Subgroups — C₂₃×#33922 #19181548

Order 5888.

488 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3339	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2234	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2232	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2228	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2218	C₂	1	2	yes	maximal, complemented
7	C₄×C₂₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2159	C₂	1	2	yes	maximal, complemented
10	C₂₃×#553	—	2	2	no	—
11	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
13	C₂₃×#499	C₂²	1	2	yes	—
14	C₂₃×#482	—	2	2	no	—
15	C₂₃×#481	—	2	2	no	—
16	C₂₃×#478	—	2	2	no	—
17	C₂₃×#466	C₂²	1	2	yes	—
18	C₂₃×#466	C₂²	1	2	yes	—
19	C₄×D₈×C₂₃	C₂²	1	2	yes	—
20	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#427	C₂²	1	2	yes	—
22	C₂×C₂₃×#107	C₂²	1	2	yes	—
23	C₂₃×#367	C₂²	1	2	yes	—
24	C₂₃×#367	C₂²	1	2	yes	—
25	C₂₃×#366	C₂²	1	1	yes	characteristic
26	C₂₃×#366	C₂²	1	2	yes	—
27	C₂₃×#366	C₂²	1	1	yes	characteristic
28	C₂₃×#366	C₂²	1	1	yes	characteristic
29	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₂₃×#357	C₂²	1	2	yes	—
31	C₂₃×#357	C₂²	1	2	yes	—
32	C₂₃×#357	C₂²	1	2	yes	—
33	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂₃×#126	—	4	2	no	—
38	C₂₃×#125	—	2	2	no	—
39	C₂₃×#123	—	4	2	no	—
40	C₂₃×#121	—	2	2	no	—
41	C₂₃×#121	—	2	2	no	—
42	C₂₃×#121	—	4	2	no	—
43	C₄×D₄×C₂₃	—	2	2	no	—
44	C₄×D₄×C₂₃	—	4	2	no	—
45	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂₃×#117	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₂₃×D₁₆	—	2	2	no	—
49	C₂₃×M₃₂	—	2	4	no	—
50	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
52	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×#107	—	2	2	no	—
54	C₂₃×#107	C₂³	1	2	yes	—
55	C₂₃×#107	C₂³	1	2	yes	—
56	C₂₃×#107	C₂³	1	2	yes	—
57	C₂₃×#106	—	2	2	no	—
58	C₂₃×#106	—	2	2	no	—
59	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₂₃×#102	—	2	2	no	—
61	C₂₃×#102	—	2	2	no	—
62	C₂₃×#102	—	2	2	no	—
63	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
64	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₂₃	—	2	2	no	—
66	C₂×D₄×C₂₃	—	4	2	no	—
67	C₂×D₄×C₂₃	—	4	2	no	—
68	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	—	4	2	no	—
70	C₂²×C₄×C₂₃	—	4	2	no	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	D₈×C₂₃	—	2	2	no	—
73	D₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	—	4	2	no	—
83	C₄⋊C₄×C₂₃	—	4	2	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₂²⋊C₄×C₂₃	—	4	2	no	—
88	C₂²⋊C₄×C₂₃	—	4	2	no	—
89	C₂²⋊C₄×C₂₃	—	4	2	no	—
90	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
91	C₄²×C₂₃	—	4	2	no	—
92	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₂₃	—	2	2	no	—
94	C₁₆×C₂₃	—	2	1	no	—
95	C₁₆×C₂₃	#41	1	2	yes	—
96	#33922	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	4	2	no	—
98	C₂³×C₂₃	C₂×D₈	1	2	yes	—
99	D₄×C₂₃	—	4	1	no	—
100	D₄×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	4	2	no	—
102	D₄×C₂₃	—	8	4	no	—
103	D₄×C₂₃	—	16	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	16	2	no	—
110	C₂×C₄×C₂₃	—	16	2	no	—
111	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₂₃	—	2	2	no	—
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₈×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
119	#3339	C₂×C₂₃	1	2	yes	complemented
120	#2253	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#2234	C₂×C₂₃	1	2	yes	complemented
122	#2232	C₂×C₂₃	1	2	yes	complemented
123	#2228	C₂×C₂₃	1	2	yes	complemented
124	#2218	C₂×C₂₃	1	2	yes	complemented
125	C₄×D₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#2169	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#2159	C₂×C₂₃	1	2	yes	complemented
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	2	2	no	—
130	C₂²×C₂₃	—	2	2	no	—
131	C₂²×C₂₃	—	8	2	no	—
132	C₂²×C₂₃	—	8	2	no	—
133	C₂²×C₂₃	—	8	2	no	—
134	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	8	2	no	—
139	C₄×C₂₃	—	8	2	no	—
140	C₄×C₂₃	—	8	2	no	—
