Subgroups — C₂₃×#8122 #19155748

Order 5888.

520 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2920	C₂	1	2	yes	maximal
2	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1307	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#432	—	2	2	no	—
6	C₂₃×#432	—	2	2	no	—
7	C₂₃×#432	C₂²	1	4	yes	—
8	C₂₃×#431	—	2	2	no	—
9	C₂₃×#431	—	2	2	no	—
10	C₂₃×#431	C₄	1	4	yes	—
11	C₂₃×#423	—	2	2	no	—
12	C₂₃×#423	C₂²	1	2	yes	—
13	C₂₃×#407	C₄	1	4	yes	—
14	C₂₃×#407	C₄	1	2	yes	—
15	C₄×M₁₆×C₂₃	—	2	2	no	—
16	C₄×M₁₆×C₂₃	C₄	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	—	2	4	no	—
20	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₂₃	D₄	1	2	yes	—
24	C₂×C₄²×C₂₃	D₄	1	2	yes	—
25	C₂×C₄²×C₂₃	D₄	1	2	yes	—
26	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	—	2	4	no	—
29	C₂₃×C₄⋊C₈	—	2	4	no	—
30	C₂₃×C₄⋊C₈	—	2	4	no	—
31	C₂₃×C₂²⋊C₈	—	2	4	no	—
32	C₂₃×C₂²⋊C₈	—	2	4	no	—
33	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
34	C₂₃×#97	—	2	2	no	—
35	C₂₃×#97	—	2	4	no	—
36	C₄×C₈×C₂₃	—	2	4	no	—
37	C₄×C₈×C₂₃	—	2	2	no	—
38	C₂⁴×C₂₃	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	—	4	4	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
51	M₁₆×C₂₃	—	4	8	no	—
52	M₁₆×C₂₃	—	4	4	no	—
53	C₂×C₈×C₂₃	—	2	8	no	—
54	C₂×C₈×C₂₃	—	2	4	no	—
55	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
60	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
61	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
62	C₄²×C₂₃	C₂×D₄	1	2	yes	—
63	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
65	#8122	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	4	1	no	—
67	C₂³×C₂₃	—	4	2	no	—
68	C₂³×C₂₃	#95	1	2	yes	—
69	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	4	1	no	—
76	C₂×C₄×C₂₃	—	4	2	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	#95	1	2	yes	—
81	C₂×C₄×C₂₃	#95	1	2	yes	—
82	C₂×C₄×C₂₃	#95	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₈×C₂₃	—	4	8	no	—
88	C₈×C₂₃	—	4	4	no	—
89	#2920	C₂×C₂₃	1	2	yes	—
90	#2920	C₂×C₂₃	1	1	yes	characteristic
91	#1307	C₂×C₂₃	1	4	yes	complemented
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	2	1	no	—
94	C₂²×C₂₃	—	2	2	no	—
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	—	4	4	no	—
98	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	—	2	2	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	#344	1	4	yes	—
105	C₄×C₂₃	#413	1	2	yes	—
106	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
107	#432	—	2	2	no	—
108	#432	—	2	2	no	—
109	#432	C₂²×C₂₃	1	4	yes	—
110	#431	—	2	2	no	—
111	#431	—	2	2	no	—
112	#431	C₄×C₂₃	1	4	yes	—
113	#423	—	2	2	no	—
114	#423	C₂²×C₂₃	1	2	yes	—
115	#407	C₄×C₂₃	1	2	yes	—
116	#407	C₄×C₂₃	1	4	yes	—
117	C₄×M₁₆	—	2	2	no	—
118	C₄×M₁₆	C₄×C₂₃	1	2	yes	—
119	C₂×C₂₃	—	4	2	no	—
120	C₂×C₂₃	—	4	1	no	—
121	C₂×C₂₃	#1742	1	1	yes	characteristic
122	C₂×C₂₃	C₂×#344	1	2	yes	—
123	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
124	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
127	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
129	C₂×C₄²	D₄×C₂₃	1	2	yes	—
130	C₂×C₄²	D₄×C₂₃	1	2	yes	—
131	C₂×C₄²	D₄×C₂₃	1	2	yes	—
132	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
133	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
134	C₄⋊C₈	—	2	4	no	—
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	2	4	no	—
137	C₂²⋊C₈	—	2	4	no	—
138	C₂²⋊C₈	—	2	4	no	—
139	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
140	#97	—	2	2	no	—
141	#97	—	2	4	no	—
142	C₄×C₈	—	2	4	no	—
143	C₄×C₈	—	2	2	no	—
144	C₂₃	#8122	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁴	C₄²×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	—	4	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
149	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
151	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
152	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
153	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
154	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
155	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
158	M₁₆	—	4	8	no	—
159	M₁₆	—	4	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	8	no	—
162	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
163	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
164	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
165	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
166	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
167	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
168	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
169	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
170	C₄²	C₂×D₄×C₂₃	1	2	yes	—
171	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
172	C₂³	—	4	1	no	—
173	C₂³	—	4	2	no	—
174	C₂³	C₂₃×#95	1	2	yes	—
175	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	C₂₃×#95	1	2	yes	—
187	C₂×C₄	C₂₃×#95	1	2	yes	—
188	C₂×C₄	C₂₃×#95	1	2	yes	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
193	C₈	—	4	8	no	—
194	C₈	—	4	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	C₂₃×#344	1	4	yes	—
208	C₄	C₂₃×#413	1	2	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	4	2	no	—
211	C₂	C₂₃×#1742	1	1	yes	characteristic
212	C₂	C₂×C₂₃×#344	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#95
commutator	C₂³	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1307	C₂	semidirect
65. #8122	C₂₃	direct (N × H)
91. #1307	C₂×C₂₃	semidirect
144. C₂₃	#8122	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×#2920	2
C₂₃×#1307	2
#8122	23

Subgroups — C23×#8122 #19155748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8122 #19155748