Subgroups — #28310790

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811545	C₂	1	1	yes	maximal, characteristic
2	#20290926	C₂	1	2	yes	maximal
3	C₂×#16811509	C₂	1	1	yes	maximal, characteristic
4	#20289770	C₂	1	2	yes	maximal
5	#20274597	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂²×#11719910	C₂²	1	1	yes	characteristic
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	#16811545	C₄	1	4	yes	complemented
10	#16811512	—	2	2	no	—
11	#16811509	—	2	2	no	—
12	#16811509	C₄	1	4	yes	complemented
13	#16810661	C₄	1	1	yes	characteristic
14	#16810581	C₂²	1	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810384	C₄	1	2	yes	—
17	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
18	#13133291	—	2	2	no	—
19	#13133291	C₂×C₄	1	4	yes	—
20	C₂×#11719910	—	2	2	no	—
21	C₂×#11719910	—	2	1	no	—
22	C₂×#11719910	—	2	1	no	—
23	C₂×#11719910	C₂×C₄	1	2	yes	—
24	C₂×#11719910	C₂×C₄	1	2	yes	—
25	C₂×#11719910	D₄	1	1	yes	characteristic
26	C₂×#11719910	D₄	1	1	yes	characteristic
27	C₂×#11719910	Q₈	1	2	yes	—
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	C₈	1	2	yes	—
32	#13133149	C₈	1	2	yes	—
33	#13133127	—	2	2	no	—
34	C₈×Dic₅₃	—	2	2	no	—
35	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	#108×C₅₃	C₈	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	1	no	—
42	M₁₆×C₅₃	—	2	2	no	—
43	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₄²	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²⋊Dic₅₃	—	2	2	no	—
50	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
51	C₄×Dic₅₃	—	2	2	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	C₄²	1	2	yes	—
54	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
55	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
56	C₂×#88322	C₂×D₄	1	1	yes	characteristic
57	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
58	C₂×#88322	#41	1	1	yes	characteristic
59	C₂×#88322	#41	1	1	yes	characteristic
60	#11719910	—	2	2	no	—
61	#11719910	—	2	4	no	—
62	#11719910	C₄⋊C₄	1	2	yes	—
63	#11719910	C₄⋊C₄	1	2	yes	—
64	C₂³×C₅₃	#131	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
71	C₂×Dic₅₃	—	2	4	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	2	4	no	—
74	C₈×C₅₃	—	2	4	no	—
75	C₈×C₅₃	C₄×C₈	1	2	yes	—
76	C₈×C₅₃	C₄×C₈	1	2	yes	—
77	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
78	#88322	#117	1	1	yes	characteristic
79	#88322	C₄×D₄	1	1	yes	characteristic
80	#88322	C₄×Q₈	1	1	yes	characteristic
81	#12664	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	#405	1	1	yes	characteristic
84	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
85	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
86	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
89	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
90	Dic₅₃	—	2	4	no	—
91	Dic₅₃	—	4	4	no	—
92	C₂×#364	—	53	1	no	—
93	#2108	—	53	2	no	—
94	C₂×C₄×C₁₆	—	53	1	no	—
95	#1772	—	53	1	no	—
96	#1379	—	53	2	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₅₃	#2170	1	2	yes	—
101	C₂×M₃₂	—	53	2	no	—
102	C₂²×C₁₆	—	53	1	no	—
103	C₂²×C₁₆	—	53	1	no	—
104	C₂×#108	#6157	1	1	yes	characteristic
105	#427	—	53	1	no	—
106	#405	—	53	2	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	#364	—	53	4	no	—
109	#346	—	106	2	no	—
110	C₄×C₁₆	—	53	4	no	—
111	C₄×C₁₆	—	106	2	no	—
112	#339	—	53	2	no	—
113	C₅₃	#12664	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	C₂×#6157	1	2	yes	—
115	C₂²×C₈	—	53	1	no	—
116	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
117	#117	—	53	2	no	—
118	C₂×C₄²	—	53	1	no	—
119	M₃₂	—	53	4	no	—
120	M₃₂	—	106	2	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	—	53	1	no	—
123	C₂×C₁₆	—	53	1	no	—
124	C₂×C₁₆	—	53	2	no	—
125	C₂×C₁₆	—	53	2	no	—
126	C₂×C₁₆	—	106	2	no	—
127	C₂×C₁₆	—	106	1	no	—
128	C₂×C₁₆	—	106	1	no	—
129	#108	#88324	1	4	yes	—
130	#107	—	53	2	no	—
131	#106	—	53	2	no	—
132	#97	—	106	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	106	2	no	—
136	C₂²×C₄	—	106	1	no	—
137	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	C₂×#88324	1	4	yes	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	53	1	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	53	1	no	—
145	C₂×C₈	—	53	1	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
148	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
149	C₂×C₈	C₄×#6157	1	2	yes	—
150	C₂×C₈	C₂×#88324	1	2	yes	—
151	C₄⋊C₄	—	53	4	no	—
152	C₂²⋊C₄	—	106	2	no	—
153	C₄²	—	53	2	no	—
154	C₄²	—	106	2	no	—
155	C₄²	—	106	1	no	—
156	C₁₆	—	53	2	no	—
157	C₁₆	—	53	2	no	—
158	C₁₆	—	106	2	no	—
159	C₁₆	—	106	4	no	—
160	C₂³	#13133305	1	1	yes	characteristic
161	C₂×C₄	—	106	4	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	—	106	4	no	—
164	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
165	C₂×C₄	D₄×#6157	1	1	yes	characteristic
166	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
167	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
168	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
169	C₂×C₄	#13133304	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	53	1	no	—
172	C₈	—	53	1	no	—
173	C₈	—	53	1	no	—
174	C₈	—	53	1	no	—
175	C₈	C₄×#88324	1	2	yes	—
176	C₈	C₄×#88324	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	#16811588	1	1	yes	characteristic
179	C₂²	#16811651	1	1	yes	characteristic
180	C₂²	Q₈×#88324	1	1	yes	characteristic
181	C₄	—	106	4	no	—
182	C₄	—	212	4	no	—
183	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
184	C₄	C₂×C₄×#88324	1	1	yes	characteristic
185	C₄	D₄×#88324	1	1	yes	characteristic
186	C₄	#16811672	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₄⋊C₄×#88324	1	1	yes	characteristic
190	C₂	#20291104	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	#12664
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
9. #16811545	C₄	semidirect
12. #16811509	C₄	semidirect
113. C₅₃	#12664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811545	2
#20290926	2
C₂×#16811509	2
#20289770	2
#20274597	2
#12664	53