Subgroups — C₂×#20290650 #28319440

Order 13568.

1358 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133285	C₂	1	4	yes	maximal
2	C₂×#16811569	C₂	1	2	yes	maximal
3	#20290650	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#6157	—	2	4	no	—
6	C₂²×C₄⋊#6157	C₂²	1	4	yes	—
7	C₂×#13133285	—	2	16	no	—
8	C₂×#13133285	C₂²	1	16	yes	—
9	#16811569	—	2	4	no	—
10	#16811569	C₂²	1	8	yes	—
11	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×Dic₅₃	C₄	1	2	yes	complemented
15	C₄×C₈×D₅₃	C₄	1	4	yes	complemented
16	C₄×C₈×D₅₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂³×#6157	—	2	4	no	—
19	C₂×C₄⋊#6157	—	2	16	no	—
20	C₂×C₄⋊#6157	—	2	4	no	—
21	C₂×C₄⋊#6157	—	2	16	no	—
22	C₂×C₄⋊#6157	—	4	16	no	—
23	C₂×C₄⋊#6157	C₂³	1	8	yes	—
24	#13133285	—	2	32	no	—
25	C₂²×C₄×D₅₃	D₄	1	2	yes	—
26	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₅₃	D₄	1	8	yes	—
30	C₂×C₈×D₅₃	Q₈	1	8	yes	—
31	C₂×C₈×D₅₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
34	C₄²×D₅₃	C₂×C₄	1	2	yes	—
35	C₄²×D₅₃	D₄	1	2	yes	—
36	C₄²×D₅₃	Q₈	1	2	yes	—
37	C₄²×D₅₃	C₂³	1	2	yes	—
38	C₄×#88322	C₂×C₄	1	4	yes	complemented
39	C₈×Dic₅₃	C₂×C₄	1	8	yes	complemented
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
43	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×#6157	—	2	8	no	—
45	C₂²×#6157	—	4	8	no	—
46	C₂²×#6157	—	4	16	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
50	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×D₅₃	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×D₅₃	SD₁₆	1	16	yes	—
53	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅₃	C₂×D₄	1	4	yes	—
55	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×D₅₃	C₂×D₄	1	4	yes	—
57	C₂×C₄×D₅₃	C₂×Q₈	1	4	yes	—
58	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
60	C₄⋊#6157	—	2	16	no	—
61	C₄⋊#6157	—	4	8	no	—
62	C₄⋊#6157	—	4	32	no	—
63	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	complemented
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
67	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
68	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
69	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
70	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
71	C₂×#88322	C₄⋊C₄	1	8	yes	complemented
72	C₂×#88322	C₂²×C₄	1	4	yes	—
73	C₈×D₅₃	C₄⋊C₄	1	16	yes	complemented
74	C₈×D₅₃	C₂×D₄	1	8	yes	—
75	C₈×D₅₃	C₂×Q₈	1	8	yes	—
76	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×D₅₃	#127	1	2	yes	—
79	C₂²×D₅₃	C₄⋊Q₈	1	2	yes	—
80	C₂²×D₅₃	C₂×SD₁₆	1	8	yes	—
81	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×#6157	—	4	8	no	—
83	C₂×#6157	—	4	16	no	—
84	C₂×#6157	—	8	16	no	—
85	C₂×C₄×C₅₃	#106	1	8	yes	complemented
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₅₃	#106	1	8	yes	complemented
91	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
94	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
95	C₄×D₅₃	#106	1	16	yes	complemented
