—	2	2	no	—
108	C₈×C₂₃	C₄×D₄	1	2	yes	—
109	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
111	C₂×#508	C₂×C₂₃	1	1	yes	characteristic
112	#2951	C₂×C₂₃	1	2	yes	complemented
113	#2177	C₂×C₂₃	1	2	yes	complemented
114	#2176	C₂×C₂₃	1	2	yes	complemented
115	C₄×SD₃₂	C₂×C₂₃	1	4	yes	complemented
116	C₂×#367	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#2139	C₂×C₂₃	1	2	yes	complemented
118	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	4	4	no	—
121	C₂²×C₂₃	—	8	2	no	—
122	C₂²×C₂₃	—	8	2	no	—
123	C₂²×C₂₃	C₄×D₈	1	2	yes	—
124	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₂₃	—	2	4	no	—
126	C₄×C₂₃	—	4	4	no	—
127	C₄×C₂₃	—	8	4	no	—
128	C₄×C₂₃	C₄×D₈	1	2	yes	—
129	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
131	C₂×#135	C₂²×C₂₃	1	2	yes	—
132	#520	—	2	2	no	—
133	#508	C₄×C₂₃	1	8	yes	complemented
134	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
136	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
138	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
139	#440	—	2	4	no	—
140	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	—
141	C₄×D₈	C₂²×C₂₃	1	4	yes	—
142	C₂×#107	C₂²×C₂₃	1	2	yes	—
143	#417	—	2	2	no	—
144	#405	C₂²×C₂₃	1	1	yes	characteristic
145	#367	C₂²×C₂₃	1	2	yes	—
146	#367	C₂²×C₂₃	1	2	yes	—
147	#358	C₂²×C₂₃	1	4	yes	—
148	#357	C₂²×C₂₃	1	4	yes	—
149	#346	C₂²×C₂₃	1	2	yes	—
150	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
151	C₂×C₂₃	—	2	2	no	—
152	C₂×C₂₃	—	8	4	no	—
153	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
154	C₂×C₂₃	#3416	1	2	yes	—
155	C₂×#41	—	2	2	no	—
156	#135	—	2	4	no	—
157	#135	C₂×C₄×C₂₃	1	8	yes	complemented
158	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	C₂³×C₂₃	1	2	yes	—
161	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
163	C₄×Q₈	—	2	4	no	—
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	—	4	4	no	—
166	#117	—	4	2	no	—
167	#117	D₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	4	2	no	—
170	Q₃₂	C₂×C₄×C₂₃	1	4	yes	complemented
171	SD₃₂	C₂×C₄×C₂₃	1	8	yes	complemented
172	D₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
173	M₃₂	—	2	2	no	—
174	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
175	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
176	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
178	#107	—	2	2	no	—
179	#107	C₂³×C₂₃	1	4	yes	—
180	#103	—	2	4	no	—
181	#102	—	2	4	no	—
182	#97	D₄×C₂₃	1	2	yes	—
183	C₄×C₈	C₂³×C₂₃	1	2	yes	—
184	C₂₃	#33460	1	1	yes	characteristic, direct product factor, Sylow
185	#41	—	2	8	no	—
186	#41	—	4	4	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
194	Q₁₆	—	2	2	no	—
195	Q₁₆	C₂²×C₄×C₂₃	1	4	yes	—
196	SD₁₆	—	2	8	no	—
197	D₈	—	2	2	no	—
198	D₈	C₂²×C₄×C₂₃	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₂×C₈	#41×C₂₃	1	2	yes	—
204	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
211	C₄²	—	4	4	no	—
212	C₄²	C₂×D₄×C₂₃	1	2	yes	—
213	C₁₆	—	2	2	no	—
214	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
215	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
216	C₂³	—	4	2	no	—
217	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	16	4	no	—
227	C₂×C₄	—	16	2	no	—
228	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
229	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
230	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
231	C₂×C₄	C₂₃×#135	1	2	yes	—
232	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	C₄×D₄×C₂₃	1	2	yes	—
235	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
236	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	4	4	no	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	C₄×D₈×C₂₃	1	2	yes	—
242	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	4	4	no	—
245	C₄	—	8	4	no	—
246	C₄	C₄×D₈×C₂₃	1	2	yes	—
247	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
248	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
249	C₂	—	2	2	no	—
250	C₂	—	8	4	no	—
251	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic
252	C₂	C₂₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#508	C₂
LCS	G ▷ 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
2. C₂₃×#2951	C₂	semidirect
3. C₂₃×#2177	C₂	semidirect
4. C₂₃×#2176	C₂	semidirect
5. C₄×C₂₃×SD₃₂	C₂	semidirect
6. C₂×C₂₃×#367	C₂	semidirect
7. C₂₃×#2139	C₂	semidirect
8. C₂₃×#2108	C₂	semidirect
11. C₂₃×#508	C₄	semidirect
31. C₂₃×#135	C₂×C₄	semidirect
44. C₂₃×Q₃₂	C₂×C₄	semidirect
45. C₂₃×SD₃₂	C₂×C₄	semidirect
46. C₂₃×D₁₆	C₂×C₄	semidirect
48. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
89. #33460	C₂₃	direct (N × H)
112. #2951	C₂×C₂₃	semidirect
113. #2177	C₂×C₂₃	semidirect
114. #2176	C₂×C₂₃	semidirect
115. C₄×SD₃₂	C₂×C₂₃	semidirect
116. C₂×#367	C₂×C₂₃	semidirect
117. #2139	C₂×C₂₃	semidirect
118. #2108	C₂×C₂₃	semidirect
133. #508	C₄×C₂₃	semidirect
157. #135	C₂×C₄×C₂₃	semidirect
170. Q₃₂	C₂×C₄×C₂₃	semidirect
171. SD₃₂	C₂×C₄×C₂₃	semidirect
172. D₁₆	C₂×C₄×C₂₃	semidirect
174. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
184. C₂₃	#33460	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#508	2
C₂₃×#2951	2
C₂₃×#2177	2
C₂₃×#2176	2
C₄×C₂₃×SD₃₂	2
C₂×C₂₃×#367	2
C₂₃×#2139	2
C₂₃×#2108	2
#33460	23

Subgroups — C23×#33460 #19181086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33460 #19181086