Subgroups — C₂₃×#32702 #19180328

Order 5888.

480 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2103	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2102	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2080	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2050	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1937	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1881	C₂	1	2	yes	maximal
7	C₂₃×#1861	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1858	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#575	C₂²	1	1	yes	characteristic
11	C₂₃×#568	C₂²	1	2	yes	—
12	C₂₃×#430	C₂²	1	1	yes	characteristic
13	C₂₃×#430	C₂²	1	2	yes	—
14	C₂₃×#424	C₂²	1	2	yes	—
15	C₂₃×#423	C₂²	1	1	yes	characteristic
16	C₂₃×#423	C₂²	1	1	yes	characteristic
17	C₂₃×#421	C₂²	1	2	yes	—
18	C₂×C₂₃×#104	C₂²	1	2	yes	—
19	C₂₃×#416	C₂²	1	2	yes	—
20	C₂₃×#413	C₂²	1	2	yes	—
21	C₂₃×#408	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#344	C₂²	1	2	yes	—
24	C₂₃×#344	C₂²	1	2	yes	—
25	C₂₃×#341	C₂²	1	2	yes	—
26	C₂₃×#341	C₂²	1	2	yes	—
27	C₂₃×#341	C₂²	1	2	yes	—
28	C₂₃×#339	C₂²	1	2	yes	—
29	C₂₃×#338	C₂²	1	1	yes	characteristic
30	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×#41×C₂₃	C₂³	1	2	yes	—
32	C₂₃×#137	—	2	2	no	—
33	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
34	C₂₃×#135	—	4	1	no	—
35	C₂₃×#131	—	4	2	no	—
36	C₂₃×#131	—	4	2	no	—
37	C₂×M₁₆×C₂₃	—	4	2	no	—
38	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
39	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
40	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂₃×#117	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#108	—	4	2	no	—
47	C₂₃×#108	—	4	1	no	—
48	C₂₃×C₄⋊C₈	—	4	1	no	—
49	C₂₃×C₄⋊C₈	—	4	1	no	—
50	C₂₃×C₄⋊C₈	—	4	2	no	—
51	C₂₃×#104	—	2	2	no	—
52	C₂₃×#104	—	2	2	no	—
53	C₂₃×#104	—	4	2	no	—
54	C₂₃×#103	—	2	2	no	—
55	C₂₃×#102	—	2	2	no	—
56	C₂₃×#101	—	2	2	no	—
57	C₂₃×#100	—	2	2	no	—
58	C₂₃×C₂²⋊C₈	—	4	2	no	—
59	C₄×C₈×C₂₃	—	2	1	no	—
60	C₄×C₈×C₂₃	—	2	1	no	—
61	#41×C₂₃	—	2	2	no	—
62	#41×C₂₃	—	2	2	no	—
63	#41×C₂₃	—	2	2	no	—
64	#41×C₂₃	—	2	2	no	—
65	C₂×Q₈×C₂₃	#41	1	2	yes	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	#41	1	2	yes	—
68	C₂²×C₄×C₂₃	—	2	1	no	—
69	C₂²×C₄×C₂₃	—	2	2	no	—
70	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	Q₁₆×C₂₃	—	4	1	no	—
72	SD₁₆×C₂₃	—	4	2	no	—
73	D₈×C₂₃	—	4	1	no	—
74	M₁₆×C₂₃	—	2	2	no	—
75	M₁₆×C₂₃	—	2	2	no	—
76	M₁₆×C₂₃	—	2	2	no	—
77	M₁₆×C₂₃	—	2	2	no	—
78	M₁₆×C₂₃	—	2	2	no	—
79	M₁₆×C₂₃	—	2	2	no	—
80	M₁₆×C₂₃	—	2	2	no	—
81	M₁₆×C₂₃	—	2	2	no	—
82	M₁₆×C₂₃	—	4	4	no	—
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	1	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	1	no	—
90	C₂×C₈×C₂₃	—	2	2	no	—
91	C₂×C₈×C₂₃	—	4	4	no	—
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	—	2	1	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	#41	1	1	yes	characteristic
97	C₄²×C₂₃	#41	1	1	yes	characteristic
98	#32702	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₂₃	—	2	2	no	—
100	C₂³×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
101	Q₈×C₂₃	—	2	2	no	—
102	Q₈×C₂₃	—	2	2	no	—
103	D₄×C₂₃	—	2	2	no	—
104	D₄×C₂₃	—	2	2	no	—
105	D₄×C₂₃	—	4	2	no	—
106	D₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
115	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
118	C₈×C₂₃	—	4	1	no	—
119	C₈×C₂₃	—	4	1	no	—
120	C₈×C₂₃	—	4	2	no	—
121	C₈×C₂₃	—	4	2	no	—
122	C₈×C₂₃	—	4	2	no	—
123	C₈×C₂₃	—	4	2	no	—
124	#2103	C₂×C₂₃	1	2	yes	complemented
125	#2102	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#2080	C₂×C₂₃	1	2	yes	complemented
127	#2050	C₂×C₂₃	1	1	yes	characteristic
128	#1937	C₂×C₂₃	1	2	yes	complemented
129	#1881	C₂×C₂₃	1	2	yes	—
130	#1861	C₂×C₂₃	1	2	yes	complemented
131	#1858	C₂×C₂₃	1	2	yes	complemented
132	#1837	C₂×C₂₃	1	1	yes	characteristic, complemented
133	C₂²×C₂₃	—	4	1	no	—
134	C₂²×C₂₃	—	4	2	no	—
