Subgroups — C₂₃×#8838 #19156464

Order 5888.

386 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	C₂₃×#2983	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1984	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#101	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1565	C₂	1	2	yes	maximal
5	C₂₃×#1278	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#446	—	2	2	no	—
8	C₂₃×#446	—	2	2	no	—
9	C₂₃×#432	—	2	2	no	—
10	C₂₃×#431	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#101	C₄	1	2	yes	—
13	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#343	C₄	1	1	yes	characteristic
16	C₂₃×#343	C₄	1	1	yes	characteristic
17	C₂₃×#336	—	2	2	no	—
18	C₂₃×#335	—	2	2	no	—
19	C₂₃×#334	—	2	2	no	—
20	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	—	2	1	no	—
24	C₂²×C₈×C₂₃	—	2	1	no	—
25	C₄×Q₈×C₂₃	—	2	2	no	—
26	C₄×Q₈×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₂₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	—	2	2	no	—
32	C₂₃×C₄⋊C₈	—	2	2	no	—
33	C₂₃×C₄⋊C₈	—	2	2	no	—
34	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
35	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
36	C₂₃×#101	—	2	4	no	—
37	C₂₃×C₂²⋊C₈	—	2	2	no	—
38	C₂₃×#97	—	2	4	no	—
39	C₂₃×#97	—	2	2	no	—
40	C₄×C₈×C₂₃	—	2	2	no	—
41	C₄×C₈×C₂₃	—	2	2	no	—
42	C₄×C₈×C₂₃	—	2	2	no	—
43	C₂×Q₈×C₂₃	—	2	2	no	—
44	C₂×Q₈×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	—	2	1	no	—
46	C₂×Q₈×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	M₁₆×C₂₃	—	2	2	no	—
53	M₁₆×C₂₃	—	2	2	no	—
54	M₁₆×C₂₃	—	2	2	no	—
55	M₁₆×C₂₃	M₁₆	1	4	yes	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	4	1	no	—
62	C₂×C₈×C₂₃	—	4	1	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	1	no	—
65	C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₄²×C₂₃	—	2	2	no	—
69	C₄²×C₂₃	C₂×D₄	1	2	yes	—
70	C₄²×C₂₃	#41	1	2	yes	—
71	#8838	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	Q₈×C₂₃	—	4	2	no	—
74	Q₈×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	4	1	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
90	C₈×C₂₃	—	2	4	no	—
91	C₈×C₂₃	—	4	2	no	—
92	C₈×C₂₃	—	4	2	no	—
93	C₈×C₂₃	—	4	2	no	—
94	C₈×C₂₃	—	4	2	no	—
95	C₈×C₂₃	—	8	1	no	—
96	#2983	C₂×C₂₃	1	1	yes	characteristic
97	#1984	C₂×C₂₃	1	1	yes	characteristic
98	C₄×#101	C₂×C₂₃	1	1	yes	characteristic
99	#1565	C₂×C₂₃	1	2	yes	—
100	#1278	C₂×C₂₃	1	2	yes	—
101	C₂²×C₂₃	—	2	1	no	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	#385	1	1	yes	characteristic
104	C₂²×C₂₃	#423	1	1	yes	characteristic
105	C₂²×C₂₃	#424	1	1	yes	characteristic
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	#435	1	2	yes	—
111	C₄×C₂₃	#436	1	2	yes	—
112	C₄×C₂₃	#454	1	1	yes	characteristic
113	C₄×C₂₃	#456	1	1	yes	characteristic
114	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
115	#446	—	2	2	no	—
116	#446	—	2	2	no	—
117	#432	—	2	2	no	—
118	#431	C₂²×C₂₃	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×#101	C₄×C₂₃	1	2	yes	—
121	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
123	#343	C₄×C₂₃	1	1	yes	characteristic
124	#343	C₄×C₂₃	1	1	yes	characteristic
125	#336	—	2	2	no	—
126	#335	—	2	2	no	—
127	#334	—	2	2	no	—
128	C₂×C₂₃	—	2	1	no	—
129	C₂×C₂₃	—	2	1	no	—
130	C₂×C₂₃	#1851	1	1	yes	characteristic
131	C₂×C₂₃	#1857	1	1	yes	characteristic
132	C₂×C₂₃	#1858	1	1	yes	characteristic
133	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
134	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
135	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	Q₈×C₂₃	1	2	yes	—
142	C₂×C₄²	D₄×C₂₃	1	2	yes	—
143	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
148	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
149	#101	—	2	4	no	—
150	C₂²⋊C₈	—	2	2	no	—
151	#97	—	2	4	no	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂₃	#8838	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
162	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
166	M₁₆	—	2	2	no	—
167	M₁₆	—	2	2	no	—
168	M₁₆	—	2	2	no	—
169	M₁₆	M₁₆×C₂₃	1	4	yes	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂×D₄×C₂₃	1	2	yes	—
184	C₄²	#41×C₂₃	1	2	yes	—
185	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
197	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
198	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
199	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
200	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
201	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
202	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₈	—	4	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	4	2	no	—
207	C₈	—	4	2	no	—
208	C₈	—	8	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂₃×#385	1	1	yes	characteristic
212	C₂²	C₂₃×#423	1	1	yes	characteristic
213	C₂²	C₂₃×#424	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	C₂₃×#435	1	2	yes	—
219	C₄	C₂₃×#436	1	2	yes	—
220	C₄	C₂₃×#454	1	1	yes	characteristic
221	C₄	C₂₃×#456	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	C₂₃×#1851	1	1	yes	characteristic
225	C₂	C₂₃×#1857	1	1	yes	characteristic
226	C₂	C₂₃×#1858	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#385
prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
71. #8838	C₂₃	direct (N × H)
156. C₂₃	#8838	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×#2983	2
C₂₃×#1984	2
C₄×C₂₃×#101	2
C₂₃×#1565	2
C₂₃×#1278	2
#8838	23

Subgroups — C23×#8838 #19156464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8838 #19156464