Subgroups — #28302171

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290262	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#88324	C₂	1	2	yes	maximal
3	C₂×#16811464	C₂	1	1	yes	maximal, characteristic
4	#20289548	C₂	1	2	yes	maximal
5	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	#16811486	C₄	1	2	yes	—
10	#16811484	C₄	1	2	yes	—
11	#16811464	—	2	2	no	—
12	#16811464	C₄	1	2	yes	—
13	#16811464	C₄	1	2	yes	—
14	C₈×#88324	—	2	2	no	—
15	C₂×#13133153	C₄	1	1	yes	characteristic
16	M₁₆×Dic₅₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#16810387	C₄	1	2	yes	—
19	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	—	2	1	no	—
21	C₂²×#88324	Q₈	1	2	yes	—
22	C₄⋊#88324	C₂×C₄	1	4	yes	—
23	C₂²⋊#88324	—	2	2	no	—
24	C₄×#88324	—	2	2	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	C₂×C₄	1	2	yes	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	D₄	1	2	yes	—
29	C₂×#11719917	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	#13133153	C₈	1	4	yes	complemented
33	#13133127	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	#108×C₅₃	C₈	1	4	yes	complemented
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×Dic₅₃	#41	1	1	yes	characteristic
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	—	2	4	no	—
47	C₂×#88324	—	4	2	no	—
48	C₂×#88324	—	4	2	no	—
49	C₂×#88324	C₄⋊C₄	1	4	yes	—
50	M₁₆×C₅₃	—	2	2	no	—
51	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
57	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₅₃	#41	1	1	yes	characteristic
61	#11719917	—	2	2	no	—
62	#11719917	C₂×C₈	1	4	yes	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	C₄²	1	1	yes	characteristic
67	C₂×#88322	C₄²	1	1	yes	characteristic
68	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×D₄	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	#131	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×Dic₅₃	#117	1	1	yes	characteristic
79	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
81	#88324	—	2	4	no	—
82	#88324	—	4	2	no	—
83	#88324	—	4	4	no	—
84	C₈×C₅₃	—	2	4	no	—
85	C₈×C₅₃	C₄×C₈	1	2	yes	—
86	C₈×C₅₃	C₄×C₈	1	2	yes	—
87	#88322	—	2	1	no	—
88	#88322	—	2	1	no	—
89	#88322	—	2	4	no	—
90	#8294	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
95	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
96	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₅₃	#405	1	1	yes	characteristic
98	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
99	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
100	Dic₅₃	—	2	1	no	—
101	Dic₅₃	—	2	1	no	—
102	Dic₅₃	#430	1	2	yes	—
103	Dic₅₃	#444	1	2	yes	—
104	C₄×#108	—	53	1	no	—
105	#1564	—	53	1	no	—
106	C₈×M₁₆	—	53	2	no	—
107	C₂×#322	—	53	1	no	—
108	#1278	—	53	2	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₅₃	#1773	1	1	yes	characteristic
113	C₂×C₅₃	#1778	1	1	yes	characteristic
114	#431	—	53	2	no	—
115	C₂×#108	—	53	1	no	—
116	C₂×#108	#6157	1	1	yes	characteristic
117	C₄×M₁₆	—	53	2	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	#341	—	53	2	no	—
122	#335	—	53	2	no	—
123	#334	—	53	2	no	—
124	#322	—	53	2	no	—
125	#322	—	53	2	no	—
126	#322	—	106	2	no	—
127	C₈²	—	106	2	no	—
128	C₅₃	#8294	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	—	53	2	no	—
130	C₂×M₁₆	C₂×#6157	1	2	yes	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	2	no	—
133	C₂²×C₈	—	106	1	no	—
134	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
135	C₂×C₄²	—	53	1	no	—
136	#108	—	53	4	no	—
137	#108	#88324	1	4	yes	complemented
138	C₄⋊C₈	—	53	4	no	—
139	C₂²⋊C₈	—	106	2	no	—
140	#97	—	106	2	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	—	53	2	no	—
143	C₄×C₈	—	53	1	no	—
144	C₄×C₈	—	53	1	no	—
145	C₄×C₈	—	53	1	no	—
146	C₄×C₈	—	53	1	no	—
147	C₄×C₈	—	53	2	no	—
148	C₄×C₈	—	106	2	no	—
149	C₄×C₈	—	106	2	no	—
150	C₄×C₈	—	106	1	no	—
151	C₂²×C₄	—	53	1	no	—
152	C₂²×C₄	—	53	1	no	—
153	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	53	4	no	—
156	M₁₆	—	106	2	no	—
157	M₁₆	C₂×#88324	1	4	yes	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	53	4	no	—
160	C₂×C₈	—	53	1	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	1	no	—
164	C₂×C₈	—	106	1	no	—
165	C₂×C₈	—	106	2	no	—
166	C₂×C₈	—	106	4	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
170	C₂×C₈	C₄×#6157	1	1	yes	characteristic
171	C₂×C₈	C₄×#6157	1	1	yes	characteristic
172	C₂×C₈	C₂×#88324	1	2	yes	—
173	C₄²	—	53	1	no	—
174	C₄²	—	53	1	no	—
175	C₄²	—	53	1	no	—
176	C₄²	—	53	1	no	—
177	C₂³	Q₈×#6157	1	1	yes	characteristic
178	C₂×C₄	—	53	1	no	—
179	C₂×C₄	—	53	1	no	—
180	C₂×C₄	—	53	1	no	—
181	C₂×C₄	—	53	1	no	—
182	C₂×C₄	—	106	2	no	—
183	C₂×C₄	—	106	2	no	—
184	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
185	C₂×C₄	D₄×#6157	1	1	yes	characteristic
186	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
187	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
188	C₂×C₄	#13133304	1	1	yes	characteristic
189	C₂×C₄	#13133305	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	106	4	no	—
192	C₈	—	106	1	no	—
193	C₈	—	106	1	no	—
194	C₈	—	106	4	no	—
195	C₈	—	212	4	no	—
196	C₈	—	212	2	no	—
197	C₈	C₄×#88324	1	2	yes	—
198	C₈	C₄×#88324	1	2	yes	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
202	C₂²	Q₈×#88324	1	1	yes	characteristic
203	C₂²	#16811672	1	1	yes	characteristic
204	C₄	—	53	2	no	—
205	C₄	—	53	2	no	—
206	C₄	—	106	1	no	—
207	C₄	—	106	1	no	—
208	C₄	C₂×C₄×#88324	1	1	yes	characteristic
209	C₄	#16811588	1	1	yes	characteristic
210	C₄	D₄×#88324	1	1	yes	characteristic
211	C₄	#16811651	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₄⋊C₄×#88324	1	1	yes	characteristic
215	C₂	#20290524	1	1	yes	characteristic
216	C₂	#20290536	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	#8294
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
32. #13133153	C₈	semidirect
41. #108×C₅₃	C₈	semidirect
128. C₅₃	#8294	semidirect
137. #108	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290262	2
M₁₆×#88324	2
C₂×#16811464	2
#20289548	2
#108×Dic₅₃	2
#8294	53