Subgroups — C₂×#20289785 #28310302

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊#108	C₂	1	2	yes	maximal
2	#20289785	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#13133156	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊M₁₆	—	2	2	no	—
5	C₂×#13133288	—	2	2	no	—
6	D₅₃⋊#108	—	2	2	no	—
7	D₅₃⋊#108	—	2	2	no	—
8	D₅₃⋊#108	C₄	1	4	yes	—
9	D₅₃⋊#108	C₂²	1	4	yes	—
10	C₂×#13133287	C₄	1	4	yes	—
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133156	—	2	2	no	—
13	C₂×#13133156	—	2	2	no	—
14	C₂×#13133156	C₄	1	2	yes	—
15	C₂×#13133156	C₂²	1	2	yes	—
16	C₂²×#11719908	C₄	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#11719936	—	2	4	no	—
19	D₅₃⋊M₁₆	—	2	4	no	—
20	D₅₃⋊M₁₆	—	2	4	no	—
21	D₅₃⋊M₁₆	—	4	2	no	—
22	D₅₃⋊M₁₆	—	4	2	no	—
23	C₂×#11719940	—	4	2	no	—
24	C₂×D₅₃⋊C₈	—	4	2	no	—
25	#13133288	—	2	4	no	—
26	#13133288	—	2	4	no	—
27	#13133287	—	2	4	no	—
28	#13133287	C₂×C₄	1	8	yes	—
29	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
30	C₂²×#88322	Q₈	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	D₄	1	2	yes	—
35	C₂×C₈×D₅₃	D₄	1	2	yes	—
36	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
38	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
39	#13133156	—	2	4	no	—
40	#13133156	—	2	4	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₃	—	2	1	no	—
46	C₂×C₁₆×C₅₃	—	2	1	no	—
47	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
48	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
51	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
55	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
56	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
57	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
58	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
59	#11719940	—	4	4	no	—
60	#11719940	—	4	4	no	—
61	C₂×#88324	—	4	4	no	—
62	#11719936	—	2	8	no	—
63	#11719936	—	4	4	no	—
64	D₅₃⋊C₈	—	4	4	no	—
65	D₅₃⋊C₈	—	4	4	no	—
66	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
73	C₂×#88322	C₄⋊C₄	1	2	yes	—
74	C₂×#88322	C₄⋊C₄	1	2	yes	—
75	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
76	C₈×D₅₃	C₄²	1	2	yes	—
77	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
78	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
79	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
80	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
81	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
82	C₈×D₅₃	C₂²×C₄	1	2	yes	—
83	C₈×D₅₃	C₂×D₄	1	2	yes	—
84	C₁₆×C₅₃	—	2	2	no	—
85	#11719908	—	2	2	no	—
86	C₂³×C₅₃	#107	1	1	yes	characteristic
87	C₂²×D₅₃	—	4	2	no	—
88	C₂²×D₅₃	#103	1	2	yes	—
89	C₂²×D₅₃	#103	1	2	yes	—
90	C₂²×D₅₃	#107	1	1	yes	characteristic
91	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×Dic₅₃	#102	1	2	yes	—
99	C₂×Dic₅₃	#102	1	2	yes	—
100	C₂×Dic₅₃	#107	1	1	yes	characteristic
101	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
102	C₄×D₅₃	#95	1	2	yes	—
103	C₄×D₅₃	#102	1	2	yes	—
104	C₄×D₅₃	#102	1	2	yes	—
105	C₄×D₅₃	#103	1	2	yes	—
106	C₄×D₅₃	#103	1	2	yes	—
107	C₄×D₅₃	#106	1	2	yes	—
108	C₄×D₅₃	C₂×C₂²⋊C₄	1	2	yes	—
109	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
110	#88324	—	4	8	no	—
111	C₈×C₅₃	#95	1	1	yes	characteristic
112	C₈×C₅₃	#95	1	1	yes	characteristic
113	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
114	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	#88322	#95	1	2	yes	—
116	#88322	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×#1385	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	#340	1	1	yes	characteristic
120	C₂²×C₅₃	#340	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
122	C₂×D₅₃	—	2	4	no	—
123	C₂×D₅₃	—	4	2	no	—
124	C₂×D₅₃	—	4	1	no	—
125	C₂×D₅₃	—	4	1	no	—
126	C₂×D₅₃	#340	1	2	yes	—
127	C₂×D₅₃	C₂×#103	1	2	yes	—
128	C₄×C₅₃	#340	1	1	yes	characteristic
129	C₄×C₅₃	#340	1	1	yes	characteristic
130	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
131	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
132	Dic₅₃	#340	1	2	yes	—
133	Dic₅₃	C₂×#102	1	2	yes	—
134	C₂²×M₃₂	—	53	1	no	—
135	C₂²×#108	—	53	2	no	—
136	#1385	—	53	4	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	#1385	1	2	yes	complemented
140	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
