Subgroups — C₂×#20277001 #27539589

Order 13568.

518 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×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
3	#20277001	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810451	C₂	1	2	yes	maximal
5	C₂×#16810449	C₂	1	2	yes	maximal
6	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	—	2	1	no	—
8	C₂×#13133209	—	2	1	no	—
9	#16811028	—	2	2	no	—
10	#16811028	C₂²	1	4	yes	—
11	#16811025	—	2	2	no	—
12	#16811025	C₂²	1	4	yes	—
13	C₂×#13133151	—	2	1	no	—
14	C₂×#13133151	C₂²	1	1	yes	characteristic
15	C₂×#13133151	C₂²	1	2	yes	—
16	C₂×#13133151	C₂²	1	1	yes	characteristic
17	C₂×#13133149	—	2	1	no	—
18	#16810451	C₂²	1	8	yes	—
19	#16810449	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#346	C₂²	1	4	yes	—
23	C₂²×Dic₁₀₆	—	4	1	no	—
24	#13133217	—	2	4	no	—
25	#13133217	—	4	2	no	—
26	#13133209	—	2	4	no	—
27	#13133209	—	4	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	4	1	no	—
32	C₂×#11719919	—	4	1	no	—
33	C₂×#11719919	—	4	1	no	—
34	#13133151	—	2	1	no	—
35	#13133151	—	2	1	no	—
36	#13133151	—	2	2	no	—
37	#13133151	—	2	4	no	—
38	#13133151	C₂³	1	2	yes	—
39	#13133151	C₂³	1	2	yes	—
40	#13133151	C₂³	1	4	yes	—
41	#13133149	—	2	4	no	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₃	Q₈	1	8	yes	complemented
46	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂²×Dic₅₃	—	4	1	no	—
54	C₂²×Dic₅₃	—	4	2	no	—
55	C₂×Dic₁₀₆	—	4	4	no	—
56	C₂×Dic₁₀₆	—	8	4	no	—
57	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₄²×C₅₃	D₈	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄⋊Dic₅₃	—	2	2	no	—
66	C₄⋊Dic₅₃	—	2	4	no	—
67	C₄⋊Dic₅₃	—	4	1	no	—
68	C₄⋊Dic₅₃	—	4	1	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	C₄⋊Dic₅₃	—	4	4	no	—
71	#11719919	—	4	4	no	—
72	#11719919	—	4	4	no	—
73	C₁₆×C₅₃	C₂×Q₈	1	8	yes	complemented
74	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₅₃	—	4	4	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₂×Dic₅₃	—	8	4	no	—
87	C₂×Dic₅₃	—	8	2	no	—
88	C₂×Dic₅₃	—	8	2	no	—
89	Dic₁₀₆	—	8	8	no	—
90	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
91	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
92	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
93	C₂×#2255	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#500	1	2	yes	—
95	C₂²×C₅₃	#509	1	2	yes	—
96	C₂²×C₅₃	#510	1	2	yes	—
97	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	#500	1	2	yes	—
101	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₅₃	—	8	4	no	—
104	Dic₅₃	—	8	2	no	—
105	Dic₅₃	—	8	2	no	—
106	C₂×#500	—	53	1	no	—
107	C₂×#499	—	53	1	no	—
108	#2255	—	53	8	no	—
109	C₂×#367	—	53	2	no	—
110	C₂×#366	—	53	2	no	—
111	C₂×#346	D₅₃	1	1	yes	characteristic
112	C₂×C₅₃	#2255	1	4	yes	complemented
113	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
116	C₂×C₄⋊Q₈	—	106	1	no	—
117	C₂×#125	—	106	1	no	—
118	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
119	#500	—	53	4	no	—
120	#500	—	106	2	no	—
121	#499	—	53	4	no	—
122	#499	—	106	2	no	—
123	C₂×#107	—	53	1	no	—
124	C₂×#107	—	53	1	no	—
125	C₂×#107	—	53	2	no	—
126	C₂×#107	—	106	1	no	—
127	C₂×#106	—	106	1	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	#367	—	53	8	no	—
130	#366	—	53	8	no	—
131	#346	C₂×D₅₃	1	4	yes	—
