Subgroups — C₂₃×#3153 #15670772

Order 2944.

572 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#137	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#497	C₂	1	2	yes	maximal, complemented
6	C₂₃×#496	C₂	1	2	yes	maximal, complemented
7	C₂₃×#495	C₂	1	4	yes	maximal, complemented
8	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	C₂²	1	2	yes	—
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#137	—	2	8	no	—
13	C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
14	C₂×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
15	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
16	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
17	C₂×D₈×C₂₃	C₂²	1	4	yes	—
18	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
21	C₂₃×#124	—	2	2	no	—
22	C₂₃×#124	C₂²	1	2	yes	—
23	C₂₃×#124	C₂²	1	2	yes	—
24	C₂₃×#124	C₂²	1	2	yes	complemented
25	C₂₃×#121	—	2	2	no	—
26	C₄×D₄×C₂₃	—	2	2	no	—
27	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#97	C₂²	1	2	yes	complemented
30	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
31	C₂⁴×C₂₃	—	2	1	no	—
32	#41×C₂₃	—	2	4	no	—
33	#41×C₂₃	—	2	4	no	—
34	C₂×Q₈×C₂₃	—	2	1	no	—
35	C₂×Q₈×C₂₃	—	2	4	no	—
36	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
37	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	—	2	4	no	—
40	C₂×D₄×C₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	—	2	1	no	—
42	C₂×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂×D₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	D₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	Q₁₆×C₂₃	—	2	8	no	—
49	SD₁₆×C₂₃	—	2	8	no	—
50	SD₁₆×C₂₃	—	2	8	no	—
51	D₈×C₂₃	—	2	8	no	—
52	M₁₆×C₂₃	D₄	1	8	yes	complemented
53	C₂×C₈×C₂₃	C₂³	1	4	yes	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₂²⋊C₄×C₂₃	—	2	4	no	—
56	C₂²⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	—	2	4	no	—
58	C₄²×C₂₃	D₄	1	2	yes	—
59	C₄²×C₂₃	C₂³	1	1	yes	characteristic
60	C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	4	4	no	—
64	C₂³×C₂₃	—	4	2	no	—
65	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₂₃	—	2	2	no	—
67	Q₈×C₂₃	—	2	4	no	—
68	Q₈×C₂₃	—	2	2	no	—
69	Q₈×C₂₃	—	4	2	no	—
70	D₄×C₂₃	—	2	2	no	—
71	D₄×C₂₃	—	2	4	no	—
72	D₄×C₂₃	—	2	2	no	—
73	D₄×C₂₃	—	4	4	no	—
74	D₄×C₂₃	—	4	4	no	—
75	D₄×C₂₃	—	4	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	—	4	4	no	—
81	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₈×C₂₃	C₂×D₄	1	8	yes	—
87	#3153	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₂₃	—	2	2	no	—
89	C₂²×C₂₃	—	4	2	no	—
90	C₂²×C₂₃	—	4	2	no	—
91	C₂²×C₂₃	—	4	4	no	—
92	C₂²×C₂₃	—	4	4	no	—
93	C₂²×C₂₃	—	8	4	no	—
94	C₂²×C₂₃	#127	1	2	yes	—
95	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	—	4	2	no	—
100	C₄×C₂₃	#127	1	2	yes	—
101	C₄×C₂₃	C₂²×D₄	1	2	yes	—
102	C₂×#137	C₂×C₂₃	1	2	yes	complemented
103	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
104	#532	C₂×C₂₃	1	1	yes	characteristic, complemented
105	C₂×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#497	C₂×C₂₃	1	2	yes	complemented
107	#496	C₂×C₂₃	1	2	yes	complemented
108	#495	C₂×C₂₃	1	4	yes	complemented
109	C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂×C₂₃	—	2	2	no	—
111	C₂×C₂₃	—	8	2	no	—
112	C₂×C₂₃	—	8	2	no	—
113	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
114	C₂×C₂₃	#575	1	2	yes	—
115	C₂×#41	C₂²×C₂₃	1	2	yes	—
116	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
117	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
118	#137	—	2	8	no	—
119	AGL(1,ℤ/8)	—	2	8	no	—
120	C₂×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
121	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
