Subgroups — C₂×#20272730 #27294337

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 308 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, complemented
2	C₂×#16810388	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272730	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#466	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	complemented
9	#16810388	—	2	1	no	—
10	#16810388	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#121×C₅₃	—	2	1	no	—
14	C₅₃×#466	—	2	1	no	—
15	C₅₃×#466	—	2	1	no	—
16	C₅₃×#466	C₄	1	2	yes	—
17	C₅₃×#466	C₂²	1	2	yes	—
18	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#102×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×D₄×C₅₃	—	2	1	no	—
28	C₂²×D₄×C₅₃	—	4	1	no	—
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂×D₈×C₅₃	—	2	1	no	—
31	C₂×D₈×C₅₃	—	4	2	no	—
32	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	—	4	2	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	#121×C₅₃	—	2	2	no	—
41	#121×C₅₃	—	2	2	no	—
42	#121×C₅₃	—	4	1	no	—
43	#121×C₅₃	—	4	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	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	#102×C₅₃	—	2	2	no	—
49	#102×C₅₃	—	2	2	no	—
50	C₂⁴×C₅₃	—	4	1	no	—
51	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	1	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	8	4	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	D₈×C₅₃	—	4	2	no	—
69	D₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	4	1	no	—
82	C₂²⋊C₄×C₅₃	—	4	2	no	—
83	C₂²⋊C₄×C₅₃	—	4	2	no	—
84	#11719917	—	2	4	no	—
85	#11719917	—	4	4	no	—
86	#11719917	—	4	2	no	—
87	C₂×#88322	—	4	4	no	—
88	C₂×#88322	—	4	2	no	—
89	#11719908	—	4	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	—	4	2	no	—
94	C₂³×C₅₃	—	8	2	no	—
95	C₂³×C₅₃	—	8	2	no	—
96	C₂³×C₅₃	—	8	2	no	—
97	C₂³×C₅₃	#102	1	1	yes	characteristic
98	C₂³×C₅₃	#102	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	D₄×C₅₃	—	4	2	no	—
101	D₄×C₅₃	—	4	2	no	—
102	D₄×C₅₃	—	8	4	no	—
103	D₄×C₅₃	—	8	4	no	—
104	D₄×C₅₃	—	8	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	8	2	no	—
114	C₂×C₄×C₅₃	#99	1	2	yes	—
115	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#104	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	—	2	1	no	—
123	C₈×C₅₃	—	2	1	no	—
124	#88322	—	4	4	no	—
125	C₂×#1349	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	4	4	no	—
133	C₂²×C₅₃	—	8	2	no	—
134	C₂²×C₅₃	—	8	2	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	—	16	4	no	—
137	C₂²×C₅₃	#327	1	2	yes	—
138	C₂²×C₅₃	#359	1	2	yes	—
139	C₂²×C₅₃	#361	1	2	yes	—
140	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	8	2	no	—
144	C₄×C₅₃	#327	1	2	yes	—
145	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
147	C₂×#466	D₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₁₆	—	53	1	no	—
149	C₂×#341	—	53	1	no	—
150	#1349	—	53	4	no	—
151	C₂×C₅₃	—	4	1	no	—
152	C₂×C₅₃	—	4	1	no	—
153	C₂×C₅₃	—	16	2	no	—
154	C₂×C₅₃	#1349	1	4	yes	complemented
155	C₂×C₅₃	C₂×#327	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
158	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
159	C₂²×M₁₆	—	106	1	no	—
160	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₂×#121	—	2	1	no	—
162	C₂²×C₁₆	—	106	1	no	—
163	#466	—	2	1	no	—
164	#466	—	2	1	no	—
165	#466	Dic₅₃	1	2	yes	—
166	#466	C₂×D₅₃	1	2	yes	—
167	C₂×#107	Dic₅₃	1	1	yes	characteristic
168	C₂×#102	—	2	1	no	—
169	C₂²⋊C₁₆	—	53	2	no	—
170	C₂²⋊C₁₆	—	106	1	no	—
171	#341	—	53	2	no	—
172	#341	—	106	1	no	—
173	C₅₃	C₂×#1349	1	1	yes	characteristic, complemented, Sylow
174	C₂²×D₄	—	2	1	no	—
175	C₂²×D₄	—	4	1	no	—
176	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
180	C₂×M₁₆	—	106	4	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	212	2	no	—
183	C₂×M₁₆	—	212	1	no	—
184	C₂²×C₈	—	4	2	no	—
185	C₂²×C₈	—	212	1	no	—
186	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
192	#121	—	2	2	no	—
193	#121	—	2	2	no	—
194	#121	—	4	1	no	—
195	#121	—	4	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	C₂×C₁₆	—	106	2	no	—
199	C₂×C₁₆	—	212	2	no	—
200	#107	—	2	1	no	—
201	#107	C₂×Dic₅₃	1	2	yes	—
202	#102	—	2	2	no	—
203	#102	—	2	2	no	—
204	C₂⁴	—	4	1	no	—
205	C₂⁴	#11719923	1	1	yes	characteristic
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	#11719922	1	1	yes	characteristic
217	C₂²×C₄	#11719922	1	1	yes	characteristic
218	C₂²×C₄	#11719923	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
222	D₈	—	4	2	no	—
223	D₈	—	4	2	no	—
224	M₁₆	—	106	4	no	—
225	M₁₆	—	212	2	no	—
226	M₁₆	—	212	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	212	4	no	—
234	C₂×C₈	—	212	2	no	—
235	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₁₆	—	212	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	8	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	#13133138	1	1	yes	characteristic
252	C₂³	#13133138	1	1	yes	characteristic
253	C₂³	C₂×#11719923	1	1	yes	characteristic
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	4	no	—
257	D₄	—	8	4	no	—
258	D₄	—	8	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	#13133132	1	2	yes	—
269	C₂×C₄	#13133138	1	1	yes	characteristic
270	C₂×C₄	#13133138	1	1	yes	characteristic
271	C₂×C₄	#13133144	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	212	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	16	4	no	—
290	C₂²	#16810357	1	2	yes	—
291	C₂²	#16810430	1	2	yes	—
292	C₂²	#16810436	1	2	yes	—
293	C₂²	C₂×#13133138	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	2	no	—
297	C₄	#16810357	1	2	yes	—
298	C₄	C₂×#13133132	1	1	yes	characteristic
299	C₄	C₂×#13133144	1	1	yes	characteristic
300	C₂	—	4	1	no	—
301	C₂	—	4	1	no	—
302	C₂	—	16	2	no	—
303	C₂	#20272730	1	4	yes	direct product factor
304	C₂	C₂×#16810357	1	1	yes	characteristic
305	C₂	C₂×#16810430	1	1	yes	characteristic
306	C₂	C₂×#16810436	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#466	C₂
socle	C₂³×C₅₃	#102
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#466	D₅₃
hypocenter	C₅₃	C₂×#1349
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊#11719908	C₂	semidirect
2. C₂×#16810388	C₂	semidirect
3. #20272730	C₂	direct (N × H)
8. C₂²⋊#11719908	C₂²	semidirect
10. #16810388	C₂²	semidirect
154. C₂×C₅₃	#1349	semidirect
173. C₅₃	C₂×#1349	semidirect
303. C₂	#20272730	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
#20272730	2
C₂×C₅₃×#466	2
C₂×#1349	53

Subgroups — C2×#20272730 #27294337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272730 #27294337