Subgroups — #27282883

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	#16810378	—	2	2	no	—
8	#16810378	—	2	2	no	—
9	Q₈⋊#88322	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#391	C₄	1	2	yes	—
13	C₅₃×#389	—	2	2	no	—
14	C₅₃×#384	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	D₄	1	4	yes	—
22	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
25	C₄×Q₈×C₅₃	—	2	4	no	—
26	C₄×Q₈×C₅₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#95×C₅₃	—	2	4	no	—
38	C₂×Q₈×C₅₃	—	2	2	no	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	—	2	4	no	—
41	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	—	2	8	no	—
50	C₄⋊C₄×C₅₃	—	2	4	no	—
51	C₄⋊C₄×C₅₃	—	4	4	no	—
52	C₄⋊C₄×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	—	2	2	no	—
54	C₄²×C₅₃	—	2	4	no	—
55	C₄²×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	C₂×C₈	1	2	yes	—
59	C₄²×C₅₃	M₁₆	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₄²×C₅₃	#41	1	2	yes	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	4	4	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	4	no	—
66	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	Q₈×C₅₃	—	2	8	no	—
68	Q₈×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
81	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₅₃	#121	1	2	yes	—
83	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#131	1	2	yes	—
88	C₂×C₄×C₅₃	#137	1	4	yes	—
89	#88322	—	4	4	no	—
90	#88322	—	8	2	no	—
91	#9641	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	#390	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
94	C₂²×C₅₃	#408	1	1	yes	characteristic
95	C₂²×C₅₃	#419	1	2	yes	—
96	C₂²×C₅₃	#421	1	2	yes	—
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	—	2	4	no	—
99	C₄×C₅₃	—	4	4	no	—
100	C₄×C₅₃	—	4	2	no	—
101	C₄×C₅₃	—	8	2	no	—
102	C₄×C₅₃	C₈×D₄	1	2	yes	—
103	C₄×C₅₃	#436	1	2	yes	—
104	C₄×C₅₃	#491	1	2	yes	—
105	C₄×C₅₃	#497	1	2	yes	—
106	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
107	#1831	—	53	1	no	—
108	#1744	—	53	1	no	—
109	C₂×Q₈⋊C₈	—	53	4	no	—
110	C₂×C₅₃	#1479	1	2	yes	—
111	C₂×C₅₃	#1497	1	2	yes	—
112	C₂×C₅₃	#1954	1	1	yes	characteristic
113	C₂×C₅₃	#1960	1	1	yes	characteristic
114	C₂×C₅₃	#1963	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
116	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
117	C₂×C₄⋊C₈	—	53	2	no	—
118	C₂×C₄×C₈	—	53	2	no	—
119	#391	Dic₅₃	1	2	yes	—
120	#389	—	2	2	no	—
121	#384	Dic₅₃	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	2	2	no	—
123	C₄³	C₂×D₅₃	1	1	yes	characteristic
124	#336	—	106	2	no	—
125	#336	—	106	2	no	—
126	Q₈⋊C₈	—	106	8	no	—
127	C₅₃	#9641	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
129	C₂²×C₈	—	106	2	no	—
130	C₂²×C₈	—	106	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	#88322	1	4	yes	complemented
133	C₄×Q₈	—	2	4	no	—
134	C₄×Q₈	—	2	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
143	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
144	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₄⋊C₈	—	106	4	no	—
146	C₄×C₈	—	53	4	no	—
147	C₄×C₈	—	106	2	no	—
148	#95	—	2	4	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	C₂×#88322	1	4	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
156	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
160	C₂×C₈	—	106	4	no	—
161	C₂×C₈	—	212	2	no	—
162	C₂×C₈	—	212	4	no	—
163	C₂×C₈	—	212	4	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂×#88322	1	2	yes	—
174	C₄²	#11719917	1	2	yes	—
175	C₄²	D₄×D₅₃	1	2	yes	—
176	C₄²	#11719947	1	2	yes	—
177	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
178	Q₈	—	2	8	no	—
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	C₂²⋊#88322	1	4	yes	—
192	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
193	C₂×C₄	#13133188	1	2	yes	—
194	C₂×C₄	#13133204	1	1	yes	characteristic
195	C₂×C₄	#13133213	1	1	yes	characteristic
196	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
197	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
198	C₂×C₄	#13133224	1	2	yes	—
199	C₂×C₄	#13133248	1	4	yes	—
200	C₈	—	424	2	no	—
201	C₈	—	212	4	no	—
202	C₂²	#16810526	1	1	yes	characteristic
203	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
204	C₂²	#16810590	1	1	yes	characteristic
205	C₂²	#16810632	1	2	yes	—
206	C₂²	#16810640	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	4	no	—
211	C₄	—	8	2	no	—
212	C₄	D₄×#88322	1	2	yes	—
213	C₄	#16810701	1	2	yes	—
214	C₄	#16810991	1	2	yes	—
215	C₄	#16811015	1	2	yes	—
216	C₂	#20273110	1	2	yes	—
217	C₂	#20273188	1	2	yes	—
218	C₂	#20275510	1	1	yes	characteristic
219	C₂	#20275536	1	1	yes	characteristic
220	C₂	#20275548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#9641
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊Q₈×C₅₃	C₈	semidirect
127. C₅₃	#9641	semidirect
132. C₄⋊Q₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
#20274461	2
C₂×Q₈⋊#88322	2
C₄×C₄⋊Q₈×C₅₃	2
#9641	53