Subgroups — #27354625

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	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20283649	C₂	1	2	yes	maximal
3	#20273613	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₁₂	C₂	1	4	yes	maximal, complemented
5	#20273474	C₂	1	1	yes	maximal, characteristic
6	#20273171	C₂	1	2	yes	maximal
7	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811180	C₂²	1	2	yes	—
9	#16811028	C₄	1	1	yes	characteristic
10	#16811003	C₄	1	1	yes	characteristic
11	#16810974	C₄	1	2	yes	—
12	#16810864	C₄	1	4	yes	—
13	#16810753	—	2	2	no	—
14	#16810753	C₂²	1	4	yes	—
15	C₈×Dic₁₀₆	—	2	2	no	—
16	C₈×Dic₁₀₆	C₂²	1	4	yes	—
17	C₄×Dic₂₁₂	C₂²	1	4	yes	—
18	#16810649	—	2	2	no	—
19	#16810647	C₂²	1	2	yes	—
20	#16810578	—	2	2	no	—
21	#16810376	C₂²	1	4	yes	—
22	#16810355	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₈²×C₅₃	C₂²	1	2	yes	—
27	C₈²×C₅₃	C₂²	1	2	yes	—
28	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
29	C₂×#11719917	—	2	2	no	—
30	#13133217	C₂×C₄	1	2	yes	—
31	#13133217	C₂×C₄	1	2	yes	—
32	#13133209	—	2	2	no	—
33	#13133196	—	2	2	no	—
34	#13133196	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₀₆	—	2	2	no	—
36	C₄×Dic₁₀₆	C₂³	1	4	yes	—
37	#13133176	—	2	2	no	—
38	#13133151	C₂×C₄	1	4	yes	—
39	C₄⋊#88322	—	2	2	no	—
40	C₄⋊#88322	C₂³	1	4	yes	—
41	#13133146	—	2	4	no	—
42	#13133146	—	2	4	no	—
43	#13133142	C₂×C₄	1	8	yes	—
44	#13133127	—	2	4	no	—
45	C₈×Dic₅₃	—	2	4	no	—
46	C₂²×C₈×C₅₃	—	2	1	no	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₅₃	—	2	1	no	—
52	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂³	1	2	yes	—
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂×Dic₁₀₆	—	2	2	no	—
64	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊Dic₅₃	—	2	4	no	—
81	C₄⋊Dic₅₃	—	2	2	no	—
82	C₄⋊Dic₅₃	—	2	2	no	—
83	C₄⋊Dic₅₃	—	2	2	no	—
84	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
85	#11719919	—	2	4	no	—
86	#11719919	—	2	4	no	—
87	C₄×Dic₅₃	—	2	4	no	—
88	#11719917	—	4	4	no	—
89	C₂×#88322	—	2	4	no	—
90	Dic₂₁₂	—	2	8	no	—
91	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₅₃	#135	1	2	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₅₃	—	2	4	no	—
103	C₂×Dic₅₃	—	2	4	no	—
104	Dic₁₀₆	—	2	8	no	—
105	Dic₁₀₆	—	4	4	no	—
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	#131	1	4	yes	—
111	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
112	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
113	#88322	—	4	4	no	—
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	C₄×Q₁₆	1	2	yes	—
122	C₄×C₅₃	#443	1	2	yes	—
123	C₄×C₅₃	C₂×#131	1	2	yes	—
124	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
125	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
126	Dic₅₃	—	4	4	no	—
127	Dic₅₃	—	4	4	no	—
128	#3170	—	53	1	no	—
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	D₅₃	1	1	yes	characteristic
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	—	53	2	no	—
140	#500	—	53	1	no	—
141	#494	—	53	1	no	—
142	#487	—	53	2	no	—
143	#467	—	53	4	no	—
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	—	53	4	no	—
149	#424	—	106	2	no	—
150	#423	—	53	2	no	—
151	#405	—	106	2	no	—
152	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	#335	—	53	4	no	—
156	Q₈⋊C₈	—	53	8	no	—
157	C₈²	C₂×D₅₃	1	2	yes	—
158	C₈²	C₂×D₅₃	1	2	yes	—
159	C₅₃	#17958	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₁₆	—	53	4	no	—
161	C₂×M₁₆	—	106	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₄×D₅₃	1	2	yes	—
164	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
165	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
166	C₄⋊Q₈	—	53	2	no	—
167	C₄⋊Q₈	—	53	2	no	—
168	#125	—	106	2	no	—
169	C₂²⋊Q₈	—	53	4	no	—
170	C₂²⋊Q₈	—	106	2	no	—
171	C₄×Q₈	—	53	4	no	—
172	C₄×Q₈	—	106	2	no	—
173	#117	—	106	2	no	—
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	#107	—	53	4	no	—
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	—	53	8	no	—
181	#97	—	106	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	106	4	no	—
184	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
185	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
186	C₄×C₈	D₂₁₂	1	1	yes	characteristic
187	C₄×C₈	D₂₁₂	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₅₃	1	2	yes	—
189	C₄×C₈	C₂²×D₅₃	1	2	yes	—
190	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
192	C₂×Q₈	—	53	4	no	—
193	C₂×Q₈	—	106	2	no	—
194	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
196	C₂²×C₄	#11719945	1	1	yes	characteristic
197	Q₁₆	—	106	8	no	—
198	M₁₆	—	212	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	106	4	no	—
204	C₂×C₈	Dic₂₁₂	1	4	yes	complemented
205	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
207	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
208	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
209	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
210	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
211	C₄⋊C₄	—	53	4	no	—
212	C₄⋊C₄	—	106	2	no	—
213	C₄⋊C₄	—	106	2	no	—
214	C₄⋊C₄	—	106	4	no	—
215	C₄⋊C₄	—	106	2	no	—
216	C₄⋊C₄	—	106	4	no	—
217	C₂²⋊C₄	—	106	4	no	—
218	C₄²	—	106	4	no	—
219	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
221	C₄²	#11719945	1	1	yes	characteristic
222	C₄²	C₂³×D₅₃	1	1	yes	characteristic
223	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
224	Q₈	—	106	8	no	—
225	Q₈	—	212	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	106	4	no	—
229	C₂×C₄	—	106	4	no	—
230	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
231	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
232	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
234	C₂×C₄	#13133239	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	212	4	no	—
243	C₈	#13133223	1	4	yes	—
244	C₈	C₂×Dic₂₁₂	1	2	yes	—
245	C₈	C₂×Dic₂₁₂	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Dic₂₁₂	1	2	yes	—
249	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	212	4	no	—
253	C₄	—	212	4	no	—
254	C₄	C₄×Dic₂₁₂	1	2	yes	—
255	C₄	#16810741	1	2	yes	—
256	C₄	C₂×#13133223	1	2	yes	—
257	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
258	C₄	C₂×#13133239	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#20283428	1	1	yes	characteristic
261	C₂	C₂×C₄×Dic₂₁₂	1	1	yes	characteristic
262	C₂	C₂×#16810741	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#17958
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #20273613	C₂	semidirect
4. C₈×Dic₂₁₂	C₂	semidirect
69. C₂×C₈×C₅₃	Q₁₆	semidirect
159. C₅₃	#17958	semidirect
204. C₂×C₈	Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20283649	2
#20273613	2
C₈×Dic₂₁₂	2
#20273474	2
#20273171	2
C₂×C₈²×C₅₃	2
#17958	53