Subgroups — #28253781

Order 13568.

1762 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288541	C₂	1	2	yes	maximal
2	#20285267	C₂	1	6	yes	maximal, complemented
3	#20285254	C₂	1	6	yes	maximal, complemented
4	#20285246	C₂	1	6	yes	maximal, complemented
5	#20285229	C₂	1	3	yes	maximal, complemented
6	C₂×#16811025	C₂	1	3	yes	maximal, complemented
7	#20283315	C₂	1	2	yes	maximal
8	#20283310	C₂	1	2	yes	maximal
9	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
10	#16811349	—	2	4	no	—
11	#16811332	—	2	12	no	—
12	#16811251	C₂²	1	12	yes	—
13	#16811180	C₂²	1	6	yes	—
14	C₂×#13133209	C₂²	1	6	yes	—
15	#16811106	C₂²	1	4	yes	—
16	#16811032	C₂²	1	24	yes	—
17	#16811028	C₂²	1	6	yes	—
18	#16811025	C₂²	1	3	yes	—
19	#16811025	C₂²	1	6	yes	—
20	#16811025	C₂²	1	6	yes	—
21	#16811025	C₂²	1	6	yes	—
22	#16811025	C₂²	1	1	yes	characteristic
23	#16811025	C₂²	1	6	yes	—
24	#16811021	C₂²	1	6	yes	—
25	#16810665	C₂²	1	12	yes	—
26	#16810665	C₂²	1	12	yes	—
27	#16810661	C₂²	1	6	yes	—
28	#16810661	C₂²	1	6	yes	—
29	C₂×#13133151	C₂²	1	6	yes	—
30	C₂×#13133149	C₂²	1	6	yes	—
31	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#405	C₂²	1	3	yes	—
35	C₅₃×#405	C₂²	1	3	yes	—
36	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
37	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
38	#13133217	C₂³	1	12	yes	—
39	#13133216	—	2	12	no	—
40	#13133209	—	2	12	no	—
41	#13133209	C₂³	1	6	yes	—
42	#13133209	C₂³	1	2	yes	—
43	#13133209	C₂³	1	12	yes	—
44	#13133208	—	2	12	no	—
45	#13133196	—	2	12	no	—
46	#13133194	—	2	24	no	—
47	Q₈×Dic₅₃	—	2	4	no	—
48	C₄×Dic₁₀₆	—	2	12	no	—
49	D₄×Dic₅₃	—	2	12	no	—
50	#13133176	C₂³	1	12	yes	—
51	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
52	C₂×#11719919	—	2	12	no	—
53	#13133151	C₂³	1	6	yes	—
54	#13133151	C₂³	1	12	yes	—
55	#13133151	C₂³	1	2	yes	—
56	#13133151	C₂³	1	12	yes	—
57	#13133149	C₂³	1	2	yes	—
58	#13133149	C₂³	1	6	yes	—
59	#13133149	C₂³	1	12	yes	—
60	#13133149	C₂³	1	12	yes	—
61	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
62	#131×C₅₃	Q₈	1	8	yes	complemented
63	#131×C₅₃	Q₈	1	8	yes	complemented
64	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
65	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
66	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
67	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
68	C₄×Q₈×C₅₃	D₄	1	2	yes	—
69	C₄×D₄×C₅₃	D₄	1	6	yes	—
70	#117×C₅₃	C₂³	1	3	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
72	#97×C₅₃	C₂³	1	6	yes	—
73	C₄×C₈×C₅₃	C₂³	1	3	yes	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	3	yes	—
76	C₄×C₈×C₅₃	C₂³	1	3	yes	—
77	#41×C₅₃	C₂×Q₈	1	4	yes	—
78	#41×C₅₃	C₂×Q₈	1	4	yes	—
79	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
83	C₂²×Dic₅₃	—	2	12	no	—
84	C₂×Dic₁₀₆	—	2	12	no	—
85	M₁₆×C₅₃	C₂×Q₈	1	12	yes	—
86	M₁₆×C₅₃	C₂×Q₈	1	12	yes	—
87	C₂×C₈×C₅₃	C₂×Q₈	1	12	yes	—
88	C₂×C₈×C₅₃	C₂×Q₈	1	12	yes	—
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
92	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
93	C₄⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
94	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
95	C₄²×C₅₃	C₂×D₄	1	6	yes	—
96	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₄²×C₅₃	C₂⁴	1	3	yes	—
