Subgroups — C₂×#20274120 #27366455

Order 13568.

638 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×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810919	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
4	#20274120	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810371	C₂	1	2	yes	maximal
6	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	#16811028	—	2	2	no	—
10	#16811028	C₂²	1	4	yes	—
11	#16810919	—	2	4	no	—
12	#16810919	C₂²	1	8	yes	—
13	C₈×Dic₁₀₆	—	2	2	no	—
14	C₈×Dic₁₀₆	C₂²	1	4	yes	—
15	C₂×#13133151	—	2	2	no	—
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	#16810371	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	C₂²×#88322	—	2	1	no	—
29	#13133217	—	2	2	no	—
30	#13133217	—	2	2	no	—
31	#13133217	—	2	2	no	—
32	#13133217	C₂³	1	4	yes	—
33	C₄×Dic₁₀₆	—	2	1	no	—
34	C₄×Dic₁₀₆	—	2	1	no	—
35	C₄×Dic₁₀₆	D₄	1	2	yes	—
36	C₄×Dic₁₀₆	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₅₃	—	2	2	no	—
38	C₂×C₄⋊Dic₅₃	—	2	2	no	—
39	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
40	C₂×#11719919	—	2	1	no	—
41	C₂×C₄×Dic₅₃	—	2	1	no	—
42	#13133151	—	2	4	no	—
43	#13133151	—	2	4	no	—
44	#13133151	—	2	8	no	—
45	C₄⋊#88322	—	2	1	no	—
46	C₄⋊#88322	C₂³	1	2	yes	—
47	#13133146	—	2	4	no	—
48	#13133142	—	2	8	no	—
49	C₈×Dic₅₃	—	2	4	no	—
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₃	—	2	2	no	—
55	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
56	C₄×C₈×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	2	1	no	—
62	C₂²×Dic₅₃	—	2	1	no	—
63	C₂²×Dic₅₃	—	4	2	no	—
64	C₂×Dic₁₀₆	—	2	1	no	—
65	C₂×Dic₁₀₆	—	2	2	no	—
66	C₂×Dic₁₀₆	—	2	4	no	—
67	C₂×Dic₁₀₆	—	4	2	no	—
68	C₂×Dic₁₀₆	—	8	4	no	—
69	C₂×Dic₁₀₆	Q₁₆	1	4	yes	—
70	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	#41	1	1	yes	characteristic
80	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₄⋊Dic₅₃	—	2	1	no	—
82	C₄⋊Dic₅₃	—	2	4	no	—
83	C₄⋊Dic₅₃	—	2	4	no	—
84	C₄⋊Dic₅₃	—	4	2	no	—
85	C₄⋊Dic₅₃	—	4	2	no	—
86	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
87	#11719919	—	2	2	no	—
88	#11719919	—	2	2	no	—
89	C₄×Dic₅₃	—	2	2	no	—
90	C₄×Dic₅₃	—	2	2	no	—
91	C₂×#88322	—	2	2	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂³×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₅₃	#135	1	2	yes	—
101	C₂×C₄×C₅₃	#137	1	2	yes	—
102	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
105	C₂×Dic₅₃	—	2	2	no	—
106	C₂×Dic₅₃	—	2	4	no	—
107	C₂×Dic₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	C₂×Dic₅₃	—	4	4	no	—
110	C₂×Dic₅₃	—	8	4	no	—
111	Dic₁₀₆	—	2	2	no	—
112	Dic₁₀₆	—	2	4	no	—
113	Dic₁₀₆	—	8	4	no	—
114	Dic₁₀₆	—	8	4	no	—
115	Dic₁₀₆	C₂×Q₁₆	1	4	yes	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	—	4	4	no	—
118	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
119	#88322	—	4	2	no	—
120	C₂×#1676	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	#462	1	2	yes	—
122	C₂²×C₅₃	#467	1	2	yes	—
123	C₂²×C₅₃	#472	1	2	yes	—
124	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	#462	1	2	yes	—
128	C₄×C₅₃	#467	1	2	yes	—
129	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
130	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
131	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
132	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
133	Dic₅₃	—	2	4	no	—
134	Dic₅₃	—	4	2	no	—
135	Dic₅₃	—	8	4	no	—
136	C₂×#500	—	53	1	no	—
137	C₂×#477	—	53	2	no	—
138	C₂×C₈×Q₈	—	53	1	no	—
139	#1676	—	53	8	no	—
140	C₂×#335	D₅₃	1	1	yes	characteristic
141	C₂×#333	—	53	2	no	—
142	C₂×C₅₃	#1676	1	4	yes	complemented
143	C₂×C₅₃	C₂×#462	1	1	yes	characteristic
144	C₂×C₅₃	C₂×#467	1	1	yes	characteristic
145	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
146	C₂×C₄⋊Q₈	—	53	2	no	—
147	C₂×C₄×Q₈	—	53	1	no	—
148	#500	—	53	4	no	—
149	#500	—	106	2	no	—
150	#477	—	53	8	no	—
151	#477	—	106	4	no	—
152	C₈×Q₈	—	53	4	no	—
153	C₈×Q₈	—	106	2	no	—
154	C₂×#107	—	106	2	no	—
155	C₂×#107	—	106	1	no	—
156	C₂×#107	—	106	1	no	—
157	C₂×C₄⋊C₈	—	53	1	no	—
158	C₂×C₄⋊C₈	—	106	1	no	—
159	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
