Subgroups — C₂₃×#1764 #15669383

Order 2944.

300 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#104	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#416	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#339	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#131	—	2	2	no	—
10	C₂₃×#131	C₄	1	4	yes	complemented
11	C₂×M₁₆×C₂₃	—	2	1	no	—
12	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	—	2	1	no	—
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#104	C₄	1	4	yes	complemented
20	C₂₃×#103	C₄	1	2	yes	complemented
21	C₂₃×#102	C₄	1	2	yes	complemented
22	C₂₃×#97	—	2	1	no	—
23	C₄×C₈×C₂₃	—	2	2	no	—
24	C₄×C₈×C₂₃	—	2	1	no	—
25	#41×C₂₃	—	2	1	no	—
26	#41×C₂₃	—	2	1	no	—
27	#41×C₂₃	C₂×C₄	1	2	yes	—
28	#41×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₂₃	—	2	1	no	—
31	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₃	—	2	1	no	—
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₃	—	2	1	no	—
36	M₁₆×C₂₃	—	2	1	no	—
37	M₁₆×C₂₃	—	2	4	no	—
38	M₁₆×C₂₃	C₂×C₄	1	2	yes	—
39	M₁₆×C₂₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	—	2	1	no	—
42	C₂×C₈×C₂₃	—	2	1	no	—
43	C₂×C₈×C₂₃	—	2	1	no	—
44	C₂×C₈×C₂₃	—	2	4	no	—
45	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	D₄	1	2	yes	—
49	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₂₃	—	2	1	no	—
52	C₄²×C₂₃	—	2	1	no	—
53	C₄²×C₂₃	—	2	1	no	—
54	C₄²×C₂₃	C₂×C₄	1	2	yes	—
55	C₂³×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	#41	1	1	yes	characteristic
57	Q₈×C₂₃	—	2	1	no	—
58	Q₈×C₂₃	C₄²	1	2	yes	—
59	D₄×C₂₃	—	2	2	no	—
60	D₄×C₂₃	—	2	2	no	—
61	D₄×C₂₃	—	2	1	no	—
62	D₄×C₂₃	C₄²	1	2	yes	—
63	C₂×C₄×C₂₃	—	2	4	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	1	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	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
75	C₈×C₂₃	—	2	2	no	—
76	C₈×C₂₃	—	2	2	no	—
77	C₈×C₂₃	—	2	2	no	—
78	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
79	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
80	#1764	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₂₃	—	2	2	no	—
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	—	4	2	no	—
84	C₂²×C₂₃	#117	1	1	yes	characteristic
85	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
87	C₄×C₂₃	—	2	4	no	—
88	C₄×C₂₃	—	2	2	no	—
89	C₄×C₂₃	—	4	2	no	—
90	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
91	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
93	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
94	C₂×#131	C₂×C₂₃	1	1	yes	characteristic
95	C₂×#104	C₂×C₂₃	1	1	yes	characteristic
96	#416	C₂×C₂₃	1	1	yes	characteristic
97	#405	C₂×C₂₃	1	1	yes	characteristic, complemented
98	C₂×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
99	#341	C₂×C₂₃	1	1	yes	characteristic, complemented
100	#339	C₂×C₂₃	1	1	yes	characteristic, complemented
101	C₂×C₂₃	—	2	1	no	—
102	C₂×C₂₃	—	2	1	no	—
103	C₂×C₂₃	—	4	2	no	—
104	C₂×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₂₃	#443	1	2	yes	—
106	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
107	#131	—	2	2	no	—
108	#131	C₄×C₂₃	1	4	yes	complemented
109	C₂×M₁₆	—	2	1	no	—
110	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
111	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	#117	C₂²×C₂₃	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
117	#104	C₄×C₂₃	1	4	yes	complemented
118	#103	C₄×C₂₃	1	2	yes	complemented
119	#102	C₄×C₂₃	1	2	yes	complemented
120	#97	—	2	1	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	1	no	—
123	C₂₃	#1764	1	1	yes	characteristic, direct product factor, Sylow
124	#41	—	2	1	no	—
125	#41	—	2	1	no	—
126	#41	C₂×C₄×C₂₃	1	2	yes	—
127	#41	C₂×C₄×C₂₃	1	2	yes	—
128	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
129	C₂×D₄	—	2	1	no	—
130	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
134	M₁₆	—	2	1	no	—
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	4	no	—
137	M₁₆	C₂×C₄×C₂₃	1	2	yes	—
138	M₁₆	C₂×C₄×C₂₃	1	2	yes	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
145	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
146	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
147	C₂×C₈	D₄×C₂₃	1	2	yes	—
148	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
149	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
150	C₂²⋊C₄	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	C₂×C₄×C₂₃	1	2	yes	—
154	C₂³	—	2	1	no	—
155	C₂³	#41×C₂₃	1	1	yes	characteristic
156	Q₈	—	2	1	no	—
157	Q₈	C₄²×C₂₃	1	2	yes	—
158	D₄	—	2	2	no	—
159	D₄	—	2	2	no	—
160	D₄	—	2	1	no	—
161	D₄	C₄²×C₂₃	1	2	yes	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
178	C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	C₂₃×#117	1	1	yes	characteristic
183	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
184	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
189	C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
191	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	—	4	2	no	—
195	C₂	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂	C₂₃×#443	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	D₄
commutator	C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	C₄×D₄
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#405	C₂	semidirect
5. C₂×C₄×C₈×C₂₃	C₂	semidirect
6. C₂₃×#341	C₂	semidirect
7. C₂₃×#339	C₂	semidirect
10. C₂₃×#131	C₄	semidirect
19. C₂₃×#104	C₄	semidirect
20. C₂₃×#103	C₄	semidirect
21. C₂₃×#102	C₄	semidirect
80. #1764	C₂₃	direct (N × H)
97. #405	C₂×C₂₃	semidirect
98. C₂×C₄×C₈	C₂×C₂₃	semidirect
99. #341	C₂×C₂₃	semidirect
100. #339	C₂×C₂₃	semidirect
108. #131	C₄×C₂₃	semidirect
117. #104	C₄×C₂₃	semidirect
118. #103	C₄×C₂₃	semidirect
119. #102	C₄×C₂₃	semidirect
123. C₂₃	#1764	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×#131	2
C₂×C₂₃×#104	2
C₂₃×#416	2
C₂₃×#405	2
C₂×C₄×C₈×C₂₃	2
C₂₃×#341	2
C₂₃×#339	2
#1764	23

Subgroups — C23×#1764 #15669383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1764 #15669383