Subgroups — #27354624

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283650	C₂	1	2	yes	maximal
2	#20273612	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#88322	C₂	1	4	yes	maximal, complemented
4	#20273474	C₂	1	1	yes	maximal, characteristic
5	#20273170	C₂	1	2	yes	maximal
6	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
8	#16810754	—	2	2	no	—
9	#16810754	C₂²	1	4	yes	—
10	Q₈×#88322	—	2	2	no	—
11	Q₈×#88322	C₂²	1	4	yes	—
12	#16810650	—	2	2	no	—
13	#16810647	C₂²	1	2	yes	—
14	#16810579	—	2	2	no	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	4	yes	—
18	Q₈⋊#88322	C₂²	1	8	yes	—
19	C₈×#88322	C₂²	1	2	yes	—
20	C₈×#88322	C₂²	1	2	yes	—
21	C₅₃×#533	C₂²	1	2	yes	—
22	C₅₃×#500	C₄	1	1	yes	characteristic
23	C₅₃×#494	C₄	1	1	yes	characteristic
24	C₅₃×#487	C₄	1	2	yes	—
25	C₅₃×#467	C₄	1	4	yes	—
26	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#11719917	—	2	2	no	—
29	C₂²×#88322	—	2	1	no	—
30	C₂²×#88322	D₄	1	1	yes	characteristic
31	C₂²×#88322	D₄	1	1	yes	characteristic
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	—	2	4	no	—
34	C₄⋊#88322	—	2	4	no	—
35	C₄⋊#88322	C₂³	1	4	yes	—
36	#13133128	—	2	4	no	—
37	C₄×#88322	—	2	1	no	—
38	C₄×#88322	—	2	4	no	—
39	C₄×#88322	D₄	1	1	yes	characteristic
40	C₄×#88322	D₄	1	1	yes	characteristic
41	C₄×#88322	C₂³	1	2	yes	—
42	C₄×#88322	C₂³	1	1	yes	characteristic
43	C₄×#88322	C₂³	1	1	yes	characteristic
44	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
48	#125×C₅₃	—	2	2	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
53	#117×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#107×C₅₃	C₂×C₄	1	4	yes	—
56	#103×C₅₃	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
65	Q₁₆×C₅₃	—	2	8	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂²⋊C₄×C₅₃	—	2	4	no	—
76	C₄²×C₅₃	—	2	4	no	—
77	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	#41	1	1	yes	characteristic
80	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
81	#11719917	—	4	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	2	4	no	—
86	C₂×#88322	Q₁₆	1	4	yes	complemented
87	C₂×#88322	C₂×D₄	1	1	yes	characteristic
88	C₂×#88322	C₂×D₄	1	1	yes	characteristic
89	C₂×#88322	C₂×D₄	1	1	yes	characteristic
90	C₂×#88322	C₂×D₄	1	1	yes	characteristic
91	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	8	no	—
93	Q₈×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₅₃	#135	1	2	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	#131	1	4	yes	—
108	#88322	—	2	4	no	—
109	#88322	—	2	1	no	—
110	#88322	—	2	1	no	—
111	#88322	—	4	4	no	—
112	#88322	C₂×Q₁₆	1	2	yes	—
113	#88322	C₂×Q₁₆	1	2	yes	—
114	#17958	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
118	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	4	4	no	—
122	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
124	C₄×C₅₃	#443	1	2	yes	—
125	C₄×C₅₃	C₂×#131	1	2	yes	—
126	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
127	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
128	#3170	D₅₃	1	1	yes	characteristic
129	#2960	—	53	2	no	—
130	#1591	—	53	4	no	—
131	C₈×Q₁₆	—	53	4	no	—
132	#1567	—	53	1	no	—
133	#1493	—	53	2	no	—
134	C₂×C₈²	—	53	1	no	—
135	C₂×C₅₃	—	2	2	no	—
136	C₂×C₅₃	#2931	1	1	yes	characteristic
137	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
138	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
139	#533	C₂×D₅₃	1	2	yes	—
140	#500	Dic₅₃	1	1	yes	characteristic
