Subgroups — C₈×#13133127 #27273442

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274460	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133127	C₄	1	4	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	4	yes	—
13	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810377	C₄	1	4	yes	—
15	#16810345	C₄	1	4	yes	complemented
16	#16810344	C₄	1	4	yes	complemented
17	C₈×#88322	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈²×C₅₃	C₄	1	4	yes	complemented
22	C₂²×#88322	C₂×C₄	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
28	#13133127	C₈	1	16	yes	direct product factor
29	C₄×#88322	C₂×C₄	1	2	yes	—
30	C₄×#88322	C₂×C₄	1	2	yes	—
31	C₄×#88322	C₂×C₄	1	2	yes	—
32	C₄×#88322	C₂×C₄	1	2	yes	—
33	C₄×#88322	C₂×C₄	1	2	yes	—
34	C₄×#88322	C₂×C₄	1	2	yes	—
35	C₈×Dic₅₃	—	2	8	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₄²	1	4	yes	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	C₄²	1	4	yes	—
52	C₂×C₈×C₅₃	C₄²	1	4	yes	—
53	C₂×C₈×C₅₃	C₄²	1	4	yes	—
54	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
55	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₅₃	—	2	4	no	—
61	C₄×Dic₅₃	—	2	4	no	—
62	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	—	2	4	no	—
65	C₂×#88322	C₄²	1	4	yes	—
66	C₂×#88322	C₄²	1	4	yes	—
67	C₂×#88322	C₄²	1	4	yes	—
68	C₂×#88322	C₂×C₈	1	8	yes	—
69	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
78	C₂×C₄×C₅₃	#131	1	4	yes	—
79	C₂×Dic₅₃	—	2	4	no	—
80	C₂×Dic₅₃	—	2	4	no	—
81	C₂×Dic₅₃	C₄×C₈	1	8	yes	—
82	C₈×C₅₃	—	2	4	no	—
83	C₈×C₅₃	C₄×C₈	1	8	yes	complemented
84	C₈×C₅₃	#97	1	8	yes	complemented
85	#88322	—	2	4	no	—
86	#88322	—	2	4	no	—
87	#88322	C₄×C₈	1	8	yes	complemented
88	C₈×#97	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₄³	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
91	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
92	C₂²×C₅₃	#405	1	2	yes	—
93	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
94	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
95	C₄×C₅₃	C₂×#97	1	2	yes	—
96	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
97	C₄×C₅₃	#405	1	2	yes	—
98	C₄×C₅₃	#405	1	2	yes	—
99	Dic₅₃	—	2	8	no	—
100	#1744	—	53	1	no	—
101	C₄×#97	—	53	1	no	—
102	C₄²×C₈	—	53	1	no	—
103	C₂×#322	—	53	1	no	—
104	C₂×#322	—	53	1	no	—
105	C₂×C₈²	—	53	1	no	—
106	C₂×C₈²	D₅₃	1	1	yes	characteristic
107	C₂×C₅₃	C₈×M₁₆	1	4	yes	—
108	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
109	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
110	C₂×C₅₃	#1726	1	1	yes	characteristic
111	C₂×#97	—	53	4	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₂×C₄×C₈	—	53	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	4	no	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₄³	—	53	1	no	—
120	#336	—	53	4	no	—
121	#322	—	53	4	no	—
122	#322	—	53	4	no	—
123	C₈²	—	53	4	no	—
124	C₈²	Dic₅₃	1	4	yes	complemented
125	C₅₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	53	2	no	—
127	C₂²×C₈	—	53	2	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	C₄×D₅₃	1	2	yes	—
130	C₂²×C₈	C₄×D₅₃	1	2	yes	—
131	C₂²×C₈	C₄×D₅₃	1	2	yes	—
132	C₂×C₄²	—	53	2	no	—
133	C₂×C₄²	—	53	2	no	—
134	C₂×C₄²	—	53	2	no	—
135	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	#97	—	53	16	no	—
137	C₄×C₈	—	53	2	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	53	2	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	—	53	2	no	—
143	C₄×C₈	—	106	8	no	—
144	C₄×C₈	C₄×D₅₃	1	2	yes	—
145	C₄×C₈	C₄×D₅₃	1	2	yes	—
146	C₄×C₈	C₄×D₅₃	1	2	yes	—
147	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
148	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
149	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
150	C₂²×C₄	—	53	4	no	—
151	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	53	4	no	—
156	C₂×C₈	—	53	8	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	53	4	no	—
159	C₂×C₈	—	106	4	no	—
160	C₂×C₈	—	106	4	no	—
161	C₂×C₈	C₈×D₅₃	1	8	yes	—
162	C₂×C₈	#11719912	1	8	yes	—
163	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
164	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
165	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
166	C₄²	—	53	8	no	—
167	C₄²	—	106	4	no	—
168	C₄²	—	106	4	no	—
169	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
171	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₂³	C₄²×D₅₃	1	1	yes	characteristic
174	C₂×C₄	—	53	8	no	—
175	C₂×C₄	—	106	4	no	—
176	C₂×C₄	—	106	4	no	—
177	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
178	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
179	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
184	C₂×C₄	C₂×#11719912	1	4	yes	—
185	C₂×C₄	#13133223	1	4	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	53	8	no	—
188	C₈	—	106	4	no	—
189	C₈	—	106	4	no	—
190	C₈	C₈×Dic₅₃	1	8	yes	complemented
191	C₈	#13133127	1	8	yes	direct product factor
192	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
193	C₂²	C₄×C₈×D₅₃	1	2	yes	—
194	C₂²	C₄×#11719912	1	2	yes	—
195	C₂²	#16810577	1	2	yes	—
196	C₄	—	106	8	no	—
197	C₄	C₄×C₈×D₅₃	1	2	yes	—
198	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
199	C₄	C₂×#13133127	1	2	yes	—
200	C₄	C₄×#11719912	1	2	yes	—
201	C₄	#16810577	1	2	yes	—
202	C₄	#16810578	1	2	yes	—
203	C₂	C₈×#11719912	1	4	yes	—
204	C₂	C₄×C₈×Dic₅₃	1	1	yes	characteristic
205	C₂	C₄×#13133127	1	1	yes	characteristic
206	C₂	#20274401	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	C₈×#97
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
15. #16810345	C₄	semidirect
16. #16810344	C₄	semidirect
17. C₈×#88322	C₄	semidirect
21. C₈²×C₅₃	C₄	semidirect
28. #13133127	C₈	direct (N × H)
83. C₈×C₅₃	C₄×C₈	semidirect
84. C₈×C₅₃	#97	semidirect
87. #88322	C₄×C₈	semidirect
124. C₈²	Dic₅₃	semidirect
125. C₅₃	C₈×#97	semidirect
190. C₈	C₈×Dic₅₃	semidirect
191. C₈	#13133127	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274460	2
C₄×#13133127	2
C₄×C₈×Dic₅₃	2
C₂×#16810345	2
C₂×#16810344	2
C₂×C₈×#88322	2
C₂×C₈²×C₅₃	2
C₈×#97	53

Subgroups — C8×#13133127 #27273442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133127 #27273442