Subgroups — C₂×#20274210 #27366699

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810909	C₂	1	2	yes	maximal
3	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
4	#20274210	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810368	C₂	1	2	yes	maximal
6	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	—	2	1	no	—
8	C₂²×Dic₂₁₂	—	2	1	no	—
9	C₂×#13133217	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	#16811003	—	2	2	no	—
12	#16811003	C₂²	1	4	yes	—
13	#16810909	—	2	4	no	—
14	#16810909	C₂²	1	8	yes	—
15	#16810753	—	2	2	no	—
16	#16810753	C₂²	1	4	yes	—
17	C₂×#13133149	—	2	2	no	—
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133146	—	2	1	no	—
20	C₂×#13133142	—	2	2	no	—
21	C₂×#13133127	—	2	1	no	—
22	#16810368	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×#334	C₂²	1	4	yes	—
26	C₂²×Dic₁₀₆	—	2	2	no	—
27	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
28	C₂×Dic₂₁₂	—	2	4	no	—
29	C₂×Dic₂₁₂	—	2	4	no	—
30	C₂×Dic₂₁₂	—	4	4	no	—
31	C₂×Dic₂₁₂	—	4	4	no	—
32	C₂²×#88322	—	2	1	no	—
33	#13133217	—	2	2	no	—
34	#13133217	—	2	2	no	—
35	#13133217	—	2	2	no	—
36	#13133217	C₂³	1	4	yes	—
37	C₄×Dic₁₀₆	—	2	1	no	—
38	C₄×Dic₁₀₆	—	2	1	no	—
39	C₄×Dic₁₀₆	D₄	1	2	yes	—
40	C₄×Dic₁₀₆	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₅₃	—	2	2	no	—
42	C₂×C₄⋊Dic₅₃	—	2	2	no	—
43	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
44	C₂×#11719919	—	2	1	no	—
45	C₂×C₄×Dic₅₃	—	2	1	no	—
46	#13133149	—	2	8	no	—
47	C₄⋊#88322	—	2	1	no	—
48	C₄⋊#88322	C₂³	1	2	yes	—
49	#13133146	—	2	4	no	—
50	#13133142	—	2	8	no	—
51	#13133127	—	2	4	no	—
52	C₂²×C₈×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	—	2	2	no	—
57	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
58	C₄×C₈×C₅₃	D₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	C₂²×Dic₅₃	—	2	1	no	—
64	C₂²×Dic₅₃	—	2	1	no	—
65	C₂²×Dic₅₃	—	4	2	no	—
66	C₂×Dic₁₀₆	—	2	4	no	—
67	C₂×Dic₁₀₆	—	2	1	no	—
68	C₂×Dic₁₀₆	—	2	4	no	—
69	C₂×Dic₁₀₆	—	4	4	no	—
70	C₂×Dic₁₀₆	—	4	4	no	—
71	C₂×Dic₁₀₆	—	4	2	no	—
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	SD₁₆	1	4	yes	—
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	8	no	—
96	Dic₂₁₂	—	4	8	no	—
97	C₂³×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₅₃	#137	1	4	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	2	no	—
110	C₂×Dic₅₃	—	4	2	no	—
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	4	4	no	—
113	C₂×Dic₅₃	—	8	4	no	—
114	Dic₁₀₆	—	2	4	no	—
115	Dic₁₀₆	—	4	2	no	—
116	Dic₁₀₆	—	4	4	no	—
117	Dic₁₀₆	—	4	4	no	—
118	Dic₁₀₆	—	8	4	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	4	4	no	—
121	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
122	#88322	—	4	2	no	—
123	C₂×#1690	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	#460	1	2	yes	—
125	C₂²×C₅₃	#470	1	2	yes	—
126	C₂²×C₅₃	#473	1	2	yes	—
127	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	#460	1	2	yes	—
131	C₄×C₅₃	#470	1	2	yes	—
132	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
133	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₄×C₅₃	C₂×#137	1	2	yes	—
135	Dic₅₃	—	4	2	no	—
136	Dic₅₃	—	4	2	no	—
137	Dic₅₃	—	8	4	no	—
138	C₂×#494	—	53	1	no	—
139	C₂×#475	—	53	2	no	—
140	C₂×#446	—	53	1	no	—
141	#1690	—	53	8	no	—
142	C₂×#334	D₅₃	1	1	yes	characteristic
143	C₂×#332	—	53	2	no	—
144	C₂×C₅₃	#1690	1	4	yes	complemented
145	C₂×C₅₃	C₂×#460	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#470	1	1	yes	characteristic
147	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
148	C₂²×Q₁₆	—	106	1	no	—
149	C₂²×Q₁₆	—	106	1	no	—
150	C₂×C₄⋊Q₈	—	53	2	no	—
151	C₂×C₄×Q₈	—	53	1	no	—
152	#494	—	53	4	no	—
153	#494	—	106	2	no	—
154	#475	—	53	8	no	—
