Subgroups — #27537593

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	#20276624	C₂	1	8	yes	maximal, complemented
2	#20276474	C₂	1	2	yes	maximal
3	#20276471	C₂	1	1	yes	maximal, characteristic
4	#20276312	C₂	1	2	yes	maximal
5	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133157	C₂²	1	2	yes	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₄⋊#11719908	C₂²	1	4	yes	—
11	C₄⋊#11719908	C₂²	1	8	yes	—
12	#16810399	C₂²	1	4	yes	—
13	#16810399	C₂²	1	4	yes	—
14	C₄×#11719908	C₂²	1	4	yes	—
15	C₅₃×#533	C₄	1	1	yes	characteristic
16	C₅₃×#446	C₄	1	4	yes	—
17	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
18	C₅₃×#424	C₄	1	2	yes	—
19	C₅₃×#423	C₄	1	1	yes	characteristic
20	C₅₃×#405	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#13133157	—	2	4	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	Q₈	1	4	yes	complemented
25	C₂×#11719908	C₂³	1	2	yes	—
26	C₂×#11719908	C₂³	1	4	yes	—
27	C₂×#11719908	C₂³	1	2	yes	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
32	#125×C₅₃	C₈	1	2	yes	—
33	C₂²⋊Q₈×C₅₃	C₈	1	4	yes	—
34	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
35	#117×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
38	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
39	#97×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂³	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	M₁₆×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
56	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
58	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
62	#11719908	—	2	2	no	—
63	#11719908	—	2	4	no	—
64	#11719908	C₂×Q₈	1	4	yes	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	4	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#131	1	2	yes	—
74	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	—	2	4	no	—
77	C₈×C₅₃	C₄×Q₈	1	2	yes	—
78	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
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	—	2	8	no	—
87	C₄×C₅₃	C₈×Q₈	1	2	yes	—
88	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
90	#2960	D₅₃	1	1	yes	characteristic
91	#2183	—	53	8	no	—
92	#2151	—	53	2	no	—
93	#2150	—	53	1	no	—
94	#2108	—	53	2	no	—
95	C₂×#346	—	53	1	no	—
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	Dic₅₃	1	1	yes	characteristic
100	C₂×M₃₂	—	53	2	no	—
101	C₂²×C₁₆	—	53	1	no	—
102	C₂²×C₁₆	—	53	1	no	—
103	#446	Dic₅₃	1	4	yes	—
104	C₈×Q₈	C₂×D₅₃	1	4	yes	—
105	#424	Dic₅₃	1	2	yes	—
106	#423	Dic₅₃	1	1	yes	characteristic
107	#405	C₂×D₅₃	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	53	4	no	—
110	C₄⋊C₁₆	—	53	8	no	—
111	#346	—	53	4	no	—
112	#346	—	53	4	no	—
113	C₄×C₁₆	—	53	4	no	—
114	C₅₃	#33475	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
116	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₄⋊Q₈	#88322	1	2	yes	—
119	#125	#88322	1	2	yes	—
120	C₂²⋊Q₈	#88322	1	4	yes	—
121	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
122	#117	C₂×Dic₅₃	1	2	yes	—
123	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
124	M₃₂	—	106	4	no	—
125	C₂×C₁₆	—	53	4	no	—
126	C₂×C₁₆	—	53	2	no	—
127	C₂×C₁₆	—	53	4	no	—
128	C₂×C₁₆	—	53	2	no	—
129	C₂×C₁₆	—	106	2	no	—
130	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
131	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
132	#97	C₂×Dic₅₃	1	4	yes	—
133	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
134	C₄×C₈	C₂²×D₅₃	1	4	yes	—
135	C₄×C₈	C₂²×D₅₃	1	2	yes	—
136	C₂×Q₈	C₂×#88322	1	4	yes	—
137	C₂²×C₄	C₂×#88322	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	Q₈×D₅₃	1	2	yes	—
142	C₂×C₈	#11719949	1	2	yes	—
143	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
144	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
145	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
146	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
147	C₄⋊C₄	C₂×#88322	1	4	yes	—
148	C₄⋊C₄	C₂×#88322	1	8	yes	—
149	C₄⋊C₄	C₂×#88322	1	4	yes	—
150	C₂²⋊C₄	C₂×#88322	1	4	yes	—
151	C₄²	C₂×#88322	1	1	yes	characteristic
152	C₄²	C₂×#88322	1	1	yes	characteristic
153	C₄²	C₂²×Dic₅₃	1	2	yes	—
154	C₄²	C₂²×Dic₅₃	1	4	yes	—
155	C₁₆	—	53	4	no	—
156	C₁₆	—	106	2	no	—
157	C₁₆	—	106	4	no	—
158	C₂³	C₂²×#88322	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
162	C₂×C₄	C₂²×#88322	1	2	yes	—
163	C₂×C₄	C₂²×#88322	1	2	yes	—
164	C₂×C₄	C₂²×#88322	1	8	yes	—
165	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
166	C₂×C₄	#13133224	1	2	yes	—
167	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	4	no	—
170	C₈	Q₈×Dic₅₃	1	2	yes	—
171	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
172	C₈	C₂×#11719949	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#88322	1	2	yes	—
175	C₂²	C₂³×#88322	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#88322	1	2	yes	—
180	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
181	C₄	C₂×#13133224	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#88322	1	1	yes	characteristic
184	C₂	#20287818	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2960	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#33475
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
24. C₂×#11719908	Q₈	semidirect
114. C₅₃	#33475	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
#20276474	2
#20276471	2
#20276312	2
C₂×#16810399	2
C₅₃×#2960	2
#33475	53