132	C₅₃	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	212	1	no	—
134	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	106	4	no	—
137	C₄⋊Q₈	—	212	2	no	—
138	#125	—	106	4	no	—
139	#125	—	212	2	no	—
140	C₂×C₄⋊C₄	—	106	1	no	—
141	C₂×C₄⋊C₄	—	106	1	no	—
142	C₂×C₄⋊C₄	—	106	2	no	—
143	C₂×C₄⋊C₄	—	212	1	no	—
144	C₂×C₄⋊C₄	—	212	1	no	—
145	C₂×C₄⋊C₄	—	212	1	no	—
146	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
147	C₂×C₁₆	Dic₁₀₆	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
149	#107	—	53	2	no	—
150	#107	—	53	4	no	—
151	#107	—	53	2	no	—
152	#107	—	106	1	no	—
153	#107	—	106	1	no	—
154	#107	—	106	2	no	—
155	#107	—	106	4	no	—
156	#106	—	106	4	no	—
157	C₄×C₈	D₂₁₂	1	2	yes	—
158	C₄×C₈	C₂²×D₅₃	1	2	yes	—
159	C₂×Q₈	—	424	4	no	—
160	C₂×Q₈	—	212	4	no	—
161	C₂²×C₄	—	212	1	no	—
162	C₂²×C₄	—	212	1	no	—
163	C₂²×C₄	—	212	2	no	—
164	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
165	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
167	C₂×C₈	Dic₂₁₂	1	4	yes	—
168	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
169	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
170	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
171	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	106	2	no	—
173	C₄⋊C₄	—	106	2	no	—
174	C₄⋊C₄	—	106	4	no	—
175	C₄⋊C₄	—	212	1	no	—
176	C₄⋊C₄	—	212	1	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₄⋊C₄	—	212	4	no	—
179	C₄⋊C₄	—	212	4	no	—
180	C₄⋊C₄	—	212	4	no	—
181	C₄²	D₄₂₄	1	2	yes	—
182	C₄²	C₂×D₂₁₂	1	2	yes	—
183	C₁₆	C₂×Dic₁₀₆	1	8	yes	complemented
184	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
185	Q₈	—	424	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	424	2	no	—
189	C₂×C₄	—	424	4	no	—
190	C₂×C₄	—	424	2	no	—
191	C₂×C₄	—	212	2	no	—
192	C₂×C₄	—	212	2	no	—
193	C₂×C₄	—	212	4	no	—
194	C₂×C₄	#13133217	1	2	yes	—
195	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
196	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
197	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
199	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
200	C₈	#13133217	1	2	yes	—
201	C₈	C₂×Dic₂₁₂	1	4	yes	—
202	C₈	C₂²×Dic₁₀₆	1	2	yes	—
203	C₂²	#16811028	1	2	yes	—
204	C₂²	#16811061	1	2	yes	—
205	C₂²	#16811066	1	2	yes	—
206	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	424	2	no	—
210	C₄	—	424	4	no	—
211	C₄	—	424	2	no	—
212	C₄	#16811028	1	2	yes	—
213	C₄	C₂×#13133217	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
215	C₂	#20277001	1	4	yes	direct product factor
216	C₂	C₂×#16811028	1	1	yes	characteristic
217	C₂	C₂×#16811061	1	1	yes	characteristic
218	C₂	C₂×#16811066	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#2255
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #20277001	C₂	direct (N × H)
45. C₂×C₁₆×C₅₃	Q₈	semidirect
73. C₁₆×C₅₃	C₂×Q₈	semidirect
112. C₂×C₅₃	#2255	semidirect
132. C₅₃	C₂×#2255	semidirect
147. C₂×C₁₆	Dic₁₀₆	semidirect
183. C₁₆	C₂×Dic₁₀₆	semidirect
215. C₂	#20277001	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×#16811028	2
C₂×#16811025	2
#20277001	2
C₂×#16810451	2
C₂×#16810449	2
C₂×C₅₃×#346	2
C₂×#2255	53

Subgroups — C2×#20277001 #27539589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20277001 #27539589