Subgroups — C₂²×#16811024 #28253685

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133208	C₂	1	2	yes	maximal
2	C₂×#16811024	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13133148	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂×#13133208	—	2	12	no	—
9	C₂×#13133208	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×#11719919	—	2	2	no	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16811024	C₂²	1	64	yes	direct product factor
15	C₂×#13133151	C₂²	1	12	yes	—
16	C₂×#13133148	C₂²	1	24	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	12	yes	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×#107×C₅₃	C₂²	1	12	yes	—
21	C₂³×Dic₅₃	—	2	2	no	—
22	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
23	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂²×#88322	Q₈	1	8	yes	—
25	C₂²×#88322	C₂³	1	6	yes	—
26	#13133208	—	2	48	no	—
27	#13133208	C₂³	1	32	yes	—
28	C₂×C₄⋊Dic₅₃	—	2	6	no	—
29	C₂×C₄⋊Dic₅₃	—	2	24	no	—
30	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
31	C₂×#11719919	—	2	24	no	—
32	C₂×C₄×Dic₅₃	D₄	1	6	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
34	#13133151	C₂³	1	16	yes	—
35	#13133148	C₂³	1	32	yes	—
36	C₈×Dic₅₃	C₂³	1	16	yes	—
37	C₂³×C₄×C₅₃	—	2	2	no	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
43	#107×C₅₃	C₂³	1	16	yes	—
44	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	12	no	—
46	C₂²×C₄×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
51	C₂²×Dic₅₃	—	2	4	no	—
52	C₂²×Dic₅₃	—	2	4	no	—
53	C₂²×Dic₅₃	—	2	12	no	—
54	C₂²×Dic₅₃	—	4	8	no	—
55	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
56	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	24	yes	—
58	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
59	C₄⋊C₄×C₅₃	—	2	12	no	—
60	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
61	C₄⋊Dic₅₃	—	2	12	no	—
62	C₄⋊Dic₅₃	—	2	32	no	—
63	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
64	#11719919	—	2	32	no	—
65	C₄×Dic₅₃	C₂×D₄	1	12	yes	—
66	C₄×Dic₅₃	C₂⁴	1	4	yes	—
67	C₂×#88322	C₂×Q₈	1	24	yes	—
68	C₂×#88322	C₂⁴	1	4	yes	—
69	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
70	C₂³×C₅₃	#135	1	4	yes	—
71	C₂³×C₅₃	#135	1	4	yes	—
72	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	4	24	no	—
75	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	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	12	no	—
81	C₂×Dic₅₃	—	2	12	no	—
82	C₂×Dic₅₃	—	2	8	no	—
83	C₂×Dic₅₃	—	4	24	no	—
84	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
85	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
86	C₈×C₅₃	C₂²×Q₈	1	8	yes	—
87	#88322	C₂²×Q₈	1	8	yes	—
88	C₂²×#499	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#499	1	16	yes	complemented
90	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
91	C₂²×C₅₃	C₂×#135	1	6	yes	—
92	C₂²×C₅₃	C₂×#135	1	6	yes	—
93	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
94	C₄×C₅₃	—	4	8	no	—
95	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
96	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
97	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
98	Dic₅₃	—	2	4	no	—
99	Dic₅₃	—	2	4	no	—
100	Dic₅₃	—	4	8	no	—
101	C₂²×#125	—	53	2	no	—
102	C₂×#499	—	53	24	no	—
103	C₂²×#107	—	53	1	no	—
104	C₂²×#107	D₅₃	1	1	yes	characteristic
105	C₂²×#106	—	53	2	no	—
106	C₂²×C₄×C₈	—	53	1	no	—
107	C₂×C₅₃	C₂×#499	1	12	yes	complemented
108	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
110	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
111	C₂³×C₈	—	53	1	no	—
112	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₂×#125	—	53	24	no	—
114	C₂×#125	—	106	12	no	—
