Subgroups — C₂₃×#23360 #19170986

Order 5888.

688 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1905	C₂	1	4	yes	maximal, complemented
3	C₂₃×#1902	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1886	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1884	C₂	1	2	yes	maximal
6	C₂₃×#1841	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#137	C₂²	1	4	yes	—
8	C₂₃×#568	C₂²	1	2	yes	—
9	C₂₃×#532	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂₃×#519	C₂²	1	2	yes	—
12	C₂₃×#497	C₄	1	4	yes	complemented
13	C₂₃×#494	C₄	1	2	yes	complemented
14	C₂₃×#492	C₄	1	2	yes	complemented
15	C₂₃×#421	C₂²	1	4	yes	—
16	C₂₃×#419	C₂²	1	4	yes	—
17	C₂×C₂₃×#101	C₂²	1	2	yes	—
18	C₂₃×#410	C₂²	1	2	yes	—
19	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#344	C₂²	1	4	yes	—
21	C₂₃×#339	C₂²	1	4	yes	—
22	C₂₃×#338	C₂²	1	2	yes	—
23	C₂₃×H(3,ℤ/4)	C₂²	1	2	yes	—
24	C₂×#41×C₂₃	C₂³	1	2	yes	—
25	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
26	C₂₃×#137	—	4	4	no	—
27	C₂₃×#137	—	4	4	no	—
28	C₂×Q₁₆×C₂₃	C₂×C₄	1	8	yes	—
29	C₂×SD₁₆×C₂₃	C₂×C₄	1	8	yes	—
30	C₂₃×#131	—	4	4	no	—
31	C₂×M₁₆×C₂₃	—	4	2	no	—
32	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
33	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
34	C₂₃×C₄⋊Q₈	—	4	1	no	—
35	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
39	C₂₃×#124	C₂×C₄	1	2	yes	—
40	C₂₃×#121	—	4	1	no	—
41	C₄×Q₈×C₂₃	—	4	4	no	—
42	C₄×D₄×C₂₃	—	4	1	no	—
43	C₂₃×#117	—	4	2	no	—
44	C₂₃×#117	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×#104	—	4	4	no	—
48	C₂₃×#103	—	4	4	no	—
49	C₂₃×#101	—	4	2	no	—
50	C₂₃×#99	—	4	2	no	—
51	C₂₃×#97	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
53	#41×C₂₃	—	2	4	no	—
54	#41×C₂₃	—	2	4	no	—
55	C₂×Q₈×C₂₃	—	2	2	no	—
56	C₂×Q₈×C₂₃	—	2	4	no	—
57	C₂×Q₈×C₂₃	—	2	4	no	—
58	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	4	1	no	—
62	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	—	2	2	no	—
66	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	4	8	no	—
68	SD₁₆×C₂₃	—	4	8	no	—
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	2	4	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	4	4	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	4	4	no	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₄⋊C₄×C₂₃	—	4	2	no	—
79	C₄⋊C₄×C₂₃	—	4	1	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₂²⋊C₄×C₂₃	—	2	4	no	—
82	C₂²⋊C₄×C₂₃	—	4	2	no	—
83	C₄²×C₂₃	—	2	1	no	—
84	C₄²×C₂₃	—	2	4	no	—
85	C₄²×C₂₃	—	4	4	no	—
86	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	#23360	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	2	2	no	—
90	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
91	Q₈×C₂₃	—	2	8	no	—
92	Q₈×C₂₃	—	2	4	no	—
93	Q₈×C₂₃	—	4	4	no	—
94	D₄×C₂₃	—	2	4	no	—
95	D₄×C₂₃	—	4	4	no	—
96	D₄×C₂₃	—	8	2	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
109	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₈×C₂₃	—	2	4	no	—
111	C₈×C₂₃	—	4	4	no	—
112	#3152	C₂×C₂₃	1	1	yes	characteristic
113	#1905	C₂×C₂₃	1	4	yes	complemented
114	#1902	C₂×C₂₃	1	4	yes	complemented
115	#1886	C₂×C₂₃	1	2	yes	complemented
116	#1884	C₂×C₂₃	1	2	yes	—
117	#1841	C₂×C₂₃	1	2	yes	complemented
118	C₂²×C₂₃	—	4	1	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	8	2	no	—
121	C₂²×C₂₃	#518	1	1	yes	characteristic
122	C₂²×C₂₃	#560	1	2	yes	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	—	4	4	no	—
