Subgroups — #28302062

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290264	C₂	1	2	yes	maximal
2	#20289918	C₂	1	2	yes	maximal
3	#20289543	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811639	C₂²	1	4	yes	—
6	#16811487	C₄	1	4	yes	complemented
7	#16811485	C₄	1	4	yes	complemented
8	#16811464	—	2	2	no	—
9	C₈×#88324	—	2	2	no	—
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₄⋊#88324	C₂×C₄	1	8	yes	—
18	C₂²⋊#88324	—	2	4	no	—
19	C₄×#88324	—	2	4	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	—	2	2	no	—
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
25	C₄×#88322	C₈	1	4	yes	complemented
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	C₈	1	4	yes	complemented
28	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₈×Dic₅₃	D₄	1	1	yes	characteristic
31	C₈×Dic₅₃	D₄	1	1	yes	characteristic
32	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	2	1	no	—
41	C₂²×Dic₅₃	#41	1	2	yes	—
42	C₂×#88324	—	2	8	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	C₄²	1	2	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄×Dic₅₃	—	2	8	no	—
53	C₄×Dic₅₃	—	2	1	no	—
54	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
55	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
56	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×#88322	C₄²	1	1	yes	characteristic
59	C₂×#88322	C₄²	1	1	yes	characteristic
60	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
62	C₂×#88322	C₂×C₈	1	2	yes	—
63	C₂×#88322	C₂×C₈	1	2	yes	—
64	C₂×#88322	C₂×C₈	1	2	yes	—
65	C₂×#88322	M₁₆	1	2	yes	—
66	C₂³×C₅₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	4	no	—
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
79	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
80	#88324	—	4	8	no	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
83	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
84	#88322	C₄×C₈	1	2	yes	—
85	#88322	C₄×C₈	1	2	yes	—
86	#88322	C₄⋊C₈	1	2	yes	—
87	#88322	C₄⋊C₈	1	2	yes	—
88	#8266	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₅₃	#431	1	1	yes	characteristic
92	C₂²×C₅₃	#431	1	1	yes	characteristic
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
96	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	Dic₅₃	—	2	4	no	—
99	Dic₅₃	—	2	2	no	—
100	Dic₅₃	#443	1	4	yes	—
101	C₄²×C₈	—	53	1	no	—
102	#1564	—	53	2	no	—
103	#1450	—	53	2	no	—
104	#1276	—	53	2	no	—
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
108	C₂×C₅₃	#1772	1	2	yes	—
109	#431	—	53	4	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	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	#6157	1	1	yes	characteristic
116	C₄³	—	53	1	no	—
117	#335	—	53	4	no	—
118	#334	—	53	4	no	—
119	#322	—	106	2	no	—
120	C₈²	—	106	2	no	—
121	C₅₃	#8266	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	—	53	1	no	—
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
125	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
126	C₂×C₄²	—	53	1	no	—
127	C₂×C₄²	—	53	1	no	—
128	C₂×C₄²	—	106	2	no	—
129	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
130	C₄⋊C₈	—	53	8	no	—
131	C₂²⋊C₈	—	106	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	1	no	—
136	C₄×C₈	—	53	1	no	—
137	C₄×C₈	—	53	1	no	—
138	C₄×C₈	—	53	1	no	—
139	C₄×C₈	—	53	4	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	106	4	no	—
142	C₄×C₈	—	106	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	53	2	no	—
145	C₂²×C₄	—	106	1	no	—
146	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	53	1	no	—
150	C₂×C₈	—	53	1	no	—
151	C₂×C₈	—	53	1	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	106	8	no	—
158	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
159	C₂×C₈	C₄×#6157	1	2	yes	—
160	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
161	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
162	C₂×C₈	#11719940	1	2	yes	—
163	C₄²	—	2	1	no	—
164	C₄²	—	53	2	no	—
165	C₄²	—	53	1	no	—
166	C₄²	—	53	1	no	—
167	C₄²	—	53	2	no	—
168	C₄²	—	106	8	no	—
169	C₄²	—	106	1	no	—
170	C₄²	D₅₃⋊C₈	1	2	yes	—
171	C₂³	#13133296	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	53	2	no	—
175	C₂×C₄	—	53	2	no	—
176	C₂×C₄	—	106	4	no	—
177	C₂×C₄	—	106	4	no	—
178	C₂×C₄	—	106	2	no	—
179	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
181	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
182	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
184	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	53	2	no	—
187	C₈	—	53	2	no	—
188	C₈	—	53	2	no	—
189	C₈	—	53	2	no	—
190	C₈	—	212	8	no	—
191	C₈	#13133284	1	2	yes	—
192	C₈	#13133284	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
195	C₂²	#16811640	1	1	yes	characteristic
196	C₂²	#16811641	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	53	4	no	—
199	C₄	—	106	4	no	—
200	C₄	—	106	2	no	—
201	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
202	C₄	C₄×#11719940	1	1	yes	characteristic
203	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
204	C₄	C₂×#13133284	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄×#13133284	1	1	yes	characteristic
208	C₂	#20290517	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8266
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #16811487	C₄	semidirect
7. #16811485	C₄	semidirect
25. C₄×#88322	C₈	semidirect
27. C₈×Dic₅₃	C₈	semidirect
121. C₅₃	#8266	semidirect

Maximal subgroups

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

subgroup	index
#20290264	2
#20289918	2
#20289543	2
C₄×C₈×Dic₅₃	2
#8266	53