Subgroups — C₂×#20276470 #27537070

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276470	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810441	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×#13133157	—	2	4	no	—
9	C₂×#13133157	C₂²	1	4	yes	—
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅₃	C₄	1	4	yes	—
12	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
13	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
15	#16810441	C₂²	1	16	yes	complemented
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	—	2	4	no	—
18	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂³×Dic₅₃	C₈	1	2	yes	—
21	C₂²×#88322	—	2	4	no	—
22	C₂²×#88322	C₂×C₄	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	4	yes	—
24	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	C₈	1	4	yes	—
26	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	#13133157	—	2	8	no	—
30	C₂×#11719908	—	2	4	no	—
31	C₂×#11719908	C₂³	1	4	yes	—
32	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
33	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
34	C₈×Dic₅₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	4	yes	—
39	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	8	no	—
43	C₂×C₁₆×C₅₃	—	2	4	no	—
44	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
45	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	2	8	no	—
53	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
54	C₂²×Dic₅₃	C₂×C₈	1	8	yes	—
55	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
66	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
67	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
68	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
69	C₂×#88322	—	2	8	no	—
70	C₂×#88322	—	2	4	no	—
71	C₂×#88322	C₂²×C₄	1	4	yes	—
72	C₁₆×C₅₃	—	2	4	no	—
73	#11719908	—	2	4	no	—
74	C₂³×C₅₃	—	2	4	no	—
75	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
76	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
77	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
79	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
85	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	4	no	—
88	C₂×Dic₅₃	—	2	16	no	—
89	C₂×Dic₅₃	M₃₂	1	8	yes	complemented
90	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
91	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
92	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
96	#88322	—	2	4	no	—
97	C₂×#2150	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	—	2	8	no	—
100	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
101	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
102	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
103	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
104	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
105	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
106	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
108	Dic₅₃	—	2	4	no	—
109	Dic₅₃	C₂×M₃₂	1	8	yes	complemented
110	C₂²×M₃₂	—	53	1	no	—
111	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
112	C₂²×C₄×C₈	—	53	1	no	—
113	#2150	—	53	8	no	—
114	C₂×C₄⋊C₁₆	—	53	4	no	—
115	C₂×C₅₃	—	2	4	no	—
116	C₂×C₅₃	#2150	1	4	yes	complemented
117	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
119	C₂³×C₈	—	53	1	no	—
120	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₂²×C₄²	—	53	1	no	—
122	C₂×M₃₂	—	2	4	no	—
123	C₂×M₃₂	—	53	4	no	—
124	C₂×M₃₂	—	106	4	no	—
125	C₂×M₃₂	C₂×D₅₃	1	4	yes	—
126	C₂²×C₁₆	—	53	2	no	—
127	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
128	C₂×C₄×C₈	—	53	2	no	—
129	C₂×C₄×C₈	—	53	4	no	—
130	C₂×C₄×C₈	—	53	4	no	—
131	C₂×C₄×C₈	—	53	2	no	—
132	C₄⋊C₁₆	—	53	16	no	—
133	C₅₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	—	53	2	no	—
135	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₈	—	53	4	no	—
137	C₂²×C₈	—	53	2	no	—
138	C₂²×C₈	—	106	4	no	—
139	C₂²×C₈	Dic₁₀₆	1	4	yes	—
140	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₄×D₅₃	1	2	yes	—
142	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
143	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
144	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
145	C₂×C₄²	—	53	2	no	—
146	C₂×C₄²	—	53	4	no	—
147	C₂×C₄²	—	53	1	no	—
148	C₂×C₄²	—	53	4	no	—
149	C₂×C₄²	—	53	1	no	—
150	M₃₂	—	2	8	no	—
151	M₃₂	—	106	8	no	—
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	—	53	4	no	—
154	C₂×C₁₆	—	106	4	no	—
155	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
156	C₄×C₈	—	53	8	no	—
157	C₄×C₈	—	53	4	no	—
158	C₄×C₈	—	53	4	no	—
159	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
160	C₂²×C₄	—	53	2	no	—
161	C₂²×C₄	—	53	8	no	—
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	—	106	8	no	—
164	C₂²×C₄	C₈×D₅₃	1	2	yes	—
165	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
166	C₂²×C₄	#11719912	1	4	yes	—
167	C₂²×C₄	#11719919	1	4	yes	—
168	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	106	8	no	—
172	C₂×C₈	—	106	4	no	—
173	C₂×C₈	#11719919	1	4	yes	—
174	C₂×C₈	#11719919	1	8	yes	—
175	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
176	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
180	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
181	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
182	C₄²	—	53	4	no	—
183	C₄²	—	53	4	no	—
184	C₄²	—	53	4	no	—
185	C₄²	—	53	4	no	—
186	C₁₆	—	2	4	no	—
187	C₁₆	—	106	4	no	—
188	C₂³	—	2	4	no	—
189	C₂³	#13133146	1	4	yes	—
190	C₂³	C₂×C₈×D₅₃	1	2	yes	—
191	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
192	C₂×C₄	—	53	8	no	—
193	C₂×C₄	—	53	4	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	106	16	no	—
196	C₂×C₄	—	106	4	no	—
197	C₂×C₄	#13133146	1	8	yes	—
198	C₂×C₄	#13133146	1	4	yes	—
199	C₂×C₄	C₂×#11719919	1	4	yes	—
200	C₂×C₄	C₂×#11719919	1	2	yes	—
201	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
203	C₂×C₄	C₂×#11719912	1	4	yes	—
204	C₂×C₄	C₂×#11719912	1	2	yes	—
205	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
206	C₈	—	106	4	no	—
207	C₈	C₂×#11719919	1	4	yes	—
208	C₈	C₂×#11719919	1	2	yes	—
209	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
211	C₂²	—	2	4	no	—
212	C₂²	—	2	8	no	—
213	C₂²	C₂×#13133146	1	4	yes	—
214	C₂²	C₂×#13133146	1	2	yes	—
215	C₂²	M₃₂×D₅₃	1	4	yes	—
216	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
217	C₄	—	53	8	no	—
218	C₄	—	106	4	no	—
219	C₄	C₂×#13133146	1	2	yes	—
220	C₄	C₂×#13133146	1	4	yes	—
221	C₄	C₂²×#11719919	1	1	yes	characteristic
222	C₄	C₂²×#11719912	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	#20276470	1	4	yes	direct product factor
225	C₂	C₂²×#13133146	1	1	yes	characteristic
226	C₂	C₂×M₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#2150
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20276470	C₂	direct (N × H)
4. C₂×#16810441	C₂	semidirect
15. #16810441	C₂²	semidirect
89. C₂×Dic₅₃	M₃₂	semidirect
109. Dic₅₃	C₂×M₃₂	semidirect
116. C₂×C₅₃	#2150	semidirect
133. C₅₃	C₂×#2150	semidirect
224. C₂	#20276470	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×#13133157	2
C₂²×C₈×Dic₅₃	2
#20276470	2
C₂×#16810441	2
C₂²×M₃₂×C₅₃	2
C₂×#2150	53

Subgroups — C2×#20276470 #27537070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276470 #27537070