Subgroups — C₂₃×#19603 #19167229

Order 5888.

922 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3169	C₂	1	4	yes	maximal, complemented
2	C₂₃×#3147	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1709	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1706	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	—	2	4	no	—
9	C₂₃×#532	C₂²	1	4	yes	—
10	C₂₃×#500	C₂²	1	4	yes	—
11	C₂₃×#495	—	2	4	no	—
12	C₂₃×#494	C₂²	1	2	yes	—
13	C₂₃×#493	C₂²	1	4	yes	—
14	C₂₃×#493	C₂²	1	2	yes	—
15	C₂₃×#492	C₂²	1	4	yes	—
16	C₂₃×#486	C₂²	1	8	yes	—
17	C₂₃×#466	—	2	8	no	—
18	C₄×D₈×C₂₃	—	2	8	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₈²×C₂₃	C₂²	1	2	yes	—
22	C₈²×C₂₃	C₂²	1	1	yes	characteristic
23	C₈²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×#41×C₂₃	—	2	4	no	—
25	C₂₃×#135	—	4	16	no	—
26	C₂×Q₁₆×C₂₃	—	2	4	no	—
27	C₂×SD₁₆×C₂₃	—	2	8	no	—
28	C₂×D₈×C₂₃	—	2	4	no	—
29	C₂×D₈×C₂₃	—	2	8	no	—
30	C₂²×C₈×C₂₃	D₄	1	4	yes	—
31	C₂²×C₈×C₂₃	D₄	1	2	yes	—
32	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
33	C₂₃×#127	C₂³	1	4	yes	—
34	C₂₃×#124	—	2	4	no	—
35	C₂₃×#121	—	2	8	no	—
36	C₄×D₄×C₂₃	—	2	8	no	—
37	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#107	—	2	8	no	—
39	C₂₃×#102	—	2	16	no	—
40	C₄×C₈×C₂₃	D₄	1	4	yes	—
41	C₄×C₈×C₂₃	D₄	1	2	yes	—
42	C₄×C₈×C₂₃	C₂³	1	2	yes	—
43	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₂₃	C₂³	1	2	yes	—
46	#41×C₂₃	—	4	16	no	—
47	C₂×Q₈×C₂₃	—	2	4	no	—
48	C₂×D₄×C₂₃	—	2	8	no	—
49	C₂×D₄×C₂₃	—	2	4	no	—
50	C₂×D₄×C₂₃	—	4	4	no	—
51	C₂²×C₄×C₂₃	—	4	4	no	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₂₃	—	4	8	no	—
55	SD₁₆×C₂₃	—	4	16	no	—
56	D₈×C₂₃	—	4	8	no	—
57	D₈×C₂₃	—	4	16	no	—
58	C₂×C₈×C₂₃	—	2	8	no	—
59	C₂×C₈×C₂₃	D₈	1	8	yes	complemented
60	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
64	C₄⋊C₄×C₂₃	—	2	8	no	—
65	C₂²⋊C₄×C₂₃	—	4	8	no	—
66	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₂×D₄	1	2	yes	—
68	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
69	#19603	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	—	4	4	no	—
71	C₂³×C₂₃	#127	1	1	yes	characteristic
72	Q₈×C₂₃	—	4	8	no	—
73	D₄×C₂₃	—	4	8	no	—
74	D₄×C₂₃	—	4	16	no	—
75	D₄×C₂₃	—	8	8	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	4	4	no	—
78	C₂×C₄×C₂₃	—	8	8	no	—
79	C₂×C₄×C₂₃	#127	1	2	yes	—
80	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
82	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
84	C₈×C₂₃	—	2	4	no	—
85	C₈×C₂₃	C₂×D₈	1	4	yes	—
86	C₈×C₂₃	C₂×D₈	1	4	yes	—
87	C₈×C₂₃	#135	1	8	yes	complemented
88	#3169	C₂×C₂₃	1	4	yes	complemented
89	#3147	C₂×C₂₃	1	2	yes	complemented
90	#1713	C₂×C₂₃	1	1	yes	characteristic, complemented
91	#1712	C₂×C₂₃	1	1	yes	characteristic, complemented
92	#1709	C₂×C₂₃	1	2	yes	complemented
93	#1706	C₂×C₂₃	1	4	yes	complemented
94	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	8	4	no	—
97	C₂²×C₂₃	—	8	8	no	—
98	C₂²×C₂₃	#493	1	2	yes	—
99	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	8	4	no	—
102	C₄×C₂₃	#493	1	2	yes	—
103	C₄×C₂₃	C₂²×D₈	1	2	yes	—
104	C₄×C₂₃	C₂×#135	1	4	yes	—
105	C₂×#135	—	2	4	no	—
106	#532	C₂²×C₂₃	1	4	yes	—
107	#500	C₂²×C₂₃	1	4	yes	—
108	#495	—	2	4	no	—
109	#494	C₂²×C₂₃	1	2	yes	—
110	#493	C₂²×C₂₃	1	2	yes	—
111	#493	C₂²×C₂₃	1	4	yes	—
112	#492	C₂²×C₂₃	1	4	yes	—
113	#486	C₂²×C₂₃	1	8	yes	—
114	#466	—	2	8	no	—
115	C₄×D₈	—	2	8	no	—
