Subgroups — C₂×#20274874 #27384493

Order 13568.

1358 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133150	C₂	1	4	yes	maximal
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810493	C₂	1	1	yes	maximal, characteristic
4	#20274874	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
8	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×#13133150	—	2	16	no	—
10	C₂×#13133150	C₂²	1	16	yes	—
11	C₂×C₄×#88322	C₄	1	8	yes	complemented
12	C₂×C₄×#88322	C₂²	1	4	yes	—
13	#16810493	—	2	2	no	—
14	#16810493	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#384	—	2	2	no	—
19	C₅₃×#384	C₂²	1	4	yes	—
20	C₂³×Dic₅₃	—	2	2	no	—
21	C₂²×#88322	C₂×C₄	1	8	yes	—
22	C₂²×#88322	D₄	1	8	yes	—
23	C₂²×#88322	Q₈	1	8	yes	—
24	C₂²×#88322	C₂³	1	4	yes	—
25	C₂×C₄⋊Dic₅₃	—	2	8	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	8	no	—
28	C₂×C₄⋊Dic₅₃	—	4	8	no	—
29	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
30	#13133150	—	2	32	no	—
31	C₄×#88322	C₂×C₄	1	16	yes	complemented
32	C₂³×C₄×C₅₃	—	2	2	no	—
33	C₂³×C₄×C₅₃	D₄	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
38	C₂×C₄⋊C₄×C₅₃	—	4	8	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
44	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	4	no	—
46	C₂²×C₄×C₅₃	—	4	4	no	—
47	C₂²×C₄×C₅₃	—	4	8	no	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
49	C₂²×C₄×C₅₃	D₈	1	8	yes	—
50	C₂²×C₄×C₅₃	Q₁₆	1	8	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	4	no	—
59	C₂²×Dic₅₃	—	4	4	no	—
60	C₂²×Dic₅₃	—	4	8	no	—
61	C₄⋊C₄×C₅₃	—	2	8	no	—
62	C₄⋊C₄×C₅₃	—	4	4	no	—
63	C₄⋊C₄×C₅₃	—	4	16	no	—
64	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
65	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
66	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
67	C₄⋊Dic₅₃	—	2	8	no	—
68	C₄⋊Dic₅₃	—	4	4	no	—
69	C₄⋊Dic₅₃	—	4	16	no	—
70	C₂×#88322	C₄⋊C₄	1	32	yes	complemented
71	C₂×#88322	C₂²×C₄	1	8	yes	—
72	C₂×#88322	C₂×D₄	1	8	yes	—
73	C₂×#88322	C₂×Q₈	1	8	yes	—
74	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₅₃	#127	1	2	yes	—
76	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
77	C₂³×C₅₃	C₂×D₈	1	4	yes	—
78	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
79	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	4	4	no	—
81	C₂×C₄×C₅₃	—	4	8	no	—
82	C₂×C₄×C₅₃	—	8	8	no	—
83	C₂×C₄×C₅₃	#107	1	32	yes	complemented
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₅₃	#127	1	2	yes	—
88	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
90	C₂×C₄×C₅₃	C₂×D₈	1	8	yes	—
91	C₂×C₄×C₅₃	C₂×Q₁₆	1	8	yes	—
92	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×Dic₅₃	—	4	4	no	—
97	C₂×Dic₅₃	—	4	8	no	—
98	C₂×Dic₅₃	—	8	8	no	—
99	#88322	C₂×C₄⋊C₄	1	16	yes	complemented
100	C₂×#1845	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	#384	1	4	yes	—
102	C₂²×C₅₃	C₂×#107	1	8	yes	—
103	C₂²×C₅₃	#493	1	4	yes	—
104	C₂²×C₅₃	#494	1	4	yes	—
105	C₂²×C₅₃	#500	1	8	yes	—
106	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
110	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
111	C₄×C₅₃	—	8	4	no	—
112	C₄×C₅₃	#384	1	4	yes	—
113	C₄×C₅₃	C₂×#107	1	16	yes	complemented
114	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
115	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
116	Dic₅₃	—	8	4	no	—
117	C₂²×#107	—	53	4	no	—
118	C₂²×C₄×C₈	—	53	1	no	—
119	C₂×#384	—	53	1	no	—
120	C₂×#384	D₅₃	1	1	yes	characteristic
121	#1845	—	53	8	no	—
122	C₂×C₅₃	#1845	1	8	yes	complemented
123	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
124	C₂×C₅₃	C₂²×#107	1	2	yes	—
125	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#500	1	2	yes	—
128	C₂³×C₈	—	53	2	no	—
129	C₂²×C₄⋊C₄	—	2	2	no	—
130	C₂²×C₄⋊C₄	—	53	2	no	—
131	C₂²×C₄⋊C₄	—	106	2	no	—
132	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
133	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
134	C₂×#107	—	53	16	no	—
135	C₂×#107	—	106	16	no	—
136	C₂×C₄×C₈	—	53	8	no	—
137	C₂×C₄×C₈	—	53	4	no	—
138	#384	—	2	2	no	—
