Subgroups — #27353081

Order 13568.

734 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	#20283826	C₂	1	2	yes	maximal
2	#20273467	C₂	1	6	yes	maximal, complemented
3	#20273463	C₂	1	3	yes	maximal, complemented
4	C₂×#16810374	C₂	1	3	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	—	2	2	no	—
7	D₄×#88322	—	2	6	no	—
8	#16810676	C₂²	1	6	yes	—
9	C₂×C₄⋊#88322	C₂²	1	6	yes	—
10	#16810376	C₂²	1	6	yes	—
11	#16810374	C₂²	1	3	yes	—
12	#16810374	C₂²	1	6	yes	—
13	#16810374	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
16	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#405	C₄	1	3	yes	—
18	C₅₃×#405	C₄	1	3	yes	—
19	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
21	C₂²×#88322	—	2	6	no	—
22	C₄⋊#88322	—	2	6	no	—
23	C₄⋊#88322	C₂³	1	6	yes	—
24	C₄⋊#88322	C₂³	1	2	yes	—
25	C₂²⋊#88322	—	2	6	no	—
26	C₄×#88322	—	2	6	no	—
27	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#131×C₅₃	—	2	4	no	—
29	#131×C₅₃	C₈	1	8	yes	complemented
30	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
34	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
36	C₄×D₄×C₅₃	D₄	1	3	yes	—
37	C₄×D₄×C₅₃	Q₈	1	3	yes	—
38	#117×C₅₃	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
40	#97×C₅₃	C₂×C₄	1	6	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	3	yes	—
45	#41×C₅₃	C₂×C₈	1	4	yes	—
46	#41×C₅₃	M₁₆	1	4	yes	—
47	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₅₃	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
52	M₁₆×C₅₃	—	2	6	no	—
53	M₁₆×C₅₃	C₂×C₈	1	12	yes	complemented
54	C₂×C₈×C₅₃	—	2	6	no	—
55	C₂×C₈×C₅₃	C₂×C₈	1	12	yes	complemented
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
60	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
61	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
62	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
63	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
64	C₄²×C₅₃	C₂×D₄	1	3	yes	—
65	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
66	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	—	2	6	no	—
69	C₂×#88322	—	8	6	no	—
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	Q₈×C₅₃	C₄⋊C₈	1	4	yes	—
72	D₄×C₅₃	C₄⋊C₈	1	12	yes	—
73	C₂×C₄×C₅₃	—	2	6	no	—
74	C₂×C₄×C₅₃	—	4	3	no	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
79	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	3	no	—
85	C₈×C₅₃	C₂²×C₈	1	2	yes	—
86	C₈×C₅₃	C₂²×C₈	1	6	yes	—
87	#88322	—	4	2	no	—
88	#88322	—	4	6	no	—
89	#17788	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	3	no	—
91	C₂²×C₅₃	—	2	3	no	—
92	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
93	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₅₃	—	2	3	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	3	no	—
97	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
98	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
99	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
100	C₄×C₅₃	#577	1	2	yes	—
101	#2988	—	53	2	no	—
102	C₄×#131	D₅₃	1	1	yes	characteristic
103	#1565	—	53	6	no	—
104	#1564	—	53	3	no	—
105	C₂×#334	—	53	3	no	—
106	C₂×C₅₃	—	2	6	no	—
107	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
108	C₂×C₅₃	#2912	1	1	yes	characteristic
109	C₂×C₅₃	#2916	1	1	yes	characteristic
110	C₂×#131	Dic₅₃	1	1	yes	characteristic
111	C₂×#131	Dic₅₃	1	1	yes	characteristic
112	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
113	C₈×Q₈	—	106	2	no	—
114	C₈×D₄	—	106	6	no	—
115	#431	—	53	6	no	—
116	C₂×C₄⋊C₈	—	53	6	no	—
117	#405	Dic₅₃	1	3	yes	—
118	#405	Dic₅₃	1	3	yes	—