141	D₅₃	—	2	2	no	—
142	D₅₃	—	2	2	no	—
143	C₂²×M₁₆	—	106	2	no	—
144	C₂³×C₈	—	53	1	no	—
145	C₂×M₃₂	—	53	2	no	—
146	C₂×M₃₂	—	53	2	no	—
147	C₂×M₃₂	—	106	2	no	—
148	C₂×M₃₂	—	106	2	no	—
149	C₂²×C₁₆	—	53	1	no	—
150	C₂²×C₁₆	#6157	1	1	yes	characteristic
151	C₂×#108	—	53	4	no	—
152	C₂×#108	—	53	4	no	—
153	C₂×#108	—	53	4	no	—
154	C₂×#108	—	106	2	no	—
155	C₂×#108	—	106	2	no	—
156	C₂×#108	—	106	2	no	—
157	C₅₃	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	53	1	no	—
159	C₂×M₁₆	—	106	4	no	—
160	C₂×M₁₆	—	106	4	no	—
161	C₂×M₁₆	—	106	4	no	—
162	C₂×M₁₆	—	212	2	no	—
163	C₂×M₁₆	—	212	2	no	—
164	C₂×M₁₆	—	212	2	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	2	no	—
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	53	1	no	—
169	C₂²×C₈	—	53	1	no	—
170	C₂²×C₈	—	53	2	no	—
171	C₂²×C₈	—	53	2	no	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	53	1	no	—
174	C₂²×C₈	—	212	2	no	—
175	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
176	M₃₂	—	106	4	no	—
177	M₃₂	—	106	4	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	53	2	no	—
181	C₂×C₁₆	—	106	1	no	—
182	C₂×C₁₆	—	106	1	no	—
183	C₂×C₁₆	C₂×#6157	1	2	yes	—
184	#108	—	53	8	no	—
185	#108	—	106	4	no	—
186	#108	—	106	4	no	—
187	#108	—	106	4	no	—
188	C₂⁴	—	53	1	no	—
189	C₂²×C₄	—	53	1	no	—
190	C₂²×C₄	—	53	2	no	—
191	C₂²×C₄	—	53	2	no	—
192	C₂²×C₄	—	53	2	no	—
193	C₂²×C₄	—	53	2	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
199	M₁₆	—	106	8	no	—
200	M₁₆	—	212	4	no	—
201	M₁₆	—	212	4	no	—
202	M₁₆	—	212	4	no	—
203	C₂×C₈	—	53	2	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	1	no	—
206	C₂×C₈	—	53	1	no	—
207	C₂×C₈	—	53	2	no	—
208	C₂×C₈	—	53	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	53	2	no	—
211	C₂×C₈	—	53	2	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	—	53	2	no	—
215	C₂×C₈	—	212	4	no	—
216	C₂×C₈	—	212	4	no	—
217	C₂×C₈	—	212	4	no	—
218	C₂×C₈	C₄×#6157	1	1	yes	characteristic
219	C₂×C₈	C₄×#6157	1	1	yes	characteristic
220	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
223	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
224	C₁₆	—	2	2	no	—
225	C₁₆	—	106	2	no	—
226	C₂³	—	53	2	no	—
227	C₂³	—	53	1	no	—
228	C₂³	—	53	1	no	—
229	C₂³	—	53	2	no	—
230	C₂³	—	212	2	no	—
231	C₂³	#13133286	1	1	yes	characteristic
232	C₂×C₄	—	53	1	no	—
233	C₂×C₄	—	53	1	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	53	2	no	—
238	C₂×C₄	—	53	2	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	53	2	no	—
242	C₂×C₄	—	53	2	no	—
243	C₂×C₄	—	53	2	no	—
244	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
245	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
246	C₂×C₄	#13133285	1	1	yes	characteristic
247	C₂×C₄	#13133285	1	1	yes	characteristic
248	C₂×C₄	#13133286	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
250	C₈	—	53	2	no	—
251	C₈	—	53	2	no	—
252	C₈	—	212	8	no	—
253	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
254	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
255	C₈	C₂×C₄×#6157	1	1	yes	characteristic
256	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	53	2	no	—
259	C₂²	—	53	2	no	—
260	C₂²	—	106	4	no	—
261	C₂²	—	212	1	no	—
262	C₂²	—	212	1	no	—
263	C₂²	—	212	2	no	—
264	C₂²	#16811497	1	1	yes	characteristic
265	C₂²	#16811497	1	1	yes	characteristic
266	C₂²	C₂×#13133286	1	1	yes	characteristic
267	C₄	—	53	2	no	—
268	C₄	—	53	2	no	—
269	C₄	#16811497	1	1	yes	characteristic
270	C₄	#16811497	1	1	yes	characteristic
271	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
272	C₄	C₂×#13133285	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	106	2	no	—
276	C₂	—	106	2	no	—
277	C₂	#20289785	1	2	yes	direct product factor
278	C₂	C₂×#16811497	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133285
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#1385
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20289785	C₂	direct (N × H)
139. C₂×C₅₃	#1385	semidirect
157. C₅₃	C₂×#1385	semidirect
277. C₂	#20289785	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₅₃⋊#108	2
#20289785	2
C₂²×#13133156	2
C₂×#1385	53

Subgroups — C2×#20289785 #28310302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289785 #28310302