Subgroups — #27382883

Order 13568.

1090 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×#88322	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊#88322	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×#88322	C₂	1	4	yes	maximal
6	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	2	yes	—
8	#16810676	C₄	1	8	yes	complemented
9	C₂×C₄⋊#88322	C₄	1	8	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	8	yes	—
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	#16810378	C₂²	1	8	yes	—
17	C₂×#117×C₅₃	C₄	1	2	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#376	C₄	1	2	yes	—
24	C₄³×C₅₃	C₂²	1	2	yes	—
25	C₂²×#88322	—	4	4	no	—
26	C₂²×#88322	C₂×C₄	1	8	yes	—
27	C₂²×#88322	C₂³	1	4	yes	—
28	C₂²×#88322	C₂³	1	4	yes	—
29	C₄⋊#88322	C₂×C₄	1	8	yes	—
30	C₄⋊#88322	C₂×C₄	1	8	yes	—
31	C₄⋊#88322	C₂×C₄	1	8	yes	—
32	C₂²⋊#88322	C₂×C₄	1	8	yes	—
33	C₄×#88322	—	4	8	no	—
34	C₄×#88322	C₂×C₄	1	8	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	#117×C₅₃	C₈	1	16	yes	complemented
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	#95×C₅₃	C₂×C₄	1	4	yes	—
51	#95×C₅₃	C₂×C₄	1	4	yes	—
52	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
57	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	C₂×C₈	1	16	yes	complemented
69	C₂²⋊C₄×C₅₃	C₂×C₈	1	16	yes	—
70	C₄²×C₅₃	—	2	4	no	—
71	C₄²×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	4	4	no	—
73	C₄²×C₅₃	C₄²	1	4	yes	—
74	C₄²×C₅₃	C₄²	1	4	yes	—
75	C₄²×C₅₃	C₂×C₈	1	16	yes	complemented
76	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	8	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	4	8	no	—
84	C₂×#88322	C₂²×C₄	1	8	yes	—
85	C₂×#88322	C₂²×C₄	1	8	yes	—
86	C₂³×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	—	2	2	no	—
88	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
89	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
90	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	8	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	8	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
101	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	16	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
112	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
113	#88322	—	2	8	no	—
114	#88322	—	2	8	no	—
115	#20543	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	8	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
120	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
121	C₂²×C₅₃	#520	1	2	yes	—
122	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
123	C₂²×C₅₃	#568	1	2	yes	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	16	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
129	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₅₃	#518	1	1	yes	characteristic
131	C₄×C₅₃	#519	1	1	yes	characteristic
132	C₄×C₅₃	#520	1	2	yes	—
133	C₂×#431	—	53	1	no	—
134	C₂²×C₄⋊C₈	—	53	1	no	—
135	C₄×#117	D₅₃	1	1	yes	characteristic
136	C₈×C₄⋊C₄	—	53	4	no	—
137	C₄×C₄⋊C₈	—	53	4	no	—
138	C₈×C₂²⋊C₄	—	53	4	no	—
139	C₂×C₅₃	—	2	4	no	—
140	C₂×C₅₃	#2274	1	1	yes	characteristic
141	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
142	C₂×C₅₃	#2873	1	1	yes	characteristic
143	C₂×C₅₃	#2988	1	4	yes	—
144	C₂³×C₈	—	53	2	no	—
145	C₂×#117	Dic₅₃	1	2	yes	—
146	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
147	#431	—	53	8	no	—
148	C₂×C₄⋊C₈	—	53	8	no	—
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄⋊C₈	—	53	2	no	—
152	C₂×C₂²⋊C₈	—	53	2	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₂×C₄×C₈	—	53	8	no	—
155	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
156	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
157	C₄×C₂²⋊C₄	Dic₅₃	1	2	yes	—
158	C₄×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
