Subgroups — C₂²×#16811027 #28253689

Order 13568.

2398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133216	C₂	1	2	yes	maximal
2	C₂×#16811027	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13133150	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×Dic₁₀₆	—	2	2	no	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133216	—	2	12	no	—
10	C₂×#13133216	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×#11719919	—	2	2	no	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16811027	C₂²	1	64	yes	direct product factor
15	C₂×#13133151	C₂²	1	12	yes	—
16	C₂×#13133150	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	C₂²×Dic₁₀₆	—	2	24	no	—
22	C₂²×Dic₁₀₆	—	4	16	no	—
23	C₂³×Dic₅₃	—	2	2	no	—
24	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
25	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
26	C₂²×#88322	Q₈	1	8	yes	complemented
27	C₂²×#88322	C₂³	1	6	yes	—
28	#13133216	—	2	48	no	—
29	#13133216	C₂³	1	32	yes	—
30	C₂×C₄⋊Dic₅₃	—	2	6	no	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
32	C₂×#11719919	—	2	24	no	—
33	C₂×C₄×Dic₅₃	D₄	1	6	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
35	#13133151	C₂³	1	16	yes	—
36	#13133150	C₂³	1	32	yes	—
37	C₈×Dic₅₃	C₂³	1	16	yes	—
38	C₂³×C₄×C₅₃	—	2	2	no	—
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	Q₈	1	8	yes	complemented
41	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
42	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
44	#107×C₅₃	C₂³	1	16	yes	—
45	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	12	no	—
47	C₂²×C₄×C₅₃	—	4	8	no	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
52	C₂²×Dic₅₃	—	2	12	no	—
53	C₂²×Dic₅₃	—	4	8	no	—
54	C₂²×Dic₅₃	D₈	1	8	yes	—
55	C₂²×Dic₅₃	Q₁₆	1	8	yes	—
56	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
57	C₂²×Dic₅₃	C₂×D₄	1	6	yes	—
58	C₂×Dic₁₀₆	—	2	32	no	—
59	C₂×Dic₁₀₆	—	4	96	no	—
60	C₂×C₈×C₅₃	C₂×Q₈	1	24	yes	complemented
61	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
62	C₄⋊C₄×C₅₃	—	2	12	no	—
63	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
64	C₄⋊Dic₅₃	—	2	12	no	—
65	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
66	#11719919	—	2	32	no	—
67	C₄×Dic₅₃	C₂×D₄	1	12	yes	—
68	C₄×Dic₅₃	C₂⁴	1	4	yes	—
69	C₂×#88322	C₂×Q₈	1	24	yes	complemented
70	C₂×#88322	C₂⁴	1	4	yes	—
71	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
72	C₂³×C₅₃	C₂×D₈	1	4	yes	—
73	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
74	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₅₃	—	2	8	no	—
76	C₂×C₄×C₅₃	—	4	24	no	—
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
78	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
79	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
80	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
81	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
82	C₂×Dic₅₃	—	2	8	no	—
83	C₂×Dic₅₃	—	4	24	no	—
84	C₂×Dic₅₃	C₂×D₈	1	24	yes	—
85	C₂×Dic₅₃	C₂×Q₁₆	1	24	yes	—
86	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
87	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
88	Dic₁₀₆	—	4	64	no	—
89	C₈×C₅₃	C₂²×Q₈	1	8	yes	complemented
90	#88322	C₂²×Q₈	1	8	yes	complemented
91	C₂²×#500	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	#500	1	16	yes	complemented
93	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	6	yes	—
95	C₂²×C₅₃	C₂²×Q₁₆	1	6	yes	—
96	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
97	C₄×C₅₃	—	4	8	no	—
98	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
99	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
100	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
101	Dic₅₃	—	4	8	no	—
102	Dic₅₃	C₂²×D₈	1	8	yes	—
103	Dic₅₃	C₂²×Q₁₆	1	8	yes	—
104	C₂²×C₄⋊Q₈	—	53	2	no	—
105	C₂×#500	—	53	24	no	—
106	C₂²×#107	—	53	1	no	—
107	C₂²×#107	—	53	2	no	—
108	C₂²×#107	D₅₃	1	1	yes	characteristic
109	C₂²×C₄×C₈	—	53	1	no	—
110	C₂×C₅₃	C₂×#500	1	12	yes	complemented
111	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
113	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
114	C₂³×Q₈	—	106	2	no	—
115	C₂³×C₈	—	53	1	no	—
