Subgroups — C₂₃×#8598 #19156224

Order 5888.

478 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1797	C₂	1	2	yes	maximal
3	C₂×C₂₃×#334	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1277	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#431	—	2	4	no	—
7	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#336	—	2	2	no	—
14	C₂₃×#334	—	2	4	no	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	—	2	4	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₂₃	—	2	2	no	—
22	C₂×C₄²×C₂₃	—	2	2	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	C₂₃×C₄⋊C₈	—	2	4	no	—
28	C₂₃×C₄⋊C₈	—	2	4	no	—
29	C₂₃×C₄⋊C₈	—	2	4	no	—
30	C₂₃×C₂²⋊C₈	—	2	4	no	—
31	C₂₃×C₂²⋊C₈	—	4	4	no	—
32	C₄×C₈×C₂₃	—	2	2	no	—
33	C₄×C₈×C₂₃	D₄	1	4	yes	—
34	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	1	no	—
38	C₂²×C₄×C₂₃	—	2	1	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	4	2	no	—
42	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₂₃	—	2	4	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	—	4	4	no	—
53	C₂×C₈×C₂₃	—	4	4	no	—
54	C₂×C₈×C₂₃	—	4	4	no	—
55	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₂₃	—	2	1	no	—
57	C₄²×C₂₃	—	2	1	no	—
58	C₄²×C₂₃	—	2	2	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	C₂×D₄	1	2	yes	—
61	C₄²×C₂₃	#41	1	2	yes	—
62	#8598	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	—	2	1	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	#106	1	2	yes	—
68	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	8	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	4	4	no	—
78	C₂×C₄×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	#106	1	2	yes	—
81	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₂₃	#121	1	2	yes	—
84	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₂₃	#131	1	4	yes	—
86	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₂₃	#135	1	2	yes	—
88	C₈×C₂₃	—	2	4	no	—
89	C₈×C₂₃	—	4	2	no	—
90	C₈×C₂₃	—	8	4	no	—
91	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
92	#1797	C₂×C₂₃	1	2	yes	—
93	C₂×#334	C₂×C₂₃	1	2	yes	complemented
94	#1277	C₂×C₂₃	1	2	yes	complemented
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	2	4	no	—
97	C₂²×C₂₃	—	4	1	no	—
98	C₂²×C₂₃	—	4	1	no	—
99	C₂²×C₂₃	—	4	2	no	—
100	C₂²×C₂₃	#380	1	1	yes	characteristic
101	C₂²×C₂₃	#424	1	2	yes	—
102	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
103	C₂²×C₂₃	#427	1	1	yes	characteristic
104	C₂²×C₂₃	#430	1	2	yes	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	—	4	1	no	—
109	C₄×C₂₃	#408	1	2	yes	—
110	C₄×C₂₃	#433	1	2	yes	—
111	C₄×C₂₃	#465	1	4	yes	—
112	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
113	#431	—	2	4	no	—
114	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
116	C₂×C₂²⋊C₈	—	2	2	no	—
117	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
120	#336	—	2	2	no	—
121	#334	—	2	4	no	—
122	C₂×C₂₃	—	4	2	no	—
123	C₂×C₂₃	#1570	1	2	yes	—
124	C₂×C₂₃	#1572	1	2	yes	—
125	C₂×C₂₃	#1798	1	1	yes	characteristic
126	C₂×C₂₃	#1807	1	1	yes	characteristic
127	C₂×C₂₃	#1811	1	1	yes	characteristic
128	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
129	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
130	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
131	C₂²×C₈	—	2	4	no	—
132	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
133	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
134	C₂×C₄²	—	2	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
138	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
139	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
140	C₄⋊C₈	—	2	4	no	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	4	4	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	D₄×C₂₃	1	4	yes	—
147	C₂₃	#8598	1	1	yes	characteristic, direct product factor, Sylow
148	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
160	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×D₄×C₂₃	1	2	yes	—
175	C₄²	#41×C₂₃	1	2	yes	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	2	no	—
180	C₂³	C₂₃×#106	1	2	yes	—
181	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂₃×#106	1	2	yes	—
194	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
196	C₂×C₄	C₂₃×#121	1	2	yes	—
197	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
198	C₂×C₄	C₂₃×#131	1	4	yes	—
199	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
200	C₂×C₄	C₂₃×#135	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	4	2	no	—
203	C₈	—	8	4	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	2	no	—
209	C₂²	C₂₃×#380	1	1	yes	characteristic
210	C₂²	C₂₃×#424	1	2	yes	—
211	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
212	C₂²	C₂₃×#427	1	1	yes	characteristic
213	C₂²	C₂₃×#430	1	2	yes	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	C₂₃×#408	1	2	yes	—
219	C₄	C₂₃×#433	1	2	yes	—
220	C₄	C₂₃×#465	1	4	yes	—
221	C₂	—	4	2	no	—
222	C₂	C₂₃×#1570	1	2	yes	—
223	C₂	C₂₃×#1572	1	2	yes	—
224	C₂	C₂₃×#1798	1	1	yes	characteristic
225	C₂	C₂₃×#1807	1	1	yes	characteristic
226	C₂	C₂₃×#1811	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#334	C₂	semidirect
4. C₂₃×#1277	C₂	semidirect
62. #8598	C₂₃	direct (N × H)
93. C₂×#334	C₂×C₂₃	semidirect
94. #1277	C₂×C₂₃	semidirect
147. C₂₃	#8598	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#431	2
C₂₃×#1797	2
C₂×C₂₃×#334	2
C₂₃×#1277	2
#8598	23

Subgroups — C23×#8598 #19156224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8598 #19156224