Subgroups — C₂₃×#33464 #19181090

Order 5888.

642 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×#3414	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2957	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2179	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2178	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#359	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	C₂²	1	2	yes	—
9	C₂₃×#568	—	2	2	no	—
10	C₂₃×#510	C₄	1	4	yes	complemented
11	C₂₃×#509	C₄	1	4	yes	complemented
12	C₂₃×#508	C₂²	1	4	yes	—
13	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#444	—	2	4	no	—
16	C₂₃×#443	C₂²	1	8	yes	complemented
17	C₂×C₂₃×#108	C₂²	1	2	yes	—
18	C₂₃×#421	—	2	2	no	—
19	C₂₃×#405	C₂²	1	1	yes	characteristic
20	C₂₃×#368	C₂²	1	4	yes	complemented
21	C₂₃×#359	C₂²	1	4	yes	complemented
22	C₂₃×#359	C₂²	1	4	yes	complemented
23	C₂₃×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₂₃	C₂²	1	2	yes	complemented
25	C₂×#41×C₂₃	—	2	2	no	—
26	C₂₃×#135	—	2	2	no	—
27	C₂₃×#135	—	2	2	no	—
28	C₂₃×#135	C₂×C₄	1	4	yes	—
29	C₂₃×#135	C₂³	1	4	yes	—
30	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×SD₁₆×C₂₃	—	2	2	no	—
32	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
33	C₂₃×#131	—	2	8	no	—
34	C₂₃×#131	—	4	4	no	—
35	C₂×M₁₆×C₂₃	—	2	2	no	—
36	C₂×M₁₆×C₂₃	—	4	4	no	—
37	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#117	D₄	1	1	yes	characteristic
40	C₂₃×Q₃₂	C₂×C₄	1	4	yes	—
41	C₂₃×SD₃₂	C₂×C₄	1	8	yes	—
42	C₂₃×D₁₆	C₂×C₄	1	4	yes	—
43	C₂₃×M₃₂	C₂×C₄	1	4	yes	—
44	C₂×C₁₆×C₂₃	—	2	2	no	—
45	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
48	C₂₃×#108	—	2	2	no	—
49	C₂₃×#108	C₂³	1	4	yes	—
50	C₂₃×#104	—	2	8	no	—
51	C₂₃×#97	D₄	1	2	yes	—
52	C₄×C₈×C₂₃	C₂³	1	2	yes	—
53	#41×C₂₃	—	2	4	no	—
54	#41×C₂₃	—	2	4	no	—
55	#41×C₂₃	—	4	2	no	—
56	#41×C₂₃	—	4	2	no	—
57	C₂×Q₈×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₂₃	—	2	2	no	—
63	Q₁₆×C₂₃	C₂²×C₄	1	4	yes	—
64	SD₁₆×C₂₃	—	2	4	no	—
65	SD₁₆×C₂₃	—	2	4	no	—
66	D₈×C₂₃	—	2	2	no	—
67	D₈×C₂₃	C₂²×C₄	1	4	yes	—
68	M₁₆×C₂₃	—	2	4	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	M₁₆×C₂₃	—	4	8	no	—
71	M₁₆×C₂₃	D₈	1	4	yes	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	4	8	no	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₂₃	#41	1	2	yes	—
78	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂²⋊C₄×C₂₃	—	2	1	no	—
81	C₄²×C₂₃	C₂×D₄	1	2	yes	—
82	C₁₆×C₂₃	—	2	1	no	—
83	C₁₆×C₂₃	—	2	1	no	—
84	C₁₆×C₂₃	C₂²×C₄	1	4	yes	—
85	#33464	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	#135	1	1	yes	characteristic
88	Q₈×C₂₃	—	2	4	no	—
89	Q₈×C₂₃	—	4	2	no	—
90	D₄×C₂₃	—	2	4	no	—
91	D₄×C₂₃	—	4	4	no	—
92	D₄×C₂₃	—	4	4	no	—
93	D₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₂₃	—	4	4	no	—
103	C₈×C₂₃	C₄×D₄	1	2	yes	—
104	C₈×C₂₃	C₂×D₈	1	4	yes	—
105	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
107	#3414	C₂×C₂₃	1	1	yes	characteristic
108	#2957	C₂×C₂₃	1	2	yes	complemented
109	#2179	C₂×C₂₃	1	4	yes	complemented
110	#2178	C₂×C₂₃	1	4	yes	complemented
111	#2161	C₂×C₂₃	1	1	yes	characteristic, complemented
112	C₂×#359	C₂×C₂₃	1	2	yes	complemented
113	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	4	no	—
116	C₂²×C₂₃	—	16	2	no	—
117	C₂²×C₂₃	C₄×D₈	1	2	yes	—
118	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	4	4	no	—
121	C₄×C₂₃	C₄×D₈	1	2	yes	—
122	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
124	C₂×#135	C₂²×C₂₃	1	2	yes	—
125	#568	—	2	2	no	—
126	#510	C₄×C₂₃	1	4	yes	complemented
127	#509	C₄×C₂₃	1	4	yes	complemented
128	#508	C₂²×C₂₃	1	4	yes	—
129	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
130	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
131	#444	—	2	4	no	—
132	#443	C₂²×C₂₃	1	8	yes	complemented
133	C₂×#108	C₂²×C₂₃	1	2	yes	—
134	#421	—	2	2	no	—
135	#405	C₂²×C₂₃	1	1	yes	characteristic
