Subgroups — C₂₃×#3254 #15670873

Order 2944.

386 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#477	C₂	1	2	yes	maximal, complemented
4	C₂₃×#474	C₂	1	2	yes	maximal, complemented
5	C₂₃×#468	C₂	1	4	yes	maximal, complemented
6	C₂₃×#461	C₂	1	2	yes	maximal, complemented
7	C₂₃×#460	C₂	1	2	yes	maximal, complemented
8	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	—	2	1	no	—
10	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
11	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂₃×#125	—	2	1	no	—
17	C₂₃×#124	—	2	1	no	—
18	C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂₃×#121	—	2	1	no	—
21	C₂₃×#121	C₂²	1	2	yes	—
22	C₄×Q₈×C₂₃	—	2	1	no	—
23	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
24	C₄×D₄×C₂₃	—	2	1	no	—
25	C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#117	—	2	1	no	—
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#107	C₂²	1	4	yes	—
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂₃×#103	C₂²	1	4	yes	—
32	C₂₃×#102	C₂²	1	4	yes	—
33	#41×C₂₃	—	4	4	no	—
34	C₂×Q₈×C₂₃	—	2	1	no	—
35	C₂×Q₈×C₂₃	—	2	1	no	—
36	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
37	C₂×D₄×C₂₃	—	2	2	no	—
38	C₂×D₄×C₂₃	—	2	1	no	—
39	C₂×D₄×C₂₃	—	2	1	no	—
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	SD₁₆×C₂₃	—	4	4	no	—
46	M₁₆×C₂₃	—	4	2	no	—
47	C₂×C₈×C₂₃	C₂³	1	4	yes	—
48	C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₄⋊C₄×C₂₃	—	2	1	no	—
50	C₄⋊C₄×C₂₃	—	2	1	no	—
51	C₄⋊C₄×C₂₃	—	2	1	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	D₄	1	1	yes	characteristic
61	C₄²×C₂₃	D₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₂₃	—	2	2	no	—
67	Q₈×C₂₃	—	2	2	no	—
68	Q₈×C₂₃	—	4	2	no	—
69	Q₈×C₂₃	—	4	2	no	—
70	D₄×C₂₃	—	2	2	no	—
71	D₄×C₂₃	—	2	2	no	—
72	D₄×C₂₃	—	4	2	no	—
73	D₄×C₂₃	—	4	4	no	—
74	D₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
89	C₈×C₂₃	—	2	4	no	—
90	#3254	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	—	2	1	no	—
93	C₂²×C₂₃	—	2	1	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	1	no	—
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	4	2	no	—
103	C₄×C₂₃	—	4	2	no	—
104	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₄×C₂₃	#137	1	2	yes	—
106	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
108	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
109	#532	C₂×C₂₃	1	1	yes	characteristic
110	#477	C₂×C₂₃	1	2	yes	complemented
111	#474	C₂×C₂₃	1	2	yes	complemented
112	#468	C₂×C₂₃	1	4	yes	complemented
113	#461	C₂×C₂₃	1	2	yes	complemented
114	#460	C₂×C₂₃	1	2	yes	complemented
115	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×C₂₃	—	4	1	no	—
117	C₂×C₂₃	—	8	2	no	—
118	C₂×C₂₃	#537	1	1	yes	characteristic
119	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
121	C₂×#41	—	2	1	no	—
122	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
123	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
124	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
126	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
127	#127	C₂²×C₂₃	1	1	yes	characteristic
128	#125	—	2	1	no	—
129	#124	—	2	1	no	—
130	C₂²⋊Q₈	—	2	1	no	—
131	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
132	#121	—	2	1	no	—
133	#121	C₂²×C₂₃	1	2	yes	—
134	C₄×Q₈	—	2	1	no	—
135	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	C₂²×C₂₃	1	2	yes	—
138	#117	—	2	1	no	—
139	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
140	#107	C₂²×C₂₃	1	4	yes	—
141	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
142	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
143	#103	C₂²×C₂₃	1	4	yes	—
144	#102	C₂²×C₂₃	1	4	yes	—
145	C₂₃	#3254	1	1	yes	characteristic, direct product factor, Sylow
146	#41	—	4	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂³×C₂₃	1	2	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
158	SD₁₆	—	4	4	no	—
159	M₁₆	—	4	2	no	—
160	C₂×C₈	C₂³×C₂₃	1	4	yes	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
168	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	D₄×C₂₃	1	1	yes	characteristic
174	C₄²	D₄×C₂₃	1	1	yes	characteristic
175	C₄²	C₂³×C₂₃	1	1	yes	characteristic
176	C₄²	C₂³×C₂₃	1	1	yes	characteristic
177	C₂³	—	2	2	no	—
178	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
179	Q₈	—	2	2	no	—
180	Q₈	—	2	2	no	—
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	D₄	—	2	2	no	—
184	D₄	—	2	2	no	—
185	D₄	—	4	2	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
202	C₈	—	2	4	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
217	C₄	C₂₃×#137	1	2	yes	—
218	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
219	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	C₂₃×#537	1	1	yes	characteristic
223	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂	C₂×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#533	C₂	semidirect
3. C₂₃×#477	C₂	semidirect
4. C₂₃×#474	C₂	semidirect
5. C₂₃×#468	C₂	semidirect
6. C₂₃×#461	C₂	semidirect
7. C₂₃×#460	C₂	semidirect
8. C₂₃×#423	C₂	semidirect
90. #3254	C₂₃	direct (N × H)
108. #533	C₂×C₂₃	semidirect
110. #477	C₂×C₂₃	semidirect
111. #474	C₂×C₂₃	semidirect
112. #468	C₂×C₂₃	semidirect
113. #461	C₂×C₂₃	semidirect
114. #460	C₂×C₂₃	semidirect
115. #423	C₂×C₂₃	semidirect
145. C₂₃	#3254	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×#533	2
C₂₃×#532	2
C₂₃×#477	2
C₂₃×#474	2
C₂₃×#468	2
C₂₃×#461	2
C₂₃×#460	2
C₂₃×#423	2
#3254	23

Subgroups — C23×#3254 #15670873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3254 #15670873