Subgroups — C₂×#20272753 #27295298

Order 13568.

390 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
3	#20272753	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	—	2	1	no	—
6	C₂²×Dic₂₁₂	—	2	1	no	—
7	C₂×#13133217	C₄	1	2	yes	—
8	C₂×#13133157	—	2	1	no	—
9	#16811003	—	2	1	no	—
10	#16811003	—	2	1	no	—
11	#16811003	C₄	1	2	yes	—
12	#16811003	C₂²	1	2	yes	—
13	#16810444	—	2	1	no	—
14	#16810444	C₂²	1	2	yes	—
15	C₂×M₃₂×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#364	—	2	1	no	—
18	C₅₃×#364	C₂²	1	2	yes	—
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	2	no	—
24	C₂×Dic₂₁₂	—	4	1	no	—
25	C₂×Dic₂₁₂	—	4	1	no	—
26	C₂×Dic₂₁₂	—	4	1	no	—
27	C₂×Dic₂₁₂	—	4	1	no	—
28	C₂×Dic₂₁₂	—	4	1	no	—
29	C₂×Dic₂₁₂	—	4	1	no	—
30	C₂×Dic₂₁₂	—	4	1	no	—
31	C₂×Dic₂₁₂	—	4	1	no	—
32	#13133217	—	2	2	no	—
33	#13133217	—	4	2	no	—
34	#13133217	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊Dic₅₃	—	4	2	no	—
36	#13133157	—	2	2	no	—
37	#13133157	—	4	1	no	—
38	C₂×#11719908	—	4	1	no	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	2	no	—
43	M₃₂×C₅₃	—	4	1	no	—
44	C₂×C₁₆×C₅₃	—	4	1	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	2	no	—
52	C₂×Dic₁₀₆	—	2	4	no	—
53	C₂×Dic₁₀₆	—	4	2	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
61	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
62	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
63	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄⋊Dic₅₃	—	4	4	no	—
74	C₄⋊Dic₅₃	—	4	4	no	—
75	Dic₂₁₂	—	4	2	no	—
76	Dic₂₁₂	—	4	2	no	—
77	Dic₂₁₂	—	4	2	no	—
78	Dic₂₁₂	—	4	2	no	—
79	Dic₂₁₂	—	4	2	no	—
80	Dic₂₁₂	—	4	2	no	—
81	Dic₂₁₂	—	4	2	no	—
82	Dic₂₁₂	—	4	2	no	—
83	C₁₆×C₅₃	—	4	2	no	—
84	#11719908	—	4	2	no	—
85	C₂³×C₅₃	#100	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×Dic₅₃	—	4	4	no	—
96	C₂×Dic₅₃	—	8	2	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	Dic₁₀₆	—	4	4	no	—
99	Dic₁₀₆	—	4	4	no	—
100	Dic₁₀₆	—	8	2	no	—
101	Dic₁₀₆	—	8	2	no	—
102	C₈×C₅₃	#102	1	1	yes	characteristic
103	C₈×C₅₃	#102	1	1	yes	characteristic
104	C₈×C₅₃	#102	1	1	yes	characteristic
105	C₈×C₅₃	#102	1	1	yes	characteristic
106	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
107	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
108	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×#1358	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	#331	1	1	yes	characteristic
113	C₂²×C₅₃	#331	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	#331	1	1	yes	characteristic
118	C₄×C₅₃	#331	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
120	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
121	Dic₅₃	—	8	4	no	—
122	C₂×#494	—	53	1	no	—
123	C₂×#364	—	53	1	no	—
124	C₂×#364	D₅₃	1	1	yes	characteristic
125	#1358	—	53	4	no	—
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	#1358	1	2	yes	complemented
129	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
130	C₂²×Q₁₆	—	106	1	no	—
131	C₂²×Q₁₆	—	106	1	no	—
132	C₂×C₄⋊Q₈	—	53	2	no	—
133	C₂×M₃₂	—	2	1	no	—
134	C₂×M₃₂	—	106	1	no	—
135	#494	—	53	2	no	—
136	#494	—	53	2	no	—
137	#494	—	106	1	no	—
138	#494	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	#364	—	2	1	no	—
141	#364	—	53	2	no	—
142	#364	—	106	1	no	—
143	#364	C₂×D₅₃	1	2	yes	—
144	C₅₃	C₂×#1358	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	106	2	no	—
146	C₂×Q₁₆	—	106	2	no	—
147	C₂×Q₁₆	—	106	2	no	—
148	C₂×Q₁₆	—	106	2	no	—
149	C₂×Q₁₆	—	106	2	no	—
150	C₂×Q₁₆	—	212	1	no	—
151	C₂×Q₁₆	—	212	1	no	—
152	C₂×Q₁₆	—	212	1	no	—
153	C₂×Q₁₆	—	212	1	no	—
154	C₂×Q₁₆	—	212	1	no	—
