Subgroups — #28310271

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811633	C₂	1	2	yes	maximal
2	#20289773	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133296	—	2	2	no	—
5	#16811633	—	2	4	no	—
6	#16811633	C₄	1	8	yes	complemented
7	C₂×#13133286	C₂²	1	4	yes	—
8	C₂×#13133285	—	2	2	no	—
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₆×D₅₃	—	2	4	no	—
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	complemented
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
15	#13133296	—	2	8	no	—
16	#13133296	—	4	4	no	—
17	C₂×C₄⋊#6157	—	2	4	no	—
18	C₂×C₂²⋊#6157	—	4	2	no	—
19	C₂×C₄×#6157	—	4	2	no	—
20	#13133286	—	2	4	no	—
21	#13133286	C₂×C₄	1	8	yes	—
22	#13133285	—	2	8	no	—
23	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂²×#88322	Q₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₃	D₄	1	4	yes	—
28	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	2	yes	—
33	C₁₆×D₅₃	—	2	4	no	—
34	C₁₆×D₅₃	—	2	4	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂³×D₅₃	D₈	1	1	yes	characteristic
39	C₂²×#6157	—	4	4	no	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
42	C₂×C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
45	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
46	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊#6157	—	4	8	no	—
49	C₄⋊#6157	—	2	8	no	—
50	C₄⋊#6157	—	4	4	no	—
51	C₄×#6157	—	4	8	no	—
52	C₂×C₈×C₅₃	C₄²	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#88322	C₄⋊C₄	1	4	yes	—
58	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
59	C₈×D₅₃	C₄²	1	2	yes	—
60	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
61	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
62	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
63	C₈×D₅₃	C₂²×C₄	1	2	yes	—
64	C₈×D₅₃	C₂×D₄	1	2	yes	—
65	C₁₆×C₅₃	—	2	2	no	—
66	#11719908	—	2	2	no	—
67	C₂³×C₅₃	#107	1	1	yes	characteristic
68	C₂²×D₅₃	—	2	4	no	—
69	C₂²×D₅₃	#102	1	4	yes	—
70	C₂²×D₅₃	#107	1	1	yes	characteristic
71	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
72	C₂×#6157	—	4	8	no	—
73	C₂×#6157	—	8	8	no	—
74	C₂×C₄×C₅₃	#95	1	2	yes	—
75	C₂×C₄×C₅₃	#106	1	2	yes	—
76	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×Dic₅₃	#103	1	4	yes	—
79	C₂×Dic₅₃	#107	1	1	yes	characteristic
80	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₄×D₅₃	#95	1	2	yes	—
82	C₄×D₅₃	#102	1	4	yes	—
83	C₄×D₅₃	#103	1	4	yes	—
84	C₄×D₅₃	#106	1	2	yes	—
85	C₄×D₅₃	C₂×C₂²⋊C₄	1	2	yes	—
86	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
87	C₈×C₅₃	#95	1	2	yes	—
88	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
89	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	#88322	#95	1	2	yes	—
91	#88322	C₂×C₄⋊C₄	1	2	yes	—
92	#12556	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	#340	1	2	yes	—
95	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
96	C₂×D₅₃	—	2	2	no	—
97	C₂×D₅₃	—	2	8	no	—
98	C₂×D₅₃	#340	1	2	yes	—
99	C₂×D₅₃	C₂×#102	1	2	yes	—
100	C₂×D₅₃	#508	1	4	yes	—
101	#6157	—	8	8	no	—
102	C₄×C₅₃	#340	1	2	yes	—
103	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
104	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
105	Dic₅₃	#340	1	2	yes	—
106	Dic₅₃	C₂×#103	1	2	yes	—
107	C₂³×C₁₆	—	53	1	no	—
108	C₂×#427	—	53	2	no	—
109	#1380	—	53	4	no	—
110	C₂×C₅₃	—	2	2	no	—
111	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
112	C₂×C₅₃	#2158	1	2	yes	—
