Subgroups — #27537586

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 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	#20276623	C₂	1	4	yes	maximal, complemented
3	C₁₆×Dic₁₀₆	C₂	1	4	yes	maximal, complemented
4	#20276473	C₂	1	2	yes	maximal
5	#20276471	C₂	1	1	yes	maximal, characteristic
6	#20276311	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811180	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810753	C₄	1	4	yes	—
11	C₈×Dic₁₀₆	C₂²	1	4	yes	—
12	#16810649	C₄	1	2	yes	—
13	#16810647	C₄	1	1	yes	characteristic
14	#16810578	C₂²	1	2	yes	—
15	C₄⋊#11719908	C₂²	1	4	yes	—
16	#16810441	C₂²	1	8	yes	—
17	#16810398	C₂²	1	4	yes	—
18	C₁₆×Dic₅₃	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
23	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂×#11719917	C₂×C₄	1	2	yes	—
25	#13133217	C₈	1	2	yes	—
26	#13133209	C₈	1	2	yes	—
27	#13133196	C₈	1	4	yes	—
28	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
29	#13133176	C₂×C₄	1	2	yes	—
30	#13133157	—	2	4	no	—
31	C₂×#11719908	C₂³	1	4	yes	—
32	C₄⋊#88322	C₂×C₄	1	4	yes	—
33	#13133146	C₂×C₄	1	8	yes	—
34	#13133127	C₂×C₄	1	4	yes	—
35	C₈×Dic₅₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	—	2	2	no	—
40	C₂×C₁₆×C₅₃	Q₈	1	4	yes	complemented
41	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
42	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₃	#41	1	2	yes	—
54	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
59	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
60	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
61	#11719919	C₂×C₈	1	8	yes	—
62	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
63	#11719917	—	2	4	no	—
64	C₂×#88322	C₂²×C₄	1	4	yes	—
65	C₁₆×C₅₃	—	2	2	no	—
66	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
67	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
68	#11719908	—	2	4	no	—
69	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₅₃	#131	1	2	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
78	Dic₁₀₆	—	2	8	no	—
79	C₈×C₅₃	—	2	2	no	—
80	C₈×C₅₃	C₄×Q₈	1	2	yes	—
81	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
83	#88322	—	2	4	no	—
84	#33474	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
87	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	C₈×Q₈	1	2	yes	—
90	C₄×C₅₃	#504	1	4	yes	—
91	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
93	Dic₅₃	—	2	8	no	—
94	#2960	—	53	1	no	—
95	#2183	—	53	4	no	—
96	Q₈×C₁₆	—	53	4	no	—
97	#2151	—	53	2	no	—
98	#2150	—	53	1	no	—
99	#2108	—	53	2	no	—
100	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
103	C₂×C₅₃	C₂×#504	1	2	yes	—
104	#533	—	53	1	no	—
105	C₂×M₃₂	—	53	2	no	—
106	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
107	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
108	#446	—	53	4	no	—
109	C₈×Q₈	—	53	4	no	—
110	#424	—	53	2	no	—
111	#423	—	53	1	no	—
112	#405	—	53	2	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₄⋊C₁₆	—	53	8	no	—
115	C₄⋊C₁₆	—	53	4	no	—
116	#346	—	53	4	no	—
117	C₄×C₁₆	—	53	4	no	—
118	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
119	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
120	C₅₃	#33474	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	—	53	2	no	—
122	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	53	2	no	—
125	#125	—	53	2	no	—
126	C₂²⋊Q₈	—	53	4	no	—
127	C₄×Q₈	—	53	4	no	—
128	#117	—	53	2	no	—
129	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
130	M₃₂	—	106	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	53	4	no	—
133	C₂×C₁₆	Dic₁₀₆	1	4	yes	complemented
134	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
135	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
136	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊C₈	—	53	8	no	—
138	C₄⋊C₈	—	53	4	no	—
139	#97	—	53	4	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	C₄×D₅₃	1	2	yes	—
142	C₄×C₈	C₂²×D₅₃	1	2	yes	—
143	C₂×Q₈	—	53	4	no	—
144	C₂²×C₄	C₈×D₅₃	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
146	M₁₆	—	106	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	53	4	no	—
149	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₂×C₈	#11719945	1	2	yes	—
153	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
154	C₄⋊C₄	—	53	4	no	—
155	C₄⋊C₄	—	53	8	no	—
156	C₄⋊C₄	—	53	4	no	—
157	C₂²⋊C₄	—	53	4	no	—
158	C₄²	—	53	4	no	—
159	C₄²	C₈×D₅₃	1	1	yes	characteristic
160	C₄²	C₈×D₅₃	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₅₃	1	2	yes	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	106	4	no	—
164	C₁₆	C₂×Dic₁₀₆	1	2	yes	—
165	C₁₆	C₂×Dic₁₀₆	1	2	yes	—
166	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
167	Q₈	—	106	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	53	8	no	—
170	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
173	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
174	C₂×C₄	#13133223	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	106	4	no	—
178	C₈	C₄×Dic₁₀₆	1	2	yes	—
179	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
180	C₈	C₂×#11719945	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Dic₁₀₆	1	2	yes	—
183	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	106	8	no	—
186	C₄	C₈×Dic₁₀₆	1	2	yes	—
187	C₄	#16811039	1	4	yes	—
188	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
189	C₄	C₂×#13133223	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Dic₁₀₆	1	1	yes	characteristic
192	C₂	C₂×#16811039	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#33474
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276623	C₂	semidirect
3. C₁₆×Dic₁₀₆	C₂	semidirect
40. C₂×C₁₆×C₅₃	Q₈	semidirect
120. C₅₃	#33474	semidirect
133. 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
#20276623	2
C₁₆×Dic₁₀₆	2
#20276473	2
#20276471	2
#20276311	2
C₂×C₄×C₁₆×C₅₃	2
#33474	53