98	C₂²⋊Dic₅₃	—	2	12	no	—
99	C₄⋊Dic₅₃	—	2	12	no	—
100	C₄⋊Dic₅₃	—	2	12	no	—
101	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
102	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
103	#11719919	—	2	12	no	—
104	#11719919	—	2	24	no	—
105	#11719919	—	2	4	no	—
106	C₄×Dic₅₃	—	2	12	no	—
107	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
108	Q₈×C₅₃	C₄⋊Q₈	1	4	yes	—
109	D₄×C₅₃	C₄⋊Q₈	1	12	yes	—
110	C₂×C₄×C₅₃	—	4	6	no	—
111	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
112	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
115	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
116	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
117	C₂×Dic₅₃	—	2	4	no	—
118	C₂×Dic₅₃	—	2	12	no	—
119	C₂×Dic₅₃	—	8	12	no	—
120	Dic₁₀₆	—	8	12	no	—
121	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
122	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
123	C₈×C₅₃	C₂²×Q₈	1	6	yes	—
124	C₈×C₅₃	C₂²×Q₈	1	6	yes	—
125	#60666	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	6	no	—
127	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
128	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
129	C₄×C₅₃	—	2	6	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
132	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
133	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
134	Dic₅₃	—	4	4	no	—
135	Dic₅₃	—	4	12	no	—
136	#3504	—	53	2	no	—
137	#3166	—	53	6	no	—
138	#3164	—	53	6	no	—
139	#3163	—	53	6	no	—
140	#3160	—	53	3	no	—
141	C₂×#499	—	53	3	no	—
142	#2913	—	53	2	no	—
143	#2912	—	53	2	no	—
144	C₄×#131	D₅₃	1	1	yes	characteristic
145	C₂×C₅₃	—	2	6	no	—
146	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
147	C₂×C₅₃	#3584	1	2	yes	—
148	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
149	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
150	#557	—	106	4	no	—
151	#554	—	106	12	no	—
152	#544	—	53	12	no	—
153	#533	—	53	6	no	—
154	C₂×#125	—	53	6	no	—
155	#520	—	53	4	no	—
156	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
157	#501	—	53	24	no	—
158	#500	—	53	6	no	—
159	#499	—	53	3	no	—
160	#499	—	53	6	no	—
161	#499	—	53	6	no	—
162	#499	—	53	6	no	—
163	#499	—	53	1	no	—
164	#499	—	53	6	no	—
165	#498	—	53	6	no	—
166	#428	—	53	12	no	—
167	#428	—	53	12	no	—
168	#427	—	53	6	no	—
169	#427	—	53	6	no	—
170	C₂×#107	—	53	6	no	—
171	C₂×#106	—	53	6	no	—
172	#405	C₂×D₅₃	1	3	yes	—
173	#405	C₂×D₅₃	1	3	yes	—
174	C₄×M₁₆	C₂×D₅₃	1	3	yes	—
175	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
176	C₅₃	#60666	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
178	#131	Dic₁₀₆	1	8	yes	complemented
179	#131	Dic₁₀₆	1	8	yes	complemented
180	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
181	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
182	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
183	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
184	C₄⋊Q₈	—	53	12	no	—
185	C₄⋊Q₈	—	106	12	no	—
186	#125	—	53	6	no	—
187	#125	—	53	12	no	—
188	#125	—	53	2	no	—
189	#125	—	106	12	no	—
190	#125	—	106	12	no	—
191	C₂²⋊Q₈	—	106	12	no	—
192	C₂²⋊Q₈	—	106	24	no	—
193	C₄×Q₈	—	106	4	no	—
194	C₄×Q₈	—	106	12	no	—
