Subgroups — #28316267

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291361	C₂	1	2	yes	maximal
2	#20290285	C₂	1	6	yes	maximal, complemented
3	#20290273	C₂	1	3	yes	maximal, complemented
4	D₅₃⋊#335	C₂	1	3	yes	maximal, complemented
5	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133305	—	2	2	no	—
7	#16811630	—	2	6	no	—
8	#16811627	C₂²	1	6	yes	—
9	D₅₃⋊(C₄⋊C₈)	C₂²	1	6	yes	—
10	#16811487	C₂²	1	6	yes	—
11	#16811486	C₂²	1	1	yes	characteristic
12	#16811486	C₂²	1	3	yes	—
13	#16811486	C₂²	1	6	yes	—
14	C₄×#11719949	C₂²	1	1	yes	characteristic
15	Q₈×#88322	C₄	1	1	yes	characteristic
16	C₈×D₂₁₂	C₄	1	3	yes	—
17	#16810688	C₄	1	3	yes	—
18	#16810675	C₂²	1	3	yes	—
19	C₄×C₈×D₅₃	C₂²	1	3	yes	—
20	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
21	#13133305	—	4	8	no	—
22	C₂×#11719936	—	2	6	no	—
23	C₂×D₅₃⋊C₈	—	2	6	no	—
24	#13133284	—	2	6	no	—
25	C₄⋊#88324	—	2	6	no	—
26	C₄⋊#88324	C₂³	1	2	yes	—
27	C₄⋊#88324	C₂³	1	6	yes	—
28	C₂×#11719949	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
31	Q₈×Dic₅₃	Q₈	1	1	yes	characteristic
32	C₄×D₂₁₂	C₂×C₄	1	3	yes	—
33	#13133179	Q₈	1	3	yes	—
34	#13133175	D₄	1	3	yes	—
35	C₄²×D₅₃	C₂³	1	3	yes	—
36	C₄⋊#88322	C₂×C₄	1	3	yes	—
37	#13133129	C₂×C₄	1	6	yes	—
38	C₄×#88322	C₂×C₄	1	3	yes	—
39	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₈×Dic₅₃	C₂³	1	3	yes	—
41	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
44	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	#11719949	D₈	1	4	yes	—
46	#11719949	Q₁₆	1	4	yes	—
47	C₂×D₂₁₂	C₄⋊C₄	1	3	yes	—
48	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
49	C₂×C₄×D₅₃	C₂×D₄	1	3	yes	—
50	C₂×C₄×D₅₃	C₂×Q₈	1	3	yes	—
51	C₂×#88324	—	2	6	no	—
52	C₂×#88324	—	2	2	no	—
53	#11719936	—	4	12	no	—
54	D₅₃⋊C₈	—	4	12	no	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
58	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
59	#11719921	C₄⋊C₄	1	6	yes	—
60	C₄⋊Dic₅₃	C₄⋊C₄	1	3	yes	—
61	C₄×Dic₅₃	C₂×D₄	1	3	yes	—
62	C₄×Dic₅₃	C₂×Q₈	1	3	yes	—
63	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
64	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
65	C₂×#88322	C₂²×C₄	1	3	yes	—
66	C₈×D₅₃	—	2	6	no	—
67	C₈×D₅₃	—	4	3	no	—
68	C₂²×D₅₃	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₅₃	#107	1	4	yes	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
76	D₂₁₂	#107	1	12	yes	—
77	C₄×D₅₃	—	2	3	no	—
78	C₄×D₅₃	—	2	3	no	—
79	C₄×D₅₃	—	4	3	no	—
80	C₄×D₅₃	C₂×D₈	1	6	yes	—
81	C₄×D₅₃	C₂×Q₁₆	1	6	yes	—
82	#88324	—	4	6	no	—
83	#88324	—	4	2	no	—
84	C₈×C₅₃	—	2	3	no	—
85	C₈×C₅₃	#131	1	2	yes	complemented
86	#88322	—	2	3	no	—
87	#88322	—	2	1	no	—
88	#17828	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₅₃	—	2	3	no	—
91	C₂×D₅₃	—	2	3	no	—
92	C₂×D₅₃	C₂×#107	1	6	yes	—
93	C₄×C₅₃	—	2	3	no	—
94	C₄×C₅₃	C₂×#107	1	6	yes	—
95	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
96	C₄×C₅₃	#568	1	1	yes	characteristic
97	Dic₅₃	—	2	3	no	—
98	Dic₅₃	—	2	1	no	—
99	Dic₅₃	C₂²×D₈	1	1	yes	characteristic
100	Dic₅₃	C₂²×Q₁₆	1	1	yes	characteristic
101	C₈×#41	—	53	1	no	—
102	#2906	—	53	2	no	—
103	#1571	—	53	6	no	—
104	#1567	—	53	3	no	—
105	C₂×#335	—	53	3	no	—
106	C₂×C₅₃	#2906	1	1	yes	characteristic
107	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