115	C₂²×C₄⋊C₄	—	53	2	no	—
116	C₂²×C₄⋊C₄	—	106	2	no	—
117	C₂²×C₄⋊C₄	—	106	2	no	—
118	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
119	C₂²×C₄²	—	53	1	no	—
120	#499	—	53	64	no	—
121	C₂×#107	—	53	12	no	—
122	C₂×#107	C₂×D₅₃	1	12	yes	—
123	C₂×#106	—	53	24	no	—
124	C₂×C₄×C₈	—	53	12	no	—
125	C₅₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	—	2	2	no	—
127	C₂³×C₄	—	53	1	no	—
128	C₂³×C₄	—	53	1	no	—
129	C₂³×C₄	—	106	2	no	—
130	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
131	C₂²×C₈	—	53	8	no	—
132	C₂²×C₈	—	53	6	no	—
133	C₂²×C₈	Dic₁₀₆	1	8	yes	—
134	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
135	#125	—	53	32	no	—
136	#125	—	106	48	no	—
137	C₂×C₄⋊C₄	—	2	6	no	—
138	C₂×C₄⋊C₄	—	53	12	no	—
139	C₂×C₄⋊C₄	—	106	24	no	—
140	C₂×C₄⋊C₄	—	106	24	no	—
141	C₂×C₄⋊C₄	—	106	6	no	—
142	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
143	C₂×C₄²	—	53	6	no	—
144	C₂×C₄²	—	53	6	no	—
145	#107	—	53	16	no	—
146	#107	C₂²×D₅₃	1	16	yes	—
147	#106	—	53	32	no	—
148	C₄×C₈	—	53	16	no	—
149	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	—	53	6	no	—
153	C₂²×C₄	—	53	6	no	—
154	C₂²×C₄	—	106	4	no	—
155	C₂²×C₄	—	106	4	no	—
156	C₂²×C₄	—	106	12	no	—
157	C₂²×C₄	—	212	8	no	—
158	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
159	C₂²×C₄	D₄×D₅₃	1	3	yes	—
160	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
161	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
162	C₂×C₈	—	53	24	no	—
163	C₂×C₈	—	53	4	no	—
164	C₂×C₈	C₂×Dic₁₀₆	1	24	yes	—
165	C₂×C₈	C₂³×D₅₃	1	4	yes	—
166	C₄⋊C₄	—	2	12	no	—
167	C₄⋊C₄	—	53	8	no	—
168	C₄⋊C₄	—	106	32	no	—
169	C₄⋊C₄	—	106	32	no	—
170	C₄⋊C₄	—	106	12	no	—
171	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
172	C₄²	—	53	12	no	—
173	C₄²	—	53	4	no	—
174	C₂³	#13133216	1	4	yes	—
175	C₂³	#13133236	1	4	yes	—
176	C₂³	#13133240	1	4	yes	—
177	C₂³	C₂×D₄×D₅₃	1	3	yes	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	24	no	—
180	C₂×C₄	—	53	4	no	—
181	C₂×C₄	—	53	4	no	—
182	C₂×C₄	—	106	12	no	—
183	C₂×C₄	—	106	12	no	—
184	C₂×C₄	—	106	8	no	—
185	C₂×C₄	—	212	24	no	—
186	C₂×C₄	#13133216	1	12	yes	—
187	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
188	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
189	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
190	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
191	C₈	—	53	8	no	—
192	C₈	C₂²×Dic₁₀₆	1	8	yes	—
193	C₂²	#16811024	1	16	yes	direct product factor
194	C₂²	C₂×#13133216	1	6	yes	—
195	C₂²	C₂×#13133236	1	6	yes	—
196	C₂²	C₂×#13133240	1	6	yes	—
197	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	106	4	no	—
200	C₄	—	106	4	no	—
201	C₄	—	212	8	no	—
202	C₄	C₂×#13133216	1	6	yes	—
203	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
204	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
205	C₂	C₂×#16811024	1	12	yes	direct product factor
206	C₂	C₂²×#13133216	1	1	yes	characteristic
207	C₂	C₂²×#13133236	1	1	yes	characteristic
208	C₂	C₂²×#13133240	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#107×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	C₂²×#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#16811024	C₂	direct (N × H)
14. #16811024	C₂²	direct (N × H)
89. C₂²×C₅₃	#499	semidirect
107. C₂×C₅₃	C₂×#499	semidirect
125. C₅₃	C₂²×#499	semidirect
193. C₂²	#16811024	direct (N × H)
205. C₂	C₂×#16811024	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×#13133208	2
C₂×#16811024	2
C₂²×#13133151	2
C₂²×#13133148	2
C₂²×C₈×Dic₅₃	2
C₂²×#107×C₅₃	2
C₂²×#499	53

Subgroups — C22×#16811024 #28253685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811024 #28253685