Subgroups — C₂₃×#1733 #15669352

Order 2944.

404 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#339	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₃	—	2	2	no	—
7	C₂×M₁₆×C₂₃	—	2	1	no	—
8	C₂×M₁₆×C₂₃	C₄	1	4	yes	complemented
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₃	—	2	1	no	—
11	C₂₃×#117	—	2	1	no	—
12	C₂₃×#117	—	2	2	no	—
13	C₂₃×#117	—	2	1	no	—
14	C₂₃×#117	C₄	1	2	yes	complemented
15	C₂₃×#117	C₄	1	4	yes	complemented
16	C₂₃×#117	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	complemented
19	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
20	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
21	C₂×C₄²×C₂₃	—	2	1	no	—
22	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
23	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	D₄	1	2	yes	—
32	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₃	—	2	4	no	—
37	M₁₆×C₂₃	—	2	2	no	—
38	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
39	C₂×C₈×C₂₃	—	2	4	no	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
44	C₂²⋊C₄×C₂₃	—	2	2	no	—
45	C₂²⋊C₄×C₂₃	—	2	4	no	—
46	C₂²⋊C₄×C₂₃	—	2	2	no	—
47	C₄²×C₂₃	—	2	2	no	—
48	C₄²×C₂₃	—	2	2	no	—
49	C₄²×C₂₃	—	2	2	no	—
50	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
51	C₂³×C₂₃	—	4	2	no	—
52	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
55	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	—	2	4	no	—
59	C₂×C₄×C₂₃	—	4	4	no	—
60	C₂×C₄×C₂₃	—	4	4	no	—
61	C₂×C₄×C₂₃	C₄²	1	2	yes	—
62	C₂×C₄×C₂₃	C₄²	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
79	C₈×C₂₃	—	2	4	no	—
80	#1733	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	—	2	2	no	—
84	C₂²×C₂₃	—	4	2	no	—
85	C₂²×C₂₃	—	4	2	no	—
86	C₂²×C₂₃	#95	1	2	yes	—
87	C₂²×C₂₃	#95	1	2	yes	—
88	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₂₃	—	4	4	no	—
92	C₄×C₂₃	—	4	4	no	—
93	C₄×C₂₃	#95	1	2	yes	—
94	C₄×C₂₃	#95	1	2	yes	—
95	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
96	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×M₁₆	C₂×C₂₃	1	1	yes	characteristic
100	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
101	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
102	#339	C₂×C₂₃	1	4	yes	complemented
103	C₂×C₂₃	—	2	1	no	—
104	C₂×C₂₃	—	2	2	no	—
105	C₂×C₂₃	—	2	2	no	—
106	C₂×C₂₃	—	2	1	no	—
107	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
108	C₂×C₂₃	#421	1	2	yes	—
109	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	2	1	no	—
112	C₂×M₁₆	C₄×C₂₃	1	4	yes	complemented
113	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
114	C₂²×C₈	—	2	1	no	—
115	#117	—	2	1	no	—
116	#117	—	2	1	no	—
117	#117	—	2	2	no	—
118	#117	C₄×C₂₃	1	2	yes	complemented
119	#117	C₄×C₂₃	1	4	yes	complemented
120	#117	C₂²×C₂₃	1	2	yes	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	C₄×C₂₃	1	2	yes	complemented
123	C₂×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₂²⋊C₄	—	2	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
127	C₂₃	#1733	1	1	yes	characteristic, direct product factor, Sylow
128	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
132	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
135	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
137	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₄	D₄×C₂₃	1	2	yes	—
139	C₂²×C₄	Q₈×C₂₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	—	2	2	no	—
143	M₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	C₂×C₄×C₂₃	1	4	yes	complemented
156	C₂³	—	4	2	no	—
157	C₂³	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂³	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂³	C₂²⋊C₄×C₂₃	1	2	yes	—
160	C₂³	C₄⋊C₄×C₂₃	1	2	yes	—
161	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄²×C₂₃	1	2	yes	—
167	C₂×C₄	C₄²×C₂₃	1	2	yes	—
168	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
169	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
170	C₂×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
172	C₂×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
174	C₂×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
175	C₂×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
176	C₂×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
177	C₂×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
178	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	C₂₃×#95	1	2	yes	—
191	C₂²	C₂₃×#95	1	2	yes	—
192	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
193	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
194	C₂²	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	—	4	4	no	—
197	C₄	C₂₃×#95	1	2	yes	—
198	C₄	C₂₃×#95	1	2	yes	—
199	C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
200	C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
202	C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	2	no	—
207	C₂	C₂×C₂₃×#95	1	1	yes	characteristic
208	C₂	C₂₃×#421	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₄
commutator	C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#339	C₂	semidirect
8. C₂×M₁₆×C₂₃	C₄	semidirect
14. C₂₃×#117	C₄	semidirect
15. C₂₃×#117	C₄	semidirect
18. C₂×C₄⋊C₄×C₂₃	C₄	semidirect
38. M₁₆×C₂₃	C₂×C₄	semidirect
50. C₄²×C₂₃	C₂×C₄	semidirect
80. #1733	C₂₃	direct (N × H)
102. #339	C₂×C₂₃	semidirect
112. C₂×M₁₆	C₄×C₂₃	semidirect
118. #117	C₄×C₂₃	semidirect
119. #117	C₄×C₂₃	semidirect
122. C₂×C₄⋊C₄	C₄×C₂₃	semidirect
127. C₂₃	#1733	direct (N × H)
143. M₁₆	C₂×C₄×C₂₃	semidirect
155. C₄²	C₂×C₄×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×M₁₆×C₂₃	2
C₂×C₂₃×#117	2
C₂₃×#339	2
#1733	23

Subgroups — C23×#1733 #15669352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1733 #15669352