Subgroups — C₂×#20272733 #27294341

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810388	C₂	1	1	yes	maximal, characteristic
3	#20272733	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#467	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂²×#11719908	—	2	1	no	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	C₂²	1	2	yes	—
9	#16810388	—	2	1	no	—
10	#16810388	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₅₃	—	2	1	no	—
14	C₅₃×#467	—	2	1	no	—
15	C₅₃×#467	—	2	1	no	—
16	C₅₃×#467	C₄	1	2	yes	—
17	C₅₃×#467	C₂²	1	2	yes	—
18	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#103×C₅₃	—	2	1	no	—
20	C₂×#11719917	—	2	4	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	4	2	no	—
23	C₂×#11719917	—	4	1	no	—
24	C₂²×#88322	—	4	1	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	—	4	2	no	—
27	C₂²×Q₈×C₅₃	—	2	1	no	—
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×Q₁₆×C₅₃	—	4	2	no	—
31	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	—	4	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²⋊Q₈×C₅₃	—	2	2	no	—
40	C₂²⋊Q₈×C₅₃	—	2	2	no	—
41	C₂²⋊Q₈×C₅₃	—	4	1	no	—
42	C₂²⋊Q₈×C₅₃	—	4	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
46	#107×C₅₃	—	2	1	no	—
47	#107×C₅₃	C₂×C₄	1	2	yes	—
48	#103×C₅₃	—	2	2	no	—
49	#103×C₅₃	—	2	2	no	—
50	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	4	1	no	—
58	C₂²×C₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₅₃	—	4	2	no	—
66	Q₁₆×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	4	1	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₂²⋊C₄×C₅₃	—	4	2	no	—
82	C₂²⋊C₄×C₅₃	—	4	2	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	4	4	no	—
85	#11719917	—	4	2	no	—
86	C₂×#88322	—	4	4	no	—
87	C₂×#88322	—	4	2	no	—
88	#11719908	—	4	2	no	—
89	C₂³×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	#102	1	1	yes	characteristic
93	C₂³×C₅₃	#102	1	1	yes	characteristic
94	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
95	Q₈×C₅₃	—	4	2	no	—
96	Q₈×C₅₃	—	4	2	no	—
97	Q₈×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	—	4	2	no	—
107	C₂×C₄×C₅₃	—	8	2	no	—
108	C₂×C₄×C₅₃	—	8	2	no	—
109	C₂×C₄×C₅₃	#99	1	2	yes	—
110	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#104	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	#88322	—	4	4	no	—
120	C₂×#1350	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	#327	1	2	yes	—
129	C₂²×C₅₃	#359	1	2	yes	—
130	C₂²×C₅₃	#362	1	2	yes	—
131	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	8	2	no	—
135	C₄×C₅₃	—	8	2	no	—
136	C₄×C₅₃	#327	1	2	yes	—
137	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
138	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
139	C₂×#467	D₅₃	1	1	yes	characteristic
140	C₂×C₂²⋊C₁₆	—	53	1	no	—
141	C₂×#341	—	53	1	no	—
142	#1350	—	53	4	no	—
143	C₂×C₅₃	—	4	1	no	—
144	C₂×C₅₃	—	4	1	no	—
145	C₂×C₅₃	#1350	1	4	yes	complemented
146	C₂×C₅₃	C₂×#327	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
149	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
150	C₂²×M₁₆	—	106	1	no	—
151	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
152	C₂×C₂²⋊Q₈	—	2	1	no	—
153	C₂²×C₁₆	—	106	1	no	—
154	#467	—	2	1	no	—
155	#467	—	2	1	no	—
156	#467	Dic₅₃	1	2	yes	—
157	#467	C₂×D₅₃	1	2	yes	—
158	C₂×#107	Dic₅₃	1	1	yes	characteristic
159	C₂×#103	—	2	1	no	—
160	C₂²⋊C₁₆	—	53	2	no	—
161	C₂²⋊C₁₆	—	106	1	no	—
162	#341	—	53	2	no	—
163	#341	—	106	1	no	—
164	C₅₃	C₂×#1350	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	—	2	1	no	—
166	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	—	4	2	no	—
169	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
170	C₂×M₁₆	—	106	4	no	—
171	C₂×M₁₆	—	106	2	no	—
172	C₂×M₁₆	—	212	2	no	—
173	C₂×M₁₆	—	212	1	no	—
174	C₂²×C₈	—	4	2	no	—
175	C₂²×C₈	—	212	1	no	—
176	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	2	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	—	4	1	no	—
185	C₂²⋊Q₈	—	4	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	4	1	no	—
188	C₂×C₂²⋊C₄	—	4	1	no	—
189	C₂×C₁₆	—	106	2	no	—
190	C₂×C₁₆	—	212	2	no	—
191	#107	—	2	1	no	—
192	#107	C₂×Dic₅₃	1	2	yes	—
193	#103	—	2	2	no	—
194	#103	—	2	2	no	—
195	C₂⁴	#11719923	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	#11719922	1	1	yes	characteristic
205	C₂²×C₄	#11719922	1	1	yes	characteristic
206	C₂²×C₄	#11719923	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	Q₁₆	—	4	2	no	—
211	Q₁₆	—	4	2	no	—
212	M₁₆	—	106	4	no	—
213	M₁₆	—	212	2	no	—
214	M₁₆	—	212	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	212	4	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₁₆	—	212	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	#13133138	1	1	yes	characteristic
238	C₂³	#13133138	1	1	yes	characteristic
239	C₂³	C₂×#11719923	1	1	yes	characteristic
240	Q₈	—	4	2	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	8	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	#13133132	1	2	yes	—
255	C₂×C₄	#13133138	1	1	yes	characteristic
256	C₂×C₄	#13133138	1	1	yes	characteristic
257	C₂×C₄	#13133144	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	212	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	#16810357	1	2	yes	—
273	C₂²	#16810430	1	2	yes	—
274	C₂²	#16810439	1	2	yes	—
275	C₂²	C₂×#13133138	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	#16810357	1	2	yes	—
281	C₄	C₂×#13133132	1	1	yes	characteristic
282	C₄	C₂×#13133144	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	1	no	—
285	C₂	#20272733	1	4	yes	direct product factor
286	C₂	C₂×#16810357	1	1	yes	characteristic
287	C₂	C₂×#16810430	1	1	yes	characteristic
288	C₂	C₂×#16810439	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#467	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#467	D₅₃
hypocenter	C₅₃	C₂×#1350
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
3. #20272733	C₂	direct (N × H)
145. C₂×C₅₃	#1350	semidirect
164. C₅₃	C₂×#1350	semidirect
285. C₂	#20272733	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×C₂²⋊#11719908	2
C₂×#16810388	2
#20272733	2
C₂×C₅₃×#467	2
C₂×#1350	53

Subgroups — C2×#20272733 #27294341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272733 #27294341