Subgroups — C₂×C₄×#13133157 #27536723

Order 13568.

918 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	2	yes	maximal
2	C₄×#13133157	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810399	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11719908	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₄	1	16	yes	direct product factor
7	C₂×#13133157	C₂²	1	8	yes	—
8	C₂²×#11719908	C₂²	1	4	yes	—
9	#16810399	C₂²	1	8	yes	complemented
10	C₄×#11719908	C₂²	1	8	yes	complemented
11	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	C₂×C₄	1	32	yes	direct product factor
21	C₂×#11719908	C₂×C₄	1	16	yes	complemented
22	C₂×#11719908	C₂³	1	8	yes	—
23	C₂³×C₄×C₅₃	C₈	1	2	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₈	1	8	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
34	C₂×C₄²×C₅₃	C₈	1	4	yes	—
35	C₂×C₄²×C₅₃	C₈	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂³	1	2	yes	—
44	C₄×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄²	1	8	yes	—
57	C₂×C₈×C₅₃	C₄²	1	4	yes	—
58	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	—
59	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	C₂×C₈	1	8	yes	—
70	C₄²×C₅₃	C₂×C₈	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	#11719908	C₂²×C₄	1	16	yes	complemented
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	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	16	no	—
80	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₅₃	M₃₂	1	8	yes	complemented
83	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₅₃	C₂×C₄²	1	2	yes	—
93	C₈×C₅₃	C₂×C₄²	1	4	yes	—
94	C₈×C₅₃	C₂²×C₈	1	4	yes	—
95	C₈×C₅₃	C₂²×C₈	1	4	yes	—
96	C₈×C₅₃	C₂³×C₄	1	2	yes	—
97	C₂×C₄×M₃₂	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	8	no	—
99	C₂²×C₅₃	—	2	4	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	—	2	4	no	—
105	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
106	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
107	C₄×C₅₃	C₂×M₃₂	1	8	yes	complemented
108	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
109	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
110	C₂²×M₃₂	—	53	2	no	—
111	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
112	C₄×M₃₂	—	53	8	no	—
113	C₂×#346	—	53	2	no	—
114	C₂×C₄×C₁₆	—	53	2	no	—
115	C₂×C₅₃	—	2	4	no	—
116	C₂×C₅₃	C₄×M₃₂	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	Dic₅₃	1	1	yes	characteristic
120	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
122	C₂×M₃₂	—	53	16	no	—
123	C₂×M₃₂	—	53	8	no	—
124	C₂²×C₁₆	—	53	4	no	—
125	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
126	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
127	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#346	—	53	8	no	—
132	C₄×C₁₆	—	53	8	no	—
133	C₅₃	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	#88322	1	2	yes	—
135	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂²×C₈	#88322	1	8	yes	—
137	C₂²×C₈	C₄×D₅₃	1	8	yes	—
138	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
140	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
143	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
144	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
145	C₂×C₄²	#88322	1	4	yes	—
146	C₂×C₄²	#88322	1	2	yes	—
147	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
149	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
150	M₃₂	—	53	32	no	—
151	C₂×C₁₆	—	53	16	no	—
152	C₂×C₁₆	—	53	8	no	—
153	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
154	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
155	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
156	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
157	C₄×C₈	C₂²×D₅₃	1	2	yes	—
158	C₄×C₈	C₂²×D₅₃	1	2	yes	—
159	C₂⁴	C₂×#88322	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	C₂×#88322	1	2	yes	—
162	C₂²×C₄	C₂×#88322	1	8	yes	—
163	C₂²×C₄	C₂×#88322	1	2	yes	—
164	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
165	C₂²×C₄	C₂×#88322	1	4	yes	—
166	C₂²×C₄	C₂×#88322	1	2	yes	—
167	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
168	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂×#88322	1	16	yes	—
171	C₂×C₈	C₂×#88322	1	4	yes	—
172	C₂×C₈	C₂×#88322	1	4	yes	—
173	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
174	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
175	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
176	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
177	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
178	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
180	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
181	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
182	C₄²	C₂×#88322	1	2	yes	—
183	C₄²	C₂×#88322	1	8	yes	—
184	C₄²	C₂×#88322	1	2	yes	—
185	C₄²	C₂²×Dic₅₃	1	2	yes	—
186	C₄²	C₂²×Dic₅₃	1	2	yes	—
187	C₁₆	—	53	16	no	—
188	C₂³	—	2	4	no	—
189	C₂³	C₄×#88322	1	4	yes	—
190	C₂³	C₂²×#88322	1	2	yes	—
191	C₂³	C₂²×#88322	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	C₄×#88322	1	8	yes	—
195	C₂×C₄	C₄×#88322	1	4	yes	—
196	C₂×C₄	#13133157	1	8	yes	direct product factor
197	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
198	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
199	C₂×C₄	C₂²×#88322	1	4	yes	—
200	C₂×C₄	C₂²×#88322	1	4	yes	—
201	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
202	C₂×C₄	C₂²×#88322	1	2	yes	—
203	C₂×C₄	C₂²×#88322	1	4	yes	—
204	C₂×C₄	C₂²×#88322	1	2	yes	—
205	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
206	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
207	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
208	C₈	C₂²×#88322	1	4	yes	—
209	C₈	C₂²×#88322	1	4	yes	—
210	C₈	C₂²×C₄×D₅₃	1	2	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	8	no	—
213	C₂²	C₂×C₄×#88322	1	4	yes	—
214	C₂²	C₂×C₄×#88322	1	2	yes	—
215	C₂²	C₂×#13133157	1	4	yes	—
216	C₂²	C₂³×#88322	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	C₂×C₄×#88322	1	2	yes	—
219	C₄	C₂×C₄×#88322	1	4	yes	—
220	C₄	C₂×#13133157	1	8	yes	direct product factor
221	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
222	C₄	C₂³×#88322	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₄×#13133157	1	4	yes	direct product factor
225	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
226	C₂	C₂²×#13133157	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×#13133157	C₂	direct (N × H)
3. C₂×#16810399	C₂	semidirect
4. C₂×C₄×#11719908	C₂	semidirect
6. C₂×#13133157	C₄	direct (N × H)
9. #16810399	C₂²	semidirect
10. C₄×#11719908	C₂²	semidirect
20. #13133157	C₂×C₄	direct (N × H)
21. C₂×#11719908	C₂×C₄	semidirect
73. #11719908	C₂²×C₄	semidirect
82. C₂×C₄×C₅₃	M₃₂	semidirect
107. C₄×C₅₃	C₂×M₃₂	semidirect
116. C₂×C₅₃	C₄×M₃₂	semidirect
133. C₅₃	C₂×C₄×M₃₂	semidirect
196. C₂×C₄	#13133157	direct (N × H)
220. C₄	C₂×#13133157	direct (N × H)
224. C₂	C₄×#13133157	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
C₄×#13133157	2
C₂×#16810399	2
C₂×C₄×#11719908	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₄×M₃₂	53

Subgroups — C2×C4×#13133157 #27536723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133157 #27536723