Subgroups — C₂₃×#13203 #19160829

Order 5888.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#358	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₂₃	—	2	1	no	—
6	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
8	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#494	—	2	2	no	—
10	C₂₃×#494	C₄	1	4	yes	complemented
11	C₂×C₂₃×#107	—	2	1	no	—
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#384	—	2	1	no	—
15	C₂₃×#358	—	2	8	no	—
16	C₄×C₁₆×C₂₃	—	2	2	no	—
17	C₂²×Q₈×C₂₃	—	2	2	no	—
18	C₂×Q₁₆×C₂₃	—	2	4	no	—
19	C₂×Q₁₆×C₂₃	—	2	4	no	—
20	C₂×Q₁₆×C₂₃	—	2	2	no	—
21	C₂×Q₁₆×C₂₃	—	2	4	no	—
22	C₂×Q₁₆×C₂₃	—	4	2	no	—
23	C₂×Q₁₆×C₂₃	—	4	2	no	—
24	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
27	C₂₃×C₄⋊Q₈	—	2	4	no	—
28	C₂₃×C₄⋊Q₈	—	4	2	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
31	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
32	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₂₃	—	2	2	no	—
34	C₂×C₁₆×C₂₃	—	2	2	no	—
35	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
36	C₂₃×#107	—	2	4	no	—
37	C₂₃×#107	—	4	2	no	—
38	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₂₃	D₄	1	2	yes	—
40	C₂×Q₈×C₂₃	—	2	4	no	—
41	C₂×Q₈×C₂₃	—	4	2	no	—
42	C₂×Q₈×C₂₃	—	4	2	no	—
43	C₂×Q₈×C₂₃	—	4	2	no	—
44	C₂×Q₈×C₂₃	—	4	2	no	—
45	C₂×Q₈×C₂₃	—	4	2	no	—
46	C₂²×C₄×C₂₃	—	4	2	no	—
47	C₂²×C₄×C₂₃	—	4	2	no	—
48	C₂²×C₄×C₂₃	D₈	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	Q₁₆×C₂₃	—	2	8	no	—
51	Q₁₆×C₂₃	—	4	8	no	—
52	Q₁₆×C₂₃	—	4	4	no	—
53	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₂₃	D₈	1	2	yes	—
55	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	#41	1	2	yes	—
60	C₄⋊C₄×C₂₃	—	4	8	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	—	4	2	no	—
63	C₄⋊C₄×C₂₃	—	8	4	no	—
64	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₂₃	SD₁₆	1	2	yes	—
66	C₁₆×C₂₃	—	2	4	no	—
67	#13203	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
69	Q₈×C₂₃	—	4	4	no	—
70	Q₈×C₂₃	—	4	4	no	—
71	Q₈×C₂₃	—	8	2	no	—
72	Q₈×C₂₃	—	8	2	no	—
73	C₂×C₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	8	2	no	—
75	C₂×C₄×C₂₃	—	8	2	no	—
76	C₂×C₄×C₂₃	—	8	2	no	—
77	C₂×C₄×C₂₃	—	8	4	no	—
78	C₂×C₄×C₂₃	#102	1	4	yes	—
79	C₂×C₄×C₂₃	SD₃₂	1	4	yes	—
80	C₂×C₄×C₂₃	Q₃₂	1	4	yes	—
81	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₂₃	#102	1	4	yes	—
88	C₈×C₂₃	C₄×D₄	1	2	yes	—
89	C₈×C₂₃	#121	1	2	yes	—
90	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
91	C₂×#358	C₂×C₂₃	1	4	yes	—
92	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
93	#1845	C₂×C₂₃	1	1	yes	characteristic
94	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
95	C₂²×C₂₃	#486	1	1	yes	characteristic
96	C₂²×C₂₃	#493	1	1	yes	characteristic
97	C₂²×C₂₃	C₂×SD₃₂	1	2	yes	—
98	C₂²×C₂₃	C₂×Q₃₂	1	2	yes	—
99	C₄×C₂₃	—	8	4	no	—
100	C₄×C₂₃	—	16	2	no	—
101	C₄×C₂₃	#358	1	8	yes	complemented
102	C₄×C₂₃	#390	1	1	yes	characteristic
103	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
104	C₄×C₂₃	#486	1	1	yes	characteristic
105	C₄×C₂₃	#492	1	1	yes	characteristic
106	C₂²×Q₁₆	—	2	1	no	—
107	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
108	C₂×C₄⋊Q₈	—	2	1	no	—
109	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
110	#494	—	2	2	no	—
111	#494	C₄×C₂₃	1	4	yes	complemented