116	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₂×C₄⋊Q₈	—	53	24	no	—
118	C₂×C₄⋊Q₈	—	106	12	no	—
119	C₂²×C₄⋊C₄	—	53	2	no	—
120	C₂²×C₄⋊C₄	—	106	2	no	—
121	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
122	C₂²×C₄²	—	53	1	no	—
123	#500	—	53	64	no	—
124	C₂×#107	—	53	12	no	—
125	C₂×#107	—	53	24	no	—
126	C₂×#107	C₂×D₅₃	1	12	yes	—
127	C₂×C₄×C₈	—	53	12	no	—
128	C₅₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	106	24	no	—
130	C₂²×Q₈	—	212	16	no	—
131	C₂³×C₄	—	2	2	no	—
132	C₂³×C₄	—	53	1	no	—
133	C₂³×C₄	—	53	1	no	—
134	C₂³×C₄	—	106	2	no	—
135	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
136	C₂²×C₈	—	53	8	no	—
137	C₂²×C₈	—	53	6	no	—
138	C₂²×C₈	Dic₁₀₆	1	8	yes	complemented
139	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
140	C₄⋊Q₈	—	53	32	no	—
141	C₄⋊Q₈	—	106	48	no	—
142	C₂×C₄⋊C₄	—	2	6	no	—
143	C₂×C₄⋊C₄	—	53	12	no	—
144	C₂×C₄⋊C₄	—	106	24	no	—
145	C₂×C₄⋊C₄	—	106	6	no	—
146	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
147	C₂×C₄²	—	53	6	no	—
148	C₂×C₄²	—	53	6	no	—
149	#107	—	53	16	no	—
150	#107	—	53	32	no	—
151	#107	C₂²×D₅₃	1	16	yes	—
152	C₄×C₈	—	53	16	no	—
153	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
154	C₂×Q₈	—	106	32	no	—
155	C₂×Q₈	—	212	96	no	—
156	C₂²×C₄	—	2	12	no	—
157	C₂²×C₄	—	4	8	no	—
158	C₂²×C₄	—	53	8	no	—
159	C₂²×C₄	—	53	8	no	—
160	C₂²×C₄	—	53	6	no	—
161	C₂²×C₄	—	53	6	no	—
162	C₂²×C₄	—	106	12	no	—
163	C₂²×C₄	—	212	8	no	—
164	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
165	C₂²×C₄	D₄×D₅₃	1	3	yes	—
166	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
167	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
168	C₂×C₈	—	53	24	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	C₂×Dic₁₀₆	1	24	yes	complemented
171	C₂×C₈	C₂³×D₅₃	1	4	yes	—
172	C₄⋊C₄	—	2	12	no	—
173	C₄⋊C₄	—	53	8	no	—
174	C₄⋊C₄	—	106	32	no	—
175	C₄⋊C₄	—	106	12	no	—
176	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
177	C₄²	—	53	12	no	—
178	C₄²	—	53	4	no	—
179	C₂³	#13133216	1	4	yes	—
180	C₂³	D₈×D₅₃	1	4	yes	—
181	C₂³	Q₁₆×D₅₃	1	4	yes	—
182	C₂³	C₂×D₄×D₅₃	1	3	yes	—
183	Q₈	—	212	64	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	4	24	no	—
186	C₂×C₄	—	53	24	no	—
187	C₂×C₄	—	53	24	no	—
188	C₂×C₄	—	53	4	no	—
189	C₂×C₄	—	53	4	no	—
190	C₂×C₄	—	106	8	no	—
191	C₂×C₄	—	212	24	no	—
192	C₂×C₄	#13133216	1	12	yes	—
193	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
194	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
195	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
196	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
197	C₈	—	53	8	no	—
198	C₈	C₂²×Dic₁₀₆	1	8	yes	complemented
199	C₂²	#16811027	1	16	yes	direct product factor
200	C₂²	C₂×#13133216	1	6	yes	—
201	C₂²	C₂×D₈×D₅₃	1	6	yes	—
202	C₂²	C₂×Q₁₆×D₅₃	1	6	yes	—
203	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
204	C₄	—	4	8	no	—
205	C₄	—	53	8	no	—
206	C₄	—	53	8	no	—
207	C₄	—	212	8	no	—
208	C₄	C₂×#13133216	1	6	yes	—
209	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
210	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
211	C₂	C₂×#16811027	1	12	yes	direct product factor
212	C₂	C₂²×#13133216	1	1	yes	characteristic
213	C₂	C₂²×D₈×D₅₃	1	1	yes	characteristic
214	C₂	C₂²×Q₁₆×D₅₃	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×#500
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×#16811027	C₂	direct (N × H)
14. #16811027	C₂²	direct (N × H)
26. C₂²×#88322	Q₈	semidirect
40. C₂²×C₈×C₅₃	Q₈	semidirect
60. C₂×C₈×C₅₃	C₂×Q₈	semidirect
69. C₂×#88322	C₂×Q₈	semidirect
89. C₈×C₅₃	C₂²×Q₈	semidirect
90. #88322	C₂²×Q₈	semidirect
92. C₂²×C₅₃	#500	semidirect
110. C₂×C₅₃	C₂×#500	semidirect
128. C₅₃	C₂²×#500	semidirect
138. C₂²×C₈	Dic₁₀₆	semidirect
170. C₂×C₈	C₂×Dic₁₀₆	semidirect
198. C₈	C₂²×Dic₁₀₆	semidirect
199. C₂²	#16811027	direct (N × H)
211. C₂	C₂×#16811027	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×#13133216	2
C₂×#16811027	2
C₂²×#13133151	2
C₂²×#13133150	2
C₂²×C₈×Dic₅₃	2
C₂²×#107×C₅₃	2
C₂²×#500	53

Subgroups — C22×#16811027 #28253689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811027 #28253689