116	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
118	C₈²	C₂²×C₂₃	1	2	yes	—
119	C₈²	C₂²×C₂₃	1	1	yes	characteristic
120	C₈²	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×C₂₃	—	2	2	no	—
122	C₂×C₂₃	—	16	4	no	—
123	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
124	C₂×C₂₃	#3147	1	2	yes	—
125	C₂×#41	—	2	4	no	—
126	#135	—	4	16	no	—
127	C₂×Q₁₆	—	2	4	no	—
128	C₂×SD₁₆	—	2	8	no	—
129	C₂×D₈	—	2	4	no	—
130	C₂×D₈	—	2	8	no	—
131	C₂²×C₈	D₄×C₂₃	1	4	yes	—
132	C₂²×C₈	D₄×C₂₃	1	2	yes	—
133	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
134	#127	C₂³×C₂₃	1	4	yes	—
135	#124	—	2	4	no	—
136	#121	—	2	8	no	—
137	C₄×D₄	—	2	8	no	—
138	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
139	#107	—	2	8	no	—
140	#102	—	2	16	no	—
141	C₄×C₈	D₄×C₂₃	1	4	yes	—
142	C₄×C₈	D₄×C₂₃	1	2	yes	—
143	C₄×C₈	C₂³×C₂₃	1	2	yes	—
144	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
145	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
146	C₄×C₈	C₂³×C₂₃	1	2	yes	—
147	C₂₃	#19603	1	1	yes	characteristic, direct product factor, Sylow
148	#41	—	4	16	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×D₄	—	2	8	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
156	Q₁₆	—	4	8	no	—
157	SD₁₆	—	4	16	no	—
158	D₈	—	4	8	no	—
159	D₈	—	4	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₈×C₂₃	1	8	yes	complemented
162	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
163	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
166	C₄⋊C₄	—	2	8	no	—
167	C₂²⋊C₄	—	4	8	no	—
168	C₄²	C₂×D₄×C₂₃	1	2	yes	—
169	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
170	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
171	C₂³	—	4	4	no	—
172	C₂³	C₂₃×#127	1	1	yes	characteristic
173	Q₈	—	4	8	no	—
174	D₄	—	4	8	no	—
175	D₄	—	4	16	no	—
176	D₄	—	8	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	8	no	—
180	C₂×C₄	C₂₃×#127	1	2	yes	—
181	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
182	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
183	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₈×C₂₃	1	4	yes	—
187	C₈	C₂×D₈×C₂₃	1	4	yes	—
188	C₈	C₂₃×#135	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	8	4	no	—
191	C₂²	—	8	8	no	—
192	C₂²	C₂₃×#493	1	2	yes	—
193	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	8	4	no	—
196	C₄	C₂₃×#493	1	2	yes	—
197	C₄	C₂²×D₈×C₂₃	1	2	yes	—
198	C₄	C₂×C₂₃×#135	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	16	4	no	—
201	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
202	C₂	C₂₃×#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3169	C₂	semidirect
2. C₂₃×#3147	C₂	semidirect
3. C₂₃×#1713	C₂	semidirect
4. C₂₃×#1712	C₂	semidirect
5. C₂₃×#1709	C₂	semidirect
6. C₂₃×#1706	C₂	semidirect
7. C₂×C₈²×C₂₃	C₂	semidirect
59. C₂×C₈×C₂₃	D₈	semidirect
69. #19603	C₂₃	direct (N × H)
87. C₈×C₂₃	#135	semidirect
88. #3169	C₂×C₂₃	semidirect
89. #3147	C₂×C₂₃	semidirect
90. #1713	C₂×C₂₃	semidirect
91. #1712	C₂×C₂₃	semidirect
92. #1709	C₂×C₂₃	semidirect
93. #1706	C₂×C₂₃	semidirect
94. C₂×C₈²	C₂×C₂₃	semidirect
147. C₂₃	#19603	direct (N × H)
161. C₂×C₈	D₈×C₂₃	semidirect
188. C₈	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3169	2
C₂₃×#3147	2
C₂₃×#1713	2
C₂₃×#1712	2
C₂₃×#1709	2
C₂₃×#1706	2
C₂×C₈²×C₂₃	2
#19603	23

Subgroups — C23×#19603 #19167229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19603 #19167229