Subgroups — #28327932

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291684	C₂	1	2	yes	maximal
2	#20290938	C₂	1	2	yes	maximal, complemented
3	#20290937	C₂	1	4	yes	maximal, complemented
4	#20290932	C₂	1	4	yes	maximal, complemented
5	C₂×#16811515	C₂	1	2	yes	maximal, complemented
6	#20283144	C₂	1	1	yes	maximal, characteristic
7	#16811682	C₄	1	16	yes	complemented
8	D₅₃⋊M₃₂	C₂²	1	8	yes	—
9	C₂×#13133291	C₂²	1	4	yes	—
10	#16811516	C₂²	1	8	yes	—
11	#16811515	C₂²	1	4	yes	—
12	#16811515	C₂²	1	4	yes	—
13	C₂×#13133225	C₄	1	1	yes	characteristic
14	C₂×#13133225	C₂²	1	1	yes	characteristic
15	C₄×#11719945	C₄	1	1	yes	characteristic
16	#16810581	C₄	1	1	yes	characteristic
17	#16810581	C₂²	1	1	yes	characteristic
18	#16810577	C₄	1	2	yes	—
19	#16810577	C₂²	1	2	yes	—
20	#16810575	C₄	1	2	yes	—
21	C₂×#13133128	C₂²	1	1	yes	characteristic
22	#97×D₅₃	C₂²	1	2	yes	—
23	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
24	#13133292	C₂×C₄	1	16	yes	complemented
25	#13133291	C₂×C₄	1	8	yes	complemented
26	#13133291	C₂³	1	8	yes	—
27	D₅₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
28	C₂×#11719910	C₂×C₄	1	8	yes	complemented
29	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
30	#13133225	—	2	4	no	—
31	#13133225	C₂×C₄	1	8	yes	—
32	C₂×#11719912	C₂×C₄	1	2	yes	—
33	C₂×#11719912	C₂×C₄	1	2	yes	—
34	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
35	C₂²×#88322	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₃	C₂³	1	2	yes	—
38	C₄×Dic₁₀₆	—	2	1	no	—
39	C₄×D₂₁₂	—	2	1	no	—
40	C₄×C₂²⋊D₅₃	—	2	2	no	—
41	#13133176	C₂×C₄	1	1	yes	characteristic
42	#13133174	C₂×C₄	1	2	yes	—
43	C₄²×D₅₃	C₂×C₄	1	2	yes	—
44	#13133128	C₂×C₄	1	1	yes	characteristic
45	#13133128	C₂×C₄	1	1	yes	characteristic
46	#13133128	C₂³	1	1	yes	characteristic
47	#13133128	C₂³	1	1	yes	characteristic
48	#13133127	C₂×C₄	1	2	yes	—
49	#13133127	C₂³	1	2	yes	—
50	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#97×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₅₃	M₁₆	1	2	yes	—
60	#11719945	C₄²	1	4	yes	—
61	#11719945	M₁₆	1	4	yes	—
62	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
63	C₂×C₄×D₅₃	—	2	2	no	—
64	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
66	M₁₆×C₅₃	—	2	2	no	—
67	M₁₆×C₅₃	C₄²	1	4	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²⋊Dic₅₃	—	2	1	no	—
74	#11719921	—	2	2	no	—
75	C₄⋊Dic₅₃	—	2	1	no	—
76	#11719919	—	2	2	no	—
77	C₄×Dic₅₃	—	2	2	no	—
78	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
81	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
82	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
83	C₂×#88322	C₂²×C₄	1	4	yes	—
84	C₂×#88322	C₂⁴	1	1	yes	characteristic
85	#11719912	—	2	4	no	—
86	#11719912	C₄²	1	8	yes	—
87	C₈×D₅₃	—	2	4	no	—
88	C₈×D₅₃	C₂²×C₄	1	8	yes	—
89	#11719910	C₂²×C₄	1	16	yes	—
90	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	8	1	no	—
95	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₅₃	#97	1	8	yes	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
102	D₂₁₂	#97	1	4	yes	—
103	C₄×D₅₃	—	8	2	no	—
104	C₄×D₅₃	C₂×C₄²	1	4	yes	—
105	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
106	Dic₁₀₆	#97	1	4	yes	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	C₂×C₄²	1	4	yes	—
109	#88322	—	2	1	no	—
110	#88322	—	2	1	no	—
111	#88322	C₂×C₄²	1	2	yes	—
112	#88322	C₂³×C₄	1	2	yes	—
113	#33329	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	4	1	no	—
115	C₂²×C₅₃	C₂×#97	1	2	yes	—
116	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×D₅₃	—	4	2	no	—
118	C₂×D₅₃	C₂×#97	1	4	yes	—
119	C₄×C₅₃	—	4	1	no	—
120	C₄×C₅₃	—	4	1	no	—
121	C₄×C₅₃	C₂×#97	1	2	yes	—
122	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
123	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
124	Dic₅₃	—	4	2	no	—
125	Dic₅₃	C₂×#97	1	4	yes	—
126	#3407	—	53	2	no	—
127	#2875	—	53	1	no	—
128	#2110	—	53	4	no	—
129	#2110	—	53	2	no	—
130	C₂×#347	—	53	2	no	—
131	C₂×#347	—	53	4	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
135	D₅₃	—	2	4	no	—
