Subgroups — C₂×#13092442 #15681954

Order 2944.

512 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67262	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625043	C₂	1	1	yes	maximal, characteristic
4	#13092442	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₂₃)⋊C₄	C₄	1	1	yes	characteristic
14	C₂×(C₂×D₂₃)⋊C₄	C₄	1	1	yes	characteristic
15	C₂×Dic₂₃⋊C₄	C₄	1	1	yes	characteristic
16	C₂×Dic₂₃⋊C₄	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×#67262	—	2	4	no	—
21	C₂×#67262	C₂²	1	4	yes	—
22	C₂×C₈×D₂₃	—	2	4	no	—
23	C₂×C₈×D₂₃	C₂²	1	4	yes	—
24	#10625043	C₄	1	4	yes	—
25	#10625043	C₂²	1	4	yes	—
26	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
27	Dic₂₃⋊C₈	C₂²	1	4	yes	complemented
28	Dic₂₃⋊C₈	C₂²	1	4	yes	complemented
29	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
30	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₂₃	D₄	1	4	yes	—
39	C₂×C₄×D₂₃	Q₈	1	4	yes	—
40	C₂×C₄×D₂₃	C₂³	1	2	yes	—
41	C₂×C₈×C₂₃	—	2	2	no	—
42	C₂×C₈×C₂₃	—	2	2	no	—
43	C₂×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂×C₈×C₂₃	C₂³	1	2	yes	—
45	C₄²×C₂₃	C₂×C₄	1	2	yes	—
46	C₄²×C₂₃	C₂³	1	2	yes	—
47	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
48	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
49	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
50	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
51	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₃	C₂³	1	2	yes	—
53	C₂×C₂₃⋊C₈	—	2	2	no	—
54	C₂×C₂₃⋊C₈	—	2	2	no	—
55	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
56	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
57	#67262	—	2	8	no	—
58	C₈×D₂₃	—	2	8	no	—
59	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₂₃	—	2	2	no	—
61	C₂²×D₂₃	—	2	2	no	—
62	C₂²×D₂₃	C₄⋊C₄	1	4	yes	—
63	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₂₃	—	2	2	no	—
73	C₂×Dic₂₃	—	2	2	no	—
74	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
75	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
78	C₄×D₂₃	C₄⋊C₄	1	8	yes	—
79	C₄×D₂₃	C₂×D₄	1	4	yes	—
80	C₄×D₂₃	C₂×Q₈	1	4	yes	—
81	C₈×C₂₃	—	2	2	no	—
82	C₈×C₂₃	—	2	2	no	—
83	C₂₃⋊C₈	—	2	2	no	—
84	C₂₃⋊C₈	—	2	2	no	—
85	C₂×#424	—	23	1	no	maximal, Sylow
86	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₂₃	#131	1	2	yes	—
88	C₂²×C₂₃	#131	1	2	yes	—
89	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₂₃	—	2	2	no	—
91	C₂×D₂₃	—	2	2	no	—
92	C₂×D₂₃	—	2	8	no	—
93	C₂×D₂₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	2	2	no	—
96	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
99	Dic₂₃	—	2	2	no	—
100	Dic₂₃	—	2	2	no	—
101	Dic₂₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂²×M₁₆	—	23	1	no	—
103	C₂³×C₈	—	23	1	no	—
104	C₂×#117	—	23	1	no	—
105	#424	—	23	8	no	—
106	C₂×C₄⋊C₈	—	23	1	no	—
107	C₂×C₄⋊C₈	—	23	1	no	—
108	C₂×C₄⋊C₈	—	23	1	no	—
109	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic, complemented
110	C₂×C₂₃	#424	1	4	yes	complemented
111	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
113	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
114	D₂₃	—	2	4	no	—
115	C₂³×C₄	—	23	1	no	—
116	C₂×M₁₆	—	23	4	no	—
117	C₂×M₁₆	—	46	4	no	—
118	C₂²×C₈	—	23	1	no	—
119	C₂²×C₈	—	23	1	no	—
120	C₂²×C₈	—	23	4	no	—
121	C₂²×C₈	—	46	4	no	—
122	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
124	#117	—	23	4	no	—
125	#117	—	23	4	no	—
126	C₂×C₄⋊C₄	—	23	1	no	—