141	C₄×C₂₃	#547	1	1	yes	characteristic
142	C₄×C₂₃	#578	1	1	yes	characteristic
143	#553	—	2	2	no	—
144	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
146	#499	C₂²×C₂₃	1	2	yes	—
147	#482	—	2	2	no	—
148	#481	—	2	2	no	—
149	#478	—	2	2	no	—
150	#466	C₂²×C₂₃	1	2	yes	—
151	#466	C₂²×C₂₃	1	2	yes	—
152	C₄×D₈	C₂²×C₂₃	1	2	yes	—
153	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
154	#427	C₂²×C₂₃	1	2	yes	—
155	C₂×#107	C₂²×C₂₃	1	2	yes	—
156	#367	C₂²×C₂₃	1	2	yes	—
157	#367	C₂²×C₂₃	1	2	yes	—
158	#366	C₂²×C₂₃	1	1	yes	characteristic
159	#366	C₂²×C₂₃	1	2	yes	—
160	#366	C₂²×C₂₃	1	1	yes	characteristic
161	#366	C₂²×C₂₃	1	1	yes	characteristic
162	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
163	#357	C₂²×C₂₃	1	2	yes	—
164	#357	C₂²×C₂₃	1	2	yes	—
165	#357	C₂²×C₂₃	1	2	yes	—
166	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
167	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₂₃	—	4	2	no	—
169	C₂×C₂₃	—	16	2	no	—
170	C₂×C₂₃	#3334	1	1	yes	characteristic
171	C₂×C₂₃	#3414	1	1	yes	characteristic
172	C₂×C₂₃	#3415	1	1	yes	characteristic
173	C₂×D₈	C₂³×C₂₃	1	2	yes	—
174	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
175	#126	—	4	2	no	—
176	#125	—	2	2	no	—
177	#123	—	4	2	no	—
178	#121	—	2	2	no	—
179	#121	—	2	2	no	—
180	#121	—	4	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	4	2	no	—
183	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
184	#117	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	D₁₆	—	2	2	no	—
187	M₃₂	—	2	4	no	—
188	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
190	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
191	#107	—	2	2	no	—
192	#107	C₂³×C₂₃	1	2	yes	—
193	#107	C₂³×C₂₃	1	2	yes	—
194	#107	C₂³×C₂₃	1	2	yes	—
195	#106	—	2	2	no	—
196	#106	—	2	2	no	—
197	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
198	#102	—	2	2	no	—
199	#102	—	2	2	no	—
200	#102	—	2	2	no	—
201	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
202	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₂₃	#33922	1	1	yes	characteristic, direct product factor, Sylow
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
211	D₈	—	2	2	no	—
212	D₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
217	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₁₆	—	2	2	no	—
233	C₁₆	—	2	1	no	—
234	C₁₆	#41×C₂₃	1	2	yes	—
235	C₂³	—	4	2	no	—
236	C₂³	C₂×D₈×C₂₃	1	2	yes	—
237	D₄	—	4	1	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	8	4	no	—
241	D₄	—	16	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	16	2	no	—
248	C₂×C₄	—	16	2	no	—
249	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
250	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	1	no	—
255	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
256	C₈	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	C₂₃×#547	1	1	yes	characteristic
271	C₄	C₂₃×#578	1	1	yes	characteristic
272	C₂	—	4	2	no	—
273	C₂	—	16	2	no	—
274	C₂	C₂₃×#3334	1	1	yes	characteristic
275	C₂	C₂₃×#3414	1	1	yes	characteristic
276	C₂	C₂₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ C₈×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3339	C₂	semidirect
2. C₂₃×#2253	C₂	semidirect
3. C₂₃×#2234	C₂	semidirect
4. C₂₃×#2232	C₂	semidirect
5. C₂₃×#2228	C₂	semidirect
6. C₂₃×#2218	C₂	semidirect
7. C₄×C₂₃×D₁₆	C₂	semidirect
8. C₂₃×#2169	C₂	semidirect
9. C₂₃×#2159	C₂	semidirect
96. #33922	C₂₃	direct (N × H)
119. #3339	C₂×C₂₃	semidirect
120. #2253	C₂×C₂₃	semidirect
121. #2234	C₂×C₂₃	semidirect
122. #2232	C₂×C₂₃	semidirect
123. #2228	C₂×C₂₃	semidirect
124. #2218	C₂×C₂₃	semidirect
125. C₄×D₁₆	C₂×C₂₃	semidirect
126. #2169	C₂×C₂₃	semidirect
127. #2159	C₂×C₂₃	semidirect
203. C₂₃	#33922	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×#3339	2
C₂₃×#2253	2
C₂₃×#2234	2
C₂₃×#2232	2
C₂₃×#2228	2
C₂₃×#2218	2
C₄×C₂₃×D₁₆	2
C₂₃×#2169	2
C₂₃×#2159	2
#33922	23

Subgroups — C23×#33922 #19181548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33922 #19181548