160	C₂×#103	—	106	2	no	—
161	C₂×C₄×C₈	—	106	1	no	—
162	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
163	#335	C₂×D₅₃	1	4	yes	—
164	#333	—	53	8	no	—
165	C₅₃	C₂×#1676	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	53	1	no	—
167	C₂²×Q₈	—	106	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	106	1	no	—
170	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
171	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
172	C₄⋊Q₈	—	53	4	no	—
173	C₄⋊Q₈	—	106	2	no	—
174	C₄⋊Q₈	—	106	2	no	—
175	C₄⋊Q₈	—	106	2	no	—
176	C₄×Q₈	—	53	2	no	—
177	C₄×Q₈	—	53	2	no	—
178	C₄×Q₈	—	106	1	no	—
179	C₄×Q₈	—	106	1	no	—
180	C₂×C₄⋊C₄	—	53	1	no	—
181	C₂×C₄⋊C₄	—	106	1	no	—
182	C₂×C₄⋊C₄	—	106	2	no	—
183	C₂×C₄⋊C₄	—	106	2	no	—
184	C₂×C₄²	—	106	1	no	—
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	#107	—	106	8	no	—
187	#107	—	106	4	no	—
188	#107	—	106	4	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	53	2	no	—
191	C₄⋊C₈	—	106	4	no	—
192	C₄⋊C₈	—	106	1	no	—
193	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
194	#103	—	106	8	no	—
195	C₄×C₈	—	106	4	no	—
196	C₄×C₈	D₂₁₂	1	2	yes	—
197	C₄×C₈	C₂²×D₅₃	1	2	yes	—
198	C₂×Q₈	—	53	4	no	—
199	C₂×Q₈	—	53	2	no	—
200	C₂×Q₈	—	106	1	no	—
201	C₂×Q₈	—	106	2	no	—
202	C₂×Q₈	—	106	4	no	—
203	C₂×Q₈	—	424	4	no	—
204	C₂×Q₈	—	212	2	no	—
205	C₂²×C₄	—	106	1	no	—
206	C₂²×C₄	—	106	1	no	—
207	C₂²×C₄	—	212	2	no	—
208	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
209	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
210	C₂²×C₄	#11719949	1	1	yes	characteristic
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	212	2	no	—
216	C₂×C₈	Dic₂₁₂	1	4	yes	—
217	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
218	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
219	C₄⋊C₄	—	53	2	no	—
220	C₄⋊C₄	—	106	1	no	—
221	C₄⋊C₄	—	106	2	no	—
222	C₄⋊C₄	—	106	2	no	—
223	C₄⋊C₄	—	106	4	no	—
224	C₄⋊C₄	—	106	4	no	—
225	C₄⋊C₄	—	212	2	no	—
226	C₄⋊C₄	—	212	2	no	—
227	C₄²	—	106	2	no	—
228	C₄²	—	106	2	no	—
229	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
230	C₄²	D₄×D₅₃	1	1	yes	characteristic
231	C₄²	#11719949	1	1	yes	characteristic
232	C₄²	C₂³×D₅₃	1	1	yes	characteristic
233	C₂³	#13133192	1	1	yes	characteristic
234	Q₈	—	53	4	no	—
235	Q₈	—	106	4	no	—
236	Q₈	—	106	2	no	—
237	Q₈	—	424	4	no	—
238	Q₈	—	424	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	106	2	no	—
241	C₂×C₄	—	106	4	no	—
242	C₂×C₄	—	106	2	no	—
243	C₂×C₄	—	424	4	no	—
244	C₂×C₄	—	212	4	no	—
245	C₂×C₄	—	212	2	no	—
246	C₂×C₄	#13133192	1	1	yes	characteristic
247	C₂×C₄	#13133192	1	1	yes	characteristic
248	C₂×C₄	#13133192	1	1	yes	characteristic
249	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
250	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
251	C₂×C₄	#13133236	1	2	yes	—
252	C₂×C₄	#13133250	1	2	yes	—
253	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	4	no	—
258	C₈	—	212	2	no	—
259	C₈	C₂×Dic₂₁₂	1	4	yes	—
260	C₂²	#16810834	1	2	yes	—
261	C₂²	#16810865	1	2	yes	—
262	C₂²	#16810894	1	2	yes	—
263	C₂²	C₂×#13133192	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	106	4	no	—
267	C₄	—	424	4	no	—
268	C₄	—	212	2	no	—
269	C₄	#16810834	1	2	yes	—
270	C₄	#16810865	1	2	yes	—
271	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
272	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
273	C₄	C₂×#13133236	1	1	yes	characteristic
274	C₄	C₂×#13133250	1	1	yes	characteristic
275	C₂	#20274120	1	4	yes	direct product factor
276	C₂	C₂×#16810834	1	1	yes	characteristic
277	C₂	C₂×#16810865	1	1	yes	characteristic
278	C₂	C₂×#16810894	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×#1676
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. #20274120	C₂	direct (N × H)
142. C₂×C₅₃	#1676	semidirect
165. C₅₃	C₂×#1676	semidirect
275. C₂	#20274120	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×#16810919	2
C₂×C₈×Dic₁₀₆	2
#20274120	2
C₂×#16810371	2
C₂×C₅₃×#335	2
C₂×#1676	53

Subgroups — C2×#20274120 #27366455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274120 #27366455