Subgroups — C₂₃×#8171 #19155797

Order 5888.

598 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₂₃	C₄	1	2	yes	—
7	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	—	2	1	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
12	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
13	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₄³×C₂₃	—	2	1	no	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	2	no	—
18	C₂₃×#336	—	2	1	no	—
19	C₂₃×Q₈⋊C₈	C₄	1	8	yes	complemented
20	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₄×Q₈×C₂₃	—	2	4	no	—
24	C₄×Q₈×C₂₃	—	2	2	no	—
25	C₄×Q₈×C₂₃	C₈	1	8	yes	complemented
26	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₂₃	—	2	2	no	—
32	C₂×C₄²×C₂₃	—	2	1	no	—
33	C₂×C₄²×C₂₃	—	2	2	no	—
34	C₂×C₄²×C₂₃	—	2	1	no	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₂₃	—	2	2	no	—
43	C₄×C₈×C₂₃	—	4	2	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
45	C₂₃×#95	—	2	2	no	—
46	C₂₃×#95	—	2	1	no	—
47	C₂×Q₈×C₂₃	—	2	1	no	—
48	C₂×Q₈×C₂₃	—	2	2	no	—
49	C₂×Q₈×C₂₃	C₄²	1	2	yes	—
50	C₂×Q₈×C₂₃	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	—	2	4	no	—
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	—	4	4	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₄⋊C₄×C₂₃	—	2	8	no	—
65	C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	2	2	no	—
68	C₄⋊C₄×C₂₃	C₄²	1	2	yes	—
69	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
70	C₄²×C₂₃	—	2	8	no	—
71	C₄²×C₂₃	—	2	4	no	—
72	C₄²×C₂₃	—	2	1	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	—	4	2	no	—
76	C₄²×C₂₃	—	4	4	no	—
77	C₄²×C₂₃	—	4	1	no	—
78	C₄²×C₂₃	—	4	1	no	—
79	C₄²×C₂₃	C₂×C₈	1	4	yes	—
80	C₄²×C₂₃	M₁₆	1	4	yes	—
81	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₂₃	#41	1	2	yes	—
83	#8171	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₂₃	—	2	2	no	—
86	Q₈×C₂₃	—	2	4	no	—
87	Q₈×C₂₃	C₄×C₈	1	4	yes	complemented
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	4	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
101	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
102	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#137	1	2	yes	—
110	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
111	C₈×C₂₃	—	4	4	no	—
112	C₈×C₂₃	—	4	4	no	—
113	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic
114	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
115	#1744	C₂×C₂₃	1	1	yes	characteristic
116	C₂×Q₈⋊C₈	C₂×C₂₃	1	2	yes	—
117	#1279	C₂×C₂₃	1	2	yes	—
118	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
120	C₂²×C₂₃	#417	1	1	yes	characteristic
121	C₂²×C₂₃	#419	1	1	yes	characteristic
122	C₂²×C₂₃	#421	1	2	yes	—
123	C₄×C₂₃	—	2	4	no	—
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	2	4	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	4	4	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
130	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
131	C₄×C₂₃	#431	1	2	yes	—
132	C₄×C₂₃	#440	1	2	yes	—
133	C₄×C₂₃	#441	1	2	yes	—
134	C₂×C₄×Q₈	C₄×C₂₃	1	2	yes	—
135	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
137	C₂×C₄×C₈	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
139	C₄×C₄⋊C₄	—	2	1	no	—
140	C₄×C₄⋊C₄	—	2	2	no	—
141	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
142	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
143	C₄³	—	2	1	no	—
144	C₄³	C₂²×C₂₃	1	1	yes	characteristic
145	#336	—	2	2	no	—
146	#336	—	2	1	no	—
147	Q₈⋊C₈	C₄×C₂₃	1	8	yes	complemented
148	C₂×C₂₃	#1477	1	2	yes	—
149	C₂×C₂₃	#1479	1	2	yes	—
150	C₂×C₂₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
151	C₂×C₂₃	#1759	1	1	yes	characteristic
152	C₂×C₂₃	#1763	1	1	yes	characteristic
153	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₄×Q₈	—	2	4	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	C₈×C₂₃	1	8	yes	complemented
159	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
169	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₂₃	1	2	yes	—
172	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	4	2	no	—
177	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
178	#95	—	2	2	no	—
179	#95	—	2	1	no	—
180	C₂₃	#8171	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	C₄²×C₂₃	1	2	yes	—
184	C₂×Q₈	C₂×C₈×C₂₃	1	4	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
189	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	C₄²×C₂₃	1	2	yes	—
203	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	8	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	4	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	C₂×C₈×C₂₃	1	4	yes	—
214	C₄²	M₁₆×C₂₃	1	4	yes	—
215	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
216	C₄²	#41×C₂₃	1	2	yes	—
217	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	—	2	4	no	—
220	Q₈	C₄×C₈×C₂₃	1	4	yes	complemented
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
234	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
235	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
238	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
239	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
240	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
241	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂×C₄	C₂₃×#137	1	2	yes	—
243	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₈	—	4	4	no	—
246	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
247	C₂²	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
248	C₂²	C₂₃×#417	1	1	yes	characteristic
249	C₂²	C₂₃×#419	1	1	yes	characteristic
250	C₂²	C₂₃×#421	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	4	no	—
256	C₄	—	4	2	no	—
257	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
258	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
259	C₄	C₂₃×#431	1	2	yes	—
260	C₄	C₂₃×#440	1	2	yes	—
261	C₄	C₂₃×#441	1	2	yes	—
262	C₂	C₂₃×#1477	1	2	yes	—
263	C₂	C₂₃×#1479	1	2	yes	—
264	C₂	C₄×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
265	C₂	C₂₃×#1759	1	1	yes	characteristic
266	C₂	C₂₃×#1763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
19. C₂₃×Q₈⋊C₈	C₄	semidirect
25. C₄×Q₈×C₂₃	C₈	semidirect
83. #8171	C₂₃	direct (N × H)
87. Q₈×C₂₃	C₄×C₈	semidirect
147. Q₈⋊C₈	C₄×C₂₃	semidirect
158. C₄×Q₈	C₈×C₂₃	semidirect
180. C₂₃	#8171	direct (N × H)
220. Q₈	C₄×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Q₈×C₂₃	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1744	2
C₂×C₂₃×Q₈⋊C₈	2
C₂₃×#1279	2
#8171	23

Subgroups — C23×#8171 #19155797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8171 #19155797