Subgroups — #27274719

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	#20274461	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	#20272556	C₂	1	2	yes	maximal
5	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133128	—	2	1	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	—	2	2	no	—
10	#16810378	—	2	1	no	—
11	Q₈⋊#88322	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₄³×C₅₃	—	2	1	no	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	C₂×C₄	1	4	yes	—
24	#13133128	—	4	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	C₂×C₄	1	4	yes	—
27	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Q₈×C₅₃	—	2	4	no	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	—	2	4	no	—
31	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#95×C₅₃	—	2	2	no	—
46	#95×C₅₃	—	2	1	no	—
47	C₂×Q₈×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
50	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	—	2	8	no	—
60	C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
64	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
65	C₄²×C₅₃	—	2	8	no	—
66	C₄²×C₅₃	—	2	4	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	—	4	4	no	—
72	C₄²×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	SD₁₆	1	2	yes	—
74	C₄²×C₅₃	Q₁₆	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	2	yes	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	4	4	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	2	no	—
84	Q₈×C₅₃	—	2	4	no	—
85	Q₈×C₅₃	#97	1	4	yes	complemented
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	#103	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#135	1	2	yes	—
109	#88322	—	4	4	no	—
110	#88322	—	4	4	no	—
111	#8175	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₅₃	#407	1	2	yes	—
114	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
115	C₂²×C₅₃	#416	1	1	yes	characteristic
116	C₂²×C₅₃	#421	1	2	yes	—
117	C₄×C₅₃	—	2	4	no	—
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	4	4	no	—
122	C₄×C₅₃	—	4	2	no	—
123	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
124	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
125	C₄×C₅₃	#432	1	2	yes	—
126	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
127	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
128	C₄²×Q₈	D₅₃	1	1	yes	characteristic
129	#1767	—	53	1	no	—
130	#1744	—	53	1	no	—
131	C₂×Q₈⋊C₈	—	53	2	no	—
132	#1279	—	53	2	no	—
133	C₂×C₅₃	#1487	1	2	yes	—
134	C₂×C₅₃	#1488	1	2	yes	—
135	C₂×C₅₃	#1749	1	1	yes	characteristic
136	C₂×C₅₃	#1759	1	1	yes	characteristic
137	C₂×C₅₃	C₄×#103	1	1	yes	characteristic
138	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
139	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×C₄⋊C₈	—	53	2	no	—
141	C₂×#97	—	106	1	no	—
142	C₂×C₄×C₈	—	53	2	no	—
143	C₄×C₄⋊C₄	—	2	2	no	—
144	C₄×C₄⋊C₄	—	2	1	no	—
145	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
146	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
147	C₄³	—	2	1	no	—
148	C₄³	C₂×D₅₃	1	1	yes	characteristic
149	#336	—	106	2	no	—
150	#336	—	106	1	no	—
151	Q₈⋊C₈	—	53	8	no	—
152	C₅₃	#8175	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₈	—	106	2	no	—
155	C₂²×C₈	—	106	2	no	—
156	C₄×Q₈	—	2	4	no	—
157	C₄×Q₈	—	2	4	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	C₄×D₅₃	1	4	yes	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
171	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
173	C₄⋊C₈	—	53	4	no	—
174	C₄⋊C₈	—	106	2	no	—
175	#97	—	212	2	no	—
176	C₄×C₈	—	53	4	no	—
177	C₄×C₈	—	106	2	no	—
178	#95	—	2	1	no	—
179	#95	—	2	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	#11719917	1	4	yes	—
183	C₂×Q₈	C₄×Dic₅₃	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
188	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	—	106	4	no	—
194	C₂×C₈	—	212	4	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	—	212	2	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	#11719917	1	4	yes	—
202	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
203	C₄²	—	2	8	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	4	2	no	—
211	C₄²	#11719924	1	2	yes	—
212	C₄²	#11719925	1	2	yes	—
213	C₄²	C₂×C₄×D₅₃	1	2	yes	—
214	C₄²	#11719945	1	2	yes	—
215	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	2	4	no	—
218	Q₈	#13133128	1	4	yes	complemented
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	#13133141	1	4	yes	—
233	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	#13133176	1	1	yes	characteristic
235	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
236	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
237	C₂×C₄	C₂×#11719917	1	2	yes	—
238	C₂×C₄	C₂×#11719917	1	2	yes	—
239	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
240	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
241	C₂×C₄	#13133237	1	2	yes	—
242	C₈	—	212	4	no	—
243	C₈	—	212	4	no	—
244	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
245	C₂²	#16810587	1	2	yes	—
246	C₂²	C₂×#13133141	1	1	yes	characteristic
247	C₂²	#16810619	1	1	yes	characteristic
248	C₂²	#16810640	1	2	yes	—
249	C₄	—	2	4	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	4	no	—
255	C₄	C₂×#13133128	1	1	yes	characteristic
256	C₄	C₄×#11719917	1	1	yes	characteristic
257	C₄	#16810680	1	2	yes	—
258	C₄	C₄×#11719924	1	2	yes	—
259	C₄	C₄×#11719925	1	2	yes	—
260	C₂	#20273144	1	2	yes	—
261	C₂	#20273149	1	2	yes	—
262	C₂	#20274477	1	1	yes	characteristic
263	C₂	#20274525	1	1	yes	characteristic
264	C₂	C₄×#13133141	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	#8175
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
11. Q₈⋊#88322	C₄	semidirect
85. Q₈×C₅₃	#97	semidirect
152. C₅₃	#8175	semidirect
218. Q₈	#13133128	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
#20274461	2
C₂×Q₈⋊#88322	2
#20272556	2
C₄²×Q₈×C₅₃	2
#8175	53