Subgroups — #27273481

Order 13568.

674 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	C₂×#16810676	C₂	1	2	yes	maximal
2	#20272542	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	4	no	—
5	#16810676	—	2	4	no	—
6	C₂×C₄⋊#88322	C₄	1	4	yes	—
7	C₂×C₂²⋊#88322	C₄	1	4	yes	—
8	C₂×C₄×#88322	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₅₃	—	2	4	no	—
12	C₂×#97×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂²×#88322	C₂×C₄	1	8	yes	—
15	C₄⋊#88322	—	2	8	no	—
16	C₂²⋊#88322	—	2	8	no	—
17	C₄×#88322	—	2	4	no	—
18	C₄×#88322	—	2	4	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	8	no	—
22	C₂×M₁₆×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	D₄	1	4	yes	—
28	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	#97×C₅₃	—	2	4	no	—
31	C₄×C₈×C₅₃	—	2	4	no	—
32	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	—	2	8	no	—
35	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	M₁₆×C₅₃	—	2	16	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
47	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
48	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×D₄	1	2	yes	—
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×#88322	—	2	8	no	—
55	C₂×#88322	—	2	16	no	—
56	C₂³×C₅₃	—	2	2	no	—
57	C₂³×C₅₃	—	2	4	no	—
58	C₂³×C₅₃	#95	1	2	yes	—
59	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	2	8	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	#95	1	2	yes	—
67	C₂×C₄×C₅₃	#95	1	4	yes	—
68	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
73	C₂×C₄×C₅₃	#131	1	8	yes	—
74	C₈×C₅₃	—	2	8	no	—
75	#88322	—	4	8	no	—
76	#7850	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	—	2	4	no	—
79	C₂²×C₅₃	—	4	4	no	—
80	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
81	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
82	C₂²×C₅₃	#405	1	4	yes	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	#407	1	2	yes	—
86	C₄×C₅₃	#408	1	4	yes	—
87	C₄×C₅₃	#423	1	2	yes	—
88	C₄×C₅₃	#424	1	4	yes	—
89	C₂×#431	—	53	2	no	—
90	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
91	#1273	—	53	4	no	—
92	C₂×C₅₃	—	4	2	no	—
93	C₂×C₅₃	#1453	1	4	yes	—
94	C₂×C₅₃	#1728	1	1	yes	characteristic
95	C₂×C₅₃	#1729	1	2	yes	—
96	C₂²×M₁₆	Dic₅₃	1	2	yes	—
97	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	#431	—	106	4	no	—
99	#431	—	106	4	no	—
100	C₂×C₄⋊C₈	—	53	4	no	—
101	C₂×C₂²⋊C₈	—	53	4	no	—
102	C₄×M₁₆	—	2	4	no	—
103	C₂×#97	Dic₅₃	1	2	yes	—
104	C₂×C₄×C₈	—	53	4	no	—
105	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
106	C₅₃	#7850	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₄	C₄×D₅₃	1	2	yes	—
108	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂×M₁₆	—	2	4	no	—
110	C₂×M₁₆	—	2	8	no	—
111	C₂²×C₈	—	53	8	no	—
112	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
113	C₂×C₄²	Dic₁₀₆	1	2	yes	—
114	C₂×C₄²	C₄×D₅₃	1	2	yes	—
115	C₂×C₄²	D₂₁₂	1	2	yes	—
116	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	106	8	no	—
120	C₂²⋊C₈	—	106	8	no	—
121	#97	—	2	4	no	—
122	C₄×C₈	—	2	4	no	—
123	C₄×C₈	—	106	4	no	—
124	C₄×C₈	—	106	4	no	—
125	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	8	no	—
128	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
130	C₂²×C₄	#11719919	1	4	yes	—
131	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
132	C₂²×C₄	#11719921	1	4	yes	—
133	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
136	M₁₆	—	2	16	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	106	8	no	—
140	C₂×C₈	—	106	16	no	—
141	C₂×C₈	#11719917	1	8	yes	—
142	C₄²	#11719919	1	4	yes	—
143	C₄²	C₄⋊Dic₅₃	1	2	yes	—
144	C₄²	#11719921	1	4	yes	—
145	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
146	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
148	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	4	no	—
151	C₂³	#13133124	1	2	yes	—
152	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	#13133124	1	2	yes	—
160	C₂×C₄	#13133124	1	4	yes	—
161	C₂×C₄	C₂×#11719921	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂×#11719919	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	C₂×#11719917	1	4	yes	—
166	C₂×C₄	#13133225	1	8	yes	—
167	C₈	—	2	8	no	—
168	C₈	—	212	8	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	4	4	no	—
172	C₂²	C₂×#13133124	1	1	yes	characteristic
173	C₂²	C₄×#11719917	1	2	yes	—
174	C₂²	#16810581	1	4	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	#16810587	1	2	yes	—
178	C₄	#16810591	1	4	yes	—
179	C₄	#16810647	1	2	yes	—
180	C₄	#16810651	1	4	yes	—
181	C₂	—	4	2	no	—
182	C₂	#20273012	1	4	yes	—
183	C₂	#20274409	1	1	yes	characteristic
184	C₂	#20274413	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂×C₄²
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#7850
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20272542	C₂	semidirect
106. C₅₃	#7850	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
#20272542	2
C₂×C₄×M₁₆×C₅₃	2
#7850	53