Subgroups — C₂₃×#2283 #15669902

Order 2944.

568 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#383	C₂	1	2	yes	maximal, complemented
3	C₂₃×#382	C₂	1	2	yes	maximal, complemented
4	C₂₃×#381	C₂	1	2	yes	maximal
5	C₂₃×#380	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
8	C₂₃×#376	C₂	1	2	yes	maximal, complemented
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
14	C₂×C₄²×C₂₃	C₄	1	8	yes	complemented
15	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#95	C₂²	1	4	yes	—
21	C₂₃×#95	C₂²	1	4	yes	—
22	C₂₃×#95	C₂²	1	4	yes	—
23	C₂₃×#95	C₂²	1	4	yes	—
24	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	—	2	4	no	—
26	C₂²×C₄×C₂₃	—	2	4	no	—
27	C₂²×C₄×C₂₃	—	2	4	no	—
28	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₃	—	2	8	no	—
39	C₂²⋊C₄×C₂₃	—	2	8	no	—
40	C₄²×C₂₃	—	2	8	no	—
41	C₄²×C₂₃	C₂×C₄	1	4	yes	—
42	C₄²×C₂₃	C₂×C₄	1	4	yes	—
43	C₄²×C₂₃	C₂×C₄	1	4	yes	—
44	C₄²×C₂₃	C₂×C₄	1	4	yes	—
45	C₂³×C₂₃	—	2	2	no	—
46	C₂³×C₂₃	—	2	2	no	—
47	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
48	C₂³×C₂₃	#41	1	4	yes	—
49	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₂₃	—	2	8	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	—	2	8	no	—
56	C₂×C₄×C₂₃	—	2	8	no	—
57	C₂×C₄×C₂₃	—	2	8	no	—
58	C₂×C₄×C₂₃	—	2	8	no	—
59	C₂×C₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₃	#41	1	4	yes	—
67	C₂×C₄×C₂₃	#41	1	4	yes	—
68	#2283	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₂₃	—	2	2	no	—
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	—	2	8	no	—
72	C₂²×C₂₃	#117	1	4	yes	—
73	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₂₃	C₂×#41	1	2	yes	—
75	C₂²×C₂₃	C₂×#41	1	2	yes	—
76	C₂²×C₂₃	C₂×#41	1	2	yes	—
77	C₄×C₂₃	—	2	2	no	—
78	C₄×C₂₃	—	2	2	no	—
79	C₄×C₂₃	—	2	2	no	—
80	C₄×C₂₃	—	2	2	no	—
81	C₄×C₂₃	—	4	8	no	—
82	C₄×C₂₃	#117	1	4	yes	—
83	C₄×C₂₃	#117	1	4	yes	—
84	C₂²×C₄²	C₂×C₂₃	1	1	yes	characteristic
85	#383	C₂×C₂₃	1	2	yes	complemented
86	#382	C₂×C₂₃	1	2	yes	complemented
87	#381	C₂×C₂₃	1	2	yes	—
88	#380	C₂×C₂₃	1	2	yes	—
89	C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
90	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
91	#376	C₂×C₂₃	1	2	yes	complemented
92	C₂×C₂₃	—	2	4	no	—
93	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
94	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
95	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
96	C₂×C₂₃	#529	1	4	yes	—
97	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
98	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
99	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
101	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
102	C₂×C₄²	C₄×C₂₃	1	8	yes	complemented
103	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
104	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
106	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
108	#95	C₂²×C₂₃	1	4	yes	—
109	#95	C₂²×C₂₃	1	4	yes	—
110	#95	C₂²×C₂₃	1	4	yes	—
111	#95	C₂²×C₂₃	1	4	yes	—
112	C₂₃	#2283	1	1	yes	characteristic, direct product factor, Sylow
113	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
118	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
119	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
120	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
121	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	8	no	—
128	C₂²⋊C₄	—	2	8	no	—
129	C₄²	—	2	8	no	—
130	C₄²	C₂×C₄×C₂₃	1	4	yes	—
131	C₄²	C₂×C₄×C₂₃	1	4	yes	—
132	C₄²	C₂×C₄×C₂₃	1	4	yes	—
133	C₄²	C₂×C₄×C₂₃	1	4	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
137	C₂³	#41×C₂₃	1	4	yes	—
138	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
150	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
151	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
152	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
153	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
154	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
155	C₂×C₄	#41×C₂₃	1	4	yes	—
156	C₂×C₄	#41×C₂₃	1	4	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	8	no	—
160	C₂²	C₂₃×#117	1	4	yes	—
161	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
162	C₂²	C₂×#41×C₂₃	1	2	yes	—
163	C₂²	C₂×#41×C₂₃	1	2	yes	—
164	C₂²	C₂×#41×C₂₃	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	4	8	no	—
170	C₄	C₂₃×#117	1	4	yes	—
171	C₄	C₂₃×#117	1	4	yes	—
172	C₂	—	2	4	no	—
173	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
174	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
175	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
176	C₂	C₂₃×#529	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#383	C₂	semidirect
3. C₂₃×#382	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
8. C₂₃×#376	C₂	semidirect
14. C₂×C₄²×C₂₃	C₄	semidirect
68. #2283	C₂₃	direct (N × H)
85. #383	C₂×C₂₃	semidirect
86. #382	C₂×C₂₃	semidirect
89. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
91. #376	C₂×C₂₃	semidirect
102. C₂×C₄²	C₄×C₂₃	semidirect
112. C₂₃	#2283	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₂₃	2
C₂₃×#383	2
C₂₃×#382	2
C₂₃×#381	2
C₂₃×#380	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
C₂₃×#376	2
#2283	23

Subgroups — C23×#2283 #15669902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2283 #15669902