Subgroups — #28328347

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811681	C₂	1	2	yes	maximal, complemented
2	#20291105	C₂	1	4	yes	maximal, complemented
3	#20291102	C₂	1	4	yes	maximal, complemented
4	#20291056	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290957	C₂	1	2	yes	maximal, complemented
6	#20290926	C₂	1	1	yes	maximal, characteristic, complemented
7	#20283618	C₂	1	1	yes	maximal, characteristic
8	#16811681	—	2	2	no	—
9	#16811681	—	2	2	no	—
10	#16811681	C₂²	1	4	yes	complemented
11	#16811681	C₂²	1	4	yes	complemented
12	C₂×#13133291	—	2	2	no	—
13	C₂×#13133291	C₂²	1	2	yes	—
14	C₂×#13133291	C₂²	1	1	yes	characteristic
15	C₂²×#11719910	—	2	2	no	—
16	C₂²×#11719910	C₂²	1	1	yes	characteristic
17	#16811545	C₂²	1	4	yes	complemented
18	#16811521	C₂²	1	8	yes	complemented
19	#16811512	C₂²	1	2	yes	complemented
20	#16811509	C₂²	1	2	yes	complemented
21	C₂×#13133224	C₂²	1	2	yes	—
22	#16810745	C₄	1	4	yes	—
23	#16810740	C₂²	1	4	yes	—
24	#16810672	C₄	1	1	yes	characteristic
25	#16810640	C₄	1	2	yes	—
26	#16810581	C₂²	1	1	yes	characteristic
27	C₅₃×#575	C₄	1	1	yes	characteristic
28	#13133291	—	2	2	no	—
29	#13133291	—	2	2	no	—
30	#13133291	—	2	4	no	—
31	#13133291	—	2	4	no	—
32	#13133291	C₂³	1	4	yes	—
33	C₂×#11719910	—	2	2	no	—
34	C₂×#11719910	—	2	4	no	—
35	C₂×#11719910	—	2	4	no	—
36	C₂×#11719910	D₄	1	2	yes	—
37	C₂×#11719910	D₄	1	2	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	1	yes	characteristic
40	C₂×#11719910	C₂³	1	1	yes	characteristic
41	#13133224	—	2	2	no	—
42	#13133224	—	2	2	no	—
43	#13133224	C₂×C₄	1	4	yes	—
44	#13133224	C₂³	1	4	yes	—
45	C₂×#11719917	—	2	2	no	—
46	C₂×#11719917	C₂×C₄	1	2	yes	—
47	C₂²×#88322	—	2	2	no	—
48	C₂²×#88322	C₂³	1	1	yes	characteristic
49	#13133176	C₂×C₄	1	1	yes	characteristic
50	#13133153	C₂×C₄	1	4	yes	—
51	#13133144	C₂×C₄	1	8	yes	—
52	#13133127	C₂×C₄	1	2	yes	—
53	C₈×Dic₅₃	C₂³	1	2	yes	—
54	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
55	#137×C₅₃	C₈	1	4	yes	—
56	AGL(1,ℤ/8)×C₅₃	C₈	1	4	yes	—
57	#135×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#41×C₅₃	—	2	2	no	—
60	#41×C₅₃	—	2	2	no	—
61	#41×C₅₃	C₂×C₈	1	4	yes	—
62	#41×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
69	SD₁₆×C₅₃	C₂×C₈	1	8	yes	—
70	D₈×C₅₃	C₂×C₈	1	4	yes	—
71	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂²⋊Dic₅₃	—	2	1	no	—
74	C₄⋊Dic₅₃	—	2	1	no	—
75	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
76	#11719917	—	2	2	no	—
77	#11719917	—	2	4	no	—
78	#11719917	—	2	4	no	—
79	#11719917	C₂²×C₄	1	4	yes	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
84	C₂×#88322	C₂×D₄	1	2	yes	—
85	C₂×#88322	#41	1	2	yes	—
86	C₂×#88322	C₂⁴	1	1	yes	characteristic
87	#11719910	—	2	4	no	—
88	#11719910	—	2	2	no	—
89	#11719910	C₂×D₄	1	1	yes	characteristic
90	#11719910	C₂×D₄	1	2	yes	—
91	#11719910	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	#131	1	1	yes	characteristic
94	Q₈×C₅₃	—	2	2	no	—
95	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
96	D₄×C₅₃	—	2	4	no	—
97	D₄×C₅₃	—	2	4	no	—
98	D₄×C₅₃	—	2	2	no	—
99	D₄×C₅₃	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₅₃	—	2	2	no	—
108	C₈×C₅₃	C₂²×C₈	1	4	yes	—
109	#88322	—	2	4	no	—
110	#88322	C₄×D₄	1	2	yes	—
111	#88322	C₂²×D₄	1	1	yes	characteristic
112	#88322	C₂×#41	1	1	yes	characteristic
113	#33441	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	4	1	no	—
116	C₂²×C₅₃	—	4	4	no	—
117	C₂²×C₅₃	C₈×D₄	1	2	yes	—
118	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	C₈×D₄	1	2	yes	—
121	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
123	Dic₅₃	—	4	2	no	—
124	C₂×#504	—	53	2	no	—
125	#2955	—	53	1	no	—
126	#2171	—	53	4	no	—
127	#2170	—	53	4	no	—
128	#2153	—	53	1	no	—
129	#2117	—	53	2	no	—
130	#2108	—	53	1	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	4	4	no	—
