Subgroups — C₂²×#16810747 #28248442

Order 13568.

2478 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810747	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133146	C₂	1	2	yes	maximal
4	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×Dic₅₃	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	C₄	1	1	yes	characteristic
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂³×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₄	1	12	yes	—
11	C₂×C₄×Dic₁₀₆	C₂²	1	12	yes	—
12	C₂²×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
13	C₂²×#11719919	C₄	1	2	yes	—
14	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
15	#16810747	C₂²	1	64	yes	direct product factor
16	C₂×#13133146	C₂²	1	24	yes	—
17	C₂×C₄×#88322	C₂²	1	12	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	24	yes	—
19	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	12	yes	—
22	C₂²×Dic₁₀₆	C₈	1	16	yes	complemented
23	C₂²×Dic₁₀₆	C₂×C₄	1	12	yes	—
24	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂²×#88322	—	2	4	no	—
27	C₂²×#88322	Q₈	1	8	yes	complemented
28	C₂²×#88322	C₂³	1	6	yes	—
29	C₂²×#88322	C₂³	1	6	yes	—
30	C₄×Dic₁₀₆	C₂×C₄	1	48	yes	—
31	C₄×Dic₁₀₆	C₂³	1	16	yes	—
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
33	C₂×#11719919	C₂×C₄	1	24	yes	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
36	#13133146	C₂³	1	32	yes	—
37	C₄×#88322	C₂³	1	16	yes	—
38	C₈×Dic₅₃	C₂³	1	32	yes	—
39	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	—	2	8	no	—
43	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
46	C₄⋊C₈×C₅₃	C₂³	1	16	yes	—
47	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₅₃	#41	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
56	C₂²×Dic₅₃	—	2	8	no	—
57	C₂²×Dic₅₃	C₂×C₈	1	16	yes	—
58	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
59	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
60	C₂×Dic₁₀₆	C₂×C₈	1	96	yes	complemented
61	C₂×Dic₁₀₆	C₂²×C₄	1	16	yes	—
62	C₂×C₈×C₅₃	—	2	24	no	—
63	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
65	C₄²×C₅₃	C₂⁴	1	4	yes	—
66	C₄⋊Dic₅₃	C₂²×C₄	1	16	yes	—
67	#11719919	C₂²×C₄	1	32	yes	—
68	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
69	C₄×Dic₅₃	C₂⁴	1	8	yes	—
70	C₂×#88322	—	2	12	no	—
71	C₂×#88322	C₂×Q₈	1	24	yes	complemented
72	C₂×#88322	C₂⁴	1	4	yes	—
73	C₂×#88322	C₂⁴	1	4	yes	—
74	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
75	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
76	C₂³×C₅₃	#131	1	4	yes	—
77	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
78	C₂×C₄×C₅₃	—	2	12	no	—
79	C₂×C₄×C₅₃	C₄×Q₈	1	12	yes	—
80	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	—
81	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
82	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
85	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	24	no	—
88	C₂×Dic₅₃	C₂²×C₈	1	48	yes	—
89	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
90	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
91	Dic₁₀₆	C₂²×C₈	1	64	yes	complemented
92	C₈×C₅₃	—	2	8	no	—
93	#88322	—	2	4	no	—
94	#88322	C₂²×Q₈	1	8	yes	complemented
95	C₂²×C₈×Q₈	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₈×Q₈	1	16	yes	complemented
97	C₂²×C₅₃	C₂×C₄×Q₈	1	6	yes	—
98	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
99	C₂²×C₅₃	C₂×#131	1	6	yes	—
100	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₅₃	—	2	4	no	—
102	C₄×C₅₃	C₂×C₄×Q₈	1	6	yes	—
103	C₄×C₅₃	C₂³×C₈	1	8	yes	—
104	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
105	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
106	Dic₅₃	—	2	8	no	—
107	Dic₅₃	C₂³×C₈	1	16	yes	—
108	C₂²×C₄×Q₈	—	53	1	no	—
109	C₂×C₈×Q₈	—	53	24	no	—
110	C₂²×C₄⋊C₈	—	53	2	no	—
111	C₂²×C₄⋊C₈	D₅₃	1	1	yes	characteristic
112	C₂²×C₄×C₈	—	53	1	no	—
113	C₂²×C₄×C₈	—	53	2	no	—
114	C₂×C₅₃	C₂×C₈×Q₈	1	12	yes	complemented
115	C₂×C₅₃	C₂²×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
117	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
118	C₂³×Q₈	—	53	1	no	—
119	C₂³×C₈	—	53	1	no	—
120	C₂³×C₈	—	53	1	no	—
