Subgroups — #28302093

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290502	C₂	1	2	yes	maximal
2	C₄×C₈×#6157	C₂	1	1	yes	maximal, characteristic
3	D₅₃⋊#335	C₂	1	1	yes	maximal, characteristic
4	#20289544	C₂	1	2	yes	maximal
5	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13133271	—	2	2	no	—
8	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×#6157	C₄	1	2	yes	—
10	C₂×C₈×#6157	C₄	1	2	yes	—
11	C₄²×#6157	C₂²	1	1	yes	characteristic
12	(C₂×#6157)⋊C₈	—	2	2	no	—
13	#16811487	C₄	1	2	yes	complemented
14	#16811486	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	#16810376	C₄	1	1	yes	characteristic, complemented
18	#16810375	C₄	1	2	yes	complemented
19	C₅₃×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×D₅₃⋊C₈	—	2	2	no	—
21	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
23	C₂×C₄×#6157	D₄	1	1	yes	characteristic
24	C₂×C₄×#6157	D₄	1	1	yes	characteristic
25	C₂×C₄×#6157	Q₈	1	2	yes	—
26	#13133284	—	2	2	no	—
27	C₄⋊#88324	C₂×C₄	1	4	yes	—
28	#13133271	—	4	4	no	—
29	C₄×#88324	—	2	1	no	—
30	C₄×#88324	—	2	1	no	—
31	C₈×#6157	—	2	4	no	—
32	C₈×#6157	—	2	4	no	—
33	C₂×C₈×D₅₃	—	2	2	no	—
34	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊#88322	C₂×C₄	1	2	yes	—
38	#13133146	—	2	2	no	—
39	C₄×#88322	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×#6157	C₄⋊C₄	1	2	yes	—
45	C₂²×#6157	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×#88324	—	2	2	no	—
50	C₂×#88324	—	2	4	no	—
51	C₄×#6157	—	2	4	no	—
52	C₄×#6157	—	2	2	no	—
53	C₄×#6157	—	2	2	no	—
54	C₄×#6157	—	2	1	no	—
55	C₄×#6157	—	2	1	no	—
56	C₄×#6157	D₈	1	2	yes	—
57	C₄×#6157	Q₁₆	1	2	yes	—
58	D₅₃⋊C₈	—	2	2	no	—
59	D₅₃⋊C₈	—	4	2	no	—
60	D₅₃⋊C₈	—	4	2	no	—
61	D₅₃⋊C₈	M₁₆	1	4	yes	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₅₃	#41	1	1	yes	characteristic
68	C₄×Dic₅₃	#41	1	1	yes	characteristic
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	C₄²	1	1	yes	characteristic
71	C₂×#88322	C₄²	1	1	yes	characteristic
72	C₈×D₅₃	—	2	2	no	—
73	C₈×D₅₃	—	4	2	no	—
74	C₈×D₅₃	—	4	2	no	—
75	C₈×D₅₃	M₁₆	1	4	yes	—
76	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×#6157	—	2	2	no	—
78	C₂×#6157	—	2	2	no	—
79	C₂×#6157	—	2	2	no	—
80	C₂×#6157	—	2	2	no	—
81	C₂×#6157	#107	1	4	yes	—
82	C₂×#6157	#108	1	4	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
86	C₂×Dic₅₃	#117	1	1	yes	characteristic
87	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
89	C₄×D₅₃	—	2	2	no	—
90	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
91	C₄×D₅₃	C₂×M₁₆	1	2	yes	—
92	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
93	C₄×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₄×D₅₃	#135	1	1	yes	characteristic
95	C₄×D₅₃	#135	1	1	yes	characteristic
96	#88324	—	2	4	no	—
97	#88324	—	4	4	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	4	2	no	—
100	C₈×C₅₃	#97	1	2	yes	complemented
101	#88322	—	2	1	no	—
102	#88322	—	4	2	no	—
103	#88322	#97	1	2	yes	complemented
104	#8274	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₂×D₅₃	#423	1	2	yes	—
107	C₂×D₅₃	C₂×#107	1	1	yes	characteristic
108	C₂×D₅₃	#427	1	1	yes	characteristic
109	C₂×D₅₃	C₂×#108	1	2	yes	—
110	#6157	—	2	4	no	—
111	#6157	—	2	4	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
114	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
115	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
116	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
117	Dic₅₃	—	2	1	no	—
118	Dic₅₃	#432	1	2	yes	—
119	Dic₅₃	C₄×D₈	1	1	yes	characteristic
120	Dic₅₃	C₄×Q₁₆	1	1	yes	characteristic
121	#1767	—	53	2	no	—
122	C₄²×C₈	—	53	1	no	—
123	C₂×#335	—	53	1	no	—
124	C₂×#335	—	53	1	no	—
125	#1277	—	53	2	no	—
126	C₂×C₅₃	#1767	1	1	yes	characteristic
127	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
128	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
