Subgroups — C₂×C₄×#13133271 #28317892

Order 13568.

1790 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133271	C₂	1	4	yes	maximal
2	C₂×C₄×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×#6157	C₂	1	1	yes	maximal, characteristic
4	C₄×#13133271	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₅₃⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×#6157	C₄	1	4	yes	—
8	C₂²×C₄×#6157	C₂²	1	2	yes	—
9	C₂×#13133271	C₄	1	32	yes	direct product factor
10	C₂×#13133271	C₂²	1	16	yes	—
11	C₄×D₅₃⋊C₈	C₄	1	4	yes	complemented
12	C₄×D₅₃⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×#88324	C₄	1	4	yes	complemented
14	C₄²×#6157	C₄	1	4	yes	—
15	C₄²×#6157	C₂²	1	4	yes	—
16	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₅₃	C₄	1	2	yes	complemented
20	C₄×C₈×D₅₃	C₄	1	4	yes	complemented
21	C₄×C₈×D₅₃	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂³×#6157	C₂×C₄	1	4	yes	—
24	C₂×D₅₃⋊C₈	C₂×C₄	1	16	yes	—
25	C₂×D₅₃⋊C₈	C₂×C₄	1	4	yes	—
26	C₂×D₅₃⋊C₈	C₂³	1	4	yes	—
27	C₂²×#88324	C₂×C₄	1	4	yes	—
28	C₂×C₄×#6157	—	2	16	no	—
29	C₂×C₄×#6157	C₂×C₄	1	16	yes	—
30	C₂×C₄×#6157	C₂×C₄	1	4	yes	—
31	C₂×C₄×#6157	C₂×C₄	1	16	yes	—
32	C₂×C₄×#6157	C₂³	1	4	yes	—
33	#13133271	C₂×C₄	1	64	yes	direct product factor
34	C₄×#88324	C₂×C₄	1	16	yes	complemented
35	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
37	C₂²×#88322	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₈×D₅₃	C₂×C₄	1	16	yes	—
40	C₂×C₈×D₅₃	C₂³	1	4	yes	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
43	C₄²×D₅₃	C₂×C₄	1	2	yes	—
44	C₄²×D₅₃	C₂×C₄	1	2	yes	—
45	C₄²×D₅₃	C₂×C₄	1	2	yes	—
46	C₄²×D₅₃	C₂³	1	2	yes	—
47	C₄×#88322	C₂×C₄	1	4	yes	complemented
48	C₈×Dic₅₃	C₂×C₄	1	8	yes	complemented
49	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
52	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×#6157	—	2	16	no	—
54	C₂²×#6157	C₄²	1	16	yes	—
55	C₂²×#6157	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₄²	1	2	yes	—
59	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×D₅₃	C₄²	1	4	yes	—
61	C₂×C₄×D₅₃	M₁₆	1	16	yes	—
62	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂²×C₄	1	8	yes	—
66	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
67	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
68	C₂×#88324	C₄²	1	16	yes	complemented
69	C₂×#88324	C₂²×C₄	1	8	yes	—
70	C₄×#6157	—	2	32	no	—
71	C₄×#6157	C₄²	1	16	yes	—
72	C₄×#6157	C₂²×C₄	1	16	yes	—
73	D₅₃⋊C₈	C₄²	1	16	yes	complemented
74	D₅₃⋊C₈	C₂²×C₄	1	16	yes	—
75	C₂×C₈×C₅₃	C₄²	1	8	yes	complemented
76	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
77	C₄²×C₅₃	C₄²	1	2	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄×Dic₅₃	C₄²	1	4	yes	—
80	C₄×Dic₅₃	C₄²	1	2	yes	—
81	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
83	C₂×#88322	C₄²	1	8	yes	complemented
84	C₂×#88322	C₂²×C₄	1	4	yes	—
85	C₈×D₅₃	C₄²	1	16	yes	complemented
86	C₈×D₅₃	C₂²×C₄	1	16	yes	—
87	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
89	C₂²×D₅₃	C₂×C₄²	1	4	yes	—
90	C₂²×D₅₃	C₂×M₁₆	1	8	yes	—
91	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×#6157	—	2	16	no	—
93	C₂×#6157	—	2	32	no	—
94	C₂×#6157	C₂×C₄²	1	16	yes	—
95	C₂×C₄×C₅₃	#97	1	8	yes	complemented
96	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
