Subgroups — C₂₃×#3394 #15671013

Order 2944.

372 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	C₂₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#495	C₂	1	2	yes	maximal, complemented
4	C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#464	C₂	1	2	yes	maximal, complemented
6	C₂₃×#463	C₂	1	2	yes	maximal, complemented
7	C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#459	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×D₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#127	C₂²	1	1	yes	characteristic
19	C₂₃×#126	—	2	2	no	—
20	C₂₃×#124	—	2	1	no	—
21	C₂₃×#124	C₂²	1	2	yes	—
22	C₂₃×#124	C₂²	1	2	yes	—
23	C₂₃×#123	—	2	2	no	—
24	C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
26	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	—	2	1	no	—
29	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₂₃	C₂²	1	2	yes	—
31	C₂₃×#117	—	2	1	no	—
32	C₂₃×#106	C₂²	1	1	yes	characteristic
33	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₂₃×#103	C₂²	1	2	yes	—
36	C₂₃×#103	C₂²	1	1	yes	characteristic
37	C₂₃×#102	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₂₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
44	C₂×D₄×C₂₃	—	2	1	no	—
45	C₂×D₄×C₂₃	—	2	1	no	—
46	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	Q₁₆×C₂₃	—	4	2	no	—
51	SD₁₆×C₂₃	—	2	4	no	—
52	SD₁₆×C₂₃	—	2	2	no	—
53	SD₁₆×C₂₃	—	4	2	no	—
54	D₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
64	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₂₃	—	2	2	no	—
67	C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂²⋊C₄×C₂₃	—	2	2	no	—
69	C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	C₂³	1	2	yes	—
75	C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂³×C₂₃	—	2	1	no	—
77	C₂³×C₂₃	—	2	2	no	—
78	Q₈×C₂₃	—	2	2	no	—
79	Q₈×C₂₃	—	2	1	no	—
80	Q₈×C₂₃	—	2	2	no	—
81	Q₈×C₂₃	—	2	2	no	—
82	Q₈×C₂₃	—	2	1	no	—
83	D₄×C₂₃	—	2	1	no	—
84	D₄×C₂₃	—	2	2	no	—
85	D₄×C₂₃	—	2	1	no	—
86	D₄×C₂₃	—	2	2	no	—
87	D₄×C₂₃	—	4	2	no	—
88	D₄×C₂₃	—	8	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	8	2	no	—
95	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₈×C₂₃	—	2	2	no	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	#41	1	2	yes	—
103	#3394	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₂₃	—	4	1	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	1	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	—	4	2	no	—
109	C₂²×C₂₃	—	4	1	no	—
110	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	4	1	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	—	4	1	no	—
117	C₄×C₂₃	—	4	1	no	—
118	C₄×C₂₃	#135	1	2	yes	—
119	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
120	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
121	#559	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#556	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#495	C₂×C₂₃	1	2	yes	complemented
124	#492	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#464	C₂×C₂₃	1	2	yes	complemented
126	#463	C₂×C₂₃	1	2	yes	complemented
127	#462	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#459	C₂×C₂₃	1	1	yes	characteristic
129	C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
130	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
131	C₄×SD₁₆	C₂×C₂₃	1	2	yes	complemented
132	C₂×C₂₃	—	8	1	no	—
133	C₂×C₂₃	—	8	2	no	—
134	C₂×C₂₃	#550	1	1	yes	characteristic
135	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
136	C₂×C₂₃	#577	1	1	yes	characteristic
137	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
138	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
139	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
141	C₂×D₈	C₂²×C₂₃	1	2	yes	—
142	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
143	#127	C₂²×C₂₃	1	1	yes	characteristic
144	#126	—	2	2	no	—
145	#124	—	2	1	no	—
146	#124	C₂²×C₂₃	1	2	yes	—
147	#124	C₂²×C₂₃	1	2	yes	—
148	#123	—	2	2	no	—
149	C₂²⋊Q₈	—	2	1	no	—
150	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
151	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₄×D₄	C₂²×C₂₃	1	2	yes	—
156	#117	—	2	1	no	—
157	#106	C₂²×C₂₃	1	1	yes	characteristic
158	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
159	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
160	#103	C₂²×C₂₃	1	2	yes	—
161	#103	C₂²×C₂₃	1	1	yes	characteristic
162	#102	C₂²×C₂₃	1	2	yes	—
163	#102	C₂²×C₂₃	1	1	yes	characteristic
164	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
165	C₄×C₈	C₂²×C₂₃	1	2	yes	—
166	C₂₃	#3394	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
168	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
173	C₂×D₄	C₂³×C₂₃	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	Q₁₆	—	4	2	no	—
177	SD₁₆	—	2	4	no	—
178	SD₁₆	—	2	2	no	—
179	SD₁₆	—	4	2	no	—
180	D₈	—	4	2	no	—
181	C₂×C₈	C₂³×C₂₃	1	2	yes	—
182	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
189	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
190	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	1	no	—
200	C₄²	C₂³×C₂₃	1	2	yes	—
201	C₄²	C₂³×C₂₃	1	1	yes	characteristic
202	C₂³	—	2	1	no	—
203	C₂³	—	2	2	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	1	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	2	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	2	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
222	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
223	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	1	no	—
228	C₈	#41×C₂₃	1	2	yes	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	1	no	—
235	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	C₂₃×#135	1	2	yes	—
244	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
245	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂	—	8	1	no	—
247	C₂	—	8	2	no	—
248	C₂	C₂₃×#550	1	1	yes	characteristic
249	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
250	C₂	C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#559	C₂	semidirect
2. C₂₃×#556	C₂	semidirect
3. C₂₃×#495	C₂	semidirect
4. C₂₃×#492	C₂	semidirect
5. C₂₃×#464	C₂	semidirect
6. C₂₃×#463	C₂	semidirect
7. C₂₃×#462	C₂	semidirect
9. C₈×Q₈×C₂₃	C₂	semidirect
10. C₄×SD₁₆×C₂₃	C₂	semidirect
11. C₄×SD₁₆×C₂₃	C₂	semidirect
103. #3394	C₂₃	direct (N × H)
121. #559	C₂×C₂₃	semidirect
122. #556	C₂×C₂₃	semidirect
123. #495	C₂×C₂₃	semidirect
124. #492	C₂×C₂₃	semidirect
125. #464	C₂×C₂₃	semidirect
126. #463	C₂×C₂₃	semidirect
127. #462	C₂×C₂₃	semidirect
129. C₈×Q₈	C₂×C₂₃	semidirect
130. C₄×SD₁₆	C₂×C₂₃	semidirect
131. C₄×SD₁₆	C₂×C₂₃	semidirect
166. C₂₃	#3394	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×#559	2
C₂₃×#556	2
C₂₃×#495	2
C₂₃×#492	2
C₂₃×#464	2
C₂₃×#463	2
C₂₃×#462	2
C₂₃×#459	2
C₈×Q₈×C₂₃	2
C₄×SD₁₆×C₂₃	2
#3394	23

Subgroups — C23×#3394 #15671013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3394 #15671013