Subgroups — #28328351

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291682	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	C₂×#16811521	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	4	no	—
9	#16811681	C₂²	1	8	yes	complemented
10	C₂×#13133291	—	2	4	no	—
11	C₂×#13133291	C₂²	1	2	yes	—
12	C₂×#13133291	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	1	yes	characteristic
14	#16811545	C₂²	1	4	yes	complemented
15	#16811521	C₂²	1	4	yes	complemented
16	#16811521	C₂²	1	4	yes	complemented
17	#16811512	C₂²	1	2	yes	complemented
18	#16811509	C₂²	1	2	yes	complemented
19	C₂×#13133224	C₂²	1	2	yes	—
20	#16810745	C₄	1	4	yes	—
21	#16810740	C₂²	1	4	yes	—
22	#16810672	C₄	1	1	yes	characteristic
23	#16810640	C₄	1	2	yes	—
24	#16810581	C₂²	1	1	yes	characteristic
25	C₅₃×#575	C₄	1	1	yes	characteristic
26	#13133291	—	2	2	no	—
27	#13133291	—	2	8	no	—
28	#13133291	D₄	1	4	yes	—
29	#13133291	C₂³	1	4	yes	—
30	C₂×#11719910	—	2	2	no	—
31	C₂×#11719910	—	2	2	no	—
32	C₂×#11719910	—	2	8	no	—
33	C₂×#11719910	C₂³	1	1	yes	characteristic
34	C₂×#11719910	C₂³	1	1	yes	characteristic
35	C₂×#11719910	C₂³	1	2	yes	—
36	#13133224	—	2	2	no	—
37	#13133224	—	2	2	no	—
38	#13133224	C₂×C₄	1	4	yes	—
39	#13133224	C₂³	1	4	yes	—
40	C₂×#11719917	—	2	2	no	—
41	C₂×#11719917	C₂×C₄	1	2	yes	—
42	C₂²×#88322	—	2	2	no	—
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	#13133176	C₂×C₄	1	1	yes	characteristic
45	#13133153	C₂×C₄	1	4	yes	—
46	#13133144	C₂×C₄	1	8	yes	—
47	#13133127	C₂×C₄	1	2	yes	—
48	C₈×Dic₅₃	C₂³	1	2	yes	—
49	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
50	#137×C₅₃	C₈	1	4	yes	—
51	AGL(1,ℤ/8)×C₅₃	C₈	1	4	yes	—
52	#135×C₅₃	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#41×C₅₃	—	2	2	no	—
55	#41×C₅₃	—	2	2	no	—
56	#41×C₅₃	C₂×C₈	1	4	yes	—
57	#41×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₅₃	—	2	2	no	—
60	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅₃	—	2	2	no	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
64	SD₁₆×C₅₃	C₂×C₈	1	8	yes	—
65	D₈×C₅₃	C₂×C₈	1	4	yes	—
66	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²⋊Dic₅₃	—	2	1	no	—
69	C₄⋊Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
71	#11719917	—	2	2	no	—
72	#11719917	—	2	4	no	—
73	#11719917	—	2	4	no	—
74	#11719917	C₂²×C₄	1	4	yes	—
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
79	C₂×#88322	C₂×D₄	1	2	yes	—
80	C₂×#88322	#41	1	2	yes	—
81	C₂×#88322	C₂⁴	1	1	yes	characteristic
82	#11719910	—	2	4	no	—
83	#11719910	—	2	1	no	—
84	#11719910	—	2	1	no	—
85	#11719910	C₂×D₄	1	4	yes	—
86	C₂³×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	#131	1	1	yes	characteristic
88	Q₈×C₅₃	—	2	2	no	—
89	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
90	D₄×C₅₃	—	2	4	no	—
91	D₄×C₅₃	—	2	4	no	—
92	D₄×C₅₃	—	2	2	no	—
93	D₄×C₅₃	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂²×C₈	1	4	yes	—
103	#88322	—	2	4	no	—
104	#88322	C₄×D₄	1	2	yes	—
105	#88322	C₂²×D₄	1	1	yes	characteristic
106	#88322	C₂×#41	1	1	yes	characteristic
107	#33442	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	4	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	8	2	no	—
111	C₂²×C₅₃	C₈×D₄	1	2	yes	—
112	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
113	C₄×C₅₃	—	2	4	no	—
114	C₄×C₅₃	C₈×D₄	1	2	yes	—
115	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
116	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
117	Dic₅₃	—	4	2	no	—
118	#3407	—	53	2	no	—
119	#2955	—	53	1	no	—
120	#2171	—	53	4	no	—
121	#2170	—	53	4	no	—
122	#2153	—	53	1	no	—
123	C₂×#350	—	53	2	no	—
124	#2108	—	53	1	no	—
