Subgroups — C₂₃×#2624 #15670243

Order 2944.

548 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×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#399	C₂	1	2	yes	maximal, complemented
4	C₂₃×#397	C₂	1	2	yes	maximal, complemented
5	C₂₃×#394	C₂	1	4	yes	maximal, complemented
6	C₂₃×#390	C₂	1	2	yes	maximal, complemented
7	C₂₃×#389	C₂	1	2	yes	maximal, complemented
8	C₂₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#121	—	2	4	no	—
13	C₂₃×#117	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
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	4	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₂₃	C₂³	1	2	yes	—
28	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₂₃	—	2	4	no	—
30	C₂×D₄×C₂₃	—	2	2	no	—
31	C₂×D₄×C₂₃	—	2	4	no	—
32	C₂×D₄×C₂₃	—	4	4	no	—
33	C₂²×C₄×C₂₃	—	2	4	no	—
34	C₂²×C₄×C₂₃	D₄	1	4	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	4	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	C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₄⋊C₄×C₂₃	—	4	4	no	—
46	C₂²⋊C₄×C₂₃	—	2	4	no	—
47	C₂²⋊C₄×C₂₃	—	2	4	no	—
48	C₂²⋊C₄×C₂₃	—	4	2	no	—
49	C₂²⋊C₄×C₂₃	—	4	4	no	—
50	C₂²⋊C₄×C₂₃	—	4	4	no	—
51	C₄²×C₂₃	—	2	4	no	—
52	C₂³×C₂₃	—	2	2	no	—
53	C₂³×C₂₃	—	2	1	no	—
54	C₂³×C₂₃	—	2	2	no	—
55	C₂³×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	—	2	4	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	4	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	C₂×D₄	1	2	yes	—
62	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
63	D₄×C₂₃	—	4	8	no	—
64	D₄×C₂₃	—	4	4	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	8	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	4	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₃	#41	1	4	yes	—
81	#2624	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	—	2	2	no	—
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	—	4	2	no	—
86	C₂²×C₂₃	—	4	2	no	—
87	C₂²×C₂₃	—	4	2	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	—	4	2	no	—
90	C₂²×C₂₃	—	4	4	no	—
91	C₂²×C₂₃	—	4	4	no	—
92	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₂₃	C₂×#41	1	2	yes	—
94	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
95	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	—	4	2	no	—
100	C₄×C₂₃	—	4	2	no	—
101	C₄×C₂₃	—	4	4	no	—
102	C₄×C₂₃	#123	1	4	yes	—
103	C₂×#121	C₂×C₂₃	1	1	yes	characteristic
104	C₂×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
105	#399	C₂×C₂₃	1	2	yes	complemented
106	#397	C₂×C₂₃	1	2	yes	complemented
107	#394	C₂×C₂₃	1	4	yes	complemented
108	#390	C₂×C₂₃	1	2	yes	complemented
109	#389	C₂×C₂₃	1	2	yes	complemented
110	#380	C₂×C₂₃	1	1	yes	characteristic, complemented
111	C₂×C₂₃	—	4	2	no	—
112	C₂×C₂₃	—	8	2	no	—
113	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
114	C₂×C₂₃	#535	1	1	yes	characteristic
115	C₂×C₂₃	#537	1	1	yes	characteristic
116	C₂×C₂₃	#540	1	2	yes	—
117	C₂×C₂₃	#542	1	2	yes	—
118	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
119	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
120	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
121	#121	—	2	4	no	—
122	#117	—	2	4	no	—
123	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
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	4	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
133	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
134	#95	C₂²×C₂₃	1	2	yes	—
135	#95	C₂²×C₂₃	1	4	yes	—
136	C₂₃	#2624	1	1	yes	characteristic, direct product factor, Sylow
137	C₂⁴	C₂³×C₂₃	1	2	yes	—
138	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
139	C₂×D₄	—	2	4	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	4	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	D₄×C₂₃	1	4	yes	—
145	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
147	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₂²⋊C₄	—	2	4	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	—	2	4	no	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	1	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₂×D₄×C₂₃	1	2	yes	—
172	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
173	D₄	—	4	8	no	—
174	D₄	—	4	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
189	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
190	C₂×C₄	#41×C₂₃	1	4	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	4	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
202	C₂²	C₂×#41×C₂₃	1	2	yes	—
203	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
204	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
205	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	4	no	—
211	C₄	C₂₃×#123	1	4	yes	—
212	C₂	—	4	2	no	—
213	C₂	—	8	2	no	—
214	C₂	C₂×C₂₃×#123	1	1	yes	characteristic
215	C₂	C₂₃×#535	1	1	yes	characteristic
216	C₂	C₂₃×#537	1	1	yes	characteristic
217	C₂	C₂₃×#540	1	2	yes	—
218	C₂	C₂₃×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂
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. C₂×C₂₃×#117	C₂	semidirect
3. C₂₃×#399	C₂	semidirect
4. C₂₃×#397	C₂	semidirect
5. C₂₃×#394	C₂	semidirect
6. C₂₃×#390	C₂	semidirect
7. C₂₃×#389	C₂	semidirect
8. C₂₃×#380	C₂	semidirect
81. #2624	C₂₃	direct (N × H)
104. C₂×#117	C₂×C₂₃	semidirect
105. #399	C₂×C₂₃	semidirect
106. #397	C₂×C₂₃	semidirect
107. #394	C₂×C₂₃	semidirect
108. #390	C₂×C₂₃	semidirect
109. #389	C₂×C₂₃	semidirect
110. #380	C₂×C₂₃	semidirect
136. C₂₃	#2624	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×#121	2
C₂×C₂₃×#117	2
C₂₃×#399	2
C₂₃×#397	2
C₂₃×#394	2
C₂₃×#390	2
C₂₃×#389	2
C₂₃×#380	2
#2624	23

Subgroups — C23×#2624 #15670243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2624 #15670243