112	C₂×#107	—	2	1	no	—
113	C₂×#107	C₂²×C₂₃	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	#384	—	2	1	no	—
116	#358	—	2	8	no	—
117	C₄×C₁₆	—	2	2	no	—
118	C₂×C₂₃	#1957	1	1	yes	characteristic
119	C₂×C₂₃	C₂×#358	1	2	yes	—
120	C₂×C₂₃	#2241	1	2	yes	—
121	C₂×C₂₃	#2247	1	1	yes	characteristic
122	C₂×C₂₃	#2248	1	1	yes	characteristic
123	C₂²×Q₈	—	2	2	no	—
124	C₂×Q₁₆	—	2	4	no	—
125	C₂×Q₁₆	—	2	4	no	—
126	C₂×Q₁₆	—	2	2	no	—
127	C₂×Q₁₆	—	2	4	no	—
128	C₂×Q₁₆	—	4	2	no	—
129	C₂×Q₁₆	—	4	2	no	—
130	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
131	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	4	no	—
134	C₄⋊Q₈	—	4	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₄⋊C₄	—	4	2	no	—
138	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
142	#107	—	2	4	no	—
143	#107	—	4	2	no	—
144	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
145	C₄×C₈	D₄×C₂₃	1	2	yes	—
146	C₂₃	#13203	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	D₈×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
157	Q₁₆	—	2	8	no	—
158	Q₁₆	—	4	8	no	—
159	Q₁₆	—	4	4	no	—
160	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
161	C₂×C₈	D₈×C₂₃	1	2	yes	—
162	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
163	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
164	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
166	C₂×C₈	#41×C₂₃	1	2	yes	—
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	8	4	no	—
171	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
172	C₄²	SD₁₆×C₂₃	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
175	Q₈	—	4	4	no	—
176	Q₈	—	4	4	no	—
177	Q₈	—	8	2	no	—
178	Q₈	—	8	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	C₂₃×#102	1	4	yes	—
185	C₂×C₄	C₂₃×SD₃₂	1	4	yes	—
186	C₂×C₄	C₂₃×Q₃₂	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
189	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
190	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
191	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
192	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
193	C₈	C₂₃×#102	1	4	yes	—
194	C₈	C₄×D₄×C₂₃	1	2	yes	—
195	C₈	C₂₃×#121	1	2	yes	—
196	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
197	C₂²	C₂₃×#486	1	1	yes	characteristic
198	C₂²	C₂₃×#493	1	1	yes	characteristic
199	C₂²	C₂×C₂₃×SD₃₂	1	2	yes	—
200	C₂²	C₂×C₂₃×Q₃₂	1	2	yes	—
201	C₄	—	8	4	no	—
202	C₄	—	16	2	no	—
203	C₄	C₂₃×#358	1	8	yes	complemented
204	C₄	C₂₃×#390	1	1	yes	characteristic
205	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
206	C₄	C₂₃×#486	1	1	yes	characteristic
207	C₄	C₂₃×#492	1	1	yes	characteristic
208	C₂	C₂₃×#1957	1	1	yes	characteristic
209	C₂	C₂×C₂₃×#358	1	2	yes	—
210	C₂	C₂₃×#2241	1	2	yes	—
211	C₂	C₂₃×#2247	1	1	yes	characteristic
212	C₂	C₂₃×#2248	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
10. C₂₃×#494	C₄	semidirect
67. #13203	C₂₃	direct (N × H)
101. C₄×C₂₃	#358	semidirect
111. #494	C₄×C₂₃	semidirect
146. C₂₃	#13203	direct (N × H)
203. C₄	C₂₃×#358	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#494	2
C₂×C₂₃×#358	2
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1845	2
#13203	23

Subgroups — C23×#13203 #19160829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13203 #19160829