98	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×Dic₅₃	#97	1	8	yes	complemented
101	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
103	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
104	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
105	C₄×D₅₃	#97	1	16	yes	complemented
106	C₄×D₅₃	C₂×C₄²	1	4	yes	—
107	C₄×D₅₃	C₂×C₄²	1	4	yes	—
108	C₄×D₅₃	C₂×C₄²	1	4	yes	—
109	C₄×D₅₃	C₂×M₁₆	1	16	yes	—
110	C₄×D₅₃	C₂³×C₄	1	4	yes	—
111	#88324	C₂×C₄²	1	16	yes	complemented
112	C₈×C₅₃	C₂×C₄²	1	8	yes	complemented
113	#88322	C₂×C₄²	1	8	yes	complemented
114	C₂×C₄×#97	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	C₄³	1	2	yes	—
116	C₂²×C₅₃	C₂×#97	1	4	yes	—
117	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
118	C₂×D₅₃	C₄³	1	2	yes	—
119	C₂×D₅₃	C₂×#97	1	4	yes	—
120	C₂×D₅₃	C₄×M₁₆	1	16	yes	—
121	C₂×D₅₃	C₂²×C₄²	1	2	yes	—
122	C₂×D₅₃	C₂²×M₁₆	1	4	yes	—
123	#6157	—	2	16	no	—
124	C₄×C₅₃	C₄³	1	2	yes	—
125	C₄×C₅₃	C₂×#97	1	8	yes	complemented
126	C₄×C₅₃	C₂²×C₄²	1	2	yes	—
127	Dic₅₃	C₄³	1	2	yes	—
128	Dic₅₃	C₂×#97	1	8	yes	complemented
129	Dic₅₃	C₂²×C₄²	1	2	yes	—
130	C₂²×#97	—	53	4	no	—
131	C₂²×C₄×C₈	—	53	1	no	—
132	C₂²×C₄×C₈	—	53	1	no	—
133	C₂×C₄³	—	53	1	no	—
134	C₄×#97	—	53	8	no	—
135	C₂×C₅₃	C₄×#97	1	4	yes	complemented
136	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
137	C₂×C₅₃	C₂²×#97	1	2	yes	—
138	D₅₃	C₄×#97	1	4	yes	complemented
139	D₅₃	C₂×C₄×M₁₆	1	4	yes	—
140	C₂³×C₈	—	53	2	no	—
141	C₂³×C₈	—	53	2	no	—
142	C₂²×C₄²	—	53	2	no	—
143	C₂²×C₄²	—	53	1	no	—
144	C₂²×C₄²	—	53	4	no	—
145	C₂×#97	—	53	32	no	—
146	C₂×#97	—	53	16	no	—
147	C₂×C₄×C₈	—	53	4	no	—
148	C₂×C₄×C₈	—	53	4	no	—
149	C₂×C₄×C₈	—	53	4	no	—
150	C₂×C₄×C₈	—	53	2	no	—
151	C₂×C₄×C₈	—	53	1	no	—
152	C₂×C₄×C₈	—	53	4	no	—
153	C₂×C₄×C₈	—	53	4	no	—
154	C₂×C₄×C₈	#6157	1	1	yes	characteristic, complemented
155	C₄³	—	53	4	no	—
156	C₄³	—	53	4	no	—
157	C₅₃	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	53	2	no	—
159	C₂³×C₄	—	53	1	no	—
160	C₂³×C₄	—	53	4	no	—
161	C₂²×C₈	—	53	4	no	—
162	C₂²×C₈	—	53	4	no	—
163	C₂²×C₈	—	53	4	no	—
164	C₂²×C₈	—	53	16	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	53	16	no	—
167	C₂²×C₈	—	53	4	no	—
168	C₂²×C₈	—	53	4	no	—
169	C₂²×C₈	C₂×#6157	1	2	yes	—
170	C₂×C₄²	—	53	4	no	—
171	C₂×C₄²	—	53	4	no	—
172	C₂×C₄²	—	53	16	no	—
173	C₂×C₄²	—	53	2	no	—
174	C₂×C₄²	—	53	1	no	—
175	C₂×C₄²	—	53	2	no	—
176	C₂×C₄²	—	53	2	no	—
177	C₂×C₄²	—	53	2	no	—
178	C₂×C₄²	—	53	2	no	—
179	C₂×C₄²	—	53	16	no	—
180	C₂×C₄²	—	106	16	no	—
181	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
182	#97	—	53	64	no	—
183	C₄×C₈	—	53	16	no	—
184	C₄×C₈	—	53	8	no	—
185	C₄×C₈	—	53	4	no	—
186	C₄×C₈	C₂×#6157	1	4	yes	complemented
187	C₂⁴	—	53	1	no	—
188	C₂²×C₄	—	53	2	no	—
189	C₂²×C₄	—	53	1	no	—
190	C₂²×C₄	—	53	16	no	—
191	C₂²×C₄	—	53	4	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	4	no	—
194	C₂²×C₄	—	53	8	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	53	16	no	—
199	C₂²×C₄	—	53	8	no	—
200	C₂²×C₄	—	106	16	no	—
201	C₂²×C₄	C₄×#6157	1	2	yes	—
202	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
203	C₂×C₈	—	53	16	no	—
204	C₂×C₈	—	53	8	no	—
205	C₂×C₈	—	53	16	no	—
206	C₂×C₈	—	53	16	no	—
207	C₂×C₈	—	53	8	no	—
