Subgroups — C₂₃×#22282 #19169908

Order 5888.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2932	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1963	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1876	C₂	1	8	yes	maximal, complemented
5	C₂₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#135	C₂²	1	4	yes	—
8	C₂×C₂₃×#131	C₂²	1	4	yes	—
9	C₂₃×#532	C₂²	1	2	yes	—
10	C₂₃×#495	C₄	1	4	yes	complemented
11	C₂₃×#494	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#493	C₄	1	1	yes	characteristic, complemented
13	C₂₃×#492	C₄	1	2	yes	complemented
14	C₂₃×#435	—	2	8	no	—
15	C₂₃×#423	C₂²	1	2	yes	—
16	C₂×C₂₃×#104	C₂²	1	8	yes	—
17	C₂₃×#408	C₂²	1	4	yes	—
18	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#376	C₂²	1	1	yes	characteristic
21	C₂₃×#341	C₂²	1	4	yes	—
22	C₂₃×#336	C₂²	1	4	yes	—
23	C₂×#41×C₂₃	C₂³	1	4	yes	—
24	C₂₃×#135	—	4	8	no	—
25	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×SD₁₆×C₂₃	C₂×C₄	1	8	yes	—
27	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₂₃×#131	—	4	8	no	—
29	C₂×M₁₆×C₂₃	—	2	4	no	—
30	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₂₃	—	2	4	no	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₂₃×#127	C₂×C₄	1	2	yes	—
36	C₂₃×#124	C₂×C₄	1	4	yes	—
37	C₂₃×#121	—	2	4	no	—
38	C₄×D₄×C₂₃	—	2	4	no	—
39	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	2	4	no	—
42	C₂₃×#104	—	4	8	no	—
43	C₂₃×C₂²⋊C₈	—	2	8	no	—
44	C₂₃×#97	D₄	1	4	yes	—
45	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
48	C₂₃×#95	—	2	2	no	—
49	#41×C₂₃	—	2	8	no	—
50	#41×C₂₃	—	2	8	no	—
51	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
52	C₂×D₄×C₂₃	—	2	4	no	—
53	C₂×D₄×C₂₃	—	2	4	no	—
54	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	#41	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₂₃	—	4	4	no	—
60	SD₁₆×C₂₃	—	4	8	no	—
61	D₈×C₂₃	—	4	4	no	—
62	M₁₆×C₂₃	—	2	8	no	—
63	M₁₆×C₂₃	—	4	8	no	—
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	2	4	no	—
67	C₂×C₈×C₂₃	—	4	8	no	—
68	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
71	C₄⋊C₄×C₂₃	—	2	4	no	—
72	C₂²⋊C₄×C₂₃	—	2	8	no	—
73	C₄²×C₂₃	—	2	4	no	—
74	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂×D₄	1	2	yes	—
77	#22282	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	—	2	4	no	—
79	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
80	Q₈×C₂₃	—	2	8	no	—
81	D₄×C₂₃	—	2	8	no	—
82	D₄×C₂₃	—	4	8	no	—
83	D₄×C₂₃	—	8	4	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	—	4	4	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
95	C₈×C₂₃	—	2	2	no	—
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	—	2	4	no	—
98	C₈×C₂₃	—	4	4	no	—
99	#3147	C₂×C₂₃	1	1	yes	characteristic
100	#2932	C₂×C₂₃	1	2	yes	complemented
101	#1963	C₂×C₂₃	1	2	yes	complemented
102	#1876	C₂×C₂₃	1	8	yes	complemented
103	#1837	C₂×C₂₃	1	1	yes	characteristic, complemented
104	#1746	C₂×C₂₃	1	1	yes	characteristic, complemented
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	—	4	4	no	—
107	C₂²×C₂₃	—	4	8	no	—
108	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
109	C₂²×C₂₃	#540	1	2	yes	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	4	4	no	—
113	C₄×C₂₃	—	8	2	no	—
114	C₄×C₂₃	#518	1	2	yes	—
115	C₄×C₂₃	#535	1	2	yes	—
116	C₂×#135	C₂²×C₂₃	1	4	yes	—
