Subgroups — C₂₃×#8178 #19155804

Order 5888.

486 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2295	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1767	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	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₂₃×#97	—	2	1	no	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂₃×#391	C₄	1	2	yes	—
11	C₂₃×#389	C₄	1	2	yes	—
12	C₂₃×#387	—	2	2	no	—
13	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
14	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#376	—	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	2	no	—
24	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	—	2	1	no	—
30	C₂×C₄²×C₂₃	—	2	1	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	—	2	2	no	—
37	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
38	C₂₃×#97	—	4	2	no	—
39	C₄×C₈×C₂₃	—	2	2	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
41	C₂₃×#95	—	2	2	no	—
42	C₂₃×#95	—	2	2	no	—
43	C₂₃×#95	—	2	2	no	—
44	C₂₃×#95	—	2	1	no	—
45	C₂×Q₈×C₂₃	—	2	1	no	—
46	C₂×Q₈×C₂₃	—	2	4	no	—
47	C₂×Q₈×C₂₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	—	4	4	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	—	4	2	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	4	no	—
63	C₄⋊C₄×C₂₃	—	2	4	no	—
64	C₄⋊C₄×C₂₃	—	4	4	no	—
65	C₄⋊C₄×C₂₃	C₄²	1	2	yes	—
66	C₄²×C₂₃	—	2	4	no	—
67	C₄²×C₂₃	—	2	4	no	—
68	C₄²×C₂₃	—	2	1	no	—
69	C₄²×C₂₃	—	2	1	no	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	—	4	1	no	—
73	C₄²×C₂₃	—	4	1	no	—
74	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₂₃	#41	1	2	yes	—
76	#8178	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₂₃	—	2	4	no	—
79	Q₈×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₂₃	#131	1	4	yes	—
99	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#137	1	2	yes	—
101	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
102	C₈×C₂₃	—	4	4	no	—
103	C₈×C₂₃	—	4	4	no	—
104	#2295	C₂×C₂₃	1	1	yes	characteristic
105	#1767	C₂×C₂₃	1	1	yes	characteristic
106	#1744	C₂×C₂₃	1	1	yes	characteristic
107	C₂×Q₈⋊C₈	C₂×C₂₃	1	2	yes	—
108	#1279	C₂×C₂₃	1	2	yes	—
109	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₂₃	#408	1	2	yes	—
111	C₂²×C₂₃	#417	1	1	yes	characteristic
112	C₂²×C₂₃	#419	1	1	yes	characteristic
113	C₂²×C₂₃	#421	1	2	yes	—
114	C₄×C₂₃	—	2	2	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	4	2	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	8	2	no	—
120	C₄×C₂₃	#405	1	2	yes	—
121	C₄×C₂₃	#433	1	2	yes	—
122	C₄×C₂₃	#440	1	2	yes	—
123	C₄×C₂₃	#441	1	2	yes	—
124	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
126	C₂×#97	—	2	1	no	—
127	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
128	#391	C₄×C₂₃	1	2	yes	—
129	#389	C₄×C₂₃	1	2	yes	—
130	#387	—	2	2	no	—
131	C₄×C₄⋊C₄	—	2	1	no	—
132	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
133	#376	—	2	1	no	—
134	C₄³	C₂²×C₂₃	1	1	yes	characteristic
135	#336	—	2	2	no	—
136	#336	—	2	1	no	—
137	Q₈⋊C₈	C₄×C₂₃	1	8	yes	complemented
138	C₂×C₂₃	#1495	1	2	yes	—
139	C₂×C₂₃	#1497	1	2	yes	—
140	C₂×C₂₃	#1750	1	1	yes	characteristic
141	C₂×C₂₃	#1759	1	1	yes	characteristic
142	C₂×C₂₃	#1763	1	1	yes	characteristic
143	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
155	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
156	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	D₄×C₂₃	1	2	yes	—
158	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
161	#97	—	4	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
164	#95	—	2	2	no	—
165	#95	—	2	2	no	—
166	#95	—	2	2	no	—
167	#95	—	2	1	no	—
168	C₂₃	#8178	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	C₄²×C₂₃	1	2	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	C₄²×C₂₃	1	2	yes	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
199	C₄²	#41×C₂₃	1	2	yes	—
200	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
219	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
220	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
221	C₂×C₄	C₂₃×#131	1	4	yes	—
222	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂×C₄	C₂₃×#137	1	2	yes	—
224	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
225	C₈	—	4	4	no	—
226	C₈	—	4	4	no	—
227	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂²	C₂₃×#408	1	2	yes	—
229	C₂²	C₂₃×#417	1	1	yes	characteristic
230	C₂²	C₂₃×#419	1	1	yes	characteristic
231	C₂²	C₂₃×#421	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	8	2	no	—
238	C₄	C₂₃×#405	1	2	yes	—
239	C₄	C₂₃×#433	1	2	yes	—
240	C₄	C₂₃×#440	1	2	yes	—
241	C₄	C₂₃×#441	1	2	yes	—
242	C₂	C₂₃×#1495	1	2	yes	—
243	C₂	C₂₃×#1497	1	2	yes	—
244	C₂	C₂₃×#1750	1	1	yes	characteristic
245	C₂	C₂₃×#1759	1	1	yes	characteristic
246	C₂	C₂₃×#1763	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂×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
76. #8178	C₂₃	direct (N × H)
137. Q₈⋊C₈	C₄×C₂₃	semidirect
168. C₂₃	#8178	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×#2295	2
C₂₃×#1767	2
C₂₃×#1744	2
C₂×C₂₃×Q₈⋊C₈	2
C₂₃×#1279	2
#8178	23

Subgroups — C23×#8178 #19155804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8178 #19155804