159	#376	Dic₅₃	1	2	yes	—
160	C₄³	C₂×D₅₃	1	2	yes	—
161	#336	—	53	8	no	—
162	C₅₃	#20543	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
166	C₂²×C₈	—	53	4	no	—
167	C₂²×C₈	—	53	4	no	—
168	C₂²×C₈	—	53	8	no	—
169	C₂²×C₈	—	212	4	no	—
170	#117	#88322	1	16	yes	complemented
171	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
172	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
173	C₂×C₄²	C₄×D₅₃	1	4	yes	—
174	C₂×C₄²	C₄×D₅₃	1	4	yes	—
175	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
176	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
177	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
180	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
182	C₄⋊C₈	—	53	8	no	—
183	C₄⋊C₈	—	53	8	no	—
184	C₄⋊C₈	—	53	8	no	—
185	C₂²⋊C₈	—	53	8	no	—
186	C₄×C₈	—	53	8	no	—
187	C₄×C₈	—	212	8	no	—
188	#95	C₂×Dic₅₃	1	4	yes	—
189	#95	C₂×Dic₅₃	1	4	yes	—
190	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×#88322	1	8	yes	—
195	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
196	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
197	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
203	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
206	C₂×C₈	—	53	8	no	—
207	C₂×C₈	—	53	8	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	106	8	no	—
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	212	8	no	—
212	C₄⋊C₄	C₂×#88322	1	16	yes	complemented
213	C₂²⋊C₄	C₂×#88322	1	16	yes	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂×#88322	1	16	yes	complemented
218	C₄²	C₄×Dic₅₃	1	4	yes	—
219	C₄²	C₄×Dic₅₃	1	4	yes	—
220	C₄²	C₂×C₄×D₅₃	1	2	yes	—
221	C₄²	C₂×C₄×D₅₃	1	2	yes	—
222	C₄²	C₂×C₄×D₅₃	1	2	yes	—
223	C₄²	C₂×C₄×D₅₃	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
227	C₂³	C₂²×#88322	1	4	yes	—
228	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₄×#88322	1	16	yes	complemented
239	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
240	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
241	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
242	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
244	C₂×C₄	C₂²×#88322	1	16	yes	—
245	C₂×C₄	C₂²×#88322	1	4	yes	—
246	C₂×C₄	C₂²×#88322	1	4	yes	—
247	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
249	C₂×C₄	#13133263	1	2	yes	—
250	C₂×C₄	#13133264	1	2	yes	—
251	C₈	—	106	8	no	—
252	C₈	—	106	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×C₄×#88322	1	4	yes	—
257	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
258	C₂²	#16811106	1	2	yes	—
259	C₂²	C₂³×#88322	1	2	yes	—
260	C₂²	#16811390	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	16	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	C₂×C₄×#88322	1	4	yes	—
266	C₄	C₂×C₄×#88322	1	4	yes	—
267	C₄	#16811097	1	1	yes	characteristic
268	C₄	#16811100	1	1	yes	characteristic
269	C₄	#16811103	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	#20277071	1	1	yes	characteristic
272	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
273	C₂	#20283134	1	1	yes	characteristic
274	C₂	#20283826	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₄×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#20543
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×#88322	C₂	semidirect
4. C₄×C₄⋊#88322	C₂	semidirect
8. #16810676	C₄	semidirect
9. C₂×C₄⋊#88322	C₄	semidirect
38. #117×C₅₃	C₈	semidirect
68. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
75. C₄²×C₅₃	C₂×C₈	semidirect
100. C₂×C₄×C₅₃	C₄×C₈	semidirect
162. C₅₃	#20543	semidirect
170. #117	#88322	semidirect
212. C₄⋊C₄	C₂×#88322	semidirect
217. C₄²	C₂×#88322	semidirect
238. C₂×C₄	C₄×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄⋊#88322	2
C₄⋊C₄×#88322	2
C₄×C₄⋊#88322	2
C₂²⋊C₄×#88322	2
C₄×#117×C₅₃	2
#20543	53