121	C₂³×C₈	C₂×D₅₃	1	2	yes	—
122	C₂×C₄×Q₈	—	53	12	no	—
123	C₂×C₄×Q₈	—	53	12	no	—
124	C₂²×C₄⋊C₄	—	53	2	no	—
125	C₂²×C₄⋊C₄	—	53	1	no	—
126	C₂²×C₄²	—	53	2	no	—
127	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
128	C₈×Q₈	—	53	64	no	—
129	C₂×C₄⋊C₈	—	53	24	no	—
130	C₂×C₄⋊C₈	C₂×D₅₃	1	12	yes	—
131	C₂×C₄×C₈	—	53	24	no	—
132	C₂×C₄×C₈	—	53	12	no	—
133	C₅₃	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₈	—	53	12	no	—
135	C₂²×Q₈	—	53	16	no	—
136	C₂³×C₄	—	53	2	no	—
137	C₂³×C₄	—	53	2	no	—
138	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
139	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	8	no	—
142	C₂²×C₈	—	53	8	no	—
143	C₂²×C₈	—	53	6	no	—
144	C₂²×C₈	—	53	6	no	—
145	C₂²×C₈	—	106	4	no	—
146	C₂²×C₈	C₂²×D₅₃	1	12	yes	—
147	C₄×Q₈	—	53	48	no	—
148	C₄×Q₈	—	53	16	no	—
149	C₂×C₄⋊C₄	—	53	24	no	—
150	C₂×C₄⋊C₄	—	53	12	no	—
151	C₂×C₄²	—	53	12	no	—
152	C₂×C₄²	—	53	12	no	—
153	C₂×C₄²	C₄×D₅₃	1	6	yes	—
154	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
155	C₄⋊C₈	—	53	32	no	—
156	C₄⋊C₈	C₂²×D₅₃	1	16	yes	—
157	C₄×C₈	—	53	32	no	—
158	C₄×C₈	—	53	16	no	—
159	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
160	C₂×Q₈	—	53	96	no	—
161	C₂×Q₈	—	53	16	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	53	16	no	—
164	C₂²×C₄	—	53	12	no	—
165	C₂²×C₄	—	53	12	no	—
166	C₂²×C₄	—	106	8	no	—
167	C₂²×C₄	C₈×D₅₃	1	8	yes	—
168	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
169	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
170	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
171	C₂²×C₄	#11719947	1	4	yes	—
172	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
173	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
174	C₂×C₈	—	2	24	no	—
175	C₂×C₈	—	53	24	no	—
176	C₂×C₈	—	53	4	no	—
177	C₂×C₈	—	53	4	no	—
178	C₂×C₈	—	106	12	no	—
179	C₂×C₈	C₂³×D₅₃	1	8	yes	—
180	C₄⋊C₄	—	53	32	no	—
181	C₄⋊C₄	—	53	16	no	—
182	C₄²	—	53	24	no	—
183	C₄²	—	53	8	no	—
184	C₄²	C₂×C₄×D₅₃	1	12	yes	—
185	C₄²	C₂³×D₅₃	1	4	yes	—
186	C₂³	#13133182	1	4	yes	—
187	C₂³	C₂×C₈×D₅₃	1	4	yes	—
188	C₂³	#13133225	1	4	yes	—
189	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
190	Q₈	—	53	64	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	53	48	no	—
193	C₂×C₄	—	53	8	no	—
194	C₂×C₄	—	53	8	no	—
195	C₂×C₄	—	106	24	no	—
196	C₂×C₄	#13133182	1	12	yes	—
197	C₂×C₄	C₂×C₈×D₅₃	1	24	yes	—
198	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
199	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
200	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
201	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
202	C₂×C₄	C₂×#11719947	1	6	yes	—
203	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
204	C₈	—	2	8	no	—
205	C₈	—	53	8	no	—
206	C₈	—	106	4	no	—
207	C₂²	#16810747	1	16	yes	direct product factor
208	C₂²	C₂×#13133182	1	6	yes	—
209	C₂²	C₂²×C₈×D₅₃	1	6	yes	—
210	C₂²	C₂×#13133225	1	6	yes	—
211	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	53	16	no	—
214	C₄	—	106	8	no	—
215	C₄	C₂×#13133182	1	6	yes	—
216	C₄	C₂²×C₈×D₅₃	1	8	yes	—
217	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
218	C₄	C₂²×#11719947	1	1	yes	characteristic
219	C₂	C₂×#16810747	1	12	yes	direct product factor
220	C₂	C₂²×#13133182	1	1	yes	characteristic
221	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
222	C₂	C₂²×#13133225	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810747	C₂	direct (N × H)
15. #16810747	C₂²	direct (N × H)
22. C₂²×Dic₁₀₆	C₈	semidirect
27. C₂²×#88322	Q₈	semidirect
60. C₂×Dic₁₀₆	C₂×C₈	semidirect
71. C₂×#88322	C₂×Q₈	semidirect
91. Dic₁₀₆	C₂²×C₈	semidirect
94. #88322	C₂²×Q₈	semidirect
96. C₂²×C₅₃	C₈×Q₈	semidirect
114. C₂×C₅₃	C₂×C₈×Q₈	semidirect
133. C₅₃	C₂²×C₈×Q₈	semidirect
207. C₂²	#16810747	direct (N × H)
219. C₂	C₂×#16810747	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×C₄×Dic₁₀₆	2
C₂×#16810747	2
C₂²×#13133146	2
C₂²×C₄×#88322	2
C₂²×C₈×Dic₅₃	2
C₂²×C₄⋊C₈×C₅₃	2
C₂²×C₈×Q₈	53

Subgroups — C22×#16810747 #28248442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810747 #28248442