155	C₂×Q₁₆	—	212	1	no	—
156	C₂×Q₁₆	—	212	1	no	—
157	C₂×Q₁₆	—	212	1	no	—
158	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	53	4	no	—
161	C₄⋊Q₈	—	106	2	no	—
162	C₄⋊Q₈	—	212	2	no	—
163	C₂×C₄⋊C₄	—	212	2	no	—
164	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
165	M₃₂	—	2	2	no	—
166	M₃₂	—	4	1	no	—
167	M₃₂	—	106	2	no	—
168	M₃₂	—	212	1	no	—
169	C₂×C₁₆	—	4	1	no	—
170	C₂×C₁₆	—	212	1	no	—
171	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₄×C₈	D₂₁₂	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	C₂×Q₈	—	106	4	no	—
176	C₂×Q₈	—	212	2	no	—
177	C₂×Q₈	—	212	2	no	—
178	C₂×Q₈	—	212	2	no	—
179	C₂×Q₈	—	212	2	no	—
180	C₂×Q₈	—	212	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	212	2	no	—
183	C₂²×C₄	#11719921	1	1	yes	characteristic
184	Q₁₆	—	212	2	no	—
185	Q₁₆	—	212	2	no	—
186	Q₁₆	—	212	2	no	—
187	Q₁₆	—	212	2	no	—
188	Q₁₆	—	212	2	no	—
189	Q₁₆	—	212	2	no	—
190	Q₁₆	—	212	2	no	—
191	Q₁₆	—	212	2	no	—
192	C₂×C₈	#11719913	1	1	yes	characteristic
193	C₂×C₈	#11719913	1	1	yes	characteristic
194	C₂×C₈	D₄₂₄	1	1	yes	characteristic
195	C₂×C₈	D₄₂₄	1	1	yes	characteristic
196	C₂×C₈	#11719921	1	1	yes	characteristic
197	C₂×C₈	#11719921	1	1	yes	characteristic
198	C₂×C₈	#11719922	1	1	yes	characteristic
199	C₂×C₈	#11719922	1	1	yes	characteristic
200	C₂×C₈	#11719924	1	1	yes	characteristic
201	C₂×C₈	#11719924	1	1	yes	characteristic
202	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	212	4	no	—
205	C₄⋊C₄	—	212	4	no	—
206	C₄²	—	2	1	no	—
207	C₄²	#11719921	1	1	yes	characteristic
208	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
209	C₁₆	—	4	2	no	—
210	C₁₆	—	212	2	no	—
211	C₂³	#13133133	1	1	yes	characteristic
212	Q₈	—	424	2	no	—
213	Q₈	—	424	2	no	—
214	Q₈	—	212	4	no	—
215	Q₈	—	212	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	424	2	no	—
220	C₂×C₄	—	424	2	no	—
221	C₂×C₄	—	212	4	no	—
222	C₂×C₄	#13133133	1	1	yes	characteristic
223	C₂×C₄	#13133137	1	1	yes	characteristic
224	C₂×C₄	#13133137	1	1	yes	characteristic
225	C₂×C₄	#13133139	1	1	yes	characteristic
226	C₂×C₄	#13133139	1	1	yes	characteristic
227	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
228	C₈	#13133137	1	1	yes	characteristic
229	C₈	#13133137	1	1	yes	characteristic
230	C₈	#13133139	1	1	yes	characteristic
231	C₈	#13133139	1	1	yes	characteristic
232	C₈	C₂×#11719922	1	1	yes	characteristic
233	C₈	C₂×D₄₂₄	1	1	yes	characteristic
234	C₈	C₂×#11719924	1	1	yes	characteristic
235	C₈	C₂×#11719913	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	#16810366	1	1	yes	characteristic
238	C₂²	#16810366	1	1	yes	characteristic
239	C₂²	C₂×#13133133	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	424	4	no	—
243	C₄	#16810366	1	1	yes	characteristic
244	C₄	#16810366	1	1	yes	characteristic
245	C₄	C₂×#13133137	1	1	yes	characteristic
246	C₄	C₂×#13133139	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	#20272753	1	2	yes	direct product factor
250	C₂	C₂×#16810366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810366
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#331
prime radical	C₂³×C₅₃	#100
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	C₂×#1358
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #20272753	C₂	direct (N × H)
128. C₂×C₅₃	#1358	semidirect
144. C₅₃	C₂×#1358	semidirect
249. C₂	#20272753	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×#16811003	2
C₂×#16810444	2
#20272753	2
C₂×C₅₃×#364	2
C₂×#1358	53

Subgroups — C2×#20272753 #27295298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272753 #27295298