Subgroups — C₂₃×#2722 #15670341

Order 2944.

448 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#126	C₂	1	2	yes	maximal
2	C₂₃×#398	C₂	1	2	yes	maximal, complemented
3	C₂₃×#398	C₂	1	2	yes	maximal, complemented
4	C₂₃×#397	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#397	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#389	C₂	1	2	yes	maximal, complemented
7	C₂₃×#387	C₂	1	2	yes	maximal, complemented
8	C₂₃×#385	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#380	C₂	1	2	yes	maximal
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#126	—	2	8	no	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	1	yes	characteristic
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	1	yes	characteristic
32	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	—	2	4	no	—
34	C₂²×C₄×C₂₃	—	4	1	no	—
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₄⋊C₄×C₂₃	—	2	4	no	—
46	C₄⋊C₄×C₂₃	—	2	4	no	—
47	C₄⋊C₄×C₂₃	—	2	4	no	—
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₄⋊C₄×C₂₃	—	4	1	no	—
50	C₄⋊C₄×C₂₃	—	4	1	no	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	—	2	4	no	—
53	C₂²⋊C₄×C₂₃	—	4	1	no	—
54	C₂²⋊C₄×C₂₃	D₄	1	4	yes	—
55	C₄²×C₂₃	—	2	4	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	—	2	1	no	—
58	C₂³×C₂₃	—	2	1	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	C₂×D₄	1	2	yes	—
61	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	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	—	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	—	2	2	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
84	C₂×C₄×C₂₃	#41	1	4	yes	—
85	#2722	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₂₃	—	2	2	no	—
87	C₂²×C₂₃	—	2	4	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	—	4	1	no	—
90	C₂²×C₂₃	—	4	1	no	—
91	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×#41	1	2	yes	—
93	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
95	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	4	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	—	4	2	no	—
100	C₄×C₂₃	—	4	2	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	4	1	no	—
103	C₄×C₂₃	—	4	1	no	—
104	C₂×#126	C₂×C₂₃	1	2	yes	—
105	#398	C₂×C₂₃	1	2	yes	complemented
106	#398	C₂×C₂₃	1	2	yes	complemented
107	#397	C₂×C₂₃	1	1	yes	characteristic
108	#397	C₂×C₂₃	1	1	yes	characteristic
109	#389	C₂×C₂₃	1	2	yes	complemented
110	#387	C₂×C₂₃	1	2	yes	complemented
111	#385	C₂×C₂₃	1	1	yes	characteristic
112	#380	C₂×C₂₃	1	2	yes	—
113	C₂×C₂₃	—	4	2	no	—
114	C₂×C₂₃	#537	1	1	yes	characteristic
115	C₂×C₂₃	#541	1	2	yes	—
116	C₂×C₂₃	#547	1	2	yes	—
117	C₂×C₂₃	#551	1	1	yes	characteristic
118	C₂×C₂₃	#563	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
120	#126	—	2	8	no	—
121	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
134	#95	C₂²×C₂₃	1	2	yes	—
135	#95	C₂²×C₂₃	1	2	yes	—
136	#95	C₂²×C₂₃	1	2	yes	—
137	#95	C₂²×C₂₃	1	1	yes	characteristic
138	#95	C₂²×C₂₃	1	2	yes	—
139	#95	C₂²×C₂₃	1	2	yes	—
140	#95	C₂²×C₂₃	1	1	yes	characteristic
141	C₂₃	#2722	1	1	yes	characteristic, direct product factor, Sylow
142	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	4	1	no	—
145	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
151	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₄⋊C₄	—	4	1	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₂²⋊C₄	D₄×C₂₃	1	4	yes	—
165	C₄²	—	2	4	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	1	no	—
170	C₂³	C₂×D₄×C₂₃	1	2	yes	—
171	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
194	C₂×C₄	#41×C₂₃	1	4	yes	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	4	1	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	2	no	—
200	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
201	C₂²	C₂×#41×C₂₃	1	2	yes	—
202	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
203	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
204	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	1	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂₃×#537	1	1	yes	characteristic
215	C₂	C₂₃×#541	1	2	yes	—
216	C₂	C₂₃×#547	1	2	yes	—
217	C₂	C₂₃×#551	1	1	yes	characteristic
218	C₂	C₂₃×#563	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#398	C₂	semidirect
3. C₂₃×#398	C₂	semidirect
6. C₂₃×#389	C₂	semidirect
7. C₂₃×#387	C₂	semidirect
85. #2722	C₂₃	direct (N × H)
105. #398	C₂×C₂₃	semidirect
106. #398	C₂×C₂₃	semidirect
109. #389	C₂×C₂₃	semidirect
110. #387	C₂×C₂₃	semidirect
141. C₂₃	#2722	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×#126	2
C₂₃×#398	2
C₂₃×#397	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#385	2
C₂₃×#380	2
#2722	23

Subgroups — C23×#2722 #15670341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2722 #15670341