108	C₂×C₅₃	#2916	1	1	yes	characteristic
109	D₅₃	—	2	6	no	—
110	C₂×#131	—	106	2	no	—
111	C₄×#41	—	53	1	no	—
112	C₈×Q₈	—	53	1	no	—
113	C₈×Q₈	#6157	1	1	yes	characteristic
114	C₈×D₄	—	53	3	no	—
115	C₈×D₄	—	53	3	no	—
116	#431	—	53	3	no	—
117	#424	—	106	6	no	—
118	#423	—	53	6	no	—
119	C₂×C₄⋊C₈	—	53	6	no	—
120	C₂×C₄×C₈	—	53	3	no	—
121	#335	—	53	1	no	—
122	#335	—	53	3	no	—
123	#335	—	53	6	no	—
124	#335	—	53	6	no	—
125	C₅₃	#17828	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	—	53	1	no	—
127	#131	—	212	8	no	—
128	C₂×M₁₆	—	106	6	no	—
129	C₂²×C₈	—	53	3	no	—
130	C₂²×C₈	—	53	3	no	—
131	C₂²×C₈	—	106	6	no	—
132	C₄×Q₈	—	53	1	no	—
133	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
134	C₄×D₄	—	53	3	no	—
135	C₄×D₄	—	53	3	no	—
136	#117	—	53	3	no	—
137	C₂×C₄²	—	53	3	no	—
138	C₄⋊C₈	—	53	2	no	—
139	C₄⋊C₈	—	53	6	no	—
140	C₄⋊C₈	—	53	3	no	—
141	C₄⋊C₈	—	106	6	no	—
142	C₄⋊C₈	—	106	6	no	—
143	C₄⋊C₈	C₂×#6157	1	3	yes	—
144	C₂²⋊C₈	—	53	6	no	—
145	C₄×C₈	—	53	1	no	—
146	C₄×C₈	—	53	3	no	—
147	C₄×C₈	—	53	3	no	—
148	C₄×C₈	C₂×#6157	1	3	yes	—
149	#41	—	53	4	no	—
150	#41	—	53	4	no	—
151	C₂×Q₈	C₄⋊#6157	1	1	yes	characteristic
152	C₂×D₄	—	53	3	no	—
153	C₂²×C₄	—	53	3	no	—
154	C₂²×C₄	—	53	3	no	—
155	C₂²×C₄	—	53	3	no	—
156	M₁₆	—	212	12	no	—
157	C₂×C₈	—	53	1	no	—
158	C₂×C₈	—	53	3	no	—
159	C₂×C₈	—	106	6	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	6	no	—
162	C₂×C₈	—	212	3	no	—
163	C₂×C₈	—	212	12	no	—
164	C₂×C₈	C₂²×#6157	1	3	yes	—
165	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
166	C₄⋊C₄	—	53	3	no	—
167	C₄⋊C₄	C₄⋊#6157	1	3	yes	—
168	C₂²⋊C₄	—	53	6	no	—
169	C₄²	—	53	3	no	—
170	C₄²	—	53	3	no	—
171	C₄²	—	53	1	no	—
172	C₄²	C₂²×#6157	1	3	yes	—
173	C₂³	—	53	3	no	—
174	Q₈	#13133286	1	4	yes	—
175	D₄	—	53	12	no	—
176	C₂×C₄	—	53	1	no	—
177	C₂×C₄	—	53	6	no	—
178	C₂×C₄	—	53	6	no	—
179	C₂×C₄	—	53	3	no	—
180	C₂×C₄	—	53	1	no	—
181	C₂×C₄	—	106	3	no	—
182	C₂×C₄	—	106	3	no	—
183	C₂×C₄	—	212	3	no	—
184	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
185	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
186	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	106	3	no	—
189	C₈	—	106	1	no	—
190	C₈	—	212	2	no	—
191	C₈	—	212	6	no	—
192	C₈	#13133305	1	2	yes	complemented
193	C₂²	—	53	6	no	—
194	C₂²	—	106	3	no	—
195	C₂²	—	106	3	no	—
196	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	53	1	no	—
199	C₄	—	53	1	no	—
200	C₄	—	106	3	no	—
201	C₄	—	106	1	no	—
202	C₄	C₂×#13133286	1	6	yes	—
203	C₄	C₂×#13133305	1	1	yes	characteristic
204	C₄	#16811702	1	1	yes	characteristic
205	C₂	—	106	6	no	—
206	C₂	#20291361	1	1	yes	characteristic
207	C₂	C₂²×#13133286	1	1	yes	characteristic
208	C₂	#20291377	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719949	D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#17828
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #20290285	C₂	semidirect
3. #20290273	C₂	semidirect
4. D₅₃⋊#335	C₂	semidirect
85. C₈×C₅₃	#131	semidirect
125. C₅₃	#17828	semidirect
192. C₈	#13133305	semidirect

Maximal subgroups

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

subgroup	index
#20291361	2
#20290285	2
#20290273	2
D₅₃⋊#335	2
C₈×#11719949	2
#17828	53