113	D₅₃	—	2	2	no	—
114	D₅₃	#2138	1	4	yes	—
115	C₂³×C₈	—	53	1	no	—
116	C₂×#117	—	106	2	no	—
117	C₂²×C₁₆	—	53	1	no	—
118	C₂²×C₁₆	—	53	2	no	—
119	C₂²×C₁₆	—	53	2	no	—
120	C₂²×C₁₆	—	106	4	no	—
121	C₂²×C₁₆	#6157	1	1	yes	characteristic, complemented
122	#427	—	53	8	no	—
123	#427	—	106	4	no	—
124	C₂×#107	—	53	4	no	—
125	C₂×#106	—	106	2	no	—
126	C₅₃	#12556	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₄	—	53	1	no	—
128	C₂²×C₈	—	53	1	no	—
129	C₂²×C₈	—	53	4	no	—
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	4	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
136	#117	—	106	8	no	—
137	#117	—	212	4	no	—
138	C₂×C₄⋊C₄	—	106	4	no	—
139	C₂×C₂²⋊C₄	—	212	2	no	—
140	C₂×C₄²	—	212	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	53	2	no	—
143	C₂×C₁₆	—	106	4	no	—
144	C₂×C₁₆	—	106	4	no	—
145	C₂×C₁₆	—	106	2	no	—
146	C₂×C₁₆	C₂×#6157	1	2	yes	—
147	#107	—	53	8	no	—
148	#107	—	106	4	no	—
149	#106	—	106	8	no	—
150	C₂⁴	—	53	1	no	—
151	C₂²×C₄	—	53	1	no	—
152	C₂²×C₄	—	53	4	no	—
153	C₂²×C₄	—	53	4	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	1	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	212	4	no	—
159	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
160	C₂×C₈	—	53	4	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	1	no	—
163	C₂×C₈	—	53	4	no	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	C₄×#6157	1	2	yes	—
170	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊#6157	1	2	yes	—
172	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
173	C₄⋊C₄	—	106	8	no	—
174	C₄⋊C₄	—	212	4	no	—
175	C₂²⋊C₄	—	212	8	no	—
176	C₄²	—	212	8	no	—
177	C₁₆	—	2	2	no	—
178	C₁₆	—	106	2	no	—
179	C₂³	—	53	1	no	—
180	C₂³	—	53	1	no	—
181	C₂³	—	53	4	no	—
182	C₂³	—	106	4	no	—
183	C₂³	#13133286	1	1	yes	characteristic
184	C₂×C₄	—	53	1	no	—
185	C₂×C₄	—	53	1	no	—
186	C₂×C₄	—	53	4	no	—
187	C₂×C₄	—	53	2	no	—
188	C₂×C₄	—	53	2	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	2	no	—
191	C₂×C₄	—	53	4	no	—
192	C₂×C₄	—	53	4	no	—
193	C₂×C₄	—	424	8	no	—
194	C₂×C₄	—	212	8	no	—
195	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
196	C₂×C₄	#13133285	1	2	yes	—
197	C₂×C₄	#13133286	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
199	C₈	—	53	2	no	—
200	C₈	—	53	2	no	—
201	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
202	C₈	C₂×C₄×#6157	1	1	yes	characteristic
203	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	—	53	2	no	—
206	C₂²	—	53	4	no	—
207	C₂²	—	53	2	no	—
208	C₂²	—	106	8	no	—
209	C₂²	—	106	2	no	—
210	C₂²	#16811497	1	2	yes	—
211	C₂²	C₂×#13133286	1	1	yes	characteristic
212	C₄	—	53	2	no	—
213	C₄	—	53	2	no	—
214	C₄	—	424	8	no	—
215	C₄	#16811497	1	2	yes	—
216	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
217	C₄	C₂×#13133285	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	53	4	no	—
220	C₂	—	106	2	no	—
221	C₂	C₂×#16811497	1	1	yes	characteristic
222	C₂	#20291066	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133286
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#12556
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20289773	C₂	semidirect
6. #16811633	C₄	semidirect
10. C₂²×#11719908	C₄	semidirect
12. C₂×C₁₆×D₅₃	C₄	semidirect
14. C₂²×C₁₆×C₅₃	C₄	semidirect
121. C₂²×C₁₆	#6157	semidirect
126. C₅₃	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811633	2
#20289773	2
C₂²×C₁₆×D₅₃	2
#12556	53