195	C₄×Q₈	D₂₁₂	1	2	yes	—
196	C₄×D₄	—	106	12	no	—
197	C₄×D₄	D₂₁₂	1	6	yes	—
198	#117	—	53	12	no	—
199	#117	C₂²×D₅₃	1	3	yes	—
200	C₂×C₄⋊C₄	—	53	12	no	—
201	C₂×C₄⋊C₄	—	106	12	no	—
202	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
203	#107	—	53	6	no	—
204	#107	—	53	12	no	—
205	#107	—	53	2	no	—
206	#107	—	53	12	no	—
207	#106	—	53	6	no	—
208	#106	—	53	12	no	—
209	#106	—	53	2	no	—
210	#106	—	53	12	no	—
211	#97	C₂²×D₅₃	1	6	yes	—
212	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₅₃	1	3	yes	—
214	C₄×C₈	C₂²×D₅₃	1	3	yes	—
215	C₄×C₈	C₂²×D₅₃	1	3	yes	—
216	#41	C₂×Dic₁₀₆	1	4	yes	—
217	#41	C₂×Dic₁₀₆	1	4	yes	—
218	C₂×Q₈	—	106	12	no	—
219	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
220	C₂×D₄	C₂×D₂₁₂	1	3	yes	—
221	C₂²×C₄	—	106	12	no	—
222	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
223	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
224	M₁₆	C₂×Dic₁₀₆	1	12	yes	—
225	M₁₆	C₂×Dic₁₀₆	1	12	yes	—
226	C₂×C₈	C₂×Dic₁₀₆	1	12	yes	—
227	C₂×C₈	C₂×Dic₁₀₆	1	12	yes	—
228	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂³×D₅₃	1	3	yes	—
231	C₂×C₈	C₂³×D₅₃	1	3	yes	—
232	C₄⋊C₄	—	53	12	no	—
233	C₄⋊C₄	—	53	4	no	—
234	C₄⋊C₄	—	106	12	no	—
235	C₄⋊C₄	—	106	12	no	—
236	C₄⋊C₄	—	106	24	no	—
237	C₄⋊C₄	—	106	4	no	—
238	C₄⋊C₄	—	106	12	no	—
239	C₄⋊C₄	C₂×D₂₁₂	1	6	yes	—
240	C₂²⋊C₄	—	106	12	no	—
241	C₂²⋊C₄	C₂×D₂₁₂	1	6	yes	—
242	C₄²	—	106	12	no	—
243	C₄²	C₂×D₂₁₂	1	6	yes	—
244	C₄²	C₂³×D₅₃	1	1	yes	characteristic
245	C₄²	C₂³×D₅₃	1	3	yes	—
246	C₂³	C₂²×D₂₁₂	1	3	yes	—
247	Q₈	—	424	12	no	—
248	Q₈	#13133217	1	4	yes	—
249	D₄	#13133217	1	12	yes	—
250	C₂×C₄	—	4	6	no	—
251	C₂×C₄	—	106	4	no	—
252	C₂×C₄	—	106	12	no	—
253	C₂×C₄	—	424	12	no	—
254	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
255	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
256	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
257	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
258	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
259	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
260	C₈	C₂²×Dic₁₀₆	1	2	yes	—
261	C₈	C₂²×Dic₁₀₆	1	2	yes	—
262	C₈	C₂²×Dic₁₀₆	1	6	yes	—
263	C₈	C₂²×Dic₁₀₆	1	6	yes	—
264	C₂²	—	2	6	no	—
265	C₂²	C₂×#13133217	1	6	yes	—
266	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	6	no	—
269	C₄	—	212	12	no	—
270	C₄	—	212	4	no	—
271	C₄	C₂×#13133217	1	6	yes	—
272	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
273	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
274	C₂	—	2	6	no	—
275	C₂	C₂²×#13133217	1	1	yes	characteristic
276	C₂	#20289489	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×D₄
prime radical	C₂×#41×C₅₃	C₂³
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#60666
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20285267	C₂	semidirect
3. #20285254	C₂	semidirect
4. #20285246	C₂	semidirect
5. #20285229	C₂	semidirect
6. C₂×#16811025	C₂	semidirect
62. #131×C₅₃	Q₈	semidirect
63. #131×C₅₃	Q₈	semidirect
176. C₅₃	#60666	semidirect
178. #131	Dic₁₀₆	semidirect
179. #131	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20288541	2
#20285267	2
#20285254	2
#20285246	2
#20285229	2
C₂×#16811025	2
#20283315	2
#20283310	2
C₄×#131×C₅₃	2
#60666	53