96	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₄×D₅₃	#127	1	2	yes	—
98	C₄×D₅₃	C₄⋊Q₈	1	4	yes	—
99	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
100	C₄×D₅₃	C₂×SD₁₆	1	16	yes	—
101	C₄×D₅₃	C₂²×D₄	1	2	yes	—
102	C₄×D₅₃	C₂²×Q₈	1	2	yes	—
103	C₈×C₅₃	C₂×C₄⋊C₄	1	8	yes	complemented
104	#88322	C₂×C₄⋊C₄	1	8	yes	complemented
105	C₂×#1844	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#384	1	2	yes	—
107	C₂²×C₅₃	C₂×#106	1	4	yes	—
108	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂×D₅₃	#384	1	2	yes	—
110	C₂×D₅₃	C₂×#106	1	4	yes	—
111	C₂×D₅₃	#492	1	8	yes	—
112	C₂×D₅₃	#498	1	8	yes	—
113	C₂×D₅₃	C₂×#127	1	1	yes	characteristic
114	C₂×D₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×D₅₃	C₂²×SD₁₆	1	4	yes	—
116	#6157	—	8	8	no	—
117	C₄×C₅₃	#384	1	2	yes	—
118	C₄×C₅₃	C₂×#106	1	8	yes	complemented
119	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
120	Dic₅₃	#384	1	2	yes	—
121	Dic₅₃	C₂×#106	1	8	yes	complemented
122	Dic₅₃	C₂×C₄⋊Q₈	1	2	yes	—
123	C₂²×#106	—	53	4	no	—
124	C₂²×C₄×C₈	—	53	1	no	—
125	C₂×#384	—	53	2	no	—
126	#1844	—	53	8	no	—
127	C₂×C₅₃	#1844	1	4	yes	complemented
128	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
129	C₂×C₅₃	C₂²×#106	1	2	yes	—
130	D₅₃	#1844	1	4	yes	complemented
131	D₅₃	C₂×#492	1	2	yes	—
132	D₅₃	C₂×#498	1	2	yes	—
133	C₂³×C₈	—	53	2	no	—
134	C₂²×C₄⋊C₄	—	53	4	no	—
135	C₂²×C₄⋊C₄	—	106	4	no	—
136	C₂²×C₄²	—	53	1	no	—
137	C₂×#106	—	53	16	no	—
138	C₂×#106	—	106	16	no	—
139	C₂×C₄×C₈	—	53	2	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	—	53	4	no	—
142	C₂×C₄×C₈	—	53	4	no	—
143	C₂×C₄×C₈	#6157	1	1	yes	characteristic, complemented
144	#384	—	53	8	no	—
145	#384	—	106	4	no	—
146	C₅₃	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	—	53	2	no	—
148	C₂³×C₄	—	53	1	no	—
149	C₂³×C₄	—	106	4	no	—
150	C₂²×C₈	—	53	2	no	—
151	C₂²×C₈	—	53	8	no	—
152	C₂²×C₈	—	53	8	no	—
153	C₂²×C₈	—	53	4	no	—
154	C₂²×C₈	—	53	4	no	—
155	C₂²×C₈	C₂×#6157	1	2	yes	—
156	C₂×C₄⋊C₄	—	53	8	no	—
157	C₂×C₄⋊C₄	—	106	16	no	—
158	C₂×C₄⋊C₄	—	106	4	no	—
159	C₂×C₄⋊C₄	—	106	16	no	—
160	C₂×C₄⋊C₄	—	212	16	no	—
161	C₂×C₄²	—	53	2	no	—
162	C₂×C₄²	—	53	1	no	—
163	C₂×C₄²	—	53	2	no	—
164	C₂×C₄²	—	53	2	no	—
165	C₂×C₄²	—	53	2	no	—
166	C₂×C₄²	—	53	2	no	—
167	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
168	#106	—	106	32	no	—
169	C₄×C₈	—	53	8	no	—
170	C₄×C₈	—	53	4	no	—
171	C₄×C₈	C₂×#6157	1	4	yes	complemented
172	C₂⁴	—	53	1	no	—
173	C₂²×C₄	—	53	2	no	—
174	C₂²×C₄	—	53	1	no	—
175	C₂²×C₄	—	53	16	no	—
176	C₂²×C₄	—	53	4	no	—
177	C₂²×C₄	—	53	1	no	—
178	C₂²×C₄	—	53	4	no	—
179	C₂²×C₄	—	53	4	no	—
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	4	no	—
182	C₂²×C₄	—	53	1	no	—
183	C₂²×C₄	—	53	1	no	—
184	C₂²×C₄	—	106	8	no	—
185	C₂²×C₄	—	212	8	no	—
186	C₂²×C₄	—	212	16	no	—
187	C₂²×C₄	C₄⋊#6157	1	2	yes	—
188	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
189	C₂×C₈	—	53	8	no	—
190	C₂×C₈	—	53	4	no	—
191	C₂×C₈	—	53	8	no	—
192	C₂×C₈	—	53	16	no	—
193	C₂×C₈	—	53	8	no	—
