Subgroups — #27273485

Order 13568.

802 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×C₄×#11719917	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	C₂²×#11719917	C₄	1	4	yes	—
5	C₄×#11719917	—	2	4	no	—
6	C₄×#11719917	—	2	4	no	—
7	C₂×#13133128	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	C₂×#11719917	—	2	16	no	—
15	C₂×#11719917	—	2	8	no	—
16	C₂²×#88322	C₂×C₄	1	8	yes	—
17	#13133128	—	2	8	no	—
18	C₄×#88322	—	2	4	no	—
19	C₄×#88322	—	2	4	no	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	8	no	—
23	C₂×M₁₆×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	4	yes	—
29	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#97×C₅₃	—	2	4	no	—
32	C₄×C₈×C₅₃	—	2	4	no	—
33	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	8	no	—
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	M₁₆×C₅₃	—	2	16	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
48	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
49	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×D₄	1	2	yes	—
54	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	#11719917	—	2	32	no	—
56	C₂×#88322	—	2	8	no	—
57	C₂×#88322	—	2	8	no	—
58	C₂×#88322	M₁₆	1	16	yes	complemented
59	C₂³×C₅₃	—	2	4	no	—
60	C₂³×C₅₃	—	2	2	no	—
61	C₂³×C₅₃	#95	1	2	yes	—
62	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	8	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	#95	1	2	yes	—
70	C₂×C₄×C₅₃	#95	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
76	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
77	C₈×C₅₃	—	2	8	no	—
78	#88322	—	2	16	no	—
79	#7852	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	4	no	—
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	—	4	4	no	—
83	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
84	C₂²×C₅₃	C₄×M₁₆	1	4	yes	—
85	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	#407	1	4	yes	—
89	C₄×C₅₃	#407	1	2	yes	—
90	C₄×C₅₃	#423	1	4	yes	—
91	C₄×C₅₃	#423	1	2	yes	—
92	C₂×C₄×M₁₆	—	53	2	no	—
93	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
94	#1273	—	53	4	no	—
95	C₂×C₅₃	—	4	2	no	—
96	C₂×C₅₃	#1455	1	4	yes	—
97	C₂×C₅₃	#1728	1	2	yes	—
98	C₂×C₅₃	#1728	1	1	yes	characteristic
99	C₂²×M₁₆	—	53	4	no	—
100	C₂²×M₁₆	Dic₅₃	1	2	yes	—
101	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
102	C₄×M₁₆	—	2	4	no	—
103	C₄×M₁₆	—	106	4	no	—
104	C₄×M₁₆	—	106	4	no	—
105	C₂×#97	—	53	4	no	—
106	C₂×#97	Dic₅₃	1	2	yes	—
107	C₂×C₄×C₈	—	53	4	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
109	C₅₃	#7852	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	C₄×D₅₃	1	2	yes	—
111	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
112	C₂×M₁₆	—	2	4	no	—
113	C₂×M₁₆	—	2	8	no	—
114	C₂×M₁₆	—	106	16	no	—
115	C₂×M₁₆	—	106	8	no	—
116	C₂²×C₈	—	53	8	no	—
117	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
118	C₂×C₄²	Dic₁₀₆	1	2	yes	—
119	C₂×C₄²	C₄×D₅₃	1	2	yes	—
120	C₂×C₄²	D₂₁₂	1	2	yes	—
121	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	#97	—	2	4	no	—
125	#97	—	106	8	no	—
126	C₄×C₈	—	2	4	no	—
127	C₄×C₈	—	106	4	no	—
128	C₄×C₈	—	106	4	no	—
129	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
134	C₂²×C₄	#11719919	1	4	yes	—
135	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
136	C₂²×C₄	#11719921	1	4	yes	—
137	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
140	M₁₆	—	2	16	no	—
141	M₁₆	—	106	32	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	53	16	no	—
145	C₂×C₈	—	106	8	no	—
146	C₂×C₈	—	106	8	no	—
147	C₂×C₈	#11719917	1	8	yes	complemented
148	C₄²	#11719919	1	4	yes	—
149	C₄²	C₄⋊Dic₅₃	1	2	yes	—
150	C₄²	#11719921	1	4	yes	—
151	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
152	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
153	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	#13133124	1	2	yes	—
158	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	#13133124	1	2	yes	—
166	C₂×C₄	#13133124	1	4	yes	—
167	C₂×C₄	C₂×#11719921	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
169	C₂×C₄	C₂×#11719919	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂×C₄	M₁₆×D₅₃	1	8	yes	—
172	C₂×C₄	C₂×#11719917	1	4	yes	—
173	C₈	—	2	8	no	—
174	C₈	—	106	16	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	4	4	no	—
178	C₂²	C₂×#13133124	1	1	yes	characteristic
179	C₂²	M₁₆×Dic₅₃	1	4	yes	—
180	C₂²	C₄×#11719917	1	2	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	#16810586	1	4	yes	—
184	C₄	#16810587	1	2	yes	—
185	C₄	#16810646	1	4	yes	—
186	C₄	#16810647	1	2	yes	—
187	C₂	—	4	2	no	—
188	C₂	#20273018	1	4	yes	—
189	C₂	#20274408	1	2	yes	—
190	C₂	#20274409	1	1	yes	characteristic

	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	#7852
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
47. C₂×C₈×C₅₃	M₁₆	semidirect
58. C₂×#88322	M₁₆	semidirect
109. C₅₃	#7852	semidirect
147. C₂×C₈	#11719917	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719917	2
#20272542	2
C₂×C₄×M₁₆×C₅₃	2
#7852	53