155	#475	—	106	4	no	—
156	#446	—	53	4	no	—
157	#446	—	106	2	no	—
158	C₂×#106	—	106	2	no	—
159	C₂×C₄⋊C₈	—	53	1	no	—
160	C₂×C₄⋊C₈	—	106	1	no	—
161	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
162	C₂×#103	—	106	2	no	—
163	C₂×#97	—	106	1	no	—
164	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
165	#334	C₂×D₅₃	1	4	yes	—
166	#332	—	53	8	no	—
167	C₅₃	C₂×#1690	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	—	53	1	no	—
169	C₂²×Q₈	—	106	2	no	—
170	C₂×Q₁₆	—	106	4	no	—
171	C₂×Q₁₆	—	106	4	no	—
172	C₂×Q₁₆	—	212	4	no	—
173	C₂×Q₁₆	—	212	4	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	#106	—	106	8	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	53	2	no	—
195	C₄⋊C₈	—	106	4	no	—
196	C₄⋊C₈	—	106	1	no	—
197	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
198	#103	—	106	8	no	—
199	#97	—	106	4	no	—
200	C₄×C₈	D₂₁₂	1	2	yes	—
201	C₄×C₈	C₂²×D₅₃	1	2	yes	—
202	C₂×Q₈	—	53	2	no	—
203	C₂×Q₈	—	106	1	no	—
204	C₂×Q₈	—	106	4	no	—
205	C₂×Q₈	—	106	4	no	—
206	C₂×Q₈	—	212	4	no	—
207	C₂×Q₈	—	212	4	no	—
208	C₂×Q₈	—	212	2	no	—
209	C₂²×C₄	—	106	1	no	—
210	C₂²×C₄	—	106	1	no	—
211	C₂²×C₄	—	212	2	no	—
212	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	#11719949	1	1	yes	characteristic
215	Q₁₆	—	212	8	no	—
216	Q₁₆	—	212	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	#11719913	1	4	yes	—
223	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
224	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
225	C₄⋊C₄	—	53	2	no	—
226	C₄⋊C₄	—	106	1	no	—
227	C₄⋊C₄	—	106	2	no	—
228	C₄⋊C₄	—	106	2	no	—
229	C₄⋊C₄	—	106	4	no	—
230	C₄⋊C₄	—	106	4	no	—
231	C₄⋊C₄	—	212	2	no	—
232	C₄⋊C₄	—	212	2	no	—
233	C₄²	—	106	2	no	—
234	C₄²	—	106	2	no	—
235	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
236	C₄²	D₄×D₅₃	1	1	yes	characteristic
237	C₄²	#11719949	1	1	yes	characteristic
238	C₄²	C₂³×D₅₃	1	1	yes	characteristic
239	C₂³	#13133192	1	1	yes	characteristic
240	Q₈	—	106	4	no	—
241	Q₈	—	424	4	no	—
242	Q₈	—	212	4	no	—
243	Q₈	—	212	4	no	—
244	Q₈	—	212	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	424	4	no	—
249	C₂×C₄	—	212	4	no	—
250	C₂×C₄	—	212	2	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	#13133192	1	1	yes	characteristic
253	C₂×C₄	#13133192	1	1	yes	characteristic
254	C₂×C₄	#13133192	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719913	1	2	yes	—
256	C₂×C₄	#13133245	1	2	yes	—
257	C₂×C₄	#13133246	1	4	yes	—
258	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
260	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	4	4	no	—
263	C₈	—	212	2	no	—
264	C₈	C₂×#11719913	1	4	yes	—
265	C₂²	#16810824	1	2	yes	—
266	C₂²	#16810882	1	2	yes	—
267	C₂²	#16810899	1	2	yes	—
268	C₂²	C₂×#13133192	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	424	4	no	—
272	C₄	—	212	2	no	—
273	C₄	—	212	2	no	—
274	C₄	#16810824	1	2	yes	—
275	C₄	#16810882	1	2	yes	—
276	C₄	C₂²×#11719913	1	1	yes	characteristic
277	C₄	C₂×#13133245	1	1	yes	characteristic
278	C₄	C₂×#13133246	1	2	yes	—
279	C₂	#20274210	1	4	yes	direct product factor
280	C₂	C₂×#16810824	1	1	yes	characteristic
281	C₂	C₂×#16810882	1	1	yes	characteristic
282	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×#334	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	C₂×#1690
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #20274210	C₂	direct (N × H)
144. C₂×C₅₃	#1690	semidirect
167. C₅₃	C₂×#1690	semidirect
279. C₂	#20274210	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×#16811003	2
C₂×#16810909	2
C₂×#16810753	2
#20274210	2
C₂×#16810368	2
C₂×C₅₃×#334	2
C₂×#1690	53

Subgroups — C2×#20274210 #27366699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274210 #27366699