Subgroups — C₂₃×#3077 #15670696

Order 2944.

440 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#478	C₂	1	4	yes	maximal, complemented
4	C₂₃×#476	C₂	1	2	yes	maximal, complemented
5	C₂₃×#474	C₂	1	2	yes	maximal, complemented
6	C₂₃×#428	C₂	1	2	yes	maximal, complemented
7	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#418	C₂	1	2	yes	maximal
9	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	—	2	1	no	—
11	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₂₃×#125	C₂²	1	2	yes	complemented
15	C₂₃×C₂²⋊Q₈	—	2	1	no	—
16	C₂₃×C₂²⋊Q₈	—	2	1	no	—
17	C₄×Q₈×C₂₃	—	2	1	no	—
18	C₄×Q₈×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	C₂₃×#117	C₂²	1	2	yes	—
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×#107	C₂²	1	4	yes	complemented
28	C₂₃×#106	C₂²	1	4	yes	complemented
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
31	C₂₃×#102	C₂²	1	4	yes	—
32	C₂₃×#102	C₂²	1	4	yes	—
33	C₂⁴×C₂₃	—	2	1	no	—
34	C₂×Q₈×C₂₃	—	2	1	no	—
35	C₂×Q₈×C₂₃	—	2	1	no	—
36	C₂×D₄×C₂₃	—	2	1	no	—
37	C₂×D₄×C₂₃	—	2	2	no	—
38	C₂×D₄×C₂₃	Q₈	1	4	yes	complemented
39	C₂×D₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	—	2	1	no	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	—	4	4	no	—
44	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	M₁₆×C₂₃	—	2	4	no	—
48	C₂×C₈×C₂₃	C₂³	1	4	yes	—
49	C₄⋊C₄×C₂₃	—	2	1	no	—
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	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	C₄²×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	—	4	2	no	—
68	C₂³×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₂₃	—	4	2	no	—
71	Q₈×C₂₃	—	4	2	no	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	2	no	—
74	D₄×C₂₃	C₂×Q₈	1	4	yes	—
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	—	4	1	no	—
80	C₂×C₄×C₂₃	—	4	4	no	—
81	C₂×C₄×C₂₃	—	4	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	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₂₃	#41	1	2	yes	—
90	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₂₃	—	2	4	no	—
92	#3077	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₂₃	—	2	1	no	—
94	C₂²×C₂₃	—	2	1	no	—
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	4	8	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	—	4	2	no	—
99	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
100	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	—	2	1	no	—
104	C₄×C₂₃	—	4	2	no	—
105	C₄×C₂₃	—	4	2	no	—
106	C₄×C₂₃	—	4	2	no	—
107	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
108	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
110	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
111	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
112	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
113	#478	C₂×C₂₃	1	4	yes	complemented
114	#476	C₂×C₂₃	1	2	yes	complemented
115	#474	C₂×C₂₃	1	2	yes	complemented
116	#428	C₂×C₂₃	1	2	yes	complemented
117	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#418	C₂×C₂₃	1	2	yes	—
119	C₂×C₂₃	—	2	2	no	—
120	C₂×C₂₃	—	4	4	no	—
121	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₂₃	#575	1	1	yes	characteristic
124	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
125	C₂³×C₄	—	2	1	no	—
126	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
127	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
128	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
129	#125	C₂²×C₂₃	1	2	yes	complemented
130	C₂²⋊Q₈	—	2	1	no	—
131	C₂²⋊Q₈	—	2	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₄×D₄	—	2	1	no	—
136	C₄×D₄	C₂²×C₂₃	1	2	yes	—
137	C₄×D₄	C₂²×C₂₃	1	2	yes	—
138	#117	C₂²×C₂₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
142	#107	C₂²×C₂₃	1	4	yes	complemented
143	#106	C₂²×C₂₃	1	4	yes	complemented
144	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
145	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
146	#102	C₂²×C₂₃	1	4	yes	—
147	#102	C₂²×C₂₃	1	4	yes	—
148	C₂₃	#3077	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁴	—	2	1	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	Q₈×C₂₃	1	4	yes	complemented
155	C₂×D₄	C₂³×C₂₃	1	2	yes	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	M₁₆	—	2	4	no	—
164	C₂×C₈	C₂³×C₂₃	1	4	yes	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
171	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
172	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	D₄×C₂₃	1	1	yes	characteristic
178	C₄²	D₄×C₂₃	1	1	yes	characteristic
179	C₄²	C₂³×C₂₃	1	1	yes	characteristic
180	C₄²	C₂³×C₂₃	1	1	yes	characteristic
181	C₂³	—	2	4	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	D₄	—	2	4	no	—
189	D₄	—	2	2	no	—
190	D₄	C₂×Q₈×C₂₃	1	4	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
205	C₂×C₄	#41×C₂₃	1	2	yes	—
206	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	8	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
215	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
223	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
224	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
225	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
229	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
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×#478	C₂	semidirect
4. C₂₃×#476	C₂	semidirect
5. C₂₃×#474	C₂	semidirect
6. C₂₃×#428	C₂	semidirect
7. C₂₃×#423	C₂	semidirect
12. C₂₃×C₄⋊Q₈	C₂²	semidirect
13. C₂₃×C₄⋊Q₈	C₂²	semidirect
14. C₂₃×#125	C₂²	semidirect
27. C₂₃×#107	C₂²	semidirect
28. C₂₃×#106	C₂²	semidirect
29. C₂₃×C₄⋊C₈	C₂²	semidirect
30. C₂₃×C₄⋊C₈	C₂²	semidirect
38. C₂×D₄×C₂₃	Q₈	semidirect
92. #3077	C₂₃	direct (N × H)
111. #533	C₂×C₂₃	semidirect
113. #478	C₂×C₂₃	semidirect
114. #476	C₂×C₂₃	semidirect
115. #474	C₂×C₂₃	semidirect
116. #428	C₂×C₂₃	semidirect
117. #423	C₂×C₂₃	semidirect
127. C₄⋊Q₈	C₂²×C₂₃	semidirect
128. C₄⋊Q₈	C₂²×C₂₃	semidirect
129. #125	C₂²×C₂₃	semidirect
142. #107	C₂²×C₂₃	semidirect
143. #106	C₂²×C₂₃	semidirect
144. C₄⋊C₈	C₂²×C₂₃	semidirect
145. C₄⋊C₈	C₂²×C₂₃	semidirect
148. C₂₃	#3077	direct (N × H)
154. C₂×D₄	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#533	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#478	2
C₂₃×#476	2
C₂₃×#474	2
C₂₃×#428	2
C₂₃×#423	2
C₂₃×#418	2
#3077	23

Subgroups — C23×#3077 #15670696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3077 #15670696