Subgroups — C₂₃×#3399 #15671018

Order 2944.

418 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#496	C₂	1	2	yes	maximal, complemented
4	C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#463	C₂	1	2	yes	maximal, complemented
6	C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#459	C₂	1	2	yes	maximal, complemented
8	C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#442	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	—	2	1	no	—
13	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×D₈×C₂₃	C₂²	1	2	yes	—
16	C₂×D₈×C₂₃	C₂²	1	2	yes	—
17	C₂×D₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#127	C₂²	1	2	yes	—
19	C₂₃×#127	C₂²	1	1	yes	characteristic
20	C₂₃×#127	C₂²	1	1	yes	characteristic
21	C₂₃×#124	—	2	1	no	—
22	C₂₃×#124	C₂²	1	2	yes	—
23	C₂₃×#123	—	2	2	no	—
24	C₂₃×#121	—	2	1	no	—
25	C₂₃×#121	—	2	2	no	—
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	C₄×D₄×C₂₃	C₂²	1	2	yes	—
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	1	yes	characteristic
36	C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂₃×#102	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	#41×C₂₃	—	4	4	no	—
42	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₂₃	—	2	1	no	—
45	C₂×D₄×C₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂×D₄×C₂₃	C₂³	1	2	yes	—
52	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	SD₁₆×C₂₃	—	2	2	no	—
57	SD₁₆×C₂₃	—	4	2	no	—
58	D₈×C₂₃	—	2	4	no	—
59	D₈×C₂₃	—	4	2	no	—
60	D₈×C₂₃	—	4	2	no	—
61	C₂×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	2	1	no	—
75	C₄²×C₂₃	C₂³	1	2	yes	—
76	C₄²×C₂₃	C₂³	1	1	yes	characteristic
77	C₂³×C₂₃	—	2	2	no	—
78	C₂³×C₂₃	—	2	2	no	—
79	C₂³×C₂₃	—	2	1	no	—
80	Q₈×C₂₃	—	2	2	no	—
81	Q₈×C₂₃	—	2	1	no	—
82	Q₈×C₂₃	—	2	1	no	—
83	D₄×C₂₃	—	2	2	no	—
84	D₄×C₂₃	—	2	2	no	—
85	D₄×C₂₃	—	2	1	no	—
86	D₄×C₂₃	—	2	2	no	—
87	D₄×C₂₃	—	2	1	no	—
88	D₄×C₂₃	—	2	2	no	—
89	D₄×C₂₃	—	4	4	no	—
90	D₄×C₂₃	—	4	2	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	8	2	no	—
93	D₄×C₂₃	—	8	1	no	—
94	D₄×C₂₃	—	8	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	4	4	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	8	2	no	—
101	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₈×C₂₃	—	2	1	no	—
107	C₈×C₂₃	—	2	2	no	—
108	C₈×C₂₃	#41	1	2	yes	complemented
109	#3399	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	1	no	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	2	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	—	4	2	no	—
119	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	4	1	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	1	no	—
126	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
128	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
129	#559	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#550	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#496	C₂×C₂₃	1	2	yes	complemented
132	#493	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#463	C₂×C₂₃	1	2	yes	complemented
134	#460	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#459	C₂×C₂₃	1	2	yes	complemented
136	#459	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#442	C₂×C₂₃	1	2	yes	complemented
139	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₂×C₂₃	—	8	2	no	—
141	C₂×C₂₃	—	8	1	no	—
142	C₂×C₂₃	—	8	2	no	—
143	C₂×C₂₃	#550	1	1	yes	characteristic
144	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₂₃	#576	1	1	yes	characteristic
146	C₂×#41	—	2	1	no	—
147	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