136	#368	C₂²×C₂₃	1	4	yes	complemented
137	#359	C₂²×C₂₃	1	4	yes	complemented
138	#359	C₂²×C₂₃	1	4	yes	complemented
139	#346	C₂²×C₂₃	1	2	yes	complemented
140	C₄×C₁₆	C₂²×C₂₃	1	2	yes	complemented
141	C₂×C₂₃	—	2	1	no	—
142	C₂×C₂₃	—	2	2	no	—
143	C₂×C₂₃	—	8	4	no	—
144	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
145	C₂×#41	—	2	2	no	—
146	#135	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	C₂×C₄×C₂₃	1	4	yes	—
149	#135	C₂³×C₂₃	1	4	yes	—
150	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
151	C₂×SD₁₆	—	2	2	no	—
152	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
153	#131	—	2	8	no	—
154	#131	—	4	4	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	4	4	no	—
157	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	#117	D₄×C₂₃	1	1	yes	characteristic
160	Q₃₂	C₂×C₄×C₂₃	1	4	yes	—
161	SD₃₂	C₂×C₄×C₂₃	1	8	yes	—
162	D₁₆	C₂×C₄×C₂₃	1	4	yes	—
163	M₃₂	C₂×C₄×C₂₃	1	4	yes	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
166	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
167	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
168	#108	—	2	2	no	—
169	#108	C₂³×C₂₃	1	4	yes	—
170	#104	—	2	8	no	—
171	#97	D₄×C₂₃	1	2	yes	—
172	C₄×C₈	C₂³×C₂₃	1	2	yes	—
173	C₂₃	#33464	1	1	yes	characteristic, direct product factor, Sylow
174	#41	—	2	4	no	—
175	#41	—	2	4	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	Q₁₆	—	2	2	no	—
184	Q₁₆	C₂²×C₄×C₂₃	1	4	yes	—
185	SD₁₆	—	2	4	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	2	no	—
188	D₈	C₂²×C₄×C₂₃	1	4	yes	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	8	no	—
192	M₁₆	D₈×C₂₃	1	4	yes	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	8	no	—
195	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
198	C₂×C₈	#41×C₂₃	1	2	yes	—
199	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₄×C₂₃	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂²×C₄×C₂₃	1	4	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	C₂₃×#135	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
219	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
220	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
222	C₈	—	4	4	no	—
223	C₈	C₄×D₄×C₂₃	1	2	yes	—
224	C₈	C₂×D₈×C₂₃	1	4	yes	—
225	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
226	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	C₄×D₈×C₂₃	1	2	yes	—
231	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	4	4	no	—
234	C₄	C₄×D₈×C₂₃	1	2	yes	—
235	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
236	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂×C₄×D₈
prime radical	C₂₃×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×M₁₆×C₂₃ ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2957	C₂	semidirect
3. C₂₃×#2179	C₂	semidirect
4. C₂₃×#2178	C₂	semidirect
5. C₂₃×#2161	C₂	semidirect
6. C₂×C₂₃×#359	C₂	semidirect
7. C₂₃×#2108	C₂	semidirect
10. C₂₃×#510	C₄	semidirect
11. C₂₃×#509	C₄	semidirect
16. C₂₃×#443	C₂²	semidirect
20. C₂₃×#368	C₂²	semidirect
21. C₂₃×#359	C₂²	semidirect
22. C₂₃×#359	C₂²	semidirect
23. C₂₃×#346	C₂²	semidirect
24. C₄×C₁₆×C₂₃	C₂²	semidirect
85. #33464	C₂₃	direct (N × H)
108. #2957	C₂×C₂₃	semidirect
109. #2179	C₂×C₂₃	semidirect
110. #2178	C₂×C₂₃	semidirect
111. #2161	C₂×C₂₃	semidirect
112. C₂×#359	C₂×C₂₃	semidirect
113. #2108	C₂×C₂₃	semidirect
126. #510	C₄×C₂₃	semidirect
127. #509	C₄×C₂₃	semidirect
132. #443	C₂²×C₂₃	semidirect
136. #368	C₂²×C₂₃	semidirect
137. #359	C₂²×C₂₃	semidirect
138. #359	C₂²×C₂₃	semidirect
139. #346	C₂²×C₂₃	semidirect
140. C₄×C₁₆	C₂²×C₂₃	semidirect
173. C₂₃	#33464	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×#3414	2
C₂₃×#2957	2
C₂₃×#2179	2
C₂₃×#2178	2
C₂₃×#2161	2
C₂×C₂₃×#359	2
C₂₃×#2108	2
#33464	23

Subgroups — C23×#33464 #19181090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33464 #19181090