Subgroups — C₂×#20272766 #27295353

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	#20272766	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	—	2	1	no	—
6	#16811025	—	2	1	no	—
7	#16811025	—	2	1	no	—
8	#16811025	C₄	1	2	yes	—
9	#16811025	C₂²	1	2	yes	—
10	C₂×#13133151	C₄	1	1	yes	characteristic
11	C₂×#13133151	C₄	1	1	yes	characteristic
12	C₂×#13133149	—	2	1	no	—
13	C₂×C₄⋊#88322	—	2	1	no	—
14	#16810374	—	2	1	no	—
15	#16810374	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×C₄⋊C₁₆	—	2	1	no	—
19	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
20	C₂²×#88322	—	4	1	no	—
21	#13133209	—	2	2	no	—
22	#13133209	—	2	2	no	—
23	#13133209	—	4	1	no	—
24	#13133209	—	4	1	no	—
25	C₂×C₄⋊Dic₅₃	—	2	1	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×#11719919	—	4	1	no	—
28	C₂×#11719919	—	4	1	no	—
29	#13133151	—	2	1	no	—
30	#13133151	—	2	1	no	—
31	#13133151	C₂×C₄	1	2	yes	—
32	#13133151	C₂×C₄	1	2	yes	—
33	#13133149	—	2	2	no	—
34	#13133149	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	—	4	1	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₂×C₁₆×C₅₃	—	4	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	4	1	no	—
50	C₂²×Dic₅₃	—	4	1	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₃	D₈	1	1	yes	characteristic
60	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄⋊Dic₅₃	—	2	2	no	—
63	C₄⋊Dic₅₃	—	2	2	no	—
64	C₄⋊Dic₅₃	—	4	1	no	—
65	C₄⋊Dic₅₃	—	4	1	no	—
66	#11719919	—	4	2	no	—
67	#11719919	—	4	2	no	—
68	#11719919	—	4	2	no	—
69	#11719919	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	8	2	no	—
72	C₁₆×C₅₃	—	4	2	no	—
73	C₂³×C₅₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	#101	1	2	yes	—
77	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#103	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₂×Dic₅₃	—	8	2	no	—
87	C₂×Dic₅₃	—	8	2	no	—
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	#103	1	2	yes	—
91	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
93	#88322	—	8	2	no	—
94	C₂×#1363	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	#332	1	2	yes	—
96	C₂²×C₅₃	#359	1	2	yes	—
97	C₂²×C₅₃	#360	1	2	yes	—
98	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	#332	1	2	yes	—
104	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
105	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
106	Dic₅₃	—	8	2	no	—
107	Dic₅₃	—	8	2	no	—
108	C₂×#499	—	53	1	no	—
109	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
110	C₂×#334	—	53	1	no	—
111	#1363	—	53	4	no	—
112	C₂×C₅₃	#1363	1	4	yes	complemented
113	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
116	C₂×#125	—	106	1	no	—
117	C₂²×C₁₆	—	2	1	no	—
118	#499	—	53	2	no	—
119	#499	—	53	2	no	—
120	#499	—	106	1	no	—
121	#499	—	106	1	no	—
122	C₂×#107	—	53	1	no	—
123	C₂×#107	—	53	1	no	—
124	C₂×#106	—	106	1	no	—
125	C₂×C₄⋊C₈	—	106	1	no	—
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₄⋊C₁₆	—	2	1	no	—
128	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
129	#334	—	53	2	no	—
130	#334	—	106	1	no	—
131	C₅₃	C₂×#1363	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	212	1	no	—
133	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
135	#125	—	106	2	no	—
136	#125	—	106	2	no	—
137	#125	—	212	1	no	—
138	#125	—	212	1	no	—
139	C₂×C₄⋊C₄	—	106	1	no	—
140	C₂×C₄⋊C₄	—	106	1	no	—
141	C₂×C₄⋊C₄	—	212	1	no	—
142	C₂×C₄⋊C₄	—	212	1	no	—
143	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	4	2	no	—
146	#107	—	53	2	no	—
147	#107	—	53	2	no	—
148	#107	—	106	1	no	—
149	#107	—	106	1	no	—
150	#106	—	106	2	no	—
151	#106	—	106	2	no	—
152	C₄⋊C₈	—	106	2	no	—
153	C₄⋊C₈	—	212	1	no	—
154	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
155	C₄×C₈	D₂₁₂	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	—	212	1	no	—
159	C₂²×C₄	—	212	1	no	—
160	C₂²×C₄	#11719913	1	1	yes	characteristic
161	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
162	C₂²×C₄	#11719921	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	424	2	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	#11719921	1	1	yes	characteristic
167	C₂×C₈	#11719921	1	1	yes	characteristic
168	C₂×C₈	#11719923	1	2	yes	—
169	C₂×C₈	#11719925	1	2	yes	—
170	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	106	2	no	—
173	C₄⋊C₄	—	106	2	no	—
174	C₄⋊C₄	—	212	1	no	—
175	C₄⋊C₄	—	212	1	no	—
176	C₄⋊C₄	—	212	2	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₄⋊C₄	—	212	2	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₄²	#11719913	1	1	yes	characteristic
181	C₄²	D₄₂₄	1	1	yes	characteristic
182	C₄²	#11719921	1	1	yes	characteristic
183	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
184	C₁₆	—	4	2	no	—
185	C₂³	#13133139	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	424	2	no	—
189	C₂×C₄	—	424	2	no	—
190	C₂×C₄	—	212	2	no	—
191	C₂×C₄	—	212	2	no	—
192	C₂×C₄	#13133135	1	2	yes	—
193	C₂×C₄	#13133139	1	1	yes	characteristic
194	C₂×C₄	#13133139	1	1	yes	characteristic
195	C₂×C₄	#13133139	1	1	yes	characteristic
196	C₂×C₄	#13133140	1	2	yes	—
197	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
199	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	424	2	no	—
203	C₈	#13133140	1	2	yes	—
204	C₈	C₂×#11719923	1	1	yes	characteristic
205	C₈	C₂×#11719925	1	1	yes	characteristic
206	C₂²	#16810368	1	2	yes	—
207	C₂²	#16810431	1	2	yes	—
208	C₂²	#16810434	1	2	yes	—
209	C₂²	C₂×#13133139	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	424	2	no	—
215	C₄	—	424	2	no	—
216	C₄	#16810368	1	2	yes	—
217	C₄	C₂×#13133135	1	1	yes	characteristic
218	C₄	C₂×#13133140	1	1	yes	characteristic
219	C₂	#20272766	1	4	yes	direct product factor
220	C₂	C₂×#16810368	1	1	yes	characteristic
221	C₂	C₂×#16810431	1	1	yes	characteristic
222	C₂	C₂×#16810434	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1363
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20272766	C₂	direct (N × H)
112. C₂×C₅₃	#1363	semidirect
131. C₅₃	C₂×#1363	semidirect
219. C₂	#20272766	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×#16811025	2
C₂×#16810374	2
#20272766	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#1363	53

Subgroups — C2×#20272766 #27295353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272766 #27295353