149	C₂×D₈	C₂²×C₂₃	1	2	yes	—
150	C₂×D₈	C₂²×C₂₃	1	2	yes	—
151	C₂×D₈	C₂²×C₂₃	1	2	yes	—
152	#127	C₂²×C₂₃	1	2	yes	—
153	#127	C₂²×C₂₃	1	1	yes	characteristic
154	#127	C₂²×C₂₃	1	1	yes	characteristic
155	#124	—	2	1	no	—
156	#124	C₂²×C₂₃	1	2	yes	—
157	#123	—	2	2	no	—
158	#121	—	2	1	no	—
159	#121	—	2	2	no	—
160	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
164	C₄×D₄	C₂²×C₂₃	1	2	yes	—
165	C₄×D₄	C₂²×C₂₃	1	2	yes	—
166	#106	C₂²×C₂₃	1	1	yes	characteristic
167	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
168	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
169	#103	C₂²×C₂₃	1	1	yes	characteristic
170	#102	C₂²×C₂₃	1	1	yes	characteristic
171	#102	C₂²×C₂₃	1	2	yes	—
172	#102	C₂²×C₂₃	1	2	yes	—
173	#97	C₂²×C₂₃	1	2	yes	—
174	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
175	C₂₃	#3399	1	1	yes	characteristic, direct product factor, Sylow
176	#41	—	4	4	no	—
177	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
178	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	C₂³×C₂₃	1	2	yes	—
186	C₂×D₄	C₂³×C₂₃	1	2	yes	—
187	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	SD₁₆	—	2	2	no	—
192	SD₁₆	—	4	2	no	—
193	D₈	—	2	4	no	—
194	D₈	—	4	2	no	—
195	D₈	—	4	2	no	—
196	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
197	C₂×C₈	C₂³×C₂₃	1	2	yes	—
198	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
201	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
202	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	C₂³×C₂₃	1	1	yes	characteristic
211	C₄²	C₂³×C₂₃	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	1	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	2	1	no	—
217	Q₈	—	2	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	2	1	no	—
221	D₄	—	2	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	8	2	no	—
228	D₄	—	8	1	no	—
229	D₄	—	8	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
237	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
238	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	2	no	—
243	C₈	#41×C₂₃	1	2	yes	complemented
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
261	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
262	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
263	C₂	—	8	2	no	—
264	C₂	—	8	1	no	—
265	C₂	—	8	2	no	—
266	C₂	C₂₃×#550	1	1	yes	characteristic
267	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂	C₂₃×#576	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 ◁ 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×#550	C₂	semidirect
3. C₂₃×#496	C₂	semidirect
4. C₂₃×#493	C₂	semidirect
5. C₂₃×#463	C₂	semidirect
6. C₂₃×#460	C₂	semidirect
7. C₂₃×#459	C₂	semidirect
8. C₂₃×#459	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
10. C₂₃×#442	C₂	semidirect
11. C₄×SD₁₆×C₂₃	C₂	semidirect
108. C₈×C₂₃	#41	semidirect
109. #3399	C₂₃	direct (N × H)
129. #559	C₂×C₂₃	semidirect
130. #550	C₂×C₂₃	semidirect
131. #496	C₂×C₂₃	semidirect
132. #493	C₂×C₂₃	semidirect
133. #463	C₂×C₂₃	semidirect
134. #460	C₂×C₂₃	semidirect
135. #459	C₂×C₂₃	semidirect
136. #459	C₂×C₂₃	semidirect
137. #446	C₂×C₂₃	semidirect
138. #442	C₂×C₂₃	semidirect
139. C₄×SD₁₆	C₂×C₂₃	semidirect
175. C₂₃	#3399	direct (N × H)
243. C₈	#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×#559	2
C₂₃×#550	2
C₂₃×#496	2
C₂₃×#493	2
C₂₃×#463	2
C₂₃×#460	2
C₂₃×#459	2
C₂₃×#446	2
C₂₃×#442	2
C₄×SD₁₆×C₂₃	2
#3399	23

Subgroups — C23×#3399 #15671018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3399 #15671018