Subgroups — C₂₃×#3094 #15670713

Order 2944.

462 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#482	C₂	1	4	yes	maximal, complemented
4	C₂₃×#481	C₂	1	2	yes	maximal, complemented
5	C₂₃×#480	C₂	1	2	yes	maximal, complemented
6	C₂₃×#428	C₂	1	2	yes	maximal, complemented
7	C₂₃×#418	C₂	1	2	yes	maximal, complemented
8	C₂₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₃	—	2	1	no	—
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#121	—	2	1	no	—
14	C₂₃×#121	—	2	1	no	—
15	C₂₃×#121	C₂²	1	2	yes	—
16	C₂₃×#121	C₂²	1	2	yes	—
17	C₂₃×#117	—	2	1	no	—
18	C₂₃×#117	—	2	1	no	—
19	C₂₃×#117	C₂²	1	2	yes	—
20	C₂₃×#117	C₂²	1	2	yes	complemented
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄²×C₂₃	—	2	1	no	—
27	C₂₃×#107	C₂²	1	4	yes	complemented
28	C₂₃×#106	C₂²	1	4	yes	complemented
29	C₂₃×#102	C₂²	1	4	yes	complemented
30	C₂₃×#102	C₂²	1	4	yes	complemented
31	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
32	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₃	—	2	1	no	—
34	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
35	C₂×D₄×C₂₃	—	2	2	no	—
36	C₂×D₄×C₂₃	—	2	2	no	—
37	C₂×D₄×C₂₃	—	2	1	no	—
38	C₂×D₄×C₂₃	—	2	4	no	—
39	C₂×D₄×C₂₃	—	4	4	no	—
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	M₁₆×C₂₃	—	2	4	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	—	2	1	no	—
56	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₂²⋊C₄×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	4	1	no	—
69	C₂³×C₂₃	—	4	2	no	—
70	C₂³×C₂₃	—	4	2	no	—
71	C₂³×C₂₃	—	4	2	no	—
72	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
75	D₄×C₂₃	—	2	4	no	—
76	D₄×C₂₃	—	4	4	no	—
77	D₄×C₂₃	—	4	4	no	—
78	D₄×C₂₃	—	4	2	no	—
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	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
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	#41	1	4	yes	—
94	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
95	C₈×C₂₃	—	2	4	no	—
96	#3094	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	2	1	no	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	2	1	no	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	—	4	4	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	2	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	—	8	4	no	—
109	C₂²×C₂₃	#123	1	2	yes	—
110	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
111	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	4	2	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	#123	1	2	yes	—
118	C₄×C₂₃	C₂×#41	1	2	yes	—
119	C₂×#121	C₂×C₂₃	1	1	yes	characteristic
120	C₂×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#482	C₂×C₂₃	1	4	yes	complemented
122	#481	C₂×C₂₃	1	2	yes	complemented
123	#480	C₂×C₂₃	1	2	yes	complemented
124	#428	C₂×C₂₃	1	2	yes	complemented
125	#418	C₂×C₂₃	1	2	yes	complemented
126	#407	C₂×C₂₃	1	1	yes	characteristic, complemented
127	C₂×C₂₃	—	2	2	no	—
128	C₂×C₂₃	—	2	2	no	—
129	C₂×C₂₃	—	4	1	no	—
130	C₂×C₂₃	—	8	2	no	—
131	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
132	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₂₃	#575	1	1	yes	characteristic
134	C₂²×D₄	—	2	1	no	—
135	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
136	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
138	#121	—	2	1	no	—
139	#121	—	2	1	no	—
140	#121	C₂²×C₂₃	1	2	yes	—
141	#121	C₂²×C₂₃	1	2	yes	—
142	#117	—	2	1	no	—
143	#117	—	2	1	no	—
144	#117	C₂²×C₂₃	1	2	yes	—
145	#117	C₂²×C₂₃	1	2	yes	complemented
146	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
148	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	#107	C₂²×C₂₃	1	4	yes	complemented
153	#106	C₂²×C₂₃	1	4	yes	complemented
154	#102	C₂²×C₂₃	1	4	yes	complemented
155	#102	C₂²×C₂₃	1	4	yes	complemented
156	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
157	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
158	C₂₃	#3094	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	—	2	1	no	—
160	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	4	no	—
166	C₂×D₄	C₂³×C₂₃	1	2	yes	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
177	M₁₆	—	2	4	no	—
178	C₂×C₈	C₂³×C₂₃	1	4	yes	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
183	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
184	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	4	1	no	—
195	C₂³	—	4	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
201	D₄	—	2	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂×C₄	#41×C₂₃	1	4	yes	—
220	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	4	4	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	C₂₃×#123	1	2	yes	—
235	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
236	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	C₂₃×#123	1	2	yes	—
243	C₄	C₂×#41×C₂₃	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	2	2	no	—
246	C₂	—	4	1	no	—
247	C₂	—	8	2	no	—
248	C₂	C₂×C₂₃×#123	1	1	yes	characteristic
249	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
250	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	C₂×C₂₃×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₂₃×#117	C₂	semidirect
3. C₂₃×#482	C₂	semidirect
4. C₂₃×#481	C₂	semidirect
5. C₂₃×#480	C₂	semidirect
6. C₂₃×#428	C₂	semidirect
7. C₂₃×#418	C₂	semidirect
8. C₂₃×#407	C₂	semidirect
20. C₂₃×#117	C₂²	semidirect
22. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
23. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
27. C₂₃×#107	C₂²	semidirect
28. C₂₃×#106	C₂²	semidirect
29. C₂₃×#102	C₂²	semidirect
30. C₂₃×#102	C₂²	semidirect
31. C₂₃×C₂²⋊C₈	C₂²	semidirect
32. C₂₃×C₂²⋊C₈	C₂²	semidirect
96. #3094	C₂₃	direct (N × H)
120. C₂×#117	C₂×C₂₃	semidirect
121. #482	C₂×C₂₃	semidirect
122. #481	C₂×C₂₃	semidirect
123. #480	C₂×C₂₃	semidirect
124. #428	C₂×C₂₃	semidirect
125. #418	C₂×C₂₃	semidirect
126. #407	C₂×C₂₃	semidirect
145. #117	C₂²×C₂₃	semidirect
147. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
148. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
152. #107	C₂²×C₂₃	semidirect
153. #106	C₂²×C₂₃	semidirect
154. #102	C₂²×C₂₃	semidirect
155. #102	C₂²×C₂₃	semidirect
156. C₂²⋊C₈	C₂²×C₂₃	semidirect
157. C₂²⋊C₈	C₂²×C₂₃	semidirect
158. C₂₃	#3094	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×#121	2
C₂×C₂₃×#117	2
C₂₃×#482	2
C₂₃×#481	2
C₂₃×#480	2
C₂₃×#428	2
C₂₃×#418	2
C₂₃×#407	2
#3094	23

Subgroups — C23×#3094 #15670713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3094 #15670713