Subgroups — #27274709

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 270 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	C₄×#13133128	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	2	no	—
8	C₂×#13133128	—	2	1	no	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
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	—	2	8	no	—
25	#13133128	—	4	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	C₂×C₄	1	4	yes	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₅₃	—	2	4	no	—
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₄×Q₈×C₅₃	—	2	4	no	—
32	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	#95×C₅₃	—	2	2	no	—
47	#95×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	—	2	1	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	C₄²	1	2	yes	—
51	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	—	2	8	no	—
61	C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
65	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
66	C₄²×C₅₃	—	2	8	no	—
67	C₄²×C₅₃	—	2	4	no	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	1	no	—
74	C₄²×C₅₃	—	4	4	no	—
75	C₄²×C₅₃	—	4	1	no	—
76	C₄²×C₅₃	—	4	1	no	—
77	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₃	#41	1	2	yes	—
79	C₂×#88322	—	2	8	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	2	no	—
86	Q₈×C₅₃	—	2	4	no	—
87	Q₈×C₅₃	#97	1	4	yes	complemented
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	#104	1	4	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#137	1	2	yes	—
111	#88322	—	2	8	no	—
112	#88322	—	4	4	no	—
113	#8173	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₅₃	#407	1	2	yes	—
116	C₂²×C₅₃	#417	1	1	yes	characteristic
117	C₂²×C₅₃	#419	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×#104	1	2	yes	—
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	2	4	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	2	no	—
123	C₄×C₅₃	—	4	4	no	—
124	C₄×C₅₃	—	4	2	no	—
125	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
126	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
127	C₄×C₅₃	#432	1	2	yes	—
128	C₄×C₅₃	#440	1	2	yes	—
129	C₄×C₅₃	#441	1	2	yes	—
130	C₄²×Q₈	D₅₃	1	1	yes	characteristic
131	#1767	—	53	1	no	—
132	C₄×#97	—	53	1	no	—
133	C₂×Q₈⋊C₈	—	53	2	no	—
134	#1279	—	53	2	no	—
135	C₂×C₅₃	#1489	1	2	yes	—
136	C₂×C₅₃	#1491	1	2	yes	—
137	C₂×C₅₃	#1749	1	1	yes	characteristic
138	C₂×C₅₃	C₄×#104	1	1	yes	characteristic
139	C₂×C₅₃	#1763	1	1	yes	characteristic
140	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
141	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	53	2	no	—
143	C₂×#97	—	106	2	no	—
144	C₂×#97	—	106	1	no	—
145	C₂×C₄×C₈	—	53	2	no	—
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄×C₄⋊C₄	—	2	1	no	—
148	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
149	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
150	C₄³	—	2	1	no	—
151	C₄³	C₂×D₅₃	1	1	yes	characteristic
152	#336	—	106	1	no	—
153	Q₈⋊C₈	—	53	8	no	—
154	C₅₃	#8173	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	—	106	2	no	—
158	C₄×Q₈	—	2	4	no	—
159	C₄×Q₈	—	2	4	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	C₄×D₅₃	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	—	2	1	no	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
171	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	C₄⋊C₈	—	53	4	no	—
176	C₄⋊C₈	—	106	2	no	—
177	#97	—	106	8	no	—
178	#97	—	212	2	no	—
179	C₄×C₈	—	53	4	no	—
180	C₄×C₈	—	106	2	no	—
181	#95	—	2	1	no	—
182	#95	—	2	2	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	#11719917	1	4	yes	—
186	C₂×Q₈	C₄×Dic₅₃	1	2	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
191	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂×C₈	—	106	8	no	—
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	4	no	—
198	C₂×C₈	—	212	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	#11719917	1	4	yes	—
205	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
206	C₄²	—	2	8	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	4	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	1	no	—
217	C₄²	C₂×C₄×D₅₃	1	2	yes	—
218	C₄²	#11719945	1	2	yes	—
219	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	2	4	no	—
222	Q₈	#13133128	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	#13133144	1	4	yes	—
237	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	#13133176	1	1	yes	characteristic
239	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
240	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
241	C₂×C₄	C₂×#11719917	1	2	yes	—
242	C₂×C₄	C₂×#11719917	1	2	yes	—
243	C₂×C₄	#13133242	1	1	yes	characteristic
244	C₂×C₄	#13133247	1	1	yes	characteristic
245	C₂×C₄	#13133248	1	2	yes	—
246	C₈	—	106	8	no	—
247	C₈	—	212	4	no	—
248	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₂²	#16810587	1	2	yes	—
250	C₂²	#16810624	1	1	yes	characteristic
251	C₂²	#16810632	1	1	yes	characteristic
252	C₂²	C₂×#13133144	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	2	no	—
259	C₄	C₂×#13133128	1	1	yes	characteristic
260	C₄	C₄×#11719917	1	1	yes	characteristic
261	C₄	#16810680	1	2	yes	—
262	C₄	#16810721	1	2	yes	—
263	C₄	#16810728	1	2	yes	—
264	C₂	#20273154	1	2	yes	—
265	C₂	#20273162	1	2	yes	—
266	C₂	#20274477	1	1	yes	characteristic
267	C₂	C₄×#13133144	1	1	yes	characteristic
268	C₂	#20274545	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	#8173
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
87. Q₈×C₅₃	#97	semidirect
154. C₅₃	#8173	semidirect
222. 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
C₄×#13133128	2
C₂×Q₈⋊#88322	2
#20272556	2
C₄²×Q₈×C₅₃	2
#8173	53