208	C₂×C₈	—	53	4	no	—
209	C₂×C₈	—	53	16	no	—
210	C₂×C₈	—	53	16	no	—
211	C₂×C₈	C₄×#6157	1	8	yes	complemented
212	C₂×C₈	C₂²×#6157	1	4	yes	—
213	C₄²	—	53	16	no	—
214	C₄²	—	53	16	no	—
215	C₄²	—	53	4	no	—
216	C₄²	—	53	2	no	—
217	C₄²	—	53	4	no	—
218	C₄²	—	53	2	no	—
219	C₄²	—	106	32	no	—
220	C₄²	C₄×#6157	1	2	yes	—
221	C₄²	C₂²×#6157	1	2	yes	—
222	C₂³	—	53	4	no	—
223	C₂³	—	53	8	no	—
224	C₂³	—	53	1	no	—
225	C₂³	—	53	1	no	—
226	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
227	C₂×C₄	—	53	4	no	—
228	C₂×C₄	—	53	8	no	—
229	C₂×C₄	—	53	1	no	—
230	C₂×C₄	—	53	4	no	—
231	C₂×C₄	—	53	1	no	—
232	C₂×C₄	—	53	4	no	—
233	C₂×C₄	—	53	4	no	—
234	C₂×C₄	—	53	16	no	—
235	C₂×C₄	—	53	16	no	—
236	C₂×C₄	—	53	4	no	—
237	C₂×C₄	—	53	4	no	—
238	C₂×C₄	—	53	16	no	—
239	C₂×C₄	—	106	16	no	—
240	C₂×C₄	—	106	32	no	—
241	C₂×C₄	#13133271	1	8	yes	direct product factor
242	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×#6157	1	4	yes	—
244	C₂×C₄	C₂×C₄×#6157	1	4	yes	—
245	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
246	C₈	—	53	16	no	—
247	C₈	—	53	8	no	—
248	C₈	C₂×C₄×#6157	1	8	yes	complemented
249	C₂²	—	53	16	no	—
250	C₂²	—	53	2	no	—
251	C₂²	—	53	2	no	—
252	C₂²	—	53	4	no	—
253	C₂²	—	53	4	no	—
254	C₂²	C₄²×#6157	1	2	yes	—
255	C₂²	C₂×#13133271	1	4	yes	—
256	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
257	C₄	—	53	2	no	—
258	C₄	—	53	2	no	—
259	C₄	—	53	8	no	—
260	C₄	—	106	16	no	—
261	C₄	C₄²×#6157	1	2	yes	—
262	C₄	C₂×#13133271	1	8	yes	direct product factor
263	C₄	C₂²×C₄×#6157	1	2	yes	—
264	C₂	—	53	4	no	—
265	C₂	—	53	4	no	—
266	C₂	C₄×#13133271	1	4	yes	direct product factor
267	C₂	C₂×C₄²×#6157	1	1	yes	characteristic
268	C₂	C₂²×#13133271	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄³
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×C₄×#97
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×#13133271	C₂	direct (N × H)
9. C₂×#13133271	C₄	direct (N × H)
11. C₄×D₅₃⋊C₈	C₄	semidirect
13. C₂×C₄×#88324	C₄	semidirect
18. C₂×C₄×#88322	C₄	semidirect
19. C₂×C₈×Dic₅₃	C₄	semidirect
20. C₄×C₈×D₅₃	C₄	semidirect
22. C₂×C₄×C₈×C₅₃	C₄	semidirect
33. #13133271	C₂×C₄	direct (N × H)
34. C₄×#88324	C₂×C₄	semidirect
47. C₄×#88322	C₂×C₄	semidirect
48. C₈×Dic₅₃	C₂×C₄	semidirect
51. C₄×C₈×C₅₃	C₂×C₄	semidirect
68. C₂×#88324	C₄²	semidirect
73. D₅₃⋊C₈	C₄²	semidirect
75. C₂×C₈×C₅₃	C₄²	semidirect
83. C₂×#88322	C₄²	semidirect
85. C₈×D₅₃	C₄²	semidirect
95. C₂×C₄×C₅₃	#97	semidirect
100. C₂×Dic₅₃	#97	semidirect
105. C₄×D₅₃	#97	semidirect
111. #88324	C₂×C₄²	semidirect
112. C₈×C₅₃	C₂×C₄²	semidirect
113. #88322	C₂×C₄²	semidirect
125. C₄×C₅₃	C₂×#97	semidirect
128. Dic₅₃	C₂×#97	semidirect
135. C₂×C₅₃	C₄×#97	semidirect
138. D₅₃	C₄×#97	semidirect
154. C₂×C₄×C₈	#6157	semidirect
157. C₅₃	C₂×C₄×#97	semidirect
186. C₄×C₈	C₂×#6157	semidirect
211. C₂×C₈	C₄×#6157	semidirect
241. C₂×C₄	#13133271	direct (N × H)
248. C₈	C₂×C₄×#6157	semidirect
262. C₄	C₂×#13133271	direct (N × H)
266. C₂	C₄×#13133271	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×#13133271	2
C₂×C₄×D₅₃⋊C₈	2
C₂×C₄²×#6157	2
C₄×#13133271	2
C₂×C₄×C₈×D₅₃	2
C₂×C₄×#97	53

Subgroups — C2×C4×#13133271 #28317892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133271 #28317892