117	C₂×#131	C₂²×C₂₃	1	4	yes	—
118	#532	C₂²×C₂₃	1	2	yes	—
119	#495	C₄×C₂₃	1	4	yes	complemented
120	#494	C₄×C₂₃	1	1	yes	characteristic, complemented
121	#493	C₄×C₂₃	1	1	yes	characteristic, complemented
122	#492	C₄×C₂₃	1	2	yes	complemented
123	#435	—	2	8	no	—
124	#423	C₂²×C₂₃	1	2	yes	—
125	C₂×#104	C₂²×C₂₃	1	8	yes	—
126	#408	C₂²×C₂₃	1	4	yes	—
127	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
129	#376	C₂²×C₂₃	1	1	yes	characteristic
130	#341	C₂²×C₂₃	1	4	yes	—
131	#336	C₂²×C₂₃	1	4	yes	—
132	C₂×C₂₃	—	2	2	no	—
133	C₂×C₂₃	—	8	4	no	—
134	C₂×C₂₃	#2328	1	1	yes	characteristic
135	C₂×C₂₃	#2957	1	2	yes	—
136	C₂×#41	C₂³×C₂₃	1	4	yes	—
137	#135	—	4	8	no	—
138	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
139	C₂×SD₁₆	C₂×C₄×C₂₃	1	8	yes	—
140	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
141	#131	—	4	8	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
146	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
147	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
148	#127	C₂×C₄×C₂₃	1	2	yes	—
149	#124	C₂×C₄×C₂₃	1	4	yes	—
150	#121	—	2	4	no	—
151	C₄×D₄	—	2	4	no	—
152	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
153	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	#104	—	4	8	no	—
156	C₂²⋊C₈	—	2	8	no	—
157	#97	D₄×C₂₃	1	4	yes	—
158	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
159	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
161	#95	—	2	2	no	—
162	C₂₃	#22282	1	1	yes	characteristic, direct product factor, Sylow
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	#41×C₂₃	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	8	no	—
182	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
184	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	2	4	no	—
188	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
189	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₂₃	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
204	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
205	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	4	no	—
211	C₈	—	4	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	8	no	—
214	C₂²	—	4	4	no	—
215	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
216	C₂²	C₂₃×#540	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	C₂₃×#518	1	2	yes	—
222	C₄	C₂₃×#535	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	C₂₃×#2328	1	1	yes	characteristic
226	C₂	C₂₃×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2932	C₂	semidirect
3. C₂₃×#1963	C₂	semidirect
4. C₂₃×#1876	C₂	semidirect
5. C₂₃×#1837	C₂	semidirect
6. C₂₃×#1746	C₂	semidirect
10. C₂₃×#495	C₄	semidirect
11. C₂₃×#494	C₄	semidirect
12. C₂₃×#493	C₄	semidirect
13. C₂₃×#492	C₄	semidirect
77. #22282	C₂₃	direct (N × H)
100. #2932	C₂×C₂₃	semidirect
101. #1963	C₂×C₂₃	semidirect
102. #1876	C₂×C₂₃	semidirect
103. #1837	C₂×C₂₃	semidirect
104. #1746	C₂×C₂₃	semidirect
119. #495	C₄×C₂₃	semidirect
120. #494	C₄×C₂₃	semidirect
121. #493	C₄×C₂₃	semidirect
122. #492	C₄×C₂₃	semidirect
162. C₂₃	#22282	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×#3147	2
C₂₃×#2932	2
C₂₃×#1963	2
C₂₃×#1876	2
C₂₃×#1837	2
C₂₃×#1746	2
#22282	23

Subgroups — C23×#22282 #19169908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22282 #19169908