Subgroups — C₂₃×#33708 #19181334

Order 5888.

590 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2245	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2244	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2242	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2139	C₂	1	4	yes	maximal, complemented
7	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	—	2	1	no	—
9	C₂×C₂₃×#135	C₂²	1	2	yes	—
10	C₂₃×#532	—	2	1	no	—
11	C₂×C₂₃×#125	—	2	1	no	—
12	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
13	C₂₃×#499	—	2	1	no	—
14	C₂₃×#499	—	2	1	no	—
15	C₂₃×#499	C₂²	1	2	yes	—
16	C₂₃×#499	C₂²	1	2	yes	—
17	C₂₃×#495	—	2	1	no	—
18	C₂₃×#495	C₂²	1	2	yes	—
19	C₂₃×#494	C₂²	1	1	yes	characteristic
20	C₂₃×#493	C₂²	1	1	yes	characteristic
21	C₂₃×#492	—	2	1	no	—
22	C₂×C₂₃×#107	C₂²	1	2	yes	—
23	C₂×C₂₃×#106	—	2	1	no	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#358	C₂²	1	8	yes	—
26	C₂₃×#357	C₂²	1	8	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	8	no	—
31	C₂₃×#135	—	4	4	no	—
32	C₂×Q₁₆×C₂₃	—	2	1	no	—
33	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
34	C₂×SD₁₆×C₂₃	—	2	2	no	—
35	C₂×SD₁₆×C₂₃	—	2	2	no	—
36	C₂×D₈×C₂₃	—	2	1	no	—
37	C₂×D₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊Q₈	—	2	1	no	—
41	C₂₃×#127	—	2	1	no	—
42	C₂₃×#125	—	2	2	no	—
43	C₂₃×#125	—	2	2	no	—
44	C₂₃×#125	—	4	2	no	—
45	C₂₃×#124	—	2	2	no	—
46	C₂₃×#121	—	4	2	no	—
47	C₄×D₄×C₂₃	—	4	2	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
51	C₂₃×M₃₂	—	2	4	no	—
52	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
53	C₂₃×#107	—	2	2	no	—
54	C₂₃×#107	C₂³	1	4	yes	—
55	C₂₃×#106	—	2	2	no	—
56	C₂₃×#106	—	2	2	no	—
57	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	#41×C₂₃	—	4	4	no	—
62	#41×C₂₃	—	4	4	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×D₄×C₂₃	—	2	2	no	—
65	C₂×D₄×C₂₃	—	4	2	no	—
66	C₂×D₄×C₂₃	—	4	2	no	—
67	C₂²×C₄×C₂₃	—	4	2	no	—
68	C₂²×C₄×C₂₃	—	4	2	no	—
69	C₂²×C₄×C₂₃	D₈	1	2	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	Q₁₆×C₂₃	—	2	4	no	—
72	Q₁₆×C₂₃	—	4	2	no	—
73	SD₁₆×C₂₃	—	4	4	no	—
74	SD₁₆×C₂₃	—	4	4	no	—
75	D₈×C₂₃	—	2	4	no	—
76	D₈×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
78	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	#41	1	4	yes	—
82	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₂₃	—	2	4	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	—	4	4	no	—
86	C₄⋊C₄×C₂₃	—	4	4	no	—
87	C₄⋊C₄×C₂₃	—	4	2	no	—
88	C₂²⋊C₄×C₂₃	—	4	4	no	—
89	C₄²×C₂₃	D₈	1	2	yes	—
90	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₁₆×C₂₃	—	2	4	no	—
93	#33708	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	4	2	no	—
95	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
96	Q₈×C₂₃	—	4	2	no	—
97	Q₈×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	4	2	no	—
99	D₄×C₂₃	—	4	2	no	—
100	D₄×C₂₃	—	8	4	no	—
101	D₄×C₂₃	—	8	4	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	8	4	no	—
106	C₂×C₄×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	#124	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
112	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₂₃	#124	1	2	yes	—
114	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
115	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
116	C₈×C₂₃	C₂×#41	1	2	yes	—
117	C₂×#499	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#3147	C₂×C₂₃	1	1	yes	characteristic
119	#2245	C₂×C₂₃	1	2	yes	complemented
120	#2244	C₂×C₂₃	1	2	yes	complemented
121	#2242	C₂×C₂₃	1	4	yes	complemented
122	#2139	C₂×C₂₃	1	4	yes	complemented
123	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	—	8	4	no	—
127	C₂²×C₂₃	#486	1	2	yes	—
128	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₂₃	—	2	2	no	—
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	8	2	no	—
