Subgroups — C₂×#20273486 #27353148

Order 13568.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20273486	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810374	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	—	2	1	no	—
7	C₂³×#88322	—	2	1	no	—
8	#16810647	—	2	2	no	—
9	#16810647	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	—	2	1	no	—
11	C₂×C₄⋊#88322	—	2	2	no	—
12	C₂×C₄⋊#88322	—	2	1	no	—
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×C₄⋊#88322	C₂²	1	4	yes	—
16	#16810374	C₂²	1	16	yes	complemented
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#431	C₄	1	4	yes	—
19	C₅₃×#431	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
21	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	4	4	no	—
25	C₂²×#88322	—	2	4	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₂²×#88322	—	4	4	no	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	—	2	4	no	—
31	C₄⋊#88322	—	2	4	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	C₂³	1	4	yes	—
34	C₄⋊#88322	C₂³	1	4	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
47	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
48	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₅₃	C₂³	1	4	yes	—
51	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
61	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₅₃	—	2	8	no	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
78	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
80	#11719917	—	4	8	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	4	8	no	—
84	C₂×#88322	—	4	4	no	—
85	C₂×#88322	—	4	4	no	—
86	C₂³×C₅₃	—	2	2	no	—
87	C₂³×C₅₃	—	2	2	no	—
88	C₂³×C₅₃	#106	1	4	yes	—
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	#106	1	8	yes	—
99	C₂×C₄×C₅₃	#106	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#131	1	4	yes	—
107	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
108	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
114	C₈×C₅₃	—	2	8	no	—
115	#88322	—	4	4	no	—
116	#88322	—	4	4	no	—
117	C₂×#1570	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	8	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	#424	1	2	yes	—
122	C₂²×C₅₃	C₂×#106	1	4	yes	—
123	C₂²×C₅₃	C₂×#106	1	2	yes	—
124	C₂²×C₅₃	#430	1	2	yes	—
125	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	#424	1	2	yes	—
131	C₄×C₅₃	C₂×#106	1	2	yes	—
132	C₄×C₅₃	C₂×#106	1	4	yes	—
133	C₄×C₅₃	C₂×#131	1	2	yes	—
134	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
135	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
136	C₂×#431	D₅₃	1	1	yes	characteristic
137	C₂×#423	—	53	1	no	—
138	C₂²×C₄⋊C₈	—	53	1	no	—
139	#1570	—	53	8	no	—
140	C₂×#334	—	53	4	no	—
141	C₂×C₅₃	—	2	4	no	—
142	C₂×C₅₃	#1570	1	4	yes	complemented
143	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
144	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
145	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
146	C₂²×M₁₆	—	106	1	no	—
147	C₂³×C₈	—	106	1	no	—
148	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
149	#431	Dic₅₃	1	4	yes	—
150	#431	C₂×D₅₃	1	4	yes	—
151	#423	—	53	4	no	—
152	#423	—	106	2	no	—
153	C₂×C₄⋊C₈	—	53	2	no	—
154	C₂×C₄⋊C₈	—	53	2	no	—
155	C₂×C₄⋊C₈	—	53	4	no	—
156	C₂×C₄⋊C₈	—	106	2	no	—
157	C₂×C₄⋊C₈	—	106	1	no	—
158	C₂×C₄⋊C₈	—	106	1	no	—
159	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
160	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
161	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
162	#334	—	53	16	no	—
163	C₅₃	C₂×#1570	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
165	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂×M₁₆	—	106	4	no	—
168	C₂×M₁₆	—	212	4	no	—
169	C₂²×C₈	—	106	4	no	—
170	C₂²×C₈	—	106	2	no	—
171	C₂²×C₈	—	106	2	no	—
172	C₂²×C₈	—	212	4	no	—
173	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
174	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
175	C₂×C₄²	Dic₁₀₆	1	2	yes	—
176	C₂×C₄²	D₂₁₂	1	2	yes	—
177	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
180	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
181	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
182	C₄⋊C₈	—	53	4	no	—
183	C₄⋊C₈	—	53	4	no	—
184	C₄⋊C₈	—	106	2	no	—
185	C₄⋊C₈	—	106	4	no	—
186	C₄⋊C₈	—	106	2	no	—
187	C₄⋊C₈	—	106	4	no	—
188	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
189	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
190	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
191	C₄×C₈	C₂²×D₅₃	1	4	yes	—
192	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	#11719913	1	4	yes	—
196	C₂²×C₄	#11719913	1	4	yes	—
197	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
199	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
201	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
203	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
205	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
206	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
209	M₁₆	—	212	8	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	212	8	no	—
214	C₂×C₈	—	212	4	no	—
215	C₂×C₈	—	212	4	no	—
216	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
217	C₄²	C₄⋊Dic₅₃	1	2	yes	—
218	C₄²	C₄⋊Dic₅₃	1	2	yes	—
219	C₄²	C₄⋊Dic₅₃	1	2	yes	—
220	C₄²	C₂×Dic₁₀₆	1	2	yes	—
221	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
222	C₄²	C₂×D₂₁₂	1	2	yes	—
223	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
224	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
225	C₄²	C₂²×Dic₅₃	1	2	yes	—
226	C₄²	C₂³×D₅₃	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#13133149	1	4	yes	—
230	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
231	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#13133149	1	8	yes	—
240	C₂×C₄	#13133149	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
246	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
247	C₂×C₄	#13133224	1	4	yes	—
248	C₂×C₄	C₂×#11719913	1	4	yes	—
249	C₂×C₄	C₂×#11719913	1	2	yes	—
250	C₂×C₄	C₂×#11719913	1	2	yes	—
251	C₂×C₄	C₂×#11719913	1	4	yes	—
252	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
254	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	212	4	no	—
257	C₈	—	212	4	no	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	#16810650	1	2	yes	—
262	C₂²	C₂×#13133149	1	4	yes	—
263	C₂²	C₂×#13133149	1	2	yes	—
264	C₂²	#16810672	1	2	yes	—
265	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	#16810650	1	2	yes	—
271	C₄	C₂×#13133149	1	4	yes	—
272	C₄	C₂×#13133149	1	2	yes	—
273	C₄	C₂×#13133224	1	2	yes	—
274	C₄	C₂²×#11719913	1	1	yes	characteristic
275	C₄	C₂²×#11719913	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	#20273486	1	4	yes	direct product factor
278	C₂	C₂×#16810650	1	1	yes	characteristic
279	C₂	C₂²×#13133149	1	1	yes	characteristic
280	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	C₂×#1570
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273486	C₂	direct (N × H)
4. C₂×#16810374	C₂	semidirect
16. #16810374	C₂²	semidirect
142. C₂×C₅₃	#1570	semidirect
163. C₅₃	C₂×#1570	semidirect
277. C₂	#20273486	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×#16810647	2
C₂²×C₄⋊#88322	2
#20273486	2
C₂×#16810374	2
C₂×C₅₃×#431	2
C₂×#1570	53

Subgroups — C2×#20273486 #27353148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273486 #27353148