Subgroups — C₂₃×#3397 #15671016

Order 2944.

406 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#556	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×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#463	C₂	1	2	yes	maximal, complemented
6	C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#460	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×#440	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	—	2	1	no	—
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂₃×#127	C₂²	1	1	yes	characteristic
20	C₂₃×#127	C₂²	1	2	yes	—
21	C₂₃×#126	—	2	2	no	—
22	C₂₃×#124	C₂²	1	2	yes	—
23	C₂₃×C₂²⋊Q₈	—	2	1	no	—
24	C₂₃×#121	—	2	1	no	—
25	C₂₃×#121	—	2	2	no	—
26	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₂₃	C₂²	1	2	yes	—
30	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#117	—	2	1	no	—
32	C₂₃×#107	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×#102	C₂²	1	1	yes	characteristic
37	C₂₃×#102	C₂²	1	1	yes	characteristic
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	2	yes	—
44	C₂×D₄×C₂₃	—	2	2	no	—
45	C₂×D₄×C₂₃	—	2	1	no	—
46	C₂×D₄×C₂₃	—	2	2	no	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	SD₁₆×C₂₃	—	2	4	no	—
55	SD₁₆×C₂₃	—	4	2	no	—
56	SD₁₆×C₂₃	—	4	2	no	—
57	D₈×C₂₃	—	2	2	no	—
58	D₈×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂³	1	2	yes	—
61	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
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	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₄²×C₂₃	C₂³	1	2	yes	—
77	C₂³×C₂₃	—	2	1	no	—
78	C₂³×C₂₃	—	2	1	no	—
79	C₂³×C₂₃	—	2	2	no	—
80	Q₈×C₂₃	—	2	2	no	—
81	Q₈×C₂₃	—	2	2	no	—
82	Q₈×C₂₃	—	2	2	no	—
83	D₄×C₂₃	—	2	1	no	—
84	D₄×C₂₃	—	2	1	no	—
85	D₄×C₂₃	—	2	2	no	—
86	D₄×C₂₃	—	2	1	no	—
87	D₄×C₂₃	—	2	2	no	—
88	D₄×C₂₃	—	2	1	no	—
89	D₄×C₂₃	—	4	4	no	—
90	D₄×C₂₃	—	4	2	no	—
91	D₄×C₂₃	—	8	2	no	—
92	D₄×C₂₃	—	8	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₈×C₂₃	—	2	2	no	—
105	C₈×C₂₃	—	2	1	no	—
106	C₈×C₂₃	#41	1	2	yes	complemented
107	#3397	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₂₃	—	4	1	no	—
109	C₂²×C₂₃	—	4	1	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	1	no	—
113	C₂²×C₂₃	—	4	1	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	2	no	—
116	C₂²×C₂₃	—	4	1	no	—
117	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₂₃	—	2	2	no	—
119	C₄×C₂₃	—	2	2	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	4	2	no	—
122	C₄×C₂₃	—	4	1	no	—
123	C₄×C₂₃	—	4	1	no	—
124	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
125	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
126	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
127	#556	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#550	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#496	C₂×C₂₃	1	2	yes	complemented
130	#492	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#463	C₂×C₂₃	1	2	yes	complemented
132	#461	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#460	C₂×C₂₃	1	2	yes	complemented
134	#459	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#440	C₂×C₂₃	1	2	yes	complemented
137	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₂×C₂₃	—	8	1	no	—
139	C₂×C₂₃	—	8	2	no	—
140	C₂×C₂₃	—	8	1	no	—
141	C₂×C₂₃	#550	1	1	yes	characteristic
142	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₂₃	#577	1	1	yes	characteristic
144	C₂×#41	—	2	1	no	—
145	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
146	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
147	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
148	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×D₈	C₂²×C₂₃	1	2	yes	—
150	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
151	#127	C₂²×C₂₃	1	1	yes	characteristic
152	#127	C₂²×C₂₃	1	2	yes	—
153	#126	—	2	2	no	—
154	#124	C₂²×C₂₃	1	2	yes	—
155	C₂²⋊Q₈	—	2	1	no	—
156	#121	—	2	1	no	—
157	#121	—	2	2	no	—
158	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
159	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
161	C₄×D₄	C₂²×C₂₃	1	2	yes	—
162	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
163	#117	—	2	1	no	—
164	#107	C₂²×C₂₃	1	1	yes	characteristic
165	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
167	#103	C₂²×C₂₃	1	2	yes	—
168	#102	C₂²×C₂₃	1	1	yes	characteristic
169	#102	C₂²×C₂₃	1	1	yes	characteristic
170	#102	C₂²×C₂₃	1	2	yes	—
171	#97	C₂²×C₂₃	1	2	yes	—
172	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂₃	#3397	1	1	yes	characteristic, direct product factor, Sylow
174	#41	—	4	4	no	—
175	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
176	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
183	C₂×D₄	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	SD₁₆	—	2	4	no	—
188	SD₁₆	—	4	2	no	—
189	SD₁₆	—	4	2	no	—
190	D₈	—	2	2	no	—
191	D₈	—	4	2	no	—
192	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₂₃	1	2	yes	—
194	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
199	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
200	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	C₂³×C₂₃	1	1	yes	characteristic
209	C₄²	C₂³×C₂₃	1	2	yes	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	2	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	D₄	—	2	1	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	4	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
233	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	#41×C₂₃	1	2	yes	complemented
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
257	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
258	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂	—	8	1	no	—
260	C₂	—	8	2	no	—
261	C₂	—	8	1	no	—
262	C₂	C₂₃×#550	1	1	yes	characteristic
263	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
264	C₂	C₂₃×#577	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×#556	C₂	semidirect
2. C₂₃×#550	C₂	semidirect
3. C₂₃×#496	C₂	semidirect
4. C₂₃×#492	C₂	semidirect
5. C₂₃×#463	C₂	semidirect
6. C₂₃×#461	C₂	semidirect
7. C₂₃×#460	C₂	semidirect
8. C₂₃×#459	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
10. C₂₃×#440	C₂	semidirect
11. C₄×D₈×C₂₃	C₂	semidirect
106. C₈×C₂₃	#41	semidirect
107. #3397	C₂₃	direct (N × H)
127. #556	C₂×C₂₃	semidirect
128. #550	C₂×C₂₃	semidirect
129. #496	C₂×C₂₃	semidirect
130. #492	C₂×C₂₃	semidirect
131. #463	C₂×C₂₃	semidirect
132. #461	C₂×C₂₃	semidirect
133. #460	C₂×C₂₃	semidirect
134. #459	C₂×C₂₃	semidirect
135. #446	C₂×C₂₃	semidirect
136. #440	C₂×C₂₃	semidirect
137. C₄×D₈	C₂×C₂₃	semidirect
173. C₂₃	#3397	direct (N × H)
239. 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×#556	2
C₂₃×#550	2
C₂₃×#496	2
C₂₃×#492	2
C₂₃×#463	2
C₂₃×#461	2
C₂₃×#460	2
C₂₃×#459	2
C₂₃×#446	2
C₂₃×#440	2
C₄×D₈×C₂₃	2
#3397	23

Subgroups — C23×#3397 #15671016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3397 #15671016