Subgroups — C₂×#20274110 #27366449

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810834	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
4	#20274110	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810370	C₂	1	2	yes	maximal
6	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	—	2	2	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#16811028	—	2	2	no	—
11	#16811028	C₂²	1	4	yes	—
12	#16810834	—	2	4	no	—
13	#16810834	C₂²	1	8	yes	—
14	C₈×Dic₁₀₆	—	2	2	no	—
15	C₈×Dic₁₀₆	C₂²	1	4	yes	—
16	C₂×#13133151	—	2	1	no	—
17	C₂×#13133151	—	2	1	no	—
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133146	—	2	1	no	—
20	C₂×#13133142	—	2	2	no	—
21	C₂×C₈×Dic₅₃	—	2	1	no	—
22	#16810370	C₂²	1	8	yes	—
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#335	C₂²	1	4	yes	—
26	C₂²×Dic₁₀₆	—	2	2	no	—
27	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
28	C₂×Dic₂₁₂	—	2	8	no	—
29	C₂×Dic₂₁₂	—	4	8	no	—
30	C₂²×#88322	—	2	1	no	—
31	#13133217	—	2	2	no	—
32	#13133217	—	2	2	no	—
33	#13133217	—	2	2	no	—
34	#13133217	C₂³	1	4	yes	—
35	C₄×Dic₁₀₆	—	2	1	no	—
36	C₄×Dic₁₀₆	—	2	1	no	—
37	C₄×Dic₁₀₆	D₄	1	2	yes	—
38	C₄×Dic₁₀₆	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₅₃	—	2	2	no	—
40	C₂×C₄⋊Dic₅₃	—	2	2	no	—
41	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	C₂×#11719919	—	2	1	no	—
43	C₂×C₄×Dic₅₃	—	2	1	no	—
44	#13133151	—	2	4	no	—
45	#13133151	—	2	4	no	—
46	C₄⋊#88322	—	2	1	no	—
47	C₄⋊#88322	C₂³	1	2	yes	—
48	#13133146	—	2	4	no	—
49	#13133142	—	2	8	no	—
50	C₈×Dic₅₃	—	2	4	no	—
51	C₂²×C₈×C₅₃	—	2	2	no	—
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	2	no	—
56	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
57	C₄×C₈×C₅₃	D₄	1	2	yes	—
58	C₄×C₈×C₅₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	2	1	no	—
63	C₂²×Dic₅₃	—	2	1	no	—
64	C₂²×Dic₅₃	—	4	2	no	—
65	C₂×Dic₁₀₆	—	2	1	no	—
66	C₂×Dic₁₀₆	—	2	2	no	—
67	C₂×Dic₁₀₆	—	2	4	no	—
68	C₂×Dic₁₀₆	—	4	4	no	—
69	C₂×Dic₁₀₆	—	4	4	no	—
70	C₂×Dic₁₀₆	—	4	2	no	—
71	C₂×Dic₁₀₆	D₈	1	4	yes	—
72	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	4	4	no	—
76	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
77	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	#41	1	1	yes	characteristic
82	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₄⋊Dic₅₃	—	2	1	no	—
84	C₄⋊Dic₅₃	—	2	4	no	—
85	C₄⋊Dic₅₃	—	2	4	no	—
86	C₄⋊Dic₅₃	—	4	2	no	—
87	C₄⋊Dic₅₃	—	4	2	no	—
88	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
89	#11719919	—	2	2	no	—
90	#11719919	—	2	2	no	—
91	C₄×Dic₅₃	—	2	2	no	—
92	C₄×Dic₅₃	—	2	2	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	4	2	no	—
95	Dic₂₁₂	—	4	16	no	—
96	C₂³×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₅₃	#135	1	2	yes	—
104	C₂×C₄×C₅₃	#137	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₅₃	—	2	2	no	—
109	C₂×Dic₅₃	—	2	4	no	—
110	C₂×Dic₅₃	—	2	2	no	—