139	#384	—	53	4	no	—
140	#384	—	106	2	no	—
141	#384	C₂×D₅₃	1	4	yes	—
142	C₅₃	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₄	—	2	2	no	—
144	C₂³×C₄	—	106	2	no	—
145	C₂³×C₄	C₂²⋊D₅₃	1	2	yes	—
146	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	C₂²×C₈	—	53	8	no	—
148	C₂²×C₈	—	53	8	no	—
149	C₂²×C₈	—	53	8	no	—
150	C₂²×C₈	—	53	4	no	—
151	C₂×C₄⋊C₄	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	8	no	—
155	C₂×C₄⋊C₄	—	53	4	no	—
156	C₂×C₄⋊C₄	—	106	8	no	—
157	C₂×C₄⋊C₄	—	106	2	no	—
158	C₂×C₄⋊C₄	—	106	8	no	—
159	C₂×C₄⋊C₄	—	212	8	no	—
160	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
161	C₂×C₄²	Dic₁₀₆	1	4	yes	—
162	C₂×C₄²	C₄×D₅₃	1	4	yes	—
163	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
164	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
165	#107	—	106	32	no	—
166	C₄×C₈	—	53	16	no	—
167	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	4	8	no	—
171	C₂²×C₄	—	106	4	no	—
172	C₂²×C₄	—	212	4	no	—
173	C₂²×C₄	—	212	8	no	—
174	C₂²×C₄	#11719919	1	8	yes	—
175	C₂²×C₄	#11719922	1	8	yes	—
176	C₂²×C₄	#11719925	1	8	yes	—
177	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
179	C₂²×C₄	D₄×D₅₃	1	4	yes	—
180	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
183	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
184	C₂×C₈	—	53	32	no	—
185	C₂×C₈	—	53	8	no	—
186	C₂×C₈	—	53	8	no	—
187	C₂×C₈	—	53	8	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	16	no	—
191	C₄⋊C₄	—	106	8	no	—
192	C₄⋊C₄	—	212	4	no	—
193	C₄⋊C₄	—	212	16	no	—
194	C₄²	#11719919	1	8	yes	—
195	C₄²	C₂×Dic₁₀₆	1	4	yes	—
196	C₄²	C₂×C₄×D₅₃	1	4	yes	—
197	C₂³	C₂×#11719919	1	2	yes	—
198	C₂³	#13133213	1	2	yes	—
199	C₂³	#13133215	1	2	yes	—
200	C₂³	C₂×#11719922	1	4	yes	—
201	C₂³	C₂×#11719925	1	4	yes	—
202	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	8	8	no	—
206	C₂×C₄	—	424	8	no	—
207	C₂×C₄	—	212	8	no	—
208	C₂×C₄	—	212	4	no	—
209	C₂×C₄	#13133150	1	32	yes	complemented
210	C₂×C₄	C₄⋊C₄×D₅₃	1	8	yes	—
211	C₂×C₄	C₂×#11719919	1	8	yes	—
212	C₂×C₄	C₂×#11719919	1	2	yes	—
213	C₂×C₄	#13133213	1	2	yes	—
214	C₂×C₄	#13133215	1	2	yes	—
215	C₂×C₄	#13133216	1	8	yes	—
216	C₂×C₄	C₂×#11719922	1	8	yes	—
217	C₂×C₄	C₂×#11719925	1	8	yes	—
218	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
220	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
221	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
222	C₈	—	53	16	no	—
223	C₂²	#16810492	1	4	yes	—
224	C₂²	C₂×#13133150	1	8	yes	—
225	C₂²	#16810999	1	4	yes	—
226	C₂²	#16811002	1	4	yes	—
227	C₂²	#16811026	1	8	yes	—
228	C₂²	C₂²×#11719919	1	1	yes	characteristic
229	C₂²	C₂×#13133213	1	1	yes	characteristic
230	C₂²	C₂×#13133215	1	1	yes	characteristic
231	C₂²	C₂²×#11719922	1	2	yes	—
232	C₂²	C₂²×#11719925	1	2	yes	—
233	C₄	—	8	4	no	—
234	C₄	—	424	4	no	—
235	C₄	#16810492	1	4	yes	—
236	C₄	C₂×#13133150	1	16	yes	complemented
237	C₄	C₂×C₄⋊C₄×D₅₃	1	2	yes	—
238	C₄	C₂×#13133216	1	2	yes	—
239	C₂	#20274874	1	8	yes	direct product factor
240	C₂	C₂×#16810492	1	1	yes	characteristic
241	C₂	C₂²×#13133150	1	2	yes	—
242	C₂	C₂×#16810999	1	1	yes	characteristic
243	C₂	C₂×#16811002	1	1	yes	characteristic
244	C₂	C₂×#16811026	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#384	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#384	D₅₃
hypocenter	C₅₃	C₂×#1845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
4. #20274874	C₂	direct (N × H)
11. C₂×C₄×#88322	C₄	semidirect
31. C₄×#88322	C₂×C₄	semidirect
70. C₂×#88322	C₄⋊C₄	semidirect
83. C₂×C₄×C₅₃	#107	semidirect
99. #88322	C₂×C₄⋊C₄	semidirect
113. C₄×C₅₃	C₂×#107	semidirect
122. C₂×C₅₃	#1845	semidirect
142. C₅₃	C₂×#1845	semidirect
209. C₂×C₄	#13133150	semidirect
236. C₄	C₂×#13133150	semidirect
239. C₂	#20274874	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×#13133150	2
C₂²×C₄×#88322	2
C₂×#16810493	2
#20274874	2
C₂×C₅₃×#384	2
C₂×#1845	53

Subgroups — C2×#20274874 #27384493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274874 #27384493