134	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
135	#575	#6157	1	1	yes	characteristic
136	C₂×#131	—	53	2	no	—
137	#504	—	53	4	no	—
138	#504	—	53	4	no	—
139	#504	—	106	2	no	—
140	#504	—	106	2	no	—
141	C₂×M₃₂	—	53	1	no	—
142	C₂×M₃₂	—	53	2	no	—
143	C₂×M₃₂	—	106	2	no	—
144	C₂²×C₁₆	—	53	1	no	—
145	C₂²×C₁₆	—	106	2	no	—
146	#444	—	53	4	no	—
147	#443	—	53	4	no	—
148	#430	—	53	1	no	—
149	#421	—	53	2	no	—
150	#405	—	53	1	no	—
151	#364	—	53	4	no	—
152	#350	—	53	8	no	—
153	#346	—	53	2	no	—
154	C₄×C₁₆	—	53	2	no	—
155	C₅₃	#33441	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	C₂×#6157	1	2	yes	—
157	#137	#88324	1	4	yes	—
158	AGL(1,ℤ/8)	#88324	1	4	yes	—
159	#135	C₂×#6157	1	4	yes	—
160	#131	—	53	4	no	—
161	#131	—	53	4	no	—
162	#131	—	106	2	no	—
163	#131	—	106	2	no	—
164	C₂×M₁₆	—	53	2	no	—
165	C₂×M₁₆	—	106	2	no	—
166	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
167	C₂²×C₈	—	53	1	no	—
168	C₂²×C₈	—	106	2	no	—
169	#117	—	53	1	no	—
170	M₃₂	—	53	4	no	—
171	M₃₂	—	106	4	no	—
172	M₃₂	—	106	4	no	—
173	M₃₂	—	106	2	no	—
174	M₃₂	—	106	2	no	—
175	C₂×C₁₆	—	53	1	no	—
176	C₂×C₁₆	—	53	1	no	—
177	C₂×C₁₆	—	53	2	no	—
178	C₂×C₁₆	—	53	2	no	—
179	C₂×C₁₆	—	53	2	no	—
180	C₂×C₁₆	—	106	4	no	—
181	C₂×C₁₆	—	106	4	no	—
182	C₂×C₁₆	—	106	2	no	—
183	#108	—	53	4	no	—
184	#104	—	53	8	no	—
185	#97	—	53	2	no	—
186	C₄×C₈	—	53	2	no	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×#88324	1	4	yes	—
190	#41	C₂²×#6157	1	4	yes	—
191	C₂×Q₈	C₂×#88324	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×#88324	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
196	Q₁₆	C₂×#88324	1	4	yes	—
197	SD₁₆	C₂×#88324	1	8	yes	—
198	D₈	C₂×#88324	1	4	yes	—
199	M₁₆	—	53	4	no	—
200	M₁₆	—	106	4	no	—
201	M₁₆	—	106	4	no	—
202	M₁₆	—	106	2	no	—
203	M₁₆	C₂×#88324	1	4	yes	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	1	no	—
207	C₂×C₈	—	53	1	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	106	4	no	—
210	C₂×C₈	—	106	2	no	—
211	C₂×C₈	C₂²×#6157	1	2	yes	—
212	C₄⋊C₄	—	106	1	no	—
213	C₂²⋊C₄	—	106	1	no	—
214	C₄²	—	53	2	no	—
215	C₁₆	—	53	1	no	—
216	C₁₆	—	53	2	no	—
217	C₁₆	—	53	1	no	—
218	C₁₆	—	106	2	no	—
219	C₁₆	—	106	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#13133304	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×#88324	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×#88324	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	D₄×#6157	1	2	yes	—
233	C₂×C₄	C₂²×#88324	1	2	yes	—
234	C₂×C₄	#13133304	1	1	yes	characteristic
235	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
236	C₈	—	53	2	no	—
237	C₈	—	53	1	no	—
238	C₈	—	53	1	no	—
239	C₈	—	106	4	no	—
240	C₈	C₂²×#88324	1	4	yes	—
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	D₄×#88324	1	2	yes	—
245	C₂²	C₂×#13133304	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	212	2	no	—
248	C₄	D₄×#88324	1	2	yes	—
249	C₄	C₂×D₄×#6157	1	1	yes	characteristic
250	C₄	C₂³×#88324	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×D₄×#88324	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting, prime radical	C₅₃×#575	C₄
socle	C₂×C₅₃	C₂×C₈×D₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#575	#6157
hypocenter	C₅₃	#33441
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

Complements

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

N	H ≅ G/N	type
1. C₂×#16811681	C₂	semidirect
2. #20291105	C₂	semidirect
3. #20291102	C₂	semidirect
4. #20291056	C₂	semidirect
5. #20290957	C₂	semidirect
6. #20290926	C₂	semidirect
10. #16811681	C₂²	semidirect
11. #16811681	C₂²	semidirect
17. #16811545	C₂²	semidirect
18. #16811521	C₂²	semidirect
19. #16811512	C₂²	semidirect
20. #16811509	C₂²	semidirect
155. C₅₃	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811681	2
#20291105	2
#20291102	2
#20291056	2
#20290957	2
#20290926	2
#20283618	2
#33441	53