Subgroups — C₂²×#16811022 #28253683

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133209	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811022	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133150	C₂	1	2	yes	maximal
4	C₂²×#13133148	C₂	1	2	yes	maximal
5	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133209	—	2	6	no	—
10	C₂×#13133209	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×#11719919	—	2	2	no	—
13	#16811022	C₂²	1	64	yes	direct product factor
14	C₂×#13133150	C₂²	1	24	yes	—
15	C₂×#13133148	C₂²	1	24	yes	—
16	C₂×C₄×#88322	C₂²	1	12	yes	—
17	C₂×#125×C₅₃	—	2	6	no	—
18	C₂×#125×C₅₃	C₂²	1	12	yes	—
19	C₂²×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×Dic₅₃	—	2	2	no	—
23	C₂²×#88322	Q₈	1	8	yes	—
24	C₂²×#88322	Q₈	1	8	yes	—
25	C₂²×#88322	C₂³	1	6	yes	—
26	C₂²×#88322	C₂³	1	6	yes	—
27	#13133209	—	2	24	no	—
28	#13133209	C₂³	1	16	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	6	no	—
30	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
31	C₂×#11719919	—	2	24	no	—
32	#13133150	C₂³	1	32	yes	—
33	#13133148	C₂³	1	32	yes	—
34	C₄×#88322	C₂³	1	16	yes	—
35	C₂³×C₄×C₅₃	—	2	2	no	—
36	C₂³×C₄×C₅₃	D₄	1	2	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	#125×C₅₃	—	2	24	no	—
39	#125×C₅₃	C₂³	1	16	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
43	C₂×C₄²×C₅₃	D₄	1	6	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
45	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	—	2	12	no	—
48	C₂²×C₄×C₅₃	—	4	8	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
54	C₂²×Dic₅₃	—	2	12	no	—
55	C₂²×Dic₅₃	—	4	8	no	—
56	C₄⋊C₄×C₅₃	—	2	32	no	—
57	C₄⋊C₄×C₅₃	—	2	12	no	—
58	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
59	C₄²×C₅₃	C₂×D₄	1	12	yes	—
60	C₄²×C₅₃	C₂⁴	1	4	yes	—
61	C₄⋊Dic₅₃	—	2	12	no	—
62	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
63	#11719919	—	2	32	no	—
64	C₂×#88322	C₂×Q₈	1	24	yes	—
65	C₂×#88322	C₂×Q₈	1	24	yes	—
66	C₂×#88322	C₂⁴	1	4	yes	—
67	C₂×#88322	C₂⁴	1	4	yes	—
68	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
69	C₂³×C₅₃	#135	1	8	yes	—
70	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
71	C₂×C₄×C₅₃	—	2	24	no	—
72	C₂×C₄×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	—	4	24	no	—
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
75	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
76	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
77	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
78	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
79	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	8	no	—
81	C₂×Dic₅₃	—	4	24	no	—
82	#88322	C₂²×Q₈	1	8	yes	—
83	#88322	C₂²×Q₈	1	8	yes	—
84	C₂²×#499	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	#499	1	16	yes	complemented
86	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
87	C₂²×C₅₃	C₂×#135	1	12	yes	—
88	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
89	C₄×C₅₃	—	2	8	no	—
90	C₄×C₅₃	—	4	8	no	—
91	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
92	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
93	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
94	Dic₅₃	—	4	8	no	—
95	C₂²×#125	—	53	1	no	—
96	C₂²×#125	D₅₃	1	1	yes	characteristic
97	C₂×#499	—	53	24	no	—
98	C₂²×#107	—	53	2	no	—
99	C₂²×#106	—	53	2	no	—
100	C₂²×C₄×C₈	—	53	1	no	—
101	C₂×C₅₃	C₂×#499	1	12	yes	complemented
102	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₅₃	C₂²×#135	1	2	yes	—
104	C₂³×C₈	—	53	1	no	—
