Subgroups — C₂₃×#3109 #15670728

Order 2944.

488 conjugacy classes of subgroups.

The subgroup lattice has 242 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, complemented
2	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#481	C₂	1	2	yes	maximal, complemented
4	C₂₃×#476	C₂	1	2	yes	maximal, complemented
5	C₂₃×#459	C₂	1	2	yes	maximal, complemented
6	C₂₃×#447	C₂	1	2	yes	maximal, complemented
7	C₂₃×#442	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×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	—	2	1	no	—
12	C₂×D₈×C₂₃	C₂²	1	4	yes	—
13	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#124	—	2	1	no	—
16	C₂₃×#121	—	2	1	no	—
17	C₂₃×#121	C₂²	1	2	yes	—
18	C₄×D₄×C₂₃	—	2	1	no	—
19	C₄×D₄×C₂₃	—	2	1	no	—
20	C₄×D₄×C₂₃	—	2	1	no	—
21	C₄×D₄×C₂₃	C₂²	1	2	yes	—
22	C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#106	C₂²	1	4	yes	complemented
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
29	C₂₃×#102	C₂²	1	4	yes	—
30	C₂₃×#102	C₂²	1	4	yes	complemented
31	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂₃×#97	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₃	—	2	1	no	—
34	#41×C₂₃	—	4	4	no	—
35	C₂×Q₈×C₂₃	—	2	1	no	—
36	C₂×D₄×C₂₃	—	2	4	no	—
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	—	2	1	no	—
41	C₂×D₄×C₂₃	C₂³	1	2	yes	—
42	C₂×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	—	2	2	no	—
47	C₂²×C₄×C₂₃	—	4	4	no	—
48	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	D₈×C₂₃	—	2	8	no	—
52	C₂×C₈×C₂₃	C₂³	1	4	yes	—
53	C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₄²×C₂₃	D₄	1	1	yes	characteristic
62	C₄²×C₂₃	D₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₂₃	—	2	4	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	—	4	2	no	—
70	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₂₃	—	4	2	no	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	2	no	—
74	D₄×C₂₃	—	2	4	no	—
75	D₄×C₂₃	—	4	2	no	—
76	D₄×C₂₃	—	4	4	no	—
77	D₄×C₂₃	—	4	2	no	—
78	D₄×C₂₃	#41	1	4	yes	complemented
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	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	4	4	no	—
85	C₂×C₄×C₂₃	—	4	4	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	4	no	—
95	#3109	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₂₃	—	2	1	no	—
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	2	4	no	—
99	C₂²×C₂₃	—	4	8	no	—
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₄×C₂₃	C₂×#41	1	2	yes	—
115	#532	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#481	C₂×C₂₃	1	2	yes	complemented
118	#476	C₂×C₂₃	1	2	yes	complemented
119	#459	C₂×C₂₃	1	2	yes	complemented
120	#447	C₂×C₂₃	1	2	yes	complemented
121	#442	C₂×C₂₃	1	4	yes	complemented
122	#432	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂×C₂₃	—	2	2	no	—
124	C₂×C₂₃	—	4	4	no	—
125	C₂×C₂₃	—	8	2	no	—
126	C₂×C₂₃	#525	1	1	yes	characteristic
127	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×#41	—	2	1	no	—
130	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
131	C₂³×C₄	—	2	1	no	—
132	C₂×D₈	C₂²×C₂₃	1	4	yes	—
133	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
134	#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
135	#124	—	2	1	no	—
136	#121	—	2	1	no	—
137	#121	C₂²×C₂₃	1	2	yes	—
138	C₄×D₄	—	2	1	no	—
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	C₂²×C₂₃	1	2	yes	—
142	C₄×D₄	C₂²×C₂₃	1	2	yes	—
143	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
147	#106	C₂²×C₂₃	1	4	yes	complemented
148	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
149	#102	C₂²×C₂₃	1	4	yes	—
150	#102	C₂²×C₂₃	1	4	yes	complemented
151	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
152	#97	C₂²×C₂₃	1	2	yes	complemented
153	C₂₃	#3109	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	—	2	1	no	—
155	#41	—	4	4	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	C₂³×C₂₃	1	2	yes	—
163	C₂×D₄	C₂³×C₂₃	1	2	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	D₈	—	2	8	no	—
173	C₂×C₈	C₂³×C₂₃	1	4	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
177	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₄²	D₄×C₂₃	1	1	yes	characteristic
183	C₄²	D₄×C₂₃	1	1	yes	characteristic
184	C₄²	C₂³×C₂₃	1	1	yes	characteristic
185	C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	C₂³	—	2	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	D₄	—	2	4	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	4	no	—
196	D₄	—	4	2	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	#41×C₂₃	1	4	yes	complemented
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	4	8	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
226	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
234	C₄	C₂×#41×C₂₃	1	2	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	4	4	no	—
237	C₂	—	8	2	no	—
238	C₂	C₂₃×#525	1	1	yes	characteristic
239	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	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. C₂₃×#532	C₂	semidirect
2. C₂×C₄×D₄×C₂₃	C₂	semidirect
3. C₂₃×#481	C₂	semidirect
4. C₂₃×#476	C₂	semidirect
5. C₂₃×#459	C₂	semidirect
6. C₂₃×#447	C₂	semidirect
7. C₂₃×#442	C₂	semidirect
8. C₂₃×#432	C₂	semidirect
13. C₂₃×C₄⋊Q₈	C₂²	semidirect
14. C₂₃×#127	C₂²	semidirect
23. C₄×D₄×C₂₃	C₂²	semidirect
27. C₂₃×#106	C₂²	semidirect
28. C₂₃×C₄⋊C₈	C₂²	semidirect
30. C₂₃×#102	C₂²	semidirect
32. C₂₃×#97	C₂²	semidirect
78. D₄×C₂₃	#41	semidirect
95. #3109	C₂₃	direct (N × H)
115. #532	C₂×C₂₃	semidirect
116. C₂×C₄×D₄	C₂×C₂₃	semidirect
117. #481	C₂×C₂₃	semidirect
118. #476	C₂×C₂₃	semidirect
119. #459	C₂×C₂₃	semidirect
120. #447	C₂×C₂₃	semidirect
121. #442	C₂×C₂₃	semidirect
122. #432	C₂×C₂₃	semidirect
133. C₄⋊Q₈	C₂²×C₂₃	semidirect
134. #127	C₂²×C₂₃	semidirect
143. C₄×D₄	C₂²×C₂₃	semidirect
147. #106	C₂²×C₂₃	semidirect
148. C₄⋊C₈	C₂²×C₂₃	semidirect
150. #102	C₂²×C₂₃	semidirect
152. #97	C₂²×C₂₃	semidirect
153. C₂₃	#3109	direct (N × H)
199. D₄	#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×D₄×C₂₃	2
C₂₃×#481	2
C₂₃×#476	2
C₂₃×#459	2
C₂₃×#447	2
C₂₃×#442	2
C₂₃×#432	2
#3109	23

Subgroups — C23×#3109 #15670728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3109 #15670728