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	4	4	no	—
113	C₂×Dic₅₃	—	8	4	no	—
114	Dic₁₀₆	—	2	2	no	—
115	Dic₁₀₆	—	2	4	no	—
116	Dic₁₀₆	—	4	4	no	—
117	Dic₁₀₆	—	4	4	no	—
118	Dic₁₀₆	—	8	4	no	—
119	Dic₁₀₆	C₂×D₈	1	4	yes	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	—	4	4	no	—
122	C₈×C₅₃	C₂×Q₁₆	1	4	yes	complemented
123	#88322	—	4	2	no	—
124	C₂×#1674	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	#462	1	2	yes	—
126	C₂²×C₅₃	#466	1	2	yes	—
127	C₂²×C₅₃	#473	1	2	yes	—
128	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	#462	1	2	yes	—
132	C₄×C₅₃	#466	1	2	yes	—
133	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
135	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
136	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
137	Dic₅₃	—	2	4	no	—
138	Dic₅₃	—	4	2	no	—
139	Dic₅₃	—	8	4	no	—
140	C₂×#500	—	53	1	no	—
141	C₂×#462	—	53	2	no	—
142	C₂×C₈×Q₈	—	53	1	no	—
143	#1674	—	53	8	no	—
144	C₂×#335	D₅₃	1	1	yes	characteristic
145	C₂×#332	—	53	2	no	—
146	C₂×C₅₃	#1674	1	4	yes	complemented
147	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
149	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
150	C₂²×Q₁₆	—	106	2	no	—
151	C₂×C₄⋊Q₈	—	53	2	no	—
152	C₂×C₄×Q₈	—	53	1	no	—
153	#500	—	53	4	no	—
154	#500	—	106	2	no	—
155	#462	—	53	8	no	—
156	#462	—	106	4	no	—
157	C₈×Q₈	—	53	4	no	—
158	C₈×Q₈	—	106	2	no	—
159	C₂×#107	—	106	1	no	—
160	C₂×#107	—	106	1	no	—
161	C₂×C₄⋊C₈	—	53	1	no	—
162	C₂×C₄⋊C₈	—	106	1	no	—
163	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
164	C₂×#103	—	106	2	no	—
165	C₂×C₄×C₈	—	106	1	no	—
166	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
167	#335	C₂×D₅₃	1	4	yes	—
168	#332	—	53	8	no	—
169	C₅₃	C₂×#1674	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	53	1	no	—
171	C₂²×Q₈	—	106	2	no	—
172	C₂×Q₁₆	—	106	8	no	—
173	C₂×Q₁₆	—	212	8	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	106	1	no	—
176	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
177	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
178	C₄⋊Q₈	—	53	4	no	—
179	C₄⋊Q₈	—	106	2	no	—
180	C₄⋊Q₈	—	106	2	no	—
181	C₄⋊Q₈	—	106	2	no	—
182	C₄×Q₈	—	53	2	no	—
183	C₄×Q₈	—	53	2	no	—
184	C₄×Q₈	—	106	1	no	—
185	C₄×Q₈	—	106	1	no	—
186	C₂×C₄⋊C₄	—	53	1	no	—
187	C₂×C₄⋊C₄	—	106	1	no	—
188	C₂×C₄⋊C₄	—	106	2	no	—
189	C₂×C₄⋊C₄	—	106	2	no	—
190	C₂×C₄²	—	106	1	no	—
191	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
192	#107	—	106	4	no	—
193	#107	—	106	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	53	2	no	—
196	C₄⋊C₈	—	106	4	no	—
197	C₄⋊C₈	—	106	1	no	—
198	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
199	#103	—	106	8	no	—
200	C₄×C₈	—	106	4	no	—
201	C₄×C₈	D₂₁₂	1	2	yes	—
202	C₄×C₈	C₂²×D₅₃	1	2	yes	—
203	C₂×Q₈	—	53	4	no	—
204	C₂×Q₈	—	53	2	no	—
205	C₂×Q₈	—	106	1	no	—
206	C₂×Q₈	—	106	2	no	—
207	C₂×Q₈	—	106	4	no	—
208	C₂×Q₈	—	212	4	no	—
209	C₂×Q₈	—	212	4	no	—
