Subgroups — #27295459

Order 13568.

292 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
3	#20272775	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
5	#16811180	C₄	1	2	yes	—
6	C₂×#13133157	—	2	1	no	—
7	#16811028	—	2	1	no	—
8	#16811025	C₄	1	2	yes	—
9	#16811025	C₂²	1	2	yes	—
10	#16811021	—	2	1	no	—
11	#16810661	—	2	1	no	—
12	#16810661	—	2	1	no	—
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	#13133217	—	2	2	no	—
20	#13133209	C₂×C₄	1	4	yes	—
21	#13133196	—	4	2	no	—
22	C₄×Dic₁₀₆	—	4	2	no	—
23	#13133176	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	#13133157	—	4	1	no	—
26	C₂×#11719908	—	4	1	no	—
27	#13133151	—	2	2	no	—
28	#13133151	—	2	2	no	—
29	#13133149	—	2	2	no	—
30	#13133149	—	2	2	no	—
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₅₃	—	2	2	no	—
35	M₃₂×C₅₃	—	4	1	no	—
36	C₂×C₁₆×C₅₃	—	4	1	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₆	—	4	2	no	—
44	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
51	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
52	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₃	—	2	1	no	—
57	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	#11719919	—	4	4	no	—
63	C₄×Dic₅₃	—	4	2	no	—
64	C₁₆×C₅₃	—	4	2	no	—
65	#11719908	—	4	2	no	—
66	C₂³×C₅₃	#100	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×Dic₅₃	—	4	4	no	—
77	Dic₁₀₆	—	8	4	no	—
78	C₈×C₅₃	#103	1	1	yes	characteristic
79	C₈×C₅₃	#103	1	1	yes	characteristic
80	C₈×C₅₃	#103	1	1	yes	characteristic
81	C₈×C₅₃	#103	1	1	yes	characteristic
82	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
86	#11956	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	4	1	no	—
88	C₂²×C₅₃	#333	1	1	yes	characteristic
89	C₂²×C₅₃	#333	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	—	4	1	no	—
93	C₄×C₅₃	#333	1	1	yes	characteristic
94	C₄×C₅₃	#333	1	1	yes	characteristic
95	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
96	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
97	Dic₅₃	—	8	4	no	—
98	#3160	—	53	1	no	—
99	C₂×#364	—	53	1	no	—
100	C₂×#364	D₅₃	1	1	yes	characteristic
101	#1366	—	53	4	no	—
102	C₂×C₅₃	—	2	1	no	—
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	C₂×#333	1	1	yes	characteristic
106	#533	—	53	2	no	—
107	C₂×M₃₂	—	2	1	no	—
108	C₂×M₃₂	—	106	1	no	—
109	#500	—	106	1	no	—
110	#499	—	53	2	no	—
111	#499	—	53	2	no	—
112	#498	—	106	1	no	—
113	#427	—	106	1	no	—
114	#427	—	106	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#364	—	2	1	no	—
117	#364	—	53	2	no	—
118	#364	—	106	1	no	—
119	#364	C₂×D₅₃	1	2	yes	—
120	C₅₃	#11956	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₄⋊Q₈	—	106	2	no	—
124	#125	—	53	4	no	—
125	C₂²⋊Q₈	—	212	2	no	—
126	C₄×Q₈	—	212	2	no	—
127	#117	—	106	2	no	—
128	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	4	1	no	—
131	M₃₂	—	106	2	no	—
132	M₃₂	—	212	1	no	—
133	C₂×C₁₆	—	4	1	no	—
134	C₂×C₁₆	—	212	1	no	—
135	#107	—	106	2	no	—
136	#107	—	106	2	no	—
137	#106	—	106	2	no	—
138	#106	—	106	2	no	—
139	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
140	C₄×C₈	D₂₁₂	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
143	C₂×Q₈	—	212	2	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	#11719921	1	1	yes	characteristic
146	C₂×C₈	#11719913	1	1	yes	characteristic
147	C₂×C₈	#11719913	1	1	yes	characteristic
148	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
149	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
150	C₂×C₈	#11719921	1	1	yes	characteristic
151	C₂×C₈	#11719921	1	1	yes	characteristic
152	C₂×C₈	#11719923	1	1	yes	characteristic
153	C₂×C₈	#11719923	1	1	yes	characteristic
154	C₂×C₈	#11719925	1	1	yes	characteristic
155	C₂×C₈	#11719925	1	1	yes	characteristic
156	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄⋊C₄	—	106	4	no	—
159	C₄⋊C₄	—	212	4	no	—
160	C₄⋊C₄	—	212	2	no	—
161	C₂²⋊C₄	—	212	2	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	212	2	no	—
164	C₄²	#11719921	1	1	yes	characteristic
165	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
166	C₁₆	—	4	2	no	—
167	C₁₆	—	212	2	no	—
168	C₂³	#13133133	1	1	yes	characteristic
169	Q₈	—	424	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	212	4	no	—
174	C₂×C₄	#13133133	1	1	yes	characteristic
175	C₂×C₄	#13133140	1	1	yes	characteristic
176	C₂×C₄	#13133140	1	1	yes	characteristic
177	C₂×C₄	#13133142	1	1	yes	characteristic
178	C₂×C₄	#13133142	1	1	yes	characteristic
179	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
180	C₈	#13133140	1	1	yes	characteristic
181	C₈	#13133140	1	1	yes	characteristic
182	C₈	#13133142	1	1	yes	characteristic
183	C₈	#13133142	1	1	yes	characteristic
184	C₈	C₂×#11719923	1	1	yes	characteristic
185	C₈	C₂×#11719913	1	1	yes	characteristic
186	C₈	C₂×#11719925	1	1	yes	characteristic
187	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
188	C₂²	—	4	1	no	—
189	C₂²	#16810371	1	1	yes	characteristic
190	C₂²	#16810371	1	1	yes	characteristic
191	C₂²	C₂×#13133133	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	4	1	no	—
194	C₄	—	424	4	no	—
195	C₄	#16810371	1	1	yes	characteristic
196	C₄	#16810371	1	1	yes	characteristic
197	C₄	C₂×#13133140	1	1	yes	characteristic
198	C₄	C₂×#13133142	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	C₂×#16810371	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810371
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	C₂×#333
prime radical	C₂³×C₅₃	#100
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#11956
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #20272775	C₂	semidirect
120. C₅₃	#11956	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
C₂×#16810444	2
#20272775	2
C₂×C₅₃×#364	2
#11956	53