127	C₂×C₄⋊C₄	—	23	1	no	—
128	C₂×C₂²⋊C₄	—	23	1	no	—
129	C₂×C₂²⋊C₄	—	23	1	no	—
130	C₂×C₄²	—	23	1	no	—
131	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
132	C₄⋊C₈	—	23	4	no	—
133	C₄⋊C₈	—	23	4	no	—
134	C₄⋊C₈	—	23	4	no	—
135	C₄⋊C₈	C₂×D₂₃	1	4	yes	complemented
136	C₂₃	C₂×#424	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴	—	23	1	no	—
138	C₂²×C₄	—	23	1	no	—
139	C₂²×C₄	—	23	2	no	—
140	C₂²×C₄	—	23	4	no	—
141	C₂²×C₄	—	23	4	no	—
142	C₂²×C₄	—	23	2	no	—
143	C₂²×C₄	—	23	1	no	—
144	C₂²×C₄	—	23	1	no	—
145	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
148	M₁₆	—	46	8	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	23	2	no	—
152	C₂×C₈	—	23	2	no	—
153	C₂×C₈	—	46	8	no	—
154	C₂×C₈	—	46	2	no	—
155	C₂×C₈	—	46	2	no	—
156	C₂×C₈	C₂²×D₂₃	1	2	yes	—
157	C₂×C₈	C₂²×D₂₃	1	2	yes	—
158	C₄⋊C₄	—	23	4	no	—
159	C₄⋊C₄	—	23	4	no	—
160	C₂²⋊C₄	—	23	4	no	—
161	C₂²⋊C₄	—	23	4	no	—
162	C₄²	—	23	2	no	—
163	C₄²	—	23	2	no	—
164	C₄²	C₄×D₂₃	1	2	yes	—
165	C₄²	C₂²×D₂₃	1	2	yes	—
166	C₂³	—	23	4	no	—
167	C₂³	—	23	2	no	—
168	C₂³	—	46	2	no	—
169	C₂³	—	46	2	no	—
170	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	23	4	no	—
174	C₂×C₄	—	23	2	no	—
175	C₂×C₄	—	23	8	no	—
176	C₂×C₄	—	23	4	no	—
177	C₂×C₄	—	23	4	no	—
178	C₂×C₄	—	23	2	no	—
179	C₂×C₄	—	23	2	no	—
180	C₂×C₄	—	46	2	no	—
181	C₂×C₄	—	46	2	no	—
182	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
183	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
184	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
185	C₂×C₄	D₄×D₂₃	1	2	yes	—
186	C₂×C₄	Q₈×D₂₃	1	2	yes	—
187	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	2	2	no	—
190	C₈	—	46	2	no	—
191	C₈	—	46	2	no	—
192	C₂²	—	23	4	no	—
193	C₂²	—	46	2	no	—
194	C₂²	—	46	2	no	—
195	C₂²	—	46	8	no	—
196	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
197	C₂²	#10625072	1	2	yes	—
198	C₂²	#10625078	1	2	yes	—
199	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	23	4	no	—
203	C₄	—	46	2	no	—
204	C₄	—	46	2	no	—
205	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
206	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
207	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
208	C₂	—	46	4	no	—
209	C₂	#13092442	1	4	yes	direct product factor
210	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
211	C₂	C₂×#10625072	1	1	yes	characteristic
212	C₂	C₂×#10625078	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×C₄⋊C₈	C₂
socle	C₂³×C₂₃	C₂²×C₄
prime radical	C₂³×D₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₂₃
hypocenter	C₂₃	C₂×#424
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #13092442	C₂	direct (N × H)
5. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
6. C₂×Dic₂₃⋊C₈	C₂	semidirect
7. C₂×Dic₂₃⋊C₈	C₂	semidirect
8. C₂×C₂₃×C₄⋊C₈	C₂	semidirect
26. C₂₃×C₄⋊C₈	C₂²	semidirect
27. Dic₂₃⋊C₈	C₂²	semidirect
28. Dic₂₃⋊C₈	C₂²	semidirect
29. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
109. C₂×C₄⋊C₈	D₂₃	semidirect
110. C₂×C₂₃	#424	semidirect
135. C₄⋊C₈	C₂×D₂₃	semidirect
136. C₂₃	C₂×#424	semidirect
209. C₂	#13092442	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×#67262	2
C₂²×C₈×D₂₃	2
C₂×#10625043	2
#13092442	2
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×Dic₂₃⋊C₈	2
C₂×C₂₃×C₄⋊C₈	2
C₂×#424	23

Subgroups — C2×#13092442 #15681954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092442 #15681954