Subgroups — C₂₃×#33908 #19181534

Order 5888.

492 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3341	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2234	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2226	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×C₂₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2158	C₂	1	2	yes	maximal, complemented
9	C₂₃×#555	—	2	2	no	—
10	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#500	C₂²	1	2	yes	—
13	C₂₃×#482	—	2	4	no	—
14	C₂₃×#476	—	2	2	no	—
15	C₂₃×#466	C₂²	1	4	yes	—
16	C₄×D₈×C₂₃	C₂²	1	2	yes	—
17	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#427	C₂²	1	4	yes	—
19	C₂₃×#367	C₂²	1	4	yes	—
20	C₂₃×#366	C₂²	1	1	yes	characteristic
21	C₂₃×#366	C₂²	1	2	yes	—
22	C₂₃×#366	C₂²	1	2	yes	—
23	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₂₃×#357	C₂²	1	2	yes	—
25	C₂₃×#357	C₂²	1	4	yes	—
26	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×D₈×C₂₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂₃×C₄⋊Q₈	—	2	2	no	—
31	C₂₃×#124	—	4	4	no	—
32	C₂₃×#121	—	2	4	no	—
33	C₂₃×#121	—	4	2	no	—
34	C₄×D₄×C₂₃	—	2	2	no	—
35	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
36	C₂₃×#117	—	2	4	no	—
37	C₂₃×#117	—	4	2	no	—
38	C₂₃×D₁₆	—	2	2	no	—
39	C₂×C₁₆×C₂₃	—	2	4	no	—
40	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
42	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×#107	—	2	2	no	—
44	C₂₃×#107	C₂³	1	4	yes	—
45	C₂₃×#107	C₂³	1	2	yes	—
46	C₂₃×#106	—	2	4	no	—
47	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₂₃×#102	—	2	2	no	—
49	C₂₃×#102	—	2	4	no	—
50	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₂₃	—	4	2	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	4	4	no	—
55	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	D₈×C₂₃	—	2	2	no	—
59	D₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	4	2	no	—
68	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
69	C₂²⋊C₄×C₂₃	—	4	4	no	—
70	C₂²⋊C₄×C₂₃	—	4	4	no	—
71	C₂²⋊C₄×C₂₃	—	4	2	no	—
72	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
73	C₄²×C₂₃	—	4	4	no	—
74	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₂₃	—	2	2	no	—
76	C₁₆×C₂₃	—	2	1	no	—
77	C₁₆×C₂₃	#41	1	2	yes	—
78	#33908	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₃	—	4	2	no	—
80	C₂³×C₂₃	C₂×D₈	1	2	yes	—
81	Q₈×C₂₃	—	16	2	no	—
82	D₄×C₂₃	—	2	2	no	—
83	D₄×C₂₃	—	4	2	no	—
84	D₄×C₂₃	—	4	2	no	—
85	D₄×C₂₃	—	16	4	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	4	no	—
90	C₂×C₄×C₂₃	—	8	4	no	—
91	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₂₃	—	2	2	no	—
96	C₈×C₂₃	—	2	1	no	—
97	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
98	C₈×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
99	#3341	C₂×C₂₃	1	2	yes	complemented
100	#2252	C₂×C₂₃	1	1	yes	characteristic, complemented
101	#2234	C₂×C₂₃	1	4	yes	complemented
102	#2226	C₂×C₂₃	1	2	yes	complemented
103	#2215	C₂×C₂₃	1	2	yes	complemented
104	C₄×D₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
105	D₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#2158	C₂×C₂₃	1	2	yes	complemented
107	C₂²×C₂₃	—	2	2	no	—
108	C₂²×C₂₃	—	2	2	no	—
109	C₂²×C₂₃	—	2	2	no	—
110	C₂²×C₂₃	—	8	2	no	—
111	C₂²×C₂₃	—	8	2	no	—
112	C₂²×C₂₃	—	8	2	no	—
113	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
114	C₄×C₂₃	—	2	2	no	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	8	2	no	—
117	C₄×C₂₃	—	8	4	no	—
118	C₄×C₂₃	#508	1	2	yes	—
119	C₄×C₂₃	#547	1	1	yes	characteristic
120	C₄×C₂₃	#578	1	1	yes	characteristic
121	#555	—	2	2	no	—