141	#494	Dic₅₃	1	1	yes	characteristic
142	#487	Dic₅₃	1	2	yes	—
143	#467	Dic₅₃	1	4	yes	—
144	#446	—	53	4	no	—
145	#446	—	106	2	no	—
146	C₈×Q₈	—	53	4	no	—
147	C₈×Q₈	—	106	2	no	—
148	C₄×Q₁₆	C₂×D₅₃	1	4	yes	—
149	#424	—	106	2	no	—
150	#423	—	53	2	no	—
151	#405	—	106	2	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	#335	—	53	4	no	—
156	Q₈⋊C₈	—	53	8	no	—
157	C₈²	—	53	2	no	—
158	C₈²	—	53	2	no	—
159	C₅₃	#17958	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
161	C₂×M₁₆	—	106	2	no	—
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	—	106	1	no	—
165	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
166	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
167	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
168	#125	—	2	2	no	—
169	C₂²⋊Q₈	—	2	2	no	—
170	C₂²⋊Q₈	C₂×Dic₅₃	1	4	yes	—
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
173	#117	—	2	2	no	—
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	#107	C₂×Dic₅₃	1	4	yes	—
176	C₄⋊C₈	—	53	4	no	—
177	C₄⋊C₈	—	106	4	no	—
178	C₄⋊C₈	—	106	4	no	—
179	C₄⋊C₈	—	106	2	no	—
180	#103	C₂×Dic₅₃	1	8	yes	—
181	#97	—	106	4	no	—
182	C₄×C₈	—	53	1	no	—
183	C₄×C₈	—	53	1	no	—
184	C₄×C₈	—	53	1	no	—
185	C₄×C₈	—	53	1	no	—
186	C₄×C₈	—	53	2	no	—
187	C₄×C₈	—	106	4	no	—
188	C₄×C₈	—	106	1	no	—
189	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
190	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅₃	1	2	yes	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
194	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
195	C₂²×C₄	#11719947	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
197	Q₁₆	—	2	8	no	—
198	M₁₆	—	212	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	53	4	no	—
201	C₂×C₈	—	53	1	no	—
202	C₂×C₈	—	53	1	no	—
203	C₂×C₈	—	53	1	no	—
204	C₂×C₈	—	53	1	no	—
205	C₂×C₈	—	106	4	no	—
206	C₂×C₈	—	106	2	no	—
207	C₂×C₈	—	106	2	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
210	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	D₄×D₅₃	1	1	yes	characteristic
220	C₄²	#11719947	1	1	yes	characteristic
221	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₄²	C₂³×D₅₃	1	1	yes	characteristic
223	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	Q₈	—	4	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
231	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
234	C₂×C₄	#13133240	1	2	yes	—
235	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	53	2	no	—
240	C₈	—	53	2	no	—
241	C₈	—	106	4	no	—
242	C₈	—	106	1	no	—
243	C₈	—	106	1	no	—
244	C₈	—	212	4	no	—
245	C₈	#13133224	1	4	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	Q₁₆×Dic₅₃	1	2	yes	—
249	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	Q₁₆×Dic₅₃	1	2	yes	—
255	C₄	#16810740	1	2	yes	—
256	C₄	C₂×#13133224	1	2	yes	—
257	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
258	C₄	C₂×#13133240	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#20283427	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×Dic₅₃	1	1	yes	characteristic
262	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#17958
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #20273612	C₂	semidirect
3. Q₁₆×#88322	C₂	semidirect
86. C₂×#88322	Q₁₆	semidirect
159. C₅₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#20283650	2
#20273612	2
Q₁₆×#88322	2
#20273474	2
#20273170	2
C₂×C₈×#88322	2
C₅₃×#3170	2
#17958	53