Subgroups — C₂₃×#3260 #15670879

Order 2944.

346 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#540	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#479	C₂	1	2	yes	maximal, complemented
4	C₂₃×#478	C₂	1	2	yes	maximal, complemented
5	C₂₃×#468	C₂	1	2	yes	maximal, complemented
6	C₂₃×#465	C₂	1	2	yes	maximal, complemented
7	C₂₃×#461	C₂	1	2	yes	maximal, complemented
8	C₂₃×#460	C₂	1	2	yes	maximal, complemented
9	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂₃×#126	—	2	2	no	—
17	C₂₃×#125	—	2	2	no	—
18	C₂₃×#125	C₂²	1	1	yes	characteristic
19	C₂₃×#125	C₂²	1	1	yes	characteristic
20	C₂₃×#123	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂₃×#121	—	2	2	no	—
23	C₂₃×#121	C₂²	1	2	yes	—
24	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂₃×#107	C₂²	1	2	yes	—
28	C₂₃×#106	C₂²	1	2	yes	—
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂₃×#103	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	2	yes	—
33	C₂₃×#102	C₂²	1	2	yes	—
34	C₂₃×#102	C₂²	1	2	yes	—
35	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
36	C₂×D₄×C₂₃	—	2	2	no	—
37	C₂×D₄×C₂₃	—	2	2	no	—
38	C₂×D₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	SD₁₆×C₂₃	—	4	2	no	—
42	SD₁₆×C₂₃	—	4	2	no	—
43	M₁₆×C₂₃	—	4	1	no	—
44	C₂×C₈×C₂₃	—	4	1	no	—
45	C₂×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂×C₈×C₂₃	C₂³	1	2	yes	—
47	C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
53	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₂₃	—	2	2	no	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₂₃	—	2	2	no	—
65	Q₈×C₂₃	—	2	2	no	—
66	D₄×C₂₃	—	2	2	no	—
67	D₄×C₂₃	—	2	2	no	—
68	D₄×C₂₃	—	8	2	no	—
69	D₄×C₂₃	—	8	2	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	8	2	no	—
76	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
80	C₈×C₂₃	—	2	2	no	—
81	C₈×C₂₃	—	2	2	no	—
82	#3260	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₂₃	—	2	1	no	—
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	—	4	2	no	—
86	C₂²×C₂₃	—	4	2	no	—
87	C₂²×C₂₃	—	4	2	no	—
88	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₂₃	—	2	1	no	—
90	C₄×C₂₃	—	2	2	no	—
91	C₄×C₂₃	—	2	2	no	—
92	C₄×C₂₃	—	4	2	no	—
93	C₄×C₂₃	—	4	2	no	—
94	C₄×C₂₃	—	4	2	no	—
95	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
97	#541	C₂×C₂₃	1	1	yes	characteristic, complemented
98	#540	C₂×C₂₃	1	1	yes	characteristic
99	#479	C₂×C₂₃	1	2	yes	complemented
100	#478	C₂×C₂₃	1	2	yes	complemented
101	#468	C₂×C₂₃	1	2	yes	complemented
102	#465	C₂×C₂₃	1	2	yes	complemented
103	#461	C₂×C₂₃	1	2	yes	complemented
104	#460	C₂×C₂₃	1	2	yes	complemented
105	#424	C₂×C₂₃	1	1	yes	characteristic, complemented
106	C₂×C₂₃	—	4	1	no	—
107	C₂×C₂₃	—	8	2	no	—
108	C₂×C₂₃	#537	1	1	yes	characteristic
109	C₂×C₂₃	#577	1	2	yes	—
110	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
111	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
112	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
114	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
115	#127	C₂²×C₂₃	1	1	yes	characteristic
116	#126	—	2	2	no	—
117	#125	—	2	2	no	—
118	#125	C₂²×C₂₃	1	1	yes	characteristic
119	#125	C₂²×C₂₃	1	1	yes	characteristic
120	#123	—	2	2	no	—
121	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
122	#121	—	2	2	no	—
123	#121	C₂²×C₂₃	1	2	yes	—
124	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
125	C₄×D₄	C₂²×C₂₃	1	2	yes	—
126	#117	C₂²×C₂₃	1	1	yes	characteristic
127	#107	C₂²×C₂₃	1	2	yes	—
128	#106	C₂²×C₂₃	1	2	yes	—
129	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
130	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
131	#103	C₂²×C₂₃	1	2	yes	—
132	#103	C₂²×C₂₃	1	2	yes	—
133	#102	C₂²×C₂₃	1	2	yes	—
134	#102	C₂²×C₂₃	1	2	yes	—
135	C₂₃	#3260	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	C₂³×C₂₃	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
142	SD₁₆	—	4	2	no	—
143	SD₁₆	—	4	2	no	—
144	M₁₆	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	C₂³×C₂₃	1	2	yes	—
147	C₂×C₈	C₂³×C₂₃	1	2	yes	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
154	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
155	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
160	C₄²	—	2	2	no	—
161	C₄²	C₂³×C₂₃	1	1	yes	characteristic
162	C₄²	C₂³×C₂₃	1	1	yes	characteristic
163	C₂³	—	2	2	no	—
164	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	Q₈	—	2	2	no	—
167	D₄	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	8	2	no	—
170	D₄	—	8	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
178	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
179	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
196	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	8	2	no	—
199	C₂	C₂₃×#537	1	1	yes	characteristic
200	C₂	C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#540	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#541	C₂	semidirect
3. C₂₃×#479	C₂	semidirect
4. C₂₃×#478	C₂	semidirect
5. C₂₃×#468	C₂	semidirect
6. C₂₃×#465	C₂	semidirect
7. C₂₃×#461	C₂	semidirect
8. C₂₃×#460	C₂	semidirect
9. C₂₃×#424	C₂	semidirect
82. #3260	C₂₃	direct (N × H)
97. #541	C₂×C₂₃	semidirect
99. #479	C₂×C₂₃	semidirect
100. #478	C₂×C₂₃	semidirect
101. #468	C₂×C₂₃	semidirect
102. #465	C₂×C₂₃	semidirect
103. #461	C₂×C₂₃	semidirect
104. #460	C₂×C₂₃	semidirect
105. #424	C₂×C₂₃	semidirect
135. C₂₃	#3260	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×#541	2
C₂₃×#540	2
C₂₃×#479	2
C₂₃×#478	2
C₂₃×#468	2
C₂₃×#465	2
C₂₃×#461	2
C₂₃×#460	2
C₂₃×#424	2
#3260	23

Subgroups — C23×#3260 #15670879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3260 #15670879