Subgroups — #27302113

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	2	yes	maximal
2	#20276471	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
4	#20272797	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133157	C₂²	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	1	yes	characteristic
9	#16810444	—	2	4	no	—
10	C₄⋊#11719908	—	2	1	no	—
11	C₄⋊#11719908	—	2	1	no	—
12	#16810399	—	2	1	no	—
13	C₄×#11719908	—	2	1	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
18	C₅₃×#336	C₄	1	2	yes	—
19	#13133157	—	2	1	no	—
20	#13133157	—	2	1	no	—
21	#13133157	—	2	2	no	—
22	#13133157	—	2	2	no	—
23	#13133157	D₄	1	2	yes	—
24	#13133157	Q₈	1	2	yes	—
25	C₂×#11719908	—	2	1	no	—
26	C₂×#11719908	—	2	1	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
35	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
42	#95×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
67	#11719908	—	2	2	no	—
68	#11719908	—	2	2	no	—
69	#11719908	—	4	2	no	—
70	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	4	1	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	—	2	4	no	—
89	C₈×C₅₃	#121	1	1	yes	characteristic
90	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₈×C₅₃	#125	1	1	yes	characteristic
92	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
93	#13163	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₅₃	#436	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	4	4	no	—
103	C₄×C₅₃	#364	1	2	yes	complemented
104	C₄×C₅₃	#389	1	1	yes	characteristic
105	C₄×C₅₃	#424	1	1	yes	characteristic
106	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
107	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
108	C₂×#364	—	53	2	no	—
109	#2150	—	53	1	no	—
110	C₄×M₃₂	—	53	1	no	—
111	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
112	#1376	—	53	2	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	#1939	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
117	C₂×C₅₃	#2153	1	1	yes	characteristic
118	C₂×M₃₂	—	53	1	no	—
119	C₂×M₃₂	—	53	1	no	—
120	C₂×M₃₂	—	53	2	no	—
121	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
123	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
124	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
125	#364	—	106	4	no	—
126	C₄⋊C₁₆	—	106	1	no	—
127	C₄⋊C₁₆	—	106	1	no	—
128	#346	—	106	1	no	—
129	C₄×C₁₆	—	106	1	no	—
130	#336	Dic₅₃	1	2	yes	—
131	C₅₃	#13163	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
133	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
137	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
138	M₃₂	—	53	2	no	—
139	M₃₂	—	53	2	no	—
140	M₃₂	—	106	1	no	—
141	M₃₂	—	106	1	no	—
142	M₃₂	—	106	2	no	—
143	M₃₂	—	106	2	no	—
144	C₂×C₁₆	—	106	1	no	—
145	C₂×C₁₆	—	106	1	no	—
146	C₂×C₁₆	—	106	2	no	—
147	C₄⋊C₈	#88322	1	2	yes	—
148	C₄⋊C₈	#88322	1	2	yes	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
152	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
153	C₄×C₈	D₂₁₂	1	1	yes	characteristic
154	C₄×C₈	D₂₁₂	1	1	yes	characteristic
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#11719917	1	1	yes	characteristic
158	C₂²×C₄	#11719917	1	1	yes	characteristic
159	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	C₂×#88322	1	4	yes	—
165	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
168	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
169	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
170	C₂×C₈	#11719947	1	1	yes	characteristic
171	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
172	C₂×C₈	#11719949	1	1	yes	characteristic
173	C₄⋊C₄	—	2	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂×#88322	1	1	yes	characteristic
177	C₄²	C₂×#88322	1	1	yes	characteristic
178	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₁₆	—	106	2	no	—
181	C₁₆	—	106	2	no	—
182	C₁₆	—	212	2	no	—
183	C₂³	C₂×#11719917	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
190	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
191	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
192	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
194	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
195	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
196	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
197	C₂×C₄	#13133224	1	1	yes	characteristic
198	C₂×C₄	#13133224	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	2	4	no	—
202	C₈	#13133192	1	1	yes	characteristic
203	C₈	#13133197	1	1	yes	characteristic
204	C₈	#13133208	1	1	yes	characteristic
205	C₈	#13133216	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
209	C₂²	#16810701	1	1	yes	characteristic
210	C₂²	#16810754	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	#16810444	1	2	yes	complemented
216	C₄	#16810522	1	1	yes	characteristic
217	C₄	#16810650	1	1	yes	characteristic
218	C₄	D₄×#88322	1	1	yes	characteristic
219	C₄	Q₈×#88322	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#20275434	1	1	yes	characteristic
223	C₂	C₂×#16810444	1	1	yes	characteristic
224	C₂	#20276481	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#13163
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
103. C₄×C₅₃	#364	semidirect
131. C₅₃	#13163	semidirect
215. C₄	#16810444	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
#20276471	2
C₄×#13133157	2
#20272797	2
C₈×C₄⋊C₄×C₅₃	2
#13163	53