Subgroups — C₂₃×#3112 #15670731

Order 2944.

424 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#480	C₂	1	2	yes	maximal, complemented
4	C₂₃×#477	C₂	1	2	yes	maximal, complemented
5	C₂₃×#460	C₂	1	2	yes	maximal, complemented
6	C₂₃×#448	C₂	1	2	yes	maximal, complemented
7	C₂₃×#440	C₂	1	4	yes	maximal, complemented
8	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	—	2	1	no	—
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂₃×#127	C₂²	1	1	yes	characteristic
14	C₂₃×#124	—	2	1	no	—
15	C₂₃×#121	—	2	1	no	—
16	C₂₃×#121	C₂²	1	2	yes	—
17	C₄×Q₈×C₂₃	—	2	1	no	—
18	C₄×Q₈×C₂₃	—	2	1	no	—
19	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
20	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
21	C₄×D₄×C₂₃	—	2	1	no	—
22	C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	—	2	1	no	—
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#107	C₂²	1	4	yes	—
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₂₃×#103	C₂²	1	4	yes	—
30	C₂₃×#102	C₂²	1	4	yes	—
31	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂₃×#97	C₂²	1	2	yes	—
33	#41×C₂₃	—	4	4	no	—
34	C₂×Q₈×C₂₃	—	2	4	no	—
35	C₂×Q₈×C₂₃	—	2	1	no	—
36	C₂×Q₈×C₂₃	—	2	1	no	—
37	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	—	2	1	no	—
40	C₂×D₄×C₂₃	—	2	1	no	—
41	C₂×D₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	SD₁₆×C₂₃	—	2	8	no	—
49	C₂×C₈×C₂₃	C₂³	1	4	yes	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₄⋊C₄×C₂₃	—	2	1	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	C₄²×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	4	no	—
67	Q₈×C₂₃	—	2	2	no	—
68	Q₈×C₂₃	—	4	2	no	—
69	Q₈×C₂₃	#41	1	4	yes	complemented
70	D₄×C₂₃	—	2	4	no	—
71	D₄×C₂₃	—	4	2	no	—
72	D₄×C₂₃	—	4	4	no	—
73	D₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	4	4	no	—
82	C₂×C₄×C₂₃	—	4	2	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	#3112	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	—	4	2	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
97	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₂₃	—	2	4	no	—
99	C₄×C₂₃	—	2	1	no	—
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	—	4	2	no	—
104	C₄×C₂₃	—	4	2	no	—
105	C₄×C₂₃	#137	1	2	yes	—
106	C₄×C₂₃	C₂×#41	1	2	yes	—
107	#532	C₂×C₂₃	1	1	yes	characteristic
108	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
109	#480	C₂×C₂₃	1	2	yes	complemented
110	#477	C₂×C₂₃	1	2	yes	complemented
111	#460	C₂×C₂₃	1	2	yes	complemented
112	#448	C₂×C₂₃	1	2	yes	complemented
113	#440	C₂×C₂₃	1	4	yes	complemented
114	#432	C₂×C₂₃	1	1	yes	characteristic, complemented
115	C₂×C₂₃	—	2	2	no	—
116	C₂×C₂₃	—	8	2	no	—
117	C₂×C₂₃	#525	1	1	yes	characteristic
118	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
120	C₂×#41	—	2	1	no	—
121	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
123	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
124	#127	C₂²×C₂₃	1	1	yes	characteristic
125	#124	—	2	1	no	—
126	#121	—	2	1	no	—
127	#121	C₂²×C₂₃	1	2	yes	—
128	C₄×Q₈	—	2	1	no	—
129	C₄×Q₈	—	2	1	no	—
130	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
131	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
132	C₄×D₄	—	2	1	no	—
133	C₄×D₄	C₂²×C₂₃	1	2	yes	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×C₄²	—	2	1	no	—
137	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
138	#107	C₂²×C₂₃	1	4	yes	—
139	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
140	#103	C₂²×C₂₃	1	4	yes	—
141	#102	C₂²×C₂₃	1	4	yes	—
142	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
143	#97	C₂²×C₂₃	1	2	yes	—
144	C₂₃	#3112	1	1	yes	characteristic, direct product factor, Sylow
145	#41	—	4	4	no	—
146	C₂×Q₈	—	2	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	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
160	SD₁₆	—	2	8	no	—
161	C₂×C₈	C₂³×C₂₃	1	4	yes	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
167	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	D₄×C₂₃	1	1	yes	characteristic
173	C₄²	D₄×C₂₃	1	1	yes	characteristic
174	C₄²	C₂³×C₂₃	1	1	yes	characteristic
175	C₄²	C₂³×C₂₃	1	1	yes	characteristic
176	C₂³	—	2	2	no	—
177	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
178	Q₈	—	2	4	no	—
179	Q₈	—	2	2	no	—
180	Q₈	—	4	2	no	—
181	Q₈	#41×C₂₃	1	4	yes	complemented
182	D₄	—	2	4	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	4	no	—
185	D₄	—	4	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
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	C₂⁴×C₂₃	1	1	yes	characteristic
201	C₈	—	2	4	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
208	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₂₃×#137	1	2	yes	—
217	C₄	C₂×#41×C₂₃	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	8	2	no	—
220	C₂	C₂₃×#525	1	1	yes	characteristic
221	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
222	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
2. C₂×C₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#480	C₂	semidirect
4. C₂₃×#477	C₂	semidirect
5. C₂₃×#460	C₂	semidirect
6. C₂₃×#448	C₂	semidirect
7. C₂₃×#440	C₂	semidirect
8. C₂₃×#432	C₂	semidirect
69. Q₈×C₂₃	#41	semidirect
90. #3112	C₂₃	direct (N × H)
108. C₂×C₄×Q₈	C₂×C₂₃	semidirect
109. #480	C₂×C₂₃	semidirect
110. #477	C₂×C₂₃	semidirect
111. #460	C₂×C₂₃	semidirect
112. #448	C₂×C₂₃	semidirect
113. #440	C₂×C₂₃	semidirect
114. #432	C₂×C₂₃	semidirect
144. C₂₃	#3112	direct (N × H)
181. Q₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#532	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#480	2
C₂₃×#477	2
C₂₃×#460	2
C₂₃×#448	2
C₂₃×#440	2
C₂₃×#432	2
#3112	23

Subgroups — C23×#3112 #15670731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3112 #15670731