105	C₂³×C₈	—	53	1	no	—
106	C₂×#125	—	2	6	no	—
107	C₂×#125	—	53	12	no	—
108	C₂×#125	—	106	6	no	—
109	C₂×#125	C₂×D₅₃	1	12	yes	—
110	C₂²×C₄⋊C₄	—	2	2	no	—
111	C₂²×C₄⋊C₄	—	53	2	no	—
112	C₂²×C₄⋊C₄	—	106	2	no	—
113	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
114	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
115	#499	—	53	64	no	—
116	C₂×#107	—	53	24	no	—
117	C₂×#106	—	53	24	no	—
118	C₂×C₄×C₈	—	53	12	no	—
119	C₅₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	—	2	2	no	—
121	C₂³×C₄	—	106	2	no	—
122	C₂³×C₄	C₂²⋊D₅₃	1	2	yes	—
123	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂²×C₈	—	53	8	no	—
125	C₂²×C₈	—	53	8	no	—
126	C₂²×C₈	—	53	6	no	—
127	C₂²×C₈	—	53	6	no	—
128	#125	—	2	24	no	—
129	#125	—	53	16	no	—
130	#125	—	106	24	no	—
131	#125	C₂²×D₅₃	1	16	yes	—
132	C₂×C₄⋊C₄	—	2	24	no	—
133	C₂×C₄⋊C₄	—	2	6	no	—
134	C₂×C₄⋊C₄	—	53	12	no	—
135	C₂×C₄⋊C₄	—	106	6	no	—
136	C₂×C₄⋊C₄	—	106	24	no	—
137	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
138	C₂×C₄²	C₂²⋊D₅₃	1	6	yes	—
139	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
140	#107	—	53	32	no	—
141	#106	—	53	32	no	—
142	C₄×C₈	—	53	16	no	—
143	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	4	8	no	—
147	C₂²×C₄	—	106	12	no	—
148	C₂²×C₄	—	212	8	no	—
149	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
150	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
153	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
154	C₂×C₈	—	53	24	no	—
155	C₂×C₈	—	53	24	no	—
156	C₂×C₈	—	53	4	no	—
157	C₂×C₈	—	53	4	no	—
158	C₄⋊C₄	—	2	32	no	—
159	C₄⋊C₄	—	2	12	no	—
160	C₄⋊C₄	—	53	8	no	—
161	C₄⋊C₄	—	106	12	no	—
162	C₄⋊C₄	—	106	32	no	—
163	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
164	C₄²	C₂×C₂²⋊D₅₃	1	12	yes	—
165	C₄²	C₂³×D₅₃	1	4	yes	—
166	C₂³	#13133215	1	4	yes	—
167	C₂³	#13133237	1	8	yes	—
168	C₂³	C₂²×C₂²⋊D₅₃	1	3	yes	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	24	no	—
172	C₂×C₄	—	106	8	no	—
173	C₂×C₄	—	212	24	no	—
174	C₂×C₄	#13133215	1	12	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
176	C₂×C₄	C₂²×C₂²⋊D₅₃	1	3	yes	—
177	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
178	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
179	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
180	C₈	—	53	8	no	—
181	C₈	—	53	8	no	—
182	C₂²	#16811022	1	16	yes	direct product factor
183	C₂²	C₂×#13133215	1	6	yes	—
184	C₂²	C₂×#13133237	1	12	yes	—
185	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	—	4	8	no	—
188	C₄	—	212	8	no	—
189	C₄	C₂×#13133215	1	6	yes	—
190	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
191	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
192	C₂	C₂×#16811022	1	12	yes	direct product factor
193	C₂	C₂²×#13133215	1	1	yes	characteristic
194	C₂	C₂²×#13133237	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#125×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂²×#125	D₅₃
hypocenter	C₅₃	C₂²×#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
2. C₂×#16811022	C₂	direct (N × H)
13. #16811022	C₂²	direct (N × H)
85. C₂²×C₅₃	#499	semidirect
101. C₂×C₅₃	C₂×#499	semidirect
119. C₅₃	C₂²×#499	semidirect
182. C₂²	#16811022	direct (N × H)
192. C₂	C₂×#16811022	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133209	2
C₂×#16811022	2
C₂²×#13133150	2
C₂²×#13133148	2
C₂²×C₄×#88322	2
C₂²×#125×C₅₃	2
C₂²×#499	53

Subgroups — C22×#16811022 #28253683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811022 #28253683