135	C₂²×C₂₃	—	8	2	no	—
136	C₂²×C₂₃	#539	1	1	yes	characteristic
137	C₂²×C₂₃	#539	1	2	yes	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	—	4	2	no	—
141	C₄×C₂₃	—	8	2	no	—
142	C₄×C₂₃	#551	1	1	yes	characteristic
143	C₄×C₂₃	#553	1	2	yes	—
144	C₄×C₂₃	H(3,𝔽₄)	1	1	yes	characteristic
145	#575	C₂²×C₂₃	1	1	yes	characteristic
146	#568	C₂²×C₂₃	1	2	yes	—
147	#430	C₂²×C₂₃	1	1	yes	characteristic
148	#430	C₂²×C₂₃	1	2	yes	—
149	#424	C₂²×C₂₃	1	2	yes	—
150	#423	C₂²×C₂₃	1	1	yes	characteristic
151	#423	C₂²×C₂₃	1	1	yes	characteristic
152	#421	C₂²×C₂₃	1	2	yes	—
153	C₂×#104	C₂²×C₂₃	1	2	yes	—
154	#416	C₂²×C₂₃	1	2	yes	—
155	#413	C₂²×C₂₃	1	2	yes	—
156	#408	C₂²×C₂₃	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
158	#344	C₂²×C₂₃	1	2	yes	—
159	#344	C₂²×C₂₃	1	2	yes	—
160	#341	C₂²×C₂₃	1	2	yes	—
161	#341	C₂²×C₂₃	1	2	yes	—
162	#341	C₂²×C₂₃	1	2	yes	—
163	#339	C₂²×C₂₃	1	2	yes	—
164	#338	C₂²×C₂₃	1	1	yes	characteristic
165	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	8	2	no	—
169	C₂×C₂₃	#2782	1	1	yes	characteristic
170	C₂×#41	C₂³×C₂₃	1	2	yes	—
171	#137	—	2	2	no	—
172	AGL(1,ℤ/8)	—	2	2	no	—
173	#135	—	4	1	no	—
174	#131	—	4	2	no	—
175	#131	—	4	2	no	—
176	C₂×M₁₆	—	4	2	no	—
177	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
178	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
179	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
181	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
182	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
183	#117	C₂³×C₂₃	1	2	yes	—
184	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
185	#108	—	4	2	no	—
186	#108	—	4	1	no	—
187	C₄⋊C₈	—	4	1	no	—
188	C₄⋊C₈	—	4	1	no	—
189	C₄⋊C₈	—	4	2	no	—
190	#104	—	2	2	no	—
191	#104	—	2	2	no	—
192	#104	—	4	2	no	—
193	#103	—	2	2	no	—
194	#102	—	2	2	no	—
195	#101	—	2	2	no	—
196	#100	—	2	2	no	—
197	C₂²⋊C₈	—	4	2	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₂₃	#32702	1	1	yes	characteristic, direct product factor, Sylow
201	#41	—	2	2	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	2	2	no	—
205	C₂×Q₈	#41×C₂₃	1	2	yes	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	#41×C₂₃	1	2	yes	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
211	Q₁₆	—	4	1	no	—
212	SD₁₆	—	4	2	no	—
213	D₈	—	4	1	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	2	no	—
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	4	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	4	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	#41×C₂₃	1	1	yes	characteristic
237	C₄²	#41×C₂₃	1	1	yes	characteristic
238	C₂³	—	2	2	no	—
239	C₂³	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
254	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	C₂₃×#539	1	1	yes	characteristic
267	C₂²	C₂₃×#539	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	8	2	no	—
272	C₄	C₂₃×#551	1	1	yes	characteristic
273	C₄	C₂₃×#553	1	2	yes	—
274	C₄	C₂₃×H(3,𝔽₄)	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	2	no	—
277	C₂	—	8	2	no	—
278	C₂	C₂₃×#2782	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂×C₂₃	#2782
prime radical	C₂₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2103	C₂	semidirect
2. C₂₃×#2102	C₂	semidirect
3. C₂₃×#2080	C₂	semidirect
5. C₂₃×#1937	C₂	semidirect
7. C₂₃×#1861	C₂	semidirect
8. C₂₃×#1858	C₂	semidirect
9. C₂₃×#1837	C₂	semidirect
98. #32702	C₂₃	direct (N × H)
124. #2103	C₂×C₂₃	semidirect
125. #2102	C₂×C₂₃	semidirect
126. #2080	C₂×C₂₃	semidirect
128. #1937	C₂×C₂₃	semidirect
130. #1861	C₂×C₂₃	semidirect
131. #1858	C₂×C₂₃	semidirect
132. #1837	C₂×C₂₃	semidirect
200. C₂₃	#32702	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×#2103	2
C₂₃×#2102	2
C₂₃×#2080	2
C₂₃×#2050	2
C₂₃×#1937	2
C₂₃×#1881	2
C₂₃×#1861	2
C₂₃×#1858	2
C₂₃×#1837	2
#32702	23

Subgroups — C23×#32702 #19180328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32702 #19180328