Subgroups — C₂₃×#33729 #19181355

Order 5888.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3164	C₂	1	2	yes	maximal
2	C₂₃×#2255	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2253	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2252	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2159	C₂	1	2	yes	maximal
6	C₂₃×#2158	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×#366	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#544	—	2	2	no	—
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#501	—	2	4	no	—
13	C₂₃×#500	C₂²	1	4	yes	—
14	C₂₃×#499	C₂²	1	4	yes	—
15	C₂₃×#428	—	2	2	no	—
16	C₂₃×#427	C₂²	1	2	yes	—
17	C₂×C₂₃×#107	C₂²	1	2	yes	—
18	C₂₃×#405	C₂²	1	1	yes	characteristic
19	C₂₃×#367	C₂²	1	2	yes	—
20	C₂₃×#367	C₂²	1	4	yes	—
21	C₂₃×#366	C₂²	1	2	yes	—
22	C₂₃×#366	C₂²	1	2	yes	—
23	C₂₃×#366	C₂²	1	1	yes	characteristic
24	C₂₃×#366	C₂²	1	1	yes	characteristic
25	C₂₃×#366	C₂²	1	4	yes	—
26	C₂₃×#346	C₂²	1	2	yes	—
27	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
28	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×C₄⋊Q₈	—	2	4	no	—
31	C₂₃×C₄⋊Q₈	—	4	2	no	—
32	C₂₃×#125	—	2	4	no	—
33	C₂₃×#125	—	4	2	no	—
34	C₂₃×C₂²⋊Q₈	—	4	4	no	—
35	C₂₃×#117	—	2	2	no	—
36	C₂₃×#117	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂₃×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₂₃	Q₈	1	4	yes	complemented
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	—	2	4	no	—
45	C₂₃×#107	C₂³	1	4	yes	—
46	C₂₃×#107	C₂³	1	2	yes	—
47	C₂₃×#107	C₂³	1	2	yes	—
48	C₂₃×#106	—	2	2	no	—
49	C₂₃×#106	—	2	4	no	—
50	C₂₃×#97	D₄	1	2	yes	—
51	C₄×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂×Q₈×C₂₃	—	4	4	no	—
53	C₂²×C₄×C₂₃	—	4	2	no	—
54	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₂₃	Q₁₆	1	4	yes	—
56	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	4	no	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	—	4	2	no	—
65	C₄⋊C₄×C₂₃	—	4	4	no	—
66	C₄⋊C₄×C₂₃	—	4	4	no	—
67	C₄⋊C₄×C₂₃	—	4	4	no	—
68	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₃	—	4	2	no	—
71	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
72	C₄²×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	C₂×D₄	1	2	yes	—
74	C₁₆×C₂₃	C₂×Q₈	1	2	yes	—
75	C₁₆×C₂₃	C₂×Q₈	1	2	yes	—
76	C₁₆×C₂₃	C₂×Q₈	1	4	yes	—
77	#33729	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₂₃	—	16	4	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	—	16	2	no	—
84	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₈×C₂₃	C₄⋊Q₈	1	2	yes	—
91	C₈×C₂₃	C₂×Q₁₆	1	4	yes	—
92	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
94	#3164	C₂×C₂₃	1	2	yes	—
95	#2255	C₂×C₂₃	1	4	yes	complemented
96	#2253	C₂×C₂₃	1	2	yes	complemented
97	#2252	C₂×C₂₃	1	2	yes	complemented
98	#2159	C₂×C₂₃	1	2	yes	—
99	#2158	C₂×C₂₃	1	1	yes	characteristic
100	C₂×#366	C₂×C₂₃	1	1	yes	characteristic
101	#2108	C₂×C₂₃	1	1	yes	characteristic
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	#500	1	2	yes	—
105	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	8	4	no	—
109	C₄×C₂₃	—	8	2	no	—
110	C₄×C₂₃	—	8	2	no	—
111	C₄×C₂₃	#500	1	2	yes	—
112	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
113	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
114	#544	—	2	2	no	—
115	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
116	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
117	#501	—	2	4	no	—
118	#500	C₂²×C₂₃	1	4	yes	—
119	#499	C₂²×C₂₃	1	4	yes	—
120	#428	—	2	2	no	—
121	#427	C₂²×C₂₃	1	2	yes	—
