Subgroups — #27537592

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283649	C₂	1	1	yes	maximal, characteristic
2	#20276621	C₂	1	8	yes	maximal, complemented
3	#20276473	C₂	1	2	yes	maximal
4	#20276471	C₂	1	1	yes	maximal, characteristic
5	#20276311	C₂	1	2	yes	maximal
6	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
7	#16811180	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16810753	C₄	1	4	yes	—
10	C₈×Dic₁₀₆	C₂²	1	4	yes	—
11	#16810649	C₄	1	2	yes	—
12	#16810647	C₄	1	1	yes	characteristic
13	#16810578	C₂²	1	2	yes	—
14	C₄⋊#11719908	C₂²	1	4	yes	—
15	#16810441	C₂²	1	8	yes	—
16	#16810398	C₂²	1	4	yes	—
17	C₁₆×Dic₅₃	C₂²	1	4	yes	—
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#346	C₂²	1	4	yes	—
22	C₂×#11719917	C₂×C₄	1	2	yes	—
23	#13133217	C₈	1	2	yes	—
24	#13133209	C₈	1	2	yes	—
25	#13133196	C₈	1	4	yes	—
26	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
27	#13133176	C₂×C₄	1	2	yes	—
28	#13133157	—	2	4	no	—
29	C₂×#11719908	C₂³	1	4	yes	—
30	C₄⋊#88322	C₂×C₄	1	4	yes	—
31	#13133146	C₂×C₄	1	8	yes	—
32	#13133127	C₂×C₄	1	4	yes	—
33	C₈×Dic₅₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
40	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₅₃	#41	1	2	yes	—
51	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
56	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
57	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
58	#11719919	C₂×C₈	1	8	yes	—
59	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
60	#11719917	—	2	4	no	—
61	C₂×#88322	C₂²×C₄	1	4	yes	—
62	C₁₆×C₅₃	—	2	2	no	—
63	C₁₆×C₅₃	C₂×Q₈	1	4	yes	—
64	#11719908	—	2	4	no	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#131	1	2	yes	—
72	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
74	Dic₁₀₆	—	4	4	no	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	C₄×Q₈	1	2	yes	—
77	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
79	#88322	—	2	4	no	—
80	#33475	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
83	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	C₈×Q₈	1	2	yes	—
87	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
89	Dic₅₃	—	2	8	no	—
90	#2960	—	53	1	no	—
91	#2183	—	53	8	no	—
92	#2151	—	53	2	no	—
93	#2150	—	53	1	no	—
94	#2108	—	53	2	no	—
95	C₂×#346	D₅₃	1	1	yes	characteristic
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
98	C₂×C₅₃	#3407	1	2	yes	—
99	#533	—	53	1	no	—
100	C₂×M₃₂	—	53	2	no	—
101	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
103	#446	—	53	4	no	—
104	C₈×Q₈	—	53	4	no	—
105	#424	—	53	2	no	—
106	#423	—	53	1	no	—
107	#405	—	53	2	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	53	8	no	—
110	C₄⋊C₁₆	—	53	4	no	—
111	#346	—	53	4	no	—
112	#346	C₂×D₅₃	1	4	yes	—
113	C₄×C₁₆	—	53	4	no	—
114	C₅₃	#33475	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	53	2	no	—
116	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊Q₈	—	53	2	no	—
119	#125	—	53	2	no	—
120	C₂²⋊Q₈	—	53	4	no	—
121	C₄×Q₈	—	53	4	no	—
122	#117	—	53	2	no	—
123	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
124	M₃₂	—	106	4	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	53	4	no	—
127	C₂×C₁₆	Dic₁₀₆	1	4	yes	complemented
128	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
129	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
130	C₄⋊C₈	—	53	8	no	—
131	C₄⋊C₈	—	53	4	no	—
132	#97	—	53	4	no	—
133	C₄×C₈	—	53	4	no	—
134	C₄×C₈	C₄×D₅₃	1	2	yes	—
135	C₄×C₈	C₂²×D₅₃	1	2	yes	—
136	C₂×Q₈	—	53	4	no	—
137	C₂²×C₄	C₈×D₅₃	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
139	M₁₆	—	106	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₂×C₈	#11719945	1	2	yes	—
146	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
147	C₄⋊C₄	—	53	4	no	—
148	C₄⋊C₄	—	53	8	no	—
149	C₄⋊C₄	—	53	4	no	—
150	C₂²⋊C₄	—	53	4	no	—
151	C₄²	—	53	4	no	—
152	C₄²	C₈×D₅₃	1	1	yes	characteristic
153	C₄²	C₈×D₅₃	1	1	yes	characteristic
154	C₄²	C₂×C₄×D₅₃	1	2	yes	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	106	4	no	—
157	C₁₆	C₂×Dic₁₀₆	1	4	yes	—
158	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
159	Q₈	—	212	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	53	8	no	—
162	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
163	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
165	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
166	C₂×C₄	#13133223	1	2	yes	—
167	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	106	4	no	—
170	C₈	C₄×Dic₁₀₆	1	2	yes	—
171	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
172	C₈	C₂×#11719945	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	C₈×Dic₁₀₆	1	2	yes	—
175	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	106	8	no	—
179	C₄	C₈×Dic₁₀₆	1	2	yes	—
180	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
181	C₄	C₂×#13133223	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×C₈×Dic₁₀₆	1	1	yes	characteristic
184	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×#346	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#33475
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #20276621	C₂	semidirect
38. C₂×C₁₆×C₅₃	Q₈	semidirect
114. C₅₃	#33475	semidirect
127. C₂×C₁₆	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20283649	2
#20276621	2
#20276473	2
#20276471	2
#20276311	2
C₂×C₅₃×#346	2
#33475	53