Subgroups — #28311987

Order 13568.

284 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	#20291056	C₂	1	2	yes	maximal
2	C₂×#16811545	C₂	1	1	yes	maximal, characteristic
3	#20290921	C₂	1	1	yes	maximal, characteristic
4	#20289766	C₂	1	2	yes	maximal
5	#20274597	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂×#13133291	C₂²	1	1	yes	characteristic
8	C₂×#13133291	C₂²	1	1	yes	characteristic
9	#16811545	—	2	4	no	—
10	#16811545	—	2	1	no	—
11	#16811545	—	2	1	no	—
12	#16811512	—	2	1	no	—
13	#16811509	—	2	1	no	—
14	#16810661	C₄	1	1	yes	characteristic
15	#16810581	C₂²	1	2	yes	—
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810384	C₄	1	2	yes	—
18	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
19	#13133291	—	2	1	no	—
20	#13133291	—	2	1	no	—
21	#13133291	—	2	2	no	—
22	#13133291	—	2	2	no	—
23	#13133291	D₄	1	2	yes	—
24	#13133291	Q₈	1	2	yes	—
25	C₂×#11719910	—	2	1	no	—
26	C₂×#11719910	—	2	1	no	—
27	C₂×#11719910	—	2	2	no	—
28	C₂²×#88322	C₂³	1	1	yes	characteristic
29	#13133176	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	#13133151	—	2	1	no	—
32	#13133149	—	2	1	no	—
33	#13133127	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	D₄	1	1	yes	characteristic
36	C₈×Dic₅₃	D₄	1	1	yes	characteristic
37	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
38	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	#108×C₅₃	C₈	1	2	yes	—
42	#108×C₅₃	C₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	2	1	no	—
45	M₁₆×C₅₃	—	2	2	no	—
46	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
54	C₂²⋊Dic₅₃	—	2	2	no	—
55	C₄⋊Dic₅₃	—	2	2	no	—
56	C₄×Dic₅₃	—	2	1	no	—
57	C₄×Dic₅₃	—	2	2	no	—
58	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#88322	C₂×D₄	1	1	yes	characteristic
61	C₂×#88322	C₂×D₄	1	1	yes	characteristic
62	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
63	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
64	C₂×#88322	#41	1	1	yes	characteristic
65	C₂×#88322	#41	1	1	yes	characteristic
66	#11719910	—	2	2	no	—
67	#11719910	—	2	2	no	—
68	#11719910	—	4	2	no	—
69	C₂³×C₅₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₈×C₅₃	—	2	4	no	—
81	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
85	#88322	#121	1	1	yes	characteristic
86	#88322	C₂²⋊Q₈	1	1	yes	characteristic
87	#88322	#125	1	1	yes	characteristic
88	#88322	C₄⋊Q₈	1	1	yes	characteristic
89	#13164	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	4	1	no	—
91	C₂²×C₅₃	#424	1	1	yes	characteristic
92	C₂²×C₅₃	#436	1	1	yes	characteristic
93	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
94	C₄×C₅₃	#389	1	1	yes	characteristic
95	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
97	C₄×C₅₃	#446	1	1	yes	characteristic
98	Dic₅₃	—	2	2	no	—
99	Dic₅₃	—	4	1	no	—
100	Dic₅₃	—	8	2	no	—
101	#2153	—	53	2	no	—
102	C₂×#364	—	53	1	no	—
103	C₄×M₃₂	—	53	1	no	—
104	#1772	—	53	1	no	—
105	#1378	—	53	2	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	#1939	1	1	yes	characteristic
110	C₂×M₃₂	—	53	1	no	—
111	C₂×M₃₂	—	53	1	no	—
112	C₂×M₃₂	—	53	2	no	—
113	C₂×#108	#6157	1	1	yes	characteristic
114	#427	—	53	1	no	—
115	#405	—	53	2	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	#364	—	106	1	no	—
118	#364	—	106	1	no	—
119	#364	—	106	4	no	—
120	#346	—	106	1	no	—
121	C₄×C₁₆	—	106	1	no	—
122	#339	—	53	2	no	—
123	C₅₃	#13164	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	C₂×#6157	1	2	yes	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
127	#117	—	53	2	no	—
128	C₂×C₄²	—	53	1	no	—
129	M₃₂	—	53	2	no	—
130	M₃₂	—	53	2	no	—
131	M₃₂	—	106	1	no	—
132	M₃₂	—	106	1	no	—
133	M₃₂	—	106	2	no	—
134	M₃₂	—	106	2	no	—
135	C₂×C₁₆	—	106	1	no	—
136	C₂×C₁₆	—	106	1	no	—
137	C₂×C₁₆	—	106	2	no	—
138	#108	#88324	1	2	yes	—
139	#108	#88324	1	2	yes	—
140	#107	—	106	1	no	—
141	#106	—	106	1	no	—
142	#97	—	106	2	no	—
143	C₄×C₈	—	53	1	no	—
144	C₄×C₈	—	53	1	no	—
145	C₄×C₈	—	53	1	no	—
146	C₄×C₈	—	53	1	no	—
147	C₄×C₈	—	106	2	no	—
148	C₂²×C₄	—	106	1	no	—
149	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	C₂×#88324	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	53	1	no	—
154	C₂×C₈	—	53	1	no	—
155	C₂×C₈	—	53	1	no	—
156	C₂×C₈	—	53	1	no	—
157	C₂×C₈	—	53	1	no	—
158	C₂×C₈	—	53	1	no	—
159	C₂×C₈	#11719936	1	1	yes	characteristic
160	C₂×C₈	#11719936	1	1	yes	characteristic
161	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
162	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
163	C₂×C₈	C₂×#88324	1	1	yes	characteristic
164	C₂×C₈	C₂×#88324	1	1	yes	characteristic
165	C₄⋊C₄	—	106	2	no	—
166	C₂²⋊C₄	—	106	2	no	—
167	C₄²	—	53	1	no	—
168	C₄²	—	53	1	no	—
169	C₄²	—	106	1	no	—
170	C₄²	—	106	2	no	—
171	C₁₆	—	106	2	no	—
172	C₁₆	—	106	2	no	—
173	C₁₆	—	212	2	no	—
174	C₂³	#13133305	1	1	yes	characteristic
175	C₂×C₄	—	106	1	no	—
176	C₂×C₄	—	106	1	no	—
177	C₂×C₄	—	212	2	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
180	C₂×C₄	D₄×#6157	1	1	yes	characteristic
181	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
182	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
184	C₂×C₄	#13133304	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	53	1	no	—
187	C₈	—	53	1	no	—
188	C₈	—	53	1	no	—
189	C₈	—	53	1	no	—
190	C₈	C₄⋊#88324	1	1	yes	characteristic
191	C₈	C₄⋊#88324	1	1	yes	characteristic
192	C₈	C₄⋊#88324	1	1	yes	characteristic
193	C₈	C₄⋊#88324	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#16811630	1	1	yes	characteristic
196	C₂²	#16811657	1	1	yes	characteristic
197	C₂²	Q₈×#88324	1	1	yes	characteristic
198	C₄	—	106	2	no	—
199	C₄	—	424	2	no	—
200	C₄	—	212	1	no	—
201	C₄	#16811574	1	1	yes	characteristic
202	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
203	C₄	D₄×#88324	1	1	yes	characteristic
204	C₄	#16811673	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#20290812	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811574
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂×C₅₃	#1939
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	#13164
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
123. C₅₃	#13164	semidirect

Maximal subgroups

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

subgroup	index
#20291056	2
C₂×#16811545	2
#20290921	2
#20289766	2
#20274597	2
#13164	53