Subgroups — C₂×#20272771 #27295371

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 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×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272771	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	—	2	1	no	—
6	C₂²×#11719908	—	2	1	no	—
7	#16811028	—	2	1	no	—
8	#16811028	—	2	1	no	—
9	#16811028	C₄	1	2	yes	—
10	#16811028	C₂²	1	2	yes	—
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133151	C₄	1	2	yes	—
13	C₄⋊#11719908	—	2	1	no	—
14	C₄⋊#11719908	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	—	2	1	no	—
18	C₅₃×#335	C₂²	1	2	yes	—
19	C₂²×Dic₁₀₆	—	4	1	no	—
20	#13133217	—	2	2	no	—
21	#13133217	—	2	2	no	—
22	#13133217	—	4	1	no	—
23	#13133217	—	4	1	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	4	1	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₂×#11719908	—	4	2	no	—
28	#13133151	—	2	2	no	—
29	#13133151	—	2	2	no	—
30	#13133151	—	2	2	no	—
31	#13133151	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₃	—	4	1	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	4	1	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	4	2	no	—
46	C₂×Dic₁₀₆	—	4	2	no	—
47	C₂×Dic₁₀₆	—	4	2	no	—
48	C₂×Dic₁₀₆	—	8	2	no	—
49	C₂×Dic₁₀₆	—	8	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	—	8	2	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	D₈	1	1	yes	characteristic
61	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄⋊Dic₅₃	—	2	4	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	#11719908	—	4	2	no	—
68	C₂³×C₅₃	#102	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	#101	1	2	yes	—
71	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#103	1	2	yes	—
75	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×Dic₅₃	—	4	4	no	—
81	C₂×Dic₅₃	—	8	4	no	—
82	Dic₁₀₆	—	8	4	no	—
83	Dic₁₀₆	—	8	2	no	—
84	Dic₁₀₆	—	8	2	no	—
85	C₈×C₅₃	—	2	1	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	8	2	no	—
88	C₈×C₅₃	#103	1	2	yes	—
89	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×#1364	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	#332	1	2	yes	—
93	C₂²×C₅₃	#357	1	2	yes	—
94	C₂²×C₅₃	#362	1	2	yes	—
95	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	#332	1	2	yes	—
99	C₄×C₅₃	#357	1	2	yes	—
100	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
101	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
102	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
103	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
104	Dic₅₃	—	8	4	no	—
105	C₂×#500	—	53	1	no	—
106	C₂×C₄⋊C₁₆	—	53	1	no	—
107	C₂×#335	D₅₃	1	1	yes	characteristic
108	#1364	—	53	4	no	—
109	C₂×C₅₃	#1364	1	4	yes	complemented
110	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	106	1	no	—
114	C₂²×C₁₆	—	106	1	no	—
115	#500	—	53	2	no	—
116	#500	—	53	2	no	—
117	#500	—	106	1	no	—
118	#500	—	106	1	no	—
119	C₂×#107	—	53	2	no	—
120	C₂×#107	—	106	1	no	—
121	C₂×C₄⋊C₈	—	2	1	no	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	C₄⋊C₁₆	—	53	2	no	—
124	C₄⋊C₁₆	—	106	1	no	—
125	#335	—	2	1	no	—
126	#335	C₂×D₅₃	1	2	yes	—
127	C₅₃	C₂×#1364	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	212	1	no	—
129	C₂²×C₈	—	4	1	no	—
130	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
131	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
132	C₄⋊Q₈	—	106	2	no	—
133	C₄⋊Q₈	—	106	2	no	—
134	C₄⋊Q₈	—	212	1	no	—
135	C₄⋊Q₈	—	212	1	no	—
136	C₂×C₄⋊C₄	—	106	2	no	—
137	C₂×C₄⋊C₄	—	212	1	no	—
138	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂×C₁₆	—	106	2	no	—
140	C₂×C₁₆	—	212	2	no	—
141	#107	—	53	4	no	—
142	#107	—	106	2	no	—
143	#107	—	106	2	no	—
144	#107	—	106	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	4	1	no	—
147	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
148	C₄×C₈	D₂₁₂	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
151	C₂×Q₈	—	424	2	no	—
152	C₂×Q₈	—	424	2	no	—
153	C₂×Q₈	—	212	2	no	—
154	C₂×Q₈	—	212	2	no	—
155	C₂²×C₄	—	212	2	no	—
156	C₂²×C₄	#11719921	1	1	yes	characteristic
157	C₂²×C₄	#11719922	1	1	yes	characteristic
158	C₂²×C₄	#11719924	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	8	2	no	—
162	C₂×C₈	#11719913	1	2	yes	—
163	C₂×C₈	Dic₂₁₂	1	2	yes	—
164	C₂×C₈	#11719921	1	1	yes	characteristic
165	C₂×C₈	#11719921	1	1	yes	characteristic
166	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
168	C₄⋊C₄	—	106	4	no	—
169	C₄⋊C₄	—	212	2	no	—
170	C₄⋊C₄	—	212	2	no	—
171	C₄⋊C₄	—	212	2	no	—
172	C₄²	#11719921	1	1	yes	characteristic
173	C₄²	#11719922	1	1	yes	characteristic
174	C₄²	#11719924	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
176	C₁₆	—	212	2	no	—
177	C₂³	#13133137	1	1	yes	characteristic
178	Q₈	—	424	2	no	—
179	Q₈	—	424	2	no	—
180	Q₈	—	424	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	424	4	no	—
183	C₂×C₄	—	212	4	no	—
184	C₂×C₄	#13133135	1	2	yes	—
185	C₂×C₄	#13133137	1	1	yes	characteristic
186	C₂×C₄	#13133137	1	1	yes	characteristic
187	C₂×C₄	#13133137	1	1	yes	characteristic
188	C₂×C₄	#13133142	1	2	yes	—
189	C₂×C₄	#13133158	1	2	yes	—
190	C₂×C₄	#13133160	1	2	yes	—
191	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
192	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	8	2	no	—
197	C₈	#13133142	1	2	yes	—
198	C₈	C₂×#11719913	1	1	yes	characteristic
199	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
200	C₂²	#16810370	1	2	yes	—
201	C₂²	#16810423	1	2	yes	—
202	C₂²	#16810438	1	2	yes	—
203	C₂²	C₂×#13133137	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	424	4	no	—
207	C₄	#16810370	1	2	yes	—
208	C₄	#16810423	1	2	yes	—
209	C₄	C₂×#13133135	1	1	yes	characteristic
210	C₄	C₂×#13133142	1	1	yes	characteristic
211	C₄	C₂×#13133158	1	1	yes	characteristic
212	C₄	C₂×#13133160	1	1	yes	characteristic
213	C₂	#20272771	1	4	yes	direct product factor
214	C₂	C₂×#16810370	1	1	yes	characteristic
215	C₂	C₂×#16810423	1	1	yes	characteristic
216	C₂	C₂×#16810438	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133137
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	C₂×#1364
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #20272771	C₂	direct (N × H)
109. C₂×C₅₃	#1364	semidirect
127. C₅₃	C₂×#1364	semidirect
213. C₂	#20272771	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×C₄⋊#11719908	2
#20272771	2
C₂×C₅₃×#335	2
C₂×#1364	53

Subgroups — C2×#20272771 #27295371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272771 #27295371