122	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
123	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
124	#500	C₂²×C₂₃	1	2	yes	—
125	#482	—	2	4	no	—
126	#476	—	2	2	no	—
127	#466	C₂²×C₂₃	1	4	yes	—
128	C₄×D₈	C₂²×C₂₃	1	2	yes	—
129	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
130	#427	C₂²×C₂₃	1	4	yes	—
131	#367	C₂²×C₂₃	1	4	yes	—
132	#366	C₂²×C₂₃	1	1	yes	characteristic
133	#366	C₂²×C₂₃	1	2	yes	—
134	#366	C₂²×C₂₃	1	2	yes	—
135	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
136	#357	C₂²×C₂₃	1	2	yes	—
137	#357	C₂²×C₂₃	1	4	yes	—
138	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
139	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×C₂₃	—	4	2	no	—
141	C₂×C₂₃	—	16	2	no	—
142	C₂×C₂₃	#3334	1	1	yes	characteristic
143	C₂×C₂₃	C₂×#508	1	1	yes	characteristic
144	C₂×C₂₃	#3415	1	1	yes	characteristic
145	C₂×D₈	C₂³×C₂₃	1	2	yes	—
146	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
147	C₄⋊Q₈	—	2	2	no	—
148	#124	—	4	4	no	—
149	#121	—	2	4	no	—
150	#121	—	4	2	no	—
151	C₄×D₄	—	2	2	no	—
152	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
153	#117	—	2	4	no	—
154	#117	—	4	2	no	—
155	D₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	4	no	—
157	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
158	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
159	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
160	#107	—	2	2	no	—
161	#107	C₂³×C₂₃	1	4	yes	—
162	#107	C₂³×C₂₃	1	2	yes	—
163	#106	—	2	4	no	—
164	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
165	#102	—	2	2	no	—
166	#102	—	2	4	no	—
167	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
168	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
169	C₂₃	#33908	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×Q₈	—	4	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
176	D₈	—	2	2	no	—
177	D₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
182	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
187	C₂²⋊C₄	—	4	4	no	—
188	C₂²⋊C₄	—	4	4	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
191	C₄²	—	4	4	no	—
192	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₁₆	—	2	2	no	—
194	C₁₆	—	2	1	no	—
195	C₁₆	#41×C₂₃	1	2	yes	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×D₈×C₂₃	1	2	yes	—
198	Q₈	—	16	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	16	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
215	C₈	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	8	4	no	—
227	C₄	C₂₃×#508	1	2	yes	—
228	C₄	C₂₃×#547	1	1	yes	characteristic
229	C₄	C₂₃×#578	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	16	2	no	—
232	C₂	C₂₃×#3334	1	1	yes	characteristic
233	C₂	C₂×C₂₃×#508	1	1	yes	characteristic
234	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×#3341	C₂	semidirect
2. C₂₃×#2252	C₂	semidirect
3. C₂₃×#2234	C₂	semidirect
4. C₂₃×#2226	C₂	semidirect
5. C₂₃×#2215	C₂	semidirect
6. C₄×C₂₃×D₁₆	C₂	semidirect
7. D₄×C₁₆×C₂₃	C₂	semidirect
8. C₂₃×#2158	C₂	semidirect
78. #33908	C₂₃	direct (N × H)
99. #3341	C₂×C₂₃	semidirect
100. #2252	C₂×C₂₃	semidirect
101. #2234	C₂×C₂₃	semidirect
102. #2226	C₂×C₂₃	semidirect
103. #2215	C₂×C₂₃	semidirect
104. C₄×D₁₆	C₂×C₂₃	semidirect
105. D₄×C₁₆	C₂×C₂₃	semidirect
106. #2158	C₂×C₂₃	semidirect
169. C₂₃	#33908	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×#3341	2
C₂₃×#2252	2
C₂₃×#2234	2
C₂₃×#2226	2
C₂₃×#2215	2
C₄×C₂₃×D₁₆	2
D₄×C₁₆×C₂₃	2
C₂₃×#2158	2
#33908	23

Subgroups — C23×#33908 #19181534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33908 #19181534