Subgroups — #27295690

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272784	C₂	1	2	yes	maximal, complemented
4	#20272778	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₄⋊#11719908	C₂²	1	2	yes	—
9	C₂²⋊#11719908	—	2	1	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₈×#88322	C₄	1	1	yes	characteristic
12	C₈×#88322	C₄	1	1	yes	characteristic
13	C₈×#88322	C₂²	1	1	yes	characteristic
14	C₈×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₄×C₁₆×C₅₃	—	2	1	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	C₈	1	1	yes	characteristic
21	C₂²×#88322	C₈	1	1	yes	characteristic
22	C₂×#11719908	—	2	2	no	—
23	C₄×#88322	—	2	1	no	—
24	C₄×#88322	C₈	1	1	yes	characteristic
25	C₄×#88322	C₈	1	1	yes	characteristic
26	C₄×#88322	C₂×C₄	1	1	yes	characteristic
27	C₄×#88322	C₂×C₄	1	1	yes	characteristic
28	C₄×#88322	C₂×C₄	1	1	yes	characteristic
29	C₄×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	1	no	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	C₁₆	1	4	yes	complemented
56	C₂×#88322	C₂×C₈	1	1	yes	characteristic
57	C₂×#88322	C₂×C₈	1	1	yes	characteristic
58	C₂×#88322	C₂×C₈	1	1	yes	characteristic
59	C₂×#88322	C₂×C₈	1	1	yes	characteristic
60	C₁₆×C₅₃	—	4	2	no	—
61	#11719908	—	4	2	no	—
62	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
70	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	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	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	—	2	1	no	—
76	C₈×C₅₃	#108	1	2	yes	—
77	C₈×C₅₃	#135	1	2	yes	—
78	#88322	—	2	2	no	—
79	#88322	—	2	2	no	—
80	#88322	—	2	1	no	—
81	#88322	—	2	1	no	—
82	#88322	C₂×C₁₆	1	2	yes	—
83	#88322	C₂×C₁₆	1	2	yes	—
84	#12012	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
88	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	—	2	1	no	—
91	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
92	C₄×C₅₃	#364	1	2	yes	—
93	C₄×C₅₃	#427	1	1	yes	characteristic
94	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
95	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
96	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
97	#2162	—	53	1	no	—
98	#2162	D₅₃	1	1	yes	characteristic
99	C₂×C₈²	—	53	1	no	—
100	#1370	—	53	2	no	—
101	#1367	—	53	2	no	—
102	C₂×C₅₃	—	2	2	no	—
103	C₂×C₅₃	#1564	1	1	yes	characteristic
104	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
105	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
106	C₂×C₄×C₈	—	53	1	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	53	2	no	—
110	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
111	C₂²⋊C₁₆	—	2	1	no	—
112	C₂²⋊C₁₆	—	106	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₄×C₁₆	—	106	1	no	—
115	C₈²	—	53	1	no	—
116	C₈²	—	53	1	no	—
117	C₈²	—	53	1	no	—
118	C₈²	—	53	1	no	—
119	C₅₃	#12012	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	—	53	1	no	—
122	C₂²×C₈	—	106	1	no	—
123	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	106	2	no	—
128	C₄×C₈	—	53	1	no	—
129	C₄×C₈	—	53	1	no	—
130	C₄×C₈	—	53	1	no	—
131	C₄×C₈	—	53	1	no	—
132	C₄×C₈	—	53	1	no	—
133	C₄×C₈	—	53	1	no	—
134	C₄×C₈	—	106	1	no	—
135	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
136	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	#11719912	1	1	yes	characteristic
141	C₂²×C₄	#11719919	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	53	4	no	—
145	C₂×C₈	—	53	1	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	—	53	1	no	—
148	C₂×C₈	—	53	1	no	—
149	C₂×C₈	—	106	4	no	—
150	C₂×C₈	—	106	2	no	—
151	C₂×C₈	—	106	1	no	—
152	C₂×C₈	—	106	1	no	—
153	C₂×C₈	#11719919	1	1	yes	characteristic
154	C₂×C₈	#11719919	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	C₄²	C₈×D₅₃	1	1	yes	characteristic
158	C₄²	#11719912	1	1	yes	characteristic
159	C₄²	#11719919	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	212	2	no	—
163	C₂³	#13133146	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	#13133146	1	1	yes	characteristic
168	C₂×C₄	#13133146	1	1	yes	characteristic
169	C₂×C₄	#13133146	1	1	yes	characteristic
170	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
171	C₂×C₄	#13133156	1	2	yes	—
172	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	53	2	no	—
178	C₈	—	53	2	no	—
179	C₈	—	106	2	no	—
180	C₈	—	106	2	no	—
181	C₈	—	106	1	no	—
182	C₈	—	106	1	no	—
183	C₈	#13133152	1	2	yes	—
184	C₈	#13133237	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#16810441	1	2	yes	—
188	C₂²	C₂×#13133146	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	#16810441	1	2	yes	—
192	C₄	#16810443	1	2	yes	—
193	C₄	#16810659	1	1	yes	characteristic
194	C₄	C₂×#13133152	1	1	yes	characteristic
195	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
196	C₄	C₂×#13133156	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#20273461	1	1	yes	characteristic
199	C₂	C₂×#16810441	1	1	yes	characteristic
200	C₂	C₂×#16810443	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2162	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2162	D₅₃
hypocenter	C₅₃	#12012
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #20272784	C₂	semidirect
4. #20272778	C₂	semidirect
55. C₂×#88322	C₁₆	semidirect
119. C₅₃	#12012	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
C₂×C₈×#88322	2
#20272784	2
#20272778	2
C₅₃×#2162	2
#12012	53