194	C₂×C₈	C₄⋊#6157	1	8	yes	complemented
195	C₂×C₈	C₂²×#6157	1	4	yes	—
196	C₄⋊C₄	—	106	16	no	—
197	C₄⋊C₄	—	212	8	no	—
198	C₄⋊C₄	—	212	32	no	—
199	C₄²	—	53	2	no	—
200	C₄²	—	53	4	no	—
201	C₄²	—	53	4	no	—
202	C₄²	—	53	2	no	—
203	C₄²	C₄⋊#6157	1	2	yes	—
204	C₄²	C₂²×#6157	1	2	yes	—
205	C₂³	—	53	2	no	—
206	C₂³	—	53	2	no	—
207	C₂³	—	53	8	no	—
208	C₂³	—	53	1	no	—
209	C₂³	—	53	1	no	—
210	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
211	C₂×C₄	—	53	4	no	—
212	C₂×C₄	—	53	8	no	—
213	C₂×C₄	—	53	4	no	—
214	C₂×C₄	—	53	1	no	—
215	C₂×C₄	—	53	1	no	—
216	C₂×C₄	—	53	2	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	53	2	no	—
219	C₂×C₄	—	53	16	no	—
220	C₂×C₄	—	53	16	no	—
221	C₂×C₄	—	53	4	no	—
222	C₂×C₄	—	53	2	no	—
223	C₂×C₄	—	53	2	no	—
224	C₂×C₄	—	424	16	no	—
225	C₂×C₄	—	212	8	no	—
226	C₂×C₄	—	212	16	no	—
227	C₂×C₄	#13133285	1	8	yes	complemented
228	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊#6157	1	4	yes	—
230	C₂×C₄	C₂×C₄⋊#6157	1	4	yes	—
231	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
232	C₈	—	53	8	no	—
233	C₈	C₂×C₄⋊#6157	1	8	yes	complemented
234	C₂²	—	53	4	no	—
235	C₂²	—	53	4	no	—
236	C₂²	—	53	8	no	—
237	C₂²	—	53	8	no	—
238	C₂²	—	53	1	no	—
239	C₂²	—	53	1	no	—
240	C₂²	—	53	2	no	—
241	C₂²	#16811569	1	2	yes	—
242	C₂²	C₂×#13133285	1	4	yes	—
243	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
244	C₄	—	53	8	no	—
245	C₄	—	53	2	no	—
246	C₄	—	53	2	no	—
247	C₄	—	424	8	no	—
248	C₄	#16811569	1	2	yes	—
249	C₄	C₂×#13133285	1	8	yes	complemented
250	C₄	C₂²×C₄⋊#6157	1	2	yes	—
251	C₂	—	53	2	no	—
252	C₂	—	53	4	no	—
253	C₂	—	53	2	no	—
254	C₂	#20290650	1	4	yes	direct product factor
255	C₂	C₂×#16811569	1	1	yes	characteristic
256	C₂	C₂²×#13133285	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×#1844
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20290650	C₂	direct (N × H)
13. C₂×C₄×#88322	C₄	semidirect
14. C₂×C₈×Dic₅₃	C₄	semidirect
15. C₄×C₈×D₅₃	C₄	semidirect
17. C₂×C₄×C₈×C₅₃	C₄	semidirect
38. C₄×#88322	C₂×C₄	semidirect
39. C₈×Dic₅₃	C₂×C₄	semidirect
42. C₄×C₈×C₅₃	C₂×C₄	semidirect
63. C₂×C₈×C₅₃	C₄⋊C₄	semidirect
71. C₂×#88322	C₄⋊C₄	semidirect
73. C₈×D₅₃	C₄⋊C₄	semidirect
85. C₂×C₄×C₅₃	#106	semidirect
90. C₂×Dic₅₃	#106	semidirect
95. C₄×D₅₃	#106	semidirect
103. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
104. #88322	C₂×C₄⋊C₄	semidirect
118. C₄×C₅₃	C₂×#106	semidirect
121. Dic₅₃	C₂×#106	semidirect
127. C₂×C₅₃	#1844	semidirect
130. D₅₃	#1844	semidirect
143. C₂×C₄×C₈	#6157	semidirect
146. C₅₃	C₂×#1844	semidirect
171. C₄×C₈	C₂×#6157	semidirect
194. C₂×C₈	C₄⋊#6157	semidirect
227. C₂×C₄	#13133285	semidirect
233. C₈	C₂×C₄⋊#6157	semidirect
249. C₄	C₂×#13133285	semidirect
254. C₂	#20290650	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×#13133285	2
C₂×#16811569	2
#20290650	2
C₂×C₄×C₈×D₅₃	2
C₂×#1844	53

Subgroups — C2×#20290650 #28319440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20290650 #28319440