125	C₂×C₅₃	—	2	1	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	—	4	4	no	—
128	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
129	#575	#6157	1	1	yes	characteristic
130	C₂×#131	—	53	2	no	—
131	#504	—	53	8	no	—
132	#504	—	106	4	no	—
133	C₂×M₃₂	—	53	1	no	—
134	C₂×M₃₂	—	53	2	no	—
135	C₂×M₃₂	—	106	4	no	—
136	C₂²×C₁₆	—	53	1	no	—
137	#444	—	53	4	no	—
138	#443	—	53	4	no	—
139	#430	—	53	1	no	—
140	#421	—	53	2	no	—
141	#405	—	53	1	no	—
142	#364	—	53	4	no	—
143	#350	—	53	4	no	—
144	#350	—	53	4	no	—
145	#346	—	53	2	no	—
146	C₄×C₁₆	—	53	2	no	—
147	C₅₃	#33442	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	C₂×#6157	1	2	yes	—
149	#137	#88324	1	4	yes	—
150	AGL(1,ℤ/8)	#88324	1	4	yes	—
151	#135	C₂×#6157	1	4	yes	—
152	#131	—	53	4	no	—
153	#131	—	53	4	no	—
154	#131	—	106	2	no	—
155	#131	—	106	2	no	—
156	C₂×M₁₆	—	53	2	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
159	C₂²×C₈	—	53	1	no	—
160	C₂²×C₈	—	106	2	no	—
161	#117	—	53	1	no	—
162	M₃₂	—	53	4	no	—
163	M₃₂	—	53	4	no	—
164	M₃₂	—	106	8	no	—
165	M₃₂	—	106	2	no	—
166	C₂×C₁₆	—	53	2	no	—
167	C₂×C₁₆	—	53	1	no	—
168	C₂×C₁₆	—	53	1	no	—
169	C₂×C₁₆	—	106	8	no	—
170	C₂×C₁₆	—	106	2	no	—
171	C₂×C₁₆	—	106	2	no	—
172	#108	—	53	4	no	—
173	#104	—	53	8	no	—
174	#97	—	53	2	no	—
175	C₄×C₈	—	53	2	no	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	C₂×#88324	1	4	yes	—
179	#41	C₂²×#6157	1	4	yes	—
180	C₂×Q₈	C₂×#88324	1	2	yes	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	C₂×#88324	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
185	Q₁₆	C₂×#88324	1	4	yes	—
186	SD₁₆	C₂×#88324	1	8	yes	—
187	D₈	C₂×#88324	1	4	yes	—
188	M₁₆	—	53	4	no	—
189	M₁₆	—	106	4	no	—
190	M₁₆	—	106	4	no	—
191	M₁₆	—	106	2	no	—
192	M₁₆	C₂×#88324	1	4	yes	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	1	no	—
196	C₂×C₈	—	53	1	no	—
197	C₂×C₈	—	106	4	no	—
198	C₂×C₈	—	106	4	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	C₂²×#6157	1	2	yes	—
201	C₄⋊C₄	—	106	1	no	—
202	C₂²⋊C₄	—	106	1	no	—
203	C₄²	—	53	2	no	—
204	C₁₆	—	53	4	no	—
205	C₁₆	—	106	1	no	—
206	C₁₆	—	106	1	no	—
207	C₁₆	—	106	4	no	—
208	C₂³	—	2	2	no	—
209	C₂³	#13133304	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	C₂²×#88324	1	4	yes	—
212	D₄	—	2	4	no	—
213	D₄	—	2	4	no	—
214	D₄	—	2	2	no	—
215	D₄	C₂²×#88324	1	4	yes	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	106	2	no	—
220	C₂×C₄	D₄×#6157	1	2	yes	—
221	C₂×C₄	C₂²×#88324	1	2	yes	—
222	C₂×C₄	#13133304	1	1	yes	characteristic
223	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
224	C₈	—	53	2	no	—
225	C₈	—	53	1	no	—
226	C₈	—	53	1	no	—
227	C₈	—	106	4	no	—
228	C₈	C₂²×#88324	1	4	yes	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	8	2	no	—
232	C₂²	D₄×#88324	1	2	yes	—
233	C₂²	C₂×#13133304	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	212	2	no	—
236	C₄	D₄×#88324	1	2	yes	—
237	C₄	C₂×D₄×#6157	1	1	yes	characteristic
238	C₄	C₂³×#88324	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	4	no	—
242	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	#33442
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. #20291682	C₂	semidirect
2. #20291105	C₂	semidirect
3. #20291102	C₂	semidirect
4. #20291056	C₂	semidirect
5. C₂×#16811521	C₂	semidirect
6. #20290926	C₂	semidirect
9. #16811681	C₂²	semidirect
14. #16811545	C₂²	semidirect
15. #16811521	C₂²	semidirect
16. #16811521	C₂²	semidirect
17. #16811512	C₂²	semidirect
18. #16811509	C₂²	semidirect
147. C₅₃	#33442	semidirect

Maximal subgroups

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

subgroup	index
#20291682	2
#20291105	2
#20291102	2
#20291056	2
C₂×#16811521	2
#20290926	2
#20283618	2
#33442	53