129	D₅₃	#1567	1	2	yes	—
130	D₅₃	#1568	1	2	yes	—
131	C₂×C₄⋊C₈	—	53	2	no	—
132	C₂×C₄⋊C₈	—	53	2	no	—
133	C₂×#97	—	106	2	no	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	—	53	2	no	—
136	C₂×C₄×C₈	—	53	2	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₄³	—	53	1	no	—
139	#336	—	106	2	no	—
140	#335	—	53	2	no	—
141	#335	—	53	2	no	—
142	#335	—	53	2	no	—
143	#335	—	53	1	no	—
144	#335	#6157	1	1	yes	characteristic, complemented
145	C₅₃	#8274	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₈	—	53	1	no	—
147	C₂²×C₈	—	53	2	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	106	2	no	—
150	C₂²×C₈	—	106	2	no	—
151	C₂×C₄²	—	53	1	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	—	53	1	no	—
154	C₂×C₄²	—	53	2	no	—
155	C₂×C₄²	—	53	2	no	—
156	C₄⋊C₈	—	53	4	no	—
157	C₄⋊C₈	—	53	2	no	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	C₄⋊C₈	C₂×#6157	1	2	yes	—
161	#97	—	212	4	no	—
162	C₄×C₈	—	53	1	no	—
163	C₄×C₈	—	53	1	no	—
164	C₄×C₈	—	53	1	no	—
165	C₄×C₈	—	106	1	no	—
166	C₄×C₈	—	106	1	no	—
167	C₄×C₈	—	106	4	no	—
168	C₄×C₈	—	106	4	no	—
169	C₄×C₈	C₂×#6157	1	1	yes	characteristic
170	C₂²×C₄	—	53	1	no	—
171	C₂²×C₄	—	53	1	no	—
172	C₂²×C₄	—	53	2	no	—
173	C₂²×C₄	—	53	2	no	—
174	C₂²×C₄	—	53	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	53	1	no	—
177	C₂×C₈	—	53	4	no	—
178	C₂×C₈	—	53	4	no	—
179	C₂×C₈	—	53	1	no	—
180	C₂×C₈	—	106	2	no	—
181	C₂×C₈	—	106	2	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	—	106	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	C₄×#6157	1	1	yes	characteristic
190	C₂×C₈	C₄×#6157	1	1	yes	characteristic
191	C₄²	—	53	2	no	—
192	C₄²	—	53	2	no	—
193	C₄²	—	53	1	no	—
194	C₄²	—	53	1	no	—
195	C₄²	—	53	1	no	—
196	C₄²	—	106	2	no	—
197	C₄²	—	106	2	no	—
198	C₄²	—	106	1	no	—
199	C₄²	—	106	1	no	—
200	C₄²	—	106	4	no	—
201	C₄²	C₂²×#6157	1	1	yes	characteristic
202	C₂³	—	53	1	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	1	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	53	2	no	—
207	C₂×C₄	—	53	1	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	1	no	—
211	C₂×C₄	—	53	4	no	—
212	C₂×C₄	—	53	4	no	—
213	C₂×C₄	—	53	1	no	—
214	C₂×C₄	—	106	2	no	—
215	C₂×C₄	—	106	2	no	—
216	C₂×C₄	—	106	2	no	—
217	C₂×C₄	—	106	2	no	—
218	C₂×C₄	—	106	2	no	—
219	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
220	C₂×C₄	D₄×#6157	1	1	yes	characteristic
221	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	53	2	no	—
225	C₈	—	106	4	no	—
226	C₈	—	106	1	no	—
227	C₈	—	212	4	no	—
228	C₈	—	212	2	no	—
229	C₈	#13133271	1	2	yes	complemented
230	C₂²	—	53	2	no	—
231	C₂²	—	53	1	no	—
232	C₂²	—	53	1	no	—
233	C₂²	—	53	2	no	—
234	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	53	2	no	—
237	C₄	—	53	1	no	—
238	C₄	—	53	1	no	—
239	C₄	—	106	4	no	—
240	C₄	—	106	4	no	—
241	C₄	—	106	1	no	—
242	C₄	C₂×#13133271	1	1	yes	characteristic
243	C₄	M₁₆×#6157	1	1	yes	characteristic
244	C₄	D₈×#6157	1	1	yes	characteristic
245	C₄	Q₁₆×#6157	1	1	yes	characteristic
246	C₂	—	53	2	no	—
247	C₂	—	53	2	no	—
248	C₂	#20290502	1	1	yes	characteristic
249	C₂	#107×#6157	1	1	yes	characteristic
250	C₂	#108×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₂×C₄⋊C₄
cosocle	C₄²×D₅₃	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	#8274
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
13. #16811487	C₄	semidirect
14. #16811486	C₄	semidirect
17. #16810376	C₄	semidirect
18. #16810375	C₄	semidirect
19. C₅₃×#335	C₄	semidirect
100. C₈×C₅₃	#97	semidirect
103. #88322	#97	semidirect
144. #335	#6157	semidirect
145. C₅₃	#8274	semidirect
229. C₈	#13133271	semidirect

Maximal subgroups

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

subgroup	index
#20290502	2
C₄×C₈×#6157	2
D₅₃⋊#335	2
#20289544	2
#335×D₅₃	2
#8274	53