136	C₂×#131	—	53	1	no	—
137	C₂×#131	—	53	1	no	—
138	C₄×#41	—	53	1	no	—
139	#504	—	53	16	no	—
140	C₂×M₃₂	—	53	4	no	—
141	C₂×M₃₂	—	53	8	no	—
142	#405	—	53	1	no	—
143	#405	—	53	1	no	—
144	#405	—	53	2	no	—
145	#405	—	53	2	no	—
146	C₄×M₁₆	—	53	2	no	—
147	C₄×M₁₆	#6157	1	1	yes	characteristic
148	C₂×#97	—	53	1	no	—
149	C₂×#97	—	53	2	no	—
150	#347	—	53	4	no	—
151	#347	—	53	4	no	—
152	#347	—	53	8	no	—
153	C₅₃	#33329	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	—	53	1	no	—
155	#131	—	53	8	no	—
156	#131	—	106	4	no	—
157	C₂×M₁₆	—	53	2	no	—
158	C₂×M₁₆	—	53	2	no	—
159	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
160	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	—	53	2	no	—
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	—	53	2	no	—
165	C₄×Q₈	—	106	1	no	—
166	C₄×D₄	—	106	1	no	—
167	C₄×D₄	—	106	2	no	—
168	#117	—	53	1	no	—
169	#117	—	53	2	no	—
170	C₂×C₄²	—	53	2	no	—
171	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
172	M₃₂	—	53	8	no	—
173	M₃₂	—	53	8	no	—
174	M₃₂	—	53	16	no	—
175	C₂×C₁₆	—	53	8	no	—
176	C₂×C₁₆	—	53	16	no	—
177	#97	—	53	2	no	—
178	#97	—	53	2	no	—
179	#97	—	53	1	no	—
180	#97	—	53	1	no	—
181	#97	—	53	1	no	—
182	#97	—	53	1	no	—
183	#97	C₂×#6157	1	2	yes	—
184	C₄×C₈	—	53	2	no	—
185	C₄×C₈	—	53	2	no	—
186	C₄×C₈	C₂×#6157	1	2	yes	—
187	#41	—	53	4	no	—
188	#41	—	53	4	no	—
189	C₂×Q₈	—	53	1	no	—
190	C₂×D₄	—	53	2	no	—
191	C₂×D₄	—	53	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	106	2	no	—
195	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
196	M₁₆	—	2	2	no	—
197	M₁₆	—	53	8	no	—
198	M₁₆	—	106	4	no	—
199	M₁₆	C₄×#6157	1	4	yes	—
200	C₂×C₈	—	53	4	no	—
201	C₂×C₈	—	53	1	no	—
202	C₂×C₈	—	53	1	no	—
203	C₂×C₈	—	53	8	no	—
204	C₂×C₈	—	53	1	no	—
205	C₂×C₈	—	53	1	no	—
206	C₂×C₈	—	106	4	no	—
207	C₂×C₈	—	106	2	no	—
208	C₂×C₈	C₂²×#6157	1	2	yes	—
209	C₂×C₈	C₂²×#6157	1	2	yes	—
210	C₄⋊C₄	—	106	2	no	—
211	C₄⋊C₄	—	106	1	no	—
212	C₂²⋊C₄	—	106	2	no	—
213	C₂²⋊C₄	—	106	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	53	2	no	—
216	C₄²	—	106	2	no	—
217	C₄²	C₂²×#6157	1	1	yes	characteristic
218	C₄²	C₂²×#6157	1	1	yes	characteristic
219	C₁₆	—	53	16	no	—
220	C₂³	—	53	2	no	—
221	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
222	Q₈	—	53	4	no	—
223	D₄	—	53	8	no	—
224	D₄	—	53	4	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	53	2	no	—
229	C₂×C₄	—	53	4	no	—
230	C₂×C₄	—	53	4	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	—	424	2	no	—
233	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
234	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
235	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
236	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	53	2	no	—
239	C₈	—	53	2	no	—
240	C₈	—	106	1	no	—
241	C₈	—	106	1	no	—
242	C₈	C₂×C₄×#6157	1	4	yes	—
243	C₂²	—	4	1	no	—
244	C₂²	—	53	4	no	—
245	C₂²	—	212	2	no	—
246	C₂²	#16811582	1	2	yes	—
247	C₂²	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
248	C₄	—	4	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	53	4	no	—
251	C₄	—	212	2	no	—
252	C₄	#16811582	1	2	yes	—
253	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
254	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	2	no	—
257	C₂	—	106	4	no	—
258	C₂	C₂×#16811582	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×M₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂²×#97
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	C₄×M₁₆	#6157
hypocenter	C₅₃	#33329
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20290938	C₂	semidirect
3. #20290937	C₂	semidirect
4. #20290932	C₂	semidirect
5. C₂×#16811515	C₂	semidirect
7. #16811682	C₄	semidirect
24. #13133292	C₂×C₄	semidirect
25. #13133291	C₂×C₄	semidirect
27. D₅₃⋊C₁₆	C₂×C₄	semidirect
28. C₂×#11719910	C₂×C₄	semidirect
153. C₅₃	#33329	semidirect

Maximal subgroups

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

subgroup	index
#20291684	2
#20290938	2
#20290937	2
#20290932	2
C₂×#16811515	2
#20283144	2
#33329	53