Subgroups — C₂₃×#3357 #15670976

Order 2944.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#485	C₂	1	2	yes	maximal, complemented
5	C₂₃×#480	C₂	1	2	yes	maximal, complemented
6	C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#468	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#428	C₂	1	2	yes	maximal, complemented
12	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×#125	—	2	1	no	—
17	C₂₃×#124	—	2	2	no	—
18	C₂₃×C₂²⋊Q₈	—	2	2	no	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂₃×#121	—	2	1	no	—
21	C₂₃×#121	C₂²	1	2	yes	—
22	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₂₃	—	2	1	no	—
24	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#117	—	2	1	no	—
27	C₂₃×#117	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#107	C₂²	1	2	yes	—
30	C₂₃×#107	C₂²	1	2	yes	—
31	C₂₃×#107	C₂²	1	2	yes	—
32	C₂₃×#106	C₂²	1	1	yes	characteristic
33	C₂₃×#106	C₂²	1	2	yes	—
34	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂₃×#103	C₂²	1	1	yes	characteristic
36	C₂₃×#103	C₂²	1	2	yes	—
37	C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₃	—	2	1	no	—
42	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₂₃	—	2	2	no	—
45	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	SD₁₆×C₂₃	—	2	2	no	—
51	M₁₆×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₂₃	—	2	2	no	—
66	C₂²⋊C₄×C₂₃	—	2	2	no	—
67	C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	2	1	no	—
72	C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	C₂×D₄	1	2	yes	—
75	Q₈×C₂₃	—	2	1	no	—
76	Q₈×C₂₃	—	2	1	no	—
77	Q₈×C₂₃	—	8	1	no	—
78	Q₈×C₂₃	—	8	1	no	—
79	D₄×C₂₃	—	2	2	no	—
80	D₄×C₂₃	—	2	1	no	—
81	D₄×C₂₃	—	2	1	no	—
82	D₄×C₂₃	—	8	2	no	—
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	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	8	2	no	—
92	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	#41	1	2	yes	—
99	#3357	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	4	1	no	—
104	C₂²×C₂₃	—	4	1	no	—
105	C₂²×C₂₃	—	4	1	no	—
106	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₄×C₂₃	—	4	2	no	—
114	C₄×C₂₃	#137	1	2	yes	—
115	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
117	#555	C₂×C₂₃	1	1	yes	characteristic
118	#554	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#500	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#485	C₂×C₂₃	1	2	yes	complemented
121	#480	C₂×C₂₃	1	2	yes	complemented
122	#477	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#476	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#468	C₂×C₂₃	1	2	yes	complemented
125	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#436	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#428	C₂×C₂₃	1	2	yes	complemented
128	C₂×C₂₃	—	4	2	no	—
129	C₂×C₂₃	—	8	1	no	—
130	C₂×C₂₃	#547	1	1	yes	characteristic
131	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
132	C₂×C₂₃	#576	1	1	yes	characteristic
133	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
135	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
137	#125	—	2	1	no	—
138	#124	—	2	2	no	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
141	#121	—	2	1	no	—
142	#121	C₂²×C₂₃	1	2	yes	—
143	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
147	#117	—	2	1	no	—
148	#117	C₂²×C₂₃	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
150	#107	C₂²×C₂₃	1	2	yes	—
151	#107	C₂²×C₂₃	1	2	yes	—
152	#107	C₂²×C₂₃	1	2	yes	—
153	#106	C₂²×C₂₃	1	1	yes	characteristic
154	#106	C₂²×C₂₃	1	2	yes	—
155	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
156	#103	C₂²×C₂₃	1	1	yes	characteristic
157	#103	C₂²×C₂₃	1	2	yes	—
158	#102	C₂²×C₂₃	1	1	yes	characteristic
159	#102	C₂²×C₂₃	1	2	yes	—
160	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
161	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₂₃	#3357	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
172	SD₁₆	—	2	2	no	—
173	M₁₆	—	2	4	no	—
174	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₂₃	1	2	yes	—
176	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
184	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
185	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂³×C₂₃	1	1	yes	characteristic
195	C₂³	—	2	1	no	—
196	C₂³	C₂×D₄×C₂₃	1	2	yes	—
197	Q₈	—	2	1	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	8	1	no	—
200	Q₈	—	8	1	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	1	no	—
203	D₄	—	2	1	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	1	no	—
220	C₈	#41×C₂₃	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	C₂₃×#137	1	2	yes	—
236	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
237	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	8	1	no	—
240	C₂	C₂₃×#547	1	1	yes	characteristic
241	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
242	C₂	C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#554	C₂	semidirect
3. C₂₃×#500	C₂	semidirect
4. C₂₃×#485	C₂	semidirect
5. C₂₃×#480	C₂	semidirect
6. C₂₃×#477	C₂	semidirect
7. C₂₃×#476	C₂	semidirect
8. C₂₃×#468	C₂	semidirect
9. C₄×SD₁₆×C₂₃	C₂	semidirect
10. C₂₃×#436	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
99. #3357	C₂₃	direct (N × H)
118. #554	C₂×C₂₃	semidirect
119. #500	C₂×C₂₃	semidirect
120. #485	C₂×C₂₃	semidirect
121. #480	C₂×C₂₃	semidirect
122. #477	C₂×C₂₃	semidirect
123. #476	C₂×C₂₃	semidirect
124. #468	C₂×C₂₃	semidirect
125. C₄×SD₁₆	C₂×C₂₃	semidirect
126. #436	C₂×C₂₃	semidirect
127. #428	C₂×C₂₃	semidirect
162. C₂₃	#3357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#555	2
C₂₃×#554	2
C₂₃×#500	2
C₂₃×#485	2
C₂₃×#480	2
C₂₃×#477	2
C₂₃×#476	2
C₂₃×#468	2
C₄×SD₁₆×C₂₃	2
C₂₃×#436	2
C₂₃×#428	2
#3357	23

Subgroups — C23×#3357 #15670976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3357 #15670976