122	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
123	C₂×D₈	C₂²×C₂₃	1	4	yes	—
124	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
125	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
126	#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
127	#124	—	2	2	no	—
128	#124	C₂²×C₂₃	1	2	yes	—
129	#124	C₂²×C₂₃	1	2	yes	—
130	#124	C₂²×C₂₃	1	2	yes	complemented
131	#121	—	2	2	no	—
132	C₄×D₄	—	2	2	no	—
133	C₂×C₂²⋊C₄	—	2	2	no	—
134	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
135	#97	C₂²×C₂₃	1	2	yes	complemented
136	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
137	C₂₃	#3153	1	1	yes	characteristic, direct product factor, Sylow
138	C₂⁴	—	2	1	no	—
139	#41	—	2	4	no	—
140	#41	—	2	4	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
144	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	1	no	—
149	C₂×D₄	C₂³×C₂₃	1	2	yes	—
150	C₂×D₄	C₂³×C₂₃	1	2	yes	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	D₄×C₂₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
155	Q₁₆	—	2	8	no	—
156	SD₁₆	—	2	8	no	—
157	SD₁₆	—	2	8	no	—
158	D₈	—	2	8	no	—
159	M₁₆	D₄×C₂₃	1	8	yes	complemented
160	C₂×C₈	C₂³×C₂₃	1	4	yes	—
161	C₄⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₄²	D₄×C₂₃	1	2	yes	—
166	C₄²	C₂³×C₂₃	1	1	yes	characteristic
167	C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	4	4	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	Q₈	—	2	4	no	—
175	Q₈	—	2	2	no	—
176	Q₈	—	4	2	no	—
177	D₄	—	2	2	no	—
178	D₄	—	2	4	no	—
179	D₄	—	2	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	D₄	—	4	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
189	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
192	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
193	C₈	C₂×D₄×C₂₃	1	8	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	4	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	8	4	no	—
200	C₂²	C₂₃×#127	1	2	yes	—
201	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	C₂₃×#127	1	2	yes	—
207	C₄	C₂²×D₄×C₂₃	1	2	yes	—
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	—	8	2	no	—
211	C₂	C₂×C₂₃×#127	1	1	yes	characteristic
212	C₂	C₂₃×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
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. C₂×C₂₃×#137	C₂	semidirect
2. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂₃×#532	C₂	semidirect
4. C₂×C₂₃×#124	C₂	semidirect
5. C₂₃×#497	C₂	semidirect
6. C₂₃×#496	C₂	semidirect
7. C₂₃×#495	C₂	semidirect
8. C₄×M₁₆×C₂₃	C₂	semidirect
14. C₂×Q₁₆×C₂₃	C₂²	semidirect
16. C₂×SD₁₆×C₂₃	C₂²	semidirect
19. C₂₃×C₄⋊Q₈	C₂²	semidirect
20. C₂₃×#127	C₂²	semidirect
24. C₂₃×#124	C₂²	semidirect
29. C₂₃×#97	C₂²	semidirect
30. C₄×C₈×C₂₃	C₂²	semidirect
52. M₁₆×C₂₃	D₄	semidirect
87. #3153	C₂₃	direct (N × H)
102. C₂×#137	C₂×C₂₃	semidirect
103. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
104. #532	C₂×C₂₃	semidirect
105. C₂×#124	C₂×C₂₃	semidirect
106. #497	C₂×C₂₃	semidirect
107. #496	C₂×C₂₃	semidirect
108. #495	C₂×C₂₃	semidirect
109. C₄×M₁₆	C₂×C₂₃	semidirect
120. C₂×Q₁₆	C₂²×C₂₃	semidirect
122. C₂×SD₁₆	C₂²×C₂₃	semidirect
125. C₄⋊Q₈	C₂²×C₂₃	semidirect
126. #127	C₂²×C₂₃	semidirect
130. #124	C₂²×C₂₃	semidirect
135. #97	C₂²×C₂₃	semidirect
136. C₄×C₈	C₂²×C₂₃	semidirect
137. C₂₃	#3153	direct (N × H)
159. M₁₆	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#137	2
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#532	2
C₂×C₂₃×#124	2
C₂₃×#497	2
C₂₃×#496	2
C₂₃×#495	2
C₄×M₁₆×C₂₃	2
#3153	23

Subgroups — C23×#3153 #15670772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3153 #15670772