Subgroups — C₂₃×#3042 #15670661

Order 2944.

494 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₂²×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#463	C₂	1	4	yes	maximal, complemented
6	C₂₃×#461	C₂	1	4	yes	maximal, complemented
7	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
14	C₂×SD₁₆×C₂₃	—	2	4	no	—
15	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
16	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×D₈×C₂₃	C₂²	1	2	yes	—
18	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
21	C₂₃×#124	C₂²	1	2	yes	complemented
22	C₂₃×#123	—	2	2	no	—
23	C₂₃×C₂²⋊Q₈	—	2	2	no	—
24	C₄×D₄×C₂₃	—	2	2	no	—
25	C₄×D₄×C₂₃	C₂²	1	4	yes	—
26	C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
29	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
30	C₂₃×#103	C₂²	1	2	yes	complemented
31	C₂₃×#103	C₂²	1	4	yes	complemented
32	C₂₃×#102	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₃	—	2	1	no	—
34	#41×C₂₃	—	2	2	no	—
35	#41×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	—	2	1	no	—
37	C₂×Q₈×C₂₃	—	2	2	no	—
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
40	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₂₃	—	2	1	no	—
42	C₂×D₄×C₂₃	—	2	1	no	—
43	C₂×D₄×C₂₃	—	2	2	no	—
44	C₂×D₄×C₂₃	D₄	1	4	yes	—
45	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	4	no	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	SD₁₆×C₂₃	—	2	8	no	—
52	SD₁₆×C₂₃	—	2	4	no	—
53	D₈×C₂₃	—	2	4	no	—
54	M₁₆×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₂₃	—	2	1	no	—
62	C₂²⋊C₄×C₂₃	—	2	4	no	—
63	C₂²⋊C₄×C₂₃	—	2	4	no	—
64	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
65	C₄²×C₂₃	—	2	1	no	—
66	C₄²×C₂₃	C₂³	1	2	yes	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	4	no	—
69	C₂³×C₂₃	—	2	1	no	—
70	C₂³×C₂₃	—	8	2	no	—
71	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₂₃	—	2	2	no	—
73	Q₈×C₂₃	—	2	1	no	—
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	—	2	1	no	—
76	Q₈×C₂₃	—	4	2	no	—
77	D₄×C₂₃	—	2	4	no	—
78	D₄×C₂₃	—	2	1	no	—
79	D₄×C₂₃	—	2	2	no	—
80	D₄×C₂₃	—	2	1	no	—
81	D₄×C₂₃	—	4	2	no	—
82	D₄×C₂₃	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	8	4	no	—
89	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	#41	1	2	yes	—
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	2	no	—
96	#3042	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	2	4	no	—
99	C₂²×C₂₃	—	4	4	no	—
100	C₂²×C₂₃	—	4	1	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	8	4	no	—
103	C₂²×C₂₃	#121	1	2	yes	—
104	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	—	4	1	no	—
107	C₄×C₂₃	—	4	2	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	#121	1	2	yes	—
110	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
112	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic
113	C₂²×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
114	#536	C₂×C₂₃	1	1	yes	characteristic, complemented
115	#518	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#463	C₂×C₂₃	1	4	yes	complemented
117	#461	C₂×C₂₃	1	4	yes	complemented
118	#424	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#419	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#416	C₂×C₂₃	1	1	yes	characteristic, complemented
121	C₂×C₂₃	—	2	2	no	—
122	C₂×C₂₃	—	4	4	no	—
123	C₂×C₂₃	—	8	1	no	—
124	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
125	C₂×C₂₃	#577	1	2	yes	—
126	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
127	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
128	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
129	AGL(1,ℤ/8)	—	2	4	no	—
130	C₂×SD₁₆	—	2	4	no	—
131	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
132	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
133	C₂×D₈	C₂²×C₂₃	1	2	yes	—
134	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
137	#124	C₂²×C₂₃	1	2	yes	complemented
138	#123	—	2	2	no	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₄×D₄	—	2	2	no	—
141	C₄×D₄	C₂²×C₂₃	1	4	yes	—
142	#117	C₂²×C₂₃	1	1	yes	characteristic
143	#117	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	—	2	2	no	—
145	C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
146	#103	C₂²×C₂₃	1	2	yes	complemented
147	#103	C₂²×C₂₃	1	4	yes	complemented
148	#102	C₂²×C₂₃	1	2	yes	complemented
149	C₂₃	#3042	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	—	2	1	no	—
151	#41	—	2	2	no	—
152	#41	—	2	2	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
157	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	D₄×C₂₃	1	4	yes	—
162	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂×D₄	C₂³×C₂₃	1	2	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
168	SD₁₆	—	2	8	no	—
169	SD₁₆	—	2	4	no	—
170	D₈	—	2	4	no	—
171	M₁₆	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂³×C₂₃	1	2	yes	—
174	C₂×C₈	C₂³×C₂₃	1	2	yes	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
177	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
182	C₄²	—	2	1	no	—
183	C₄²	C₂³×C₂₃	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	8	2	no	—
188	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	4	2	no	—
194	D₄	—	2	4	no	—
195	D₄	—	2	1	no	—
196	D₄	—	2	2	no	—
197	D₄	—	2	1	no	—
198	D₄	—	4	2	no	—
199	D₄	C₂×D₄×C₂₃	1	4	yes	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
207	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
209	C₂×C₄	#41×C₂₃	1	2	yes	—
210	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	8	4	no	—
219	C₂²	C₂₃×#121	1	2	yes	—
220	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	C₂₃×#121	1	2	yes	—
226	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
227	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
228	C₂	—	2	2	no	—
229	C₂	—	4	4	no	—
230	C₂	—	8	1	no	—
231	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
232	C₂	C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	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
2. C₂²×SD₁₆×C₂₃	C₂	semidirect
3. C₂₃×#536	C₂	semidirect
4. C₂₃×#518	C₂	semidirect
5. C₂₃×#463	C₂	semidirect
6. C₂₃×#461	C₂	semidirect
7. C₂₃×#424	C₂	semidirect
8. C₂₃×#419	C₂	semidirect
9. C₂₃×#416	C₂	semidirect
20. C₂₃×C₄⋊Q₈	C₂²	semidirect
21. C₂₃×#124	C₂²	semidirect
29. C₂₃×C₄⋊C₈	C₂²	semidirect
30. C₂₃×#103	C₂²	semidirect
31. C₂₃×#103	C₂²	semidirect
32. C₂₃×#102	C₂²	semidirect
96. #3042	C₂₃	direct (N × H)
113. C₂²×SD₁₆	C₂×C₂₃	semidirect
114. #536	C₂×C₂₃	semidirect
115. #518	C₂×C₂₃	semidirect
116. #463	C₂×C₂₃	semidirect
117. #461	C₂×C₂₃	semidirect
118. #424	C₂×C₂₃	semidirect
119. #419	C₂×C₂₃	semidirect
120. #416	C₂×C₂₃	semidirect
136. C₄⋊Q₈	C₂²×C₂₃	semidirect
137. #124	C₂²×C₂₃	semidirect
145. C₄⋊C₈	C₂²×C₂₃	semidirect
146. #103	C₂²×C₂₃	semidirect
147. #103	C₂²×C₂₃	semidirect
148. #102	C₂²×C₂₃	semidirect
149. C₂₃	#3042	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂²×SD₁₆×C₂₃	2
C₂₃×#536	2
C₂₃×#518	2
C₂₃×#463	2
C₂₃×#461	2
C₂₃×#424	2
C₂₃×#419	2
C₂₃×#416	2
#3042	23

Subgroups — C23×#3042 #15670661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3042 #15670661