127	C₄×C₂₃	—	8	4	no	—
128	C₄×C₂₃	#518	1	2	yes	—
129	C₄×C₂₃	#560	1	2	yes	—
130	C₂×#137	C₂²×C₂₃	1	4	yes	—
131	#568	C₂²×C₂₃	1	2	yes	—
132	#532	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
134	#519	C₂²×C₂₃	1	2	yes	—
135	#497	C₄×C₂₃	1	4	yes	complemented
136	#494	C₄×C₂₃	1	2	yes	complemented
137	#492	C₄×C₂₃	1	2	yes	complemented
138	#421	C₂²×C₂₃	1	4	yes	—
139	#419	C₂²×C₂₃	1	4	yes	—
140	C₂×#101	C₂²×C₂₃	1	2	yes	—
141	#410	C₂²×C₂₃	1	2	yes	—
142	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
143	#344	C₂²×C₂₃	1	4	yes	—
144	#339	C₂²×C₂₃	1	4	yes	—
145	#338	C₂²×C₂₃	1	2	yes	—
146	H(3,ℤ/4)	C₂²×C₂₃	1	2	yes	—
147	C₂×C₂₃	—	2	1	no	—
148	C₂×C₂₃	—	2	2	no	—
149	C₂×C₂₃	—	8	2	no	—
150	C₂×C₂₃	#2380	1	1	yes	characteristic
151	C₂×#41	C₂³×C₂₃	1	2	yes	—
152	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
153	#137	—	4	4	no	—
154	#137	—	4	4	no	—
155	C₂×Q₁₆	C₂×C₄×C₂₃	1	8	yes	—
156	C₂×SD₁₆	C₂×C₄×C₂₃	1	8	yes	—
157	#131	—	4	4	no	—
158	C₂×M₁₆	—	4	2	no	—
159	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
160	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
161	C₄⋊Q₈	—	4	1	no	—
162	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
163	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
164	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
165	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
166	#124	C₂×C₄×C₂₃	1	2	yes	—
167	#121	—	4	1	no	—
168	C₄×Q₈	—	4	4	no	—
169	C₄×D₄	—	4	1	no	—
170	#117	—	4	2	no	—
171	#117	C₂³×C₂₃	1	4	yes	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
174	#104	—	4	4	no	—
175	#103	—	4	4	no	—
176	#101	—	4	2	no	—
177	#99	—	4	2	no	—
178	#97	C₂×C₄×C₂₃	1	2	yes	—
179	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
180	C₂₃	#23360	1	1	yes	characteristic, direct product factor, Sylow
181	#41	—	2	4	no	—
182	#41	—	2	4	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
187	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
195	Q₁₆	—	4	8	no	—
196	SD₁₆	—	4	8	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₂³	—	2	2	no	—
217	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
218	Q₈	—	2	8	no	—
219	Q₈	—	2	4	no	—
220	Q₈	—	4	4	no	—
221	D₄	—	2	4	no	—
222	D₄	—	4	4	no	—
223	D₄	—	8	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
237	C₈	—	2	4	no	—
238	C₈	—	4	4	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	C₂₃×#518	1	1	yes	characteristic
243	C₂²	C₂₃×#560	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	4	no	—
249	C₄	C₂₃×#518	1	2	yes	—
250	C₄	C₂₃×#560	1	2	yes	—
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	8	2	no	—
254	C₂	C₂₃×#2380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2380
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1905	C₂	semidirect
3. C₂₃×#1902	C₂	semidirect
4. C₂₃×#1886	C₂	semidirect
6. C₂₃×#1841	C₂	semidirect
12. C₂₃×#497	C₄	semidirect
13. C₂₃×#494	C₄	semidirect
14. C₂₃×#492	C₄	semidirect
88. #23360	C₂₃	direct (N × H)
113. #1905	C₂×C₂₃	semidirect
114. #1902	C₂×C₂₃	semidirect
115. #1886	C₂×C₂₃	semidirect
117. #1841	C₂×C₂₃	semidirect
135. #497	C₄×C₂₃	semidirect
136. #494	C₄×C₂₃	semidirect
137. #492	C₄×C₂₃	semidirect
180. C₂₃	#23360	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×#3152	2
C₂₃×#1905	2
C₂₃×#1902	2
C₂₃×#1886	2
C₂₃×#1884	2
C₂₃×#1841	2
#23360	23

Subgroups — C23×#23360 #19170986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23360 #19170986