132	C₄×C₂₃	—	8	4	no	—
133	C₄×C₂₃	#486	1	2	yes	—
134	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
135	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
136	C₂×#135	—	2	1	no	—
137	C₂×#135	C₂²×C₂₃	1	2	yes	—
138	#532	—	2	1	no	—
139	C₂×#125	—	2	1	no	—
140	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
141	#499	—	2	1	no	—
142	#499	—	2	1	no	—
143	#499	C₂²×C₂₃	1	2	yes	—
144	#499	C₂²×C₂₃	1	2	yes	—
145	#495	—	2	1	no	—
146	#495	C₂²×C₂₃	1	2	yes	—
147	#494	C₂²×C₂₃	1	1	yes	characteristic
148	#493	C₂²×C₂₃	1	1	yes	characteristic
149	#492	—	2	1	no	—
150	C₂×#107	C₂²×C₂₃	1	2	yes	—
151	C₂×#106	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
153	#358	C₂²×C₂₃	1	8	yes	—
154	#357	C₂²×C₂₃	1	8	yes	—
155	#346	C₂²×C₂₃	1	2	yes	—
156	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
157	C₂×C₂₃	—	2	2	no	—
158	C₂×C₂₃	—	16	2	no	—
159	C₂×C₂₃	C₂×#486	1	1	yes	characteristic
160	C₂×C₂₃	#3414	1	2	yes	—
161	C₂×#41	—	2	2	no	—
162	#135	—	2	8	no	—
163	#135	—	4	4	no	—
164	C₂×Q₁₆	—	2	1	no	—
165	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
166	C₂×SD₁₆	—	2	2	no	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	C₂³×C₂₃	1	2	yes	—
170	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	1	no	—
173	#127	—	2	1	no	—
174	#125	—	2	2	no	—
175	#125	—	2	2	no	—
176	#125	—	4	2	no	—
177	#124	—	2	2	no	—
178	#121	—	4	2	no	—
179	C₄×D₄	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	4	2	no	—
182	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
183	M₃₂	—	2	4	no	—
184	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
185	#107	—	2	2	no	—
186	#107	C₂³×C₂₃	1	4	yes	—
187	#106	—	2	2	no	—
188	#106	—	2	2	no	—
189	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
190	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
193	C₂₃	#33708	1	1	yes	characteristic, direct product factor, Sylow
194	#41	—	4	4	no	—
195	#41	—	4	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₈×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	4	2	no	—
206	SD₁₆	—	4	4	no	—
207	SD₁₆	—	4	4	no	—
208	D₈	—	2	4	no	—
209	D₈	—	4	2	no	—
210	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
211	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₈	#41×C₂₃	1	4	yes	—
215	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	D₈×C₂₃	1	2	yes	—
223	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₁₆	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	C₂₃×#124	1	2	yes	—
240	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
241	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
245	C₈	C₂₃×#124	1	2	yes	—
246	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
247	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
248	C₈	C₂×#41×C₂₃	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₂₃×#486	1	2	yes	—
253	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	4	no	—
258	C₄	C₂₃×#486	1	2	yes	—
259	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
260	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	16	2	no	—
263	C₂	C₂×C₂₃×#486	1	1	yes	characteristic
264	C₂	C₂₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#499	C₂	semidirect
3. C₂₃×#2245	C₂	semidirect
4. C₂₃×#2244	C₂	semidirect
5. C₂₃×#2242	C₂	semidirect
6. C₂₃×#2139	C₂	semidirect
7. C₄×C₂₃×M₃₂	C₂	semidirect
93. #33708	C₂₃	direct (N × H)
117. C₂×#499	C₂×C₂₃	semidirect
119. #2245	C₂×C₂₃	semidirect
120. #2244	C₂×C₂₃	semidirect
121. #2242	C₂×C₂₃	semidirect
122. #2139	C₂×C₂₃	semidirect
123. C₄×M₃₂	C₂×C₂₃	semidirect
193. C₂₃	#33708	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×#499	2
C₂₃×#3147	2
C₂₃×#2245	2
C₂₃×#2244	2
C₂₃×#2242	2
C₂₃×#2139	2
C₄×C₂₃×M₃₂	2
#33708	23

Subgroups — C23×#33708 #19181334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33708 #19181334