210	C₂×Q₈	—	212	2	no	—
211	C₂²×C₄	—	106	1	no	—
212	C₂²×C₄	—	106	1	no	—
213	C₂²×C₄	—	212	2	no	—
214	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	#11719949	1	1	yes	characteristic
217	Q₁₆	—	212	16	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	Dic₂₁₂	1	4	yes	complemented
224	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
225	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
226	C₄⋊C₄	—	53	2	no	—
227	C₄⋊C₄	—	106	1	no	—
228	C₄⋊C₄	—	106	2	no	—
229	C₄⋊C₄	—	106	2	no	—
230	C₄⋊C₄	—	106	4	no	—
231	C₄⋊C₄	—	106	4	no	—
232	C₄⋊C₄	—	212	2	no	—
233	C₄⋊C₄	—	212	2	no	—
234	C₄²	—	106	2	no	—
235	C₄²	—	106	2	no	—
236	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
237	C₄²	D₄×D₅₃	1	1	yes	characteristic
238	C₄²	#11719949	1	1	yes	characteristic
239	C₄²	C₂³×D₅₃	1	1	yes	characteristic
240	C₂³	#13133192	1	1	yes	characteristic
241	Q₈	—	53	4	no	—
242	Q₈	—	106	4	no	—
243	Q₈	—	106	2	no	—
244	Q₈	—	424	4	no	—
245	Q₈	—	212	4	no	—
246	Q₈	—	212	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	106	4	no	—
249	C₂×C₄	—	106	2	no	—
250	C₂×C₄	—	106	2	no	—
251	C₂×C₄	—	424	4	no	—
252	C₂×C₄	—	212	4	no	—
253	C₂×C₄	—	212	2	no	—
254	C₂×C₄	#13133192	1	1	yes	characteristic
255	C₂×C₄	#13133192	1	1	yes	characteristic
256	C₂×C₄	#13133192	1	1	yes	characteristic
257	C₂×C₄	D₈×D₅₃	1	2	yes	—
258	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
259	C₂×C₄	#13133240	1	2	yes	—
260	C₂×C₄	#13133250	1	2	yes	—
261	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
263	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	4	no	—
266	C₈	—	212	2	no	—
267	C₈	C₂×Dic₂₁₂	1	4	yes	complemented
268	C₂²	#16810834	1	2	yes	—
269	C₂²	#16810860	1	2	yes	—
270	C₂²	#16810899	1	2	yes	—
271	C₂²	C₂×#13133192	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	106	4	no	—
275	C₄	—	424	4	no	—
276	C₄	—	212	2	no	—
277	C₄	#16810834	1	2	yes	—
278	C₄	#16810860	1	2	yes	—
279	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
280	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
281	C₄	C₂×#13133240	1	1	yes	characteristic
282	C₄	C₂×#13133250	1	1	yes	characteristic
283	C₂	#20274110	1	4	yes	direct product factor
284	C₂	C₂×#16810834	1	1	yes	characteristic
285	C₂	C₂×#16810860	1	1	yes	characteristic
286	C₂	C₂×#16810899	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	C₂×#1674
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #20274110	C₂	direct (N × H)
76. C₂×C₈×C₅₃	Q₁₆	semidirect
122. C₈×C₅₃	C₂×Q₁₆	semidirect
146. C₂×C₅₃	#1674	semidirect
169. C₅₃	C₂×#1674	semidirect
223. C₂×C₈	Dic₂₁₂	semidirect
267. C₈	C₂×Dic₂₁₂	semidirect
283. C₂	#20274110	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×#16811028	2
C₂×#16810834	2
C₂×C₈×Dic₁₀₆	2
#20274110	2
C₂×#16810370	2
C₂×C₅₃×#335	2
C₂×#1674	53

Subgroups — C2×#20274110 #27366449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274110 #27366449