119	C₄×M₁₆	C₂×D₅₃	1	3	yes	—
120	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
121	#335	—	53	6	no	—
122	#334	—	53	3	no	—
123	#334	—	53	6	no	—
124	#334	—	53	1	no	—
125	C₅₃	#17788	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
127	#131	—	2	4	no	—
128	#131	#88322	1	8	yes	complemented
129	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
130	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
131	C₂²×C₈	—	106	6	no	—
132	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
133	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
134	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
135	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
136	C₄×D₄	Dic₁₀₆	1	3	yes	—
137	C₄×D₄	D₂₁₂	1	3	yes	—
138	#117	C₂×Dic₅₃	1	3	yes	—
139	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
140	C₄⋊C₈	—	53	6	no	—
141	C₄⋊C₈	—	53	2	no	—
142	C₄⋊C₈	—	106	6	no	—
143	C₂²⋊C₈	—	106	6	no	—
144	#97	C₂×Dic₅₃	1	6	yes	—
145	C₄×C₈	—	106	6	no	—
146	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
147	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
148	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
149	C₄×C₈	C₂²×D₅₃	1	3	yes	—
150	#41	C₂×#88322	1	4	yes	—
151	#41	#11719917	1	4	yes	—
152	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
153	C₂×D₄	C₄⋊Dic₅₃	1	3	yes	—
154	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
155	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
156	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
157	M₁₆	—	2	6	no	—
158	M₁₆	C₂×#88322	1	12	yes	complemented
159	C₂×C₈	—	2	6	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	6	no	—
162	C₂×C₈	—	424	6	no	—
163	C₂×C₈	C₂×#88322	1	12	yes	complemented
164	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
166	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
167	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
168	C₄⋊C₄	C₄⋊Dic₅₃	1	3	yes	—
169	C₄⋊C₄	C₄⋊Dic₅₃	1	3	yes	—
170	C₂²⋊C₄	C₄⋊Dic₅₃	1	6	yes	—
171	C₄²	C₂×Dic₁₀₆	1	3	yes	—
172	C₄²	C₂×D₂₁₂	1	3	yes	—
173	C₄²	C₂²×Dic₅₃	1	3	yes	—
174	C₄²	C₂³×D₅₃	1	1	yes	characteristic
175	C₂³	C₂×C₄⋊Dic₅₃	1	3	yes	—
176	Q₈	C₄⋊#88322	1	4	yes	—
177	D₄	C₄⋊#88322	1	12	yes	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	4	3	no	—
180	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
183	C₂×C₄	C₂²×#88322	1	6	yes	—
184	C₂×C₄	C₂×#11719917	1	6	yes	—
185	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	212	2	no	—
191	C₈	—	212	6	no	—
192	C₈	C₂²×#88322	1	2	yes	—
193	C₈	C₂²×#88322	1	6	yes	—
194	C₂²	—	2	3	no	—
195	C₂²	—	2	3	no	—
196	C₂²	C₂×C₄⋊#88322	1	6	yes	—
197	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
198	C₄	—	2	3	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	3	no	—
201	C₄	C₂×C₄⋊#88322	1	6	yes	—
202	C₄	C₂³×#88322	1	1	yes	characteristic
203	C₄	C₂²×#11719917	1	1	yes	characteristic
204	C₄	#16811435	1	2	yes	—
205	C₂	—	2	6	no	—
206	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
207	C₂	#20283310	1	1	yes	characteristic
208	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#17788
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20273467	C₂	semidirect
3. #20273463	C₂	semidirect
4. C₂×#16810374	C₂	semidirect
29. #131×C₅₃	C₈	semidirect
53. M₁₆×C₅₃	C₂×C₈	semidirect
55. C₂×C₈×C₅₃	C₂×C₈	semidirect
125. C₅₃	#17788	semidirect
128. #131	#88322	semidirect
158. M₁₆	C₂×#88322	semidirect
163. C₂×C₈	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
#20283826	2
#20273467	2
#20273463	2
C₂×#16810374	2
C₄×#131×C₅₃	2
#17788	53