122	C₂×#107	C₂²×C₂₃	1	2	yes	—
123	#405	C₂²×C₂₃	1	1	yes	characteristic
124	#367	C₂²×C₂₃	1	2	yes	—
125	#367	C₂²×C₂₃	1	4	yes	—
126	#366	C₂²×C₂₃	1	2	yes	—
127	#366	C₂²×C₂₃	1	2	yes	—
128	#366	C₂²×C₂₃	1	1	yes	characteristic
129	#366	C₂²×C₂₃	1	1	yes	characteristic
130	#366	C₂²×C₂₃	1	4	yes	—
131	#346	C₂²×C₂₃	1	2	yes	—
132	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
133	C₂×C₂₃	—	2	2	no	—
134	C₂×C₂₃	C₂×#500	1	1	yes	characteristic
135	C₂×C₂₃	#3415	1	1	yes	characteristic
136	C₂×C₂₃	#3417	1	1	yes	characteristic
137	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	4	no	—
140	C₄⋊Q₈	—	4	2	no	—
141	#125	—	2	4	no	—
142	#125	—	4	2	no	—
143	C₂²⋊Q₈	—	4	4	no	—
144	#117	—	2	2	no	—
145	#117	D₄×C₂₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	2	no	—
147	M₃₂	Q₈×C₂₃	1	4	yes	complemented
148	C₂×C₁₆	Q₈×C₂₃	1	4	yes	complemented
149	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
150	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
151	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
152	#107	—	2	2	no	—
153	#107	—	2	4	no	—
154	#107	C₂³×C₂₃	1	4	yes	—
155	#107	C₂³×C₂₃	1	2	yes	—
156	#107	C₂³×C₂₃	1	2	yes	—
157	#106	—	2	2	no	—
158	#106	—	2	4	no	—
159	#97	D₄×C₂₃	1	2	yes	—
160	C₄×C₈	C₂³×C₂₃	1	2	yes	—
161	C₂₃	#33729	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×Q₈	—	4	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
165	M₁₆	Q₁₆×C₂₃	1	4	yes	—
166	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
168	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
169	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
170	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
179	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
182	C₄²	—	4	2	no	—
183	C₄²	C₂×D₄×C₂₃	1	2	yes	—
184	C₁₆	C₂×Q₈×C₂₃	1	2	yes	—
185	C₁₆	C₂×Q₈×C₂₃	1	2	yes	—
186	C₁₆	C₂×Q₈×C₂₃	1	4	yes	—
187	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
188	Q₈	—	16	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	16	2	no	—
193	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
194	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
195	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
196	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
197	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
198	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
199	C₈	C₂₃×C₄⋊Q₈	1	2	yes	—
200	C₈	C₂×Q₁₆×C₂₃	1	4	yes	—
201	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
202	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	C₂₃×#500	1	2	yes	—
206	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	2	no	—
212	C₄	C₂₃×#500	1	2	yes	—
213	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
214	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×C₂₃×#500	1	1	yes	characteristic
217	C₂	C₂₃×#3415	1	1	yes	characteristic
218	C₂	C₂₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2255	C₂	semidirect
3. C₂₃×#2253	C₂	semidirect
4. C₂₃×#2252	C₂	semidirect
38. C₂₃×M₃₂	Q₈	semidirect
39. C₂×C₁₆×C₂₃	Q₈	semidirect
77. #33729	C₂₃	direct (N × H)
95. #2255	C₂×C₂₃	semidirect
96. #2253	C₂×C₂₃	semidirect
97. #2252	C₂×C₂₃	semidirect
147. M₃₂	Q₈×C₂₃	semidirect
148. C₂×C₁₆	Q₈×C₂₃	semidirect
161. C₂₃	#33729	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×#3164	2
C₂₃×#2255	2
C₂₃×#2253	2
C₂₃×#2252	2
C₂₃×#2159	2
C₂₃×#2158	2
C₂×C₂₃×#366	2
C₂₃×#2108	2
#33729	23

Subgroups — C23×#33729 #19181355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33729 #19181355