Subgroups — C₂×#20276883 #27538819

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810741	C₂	1	1	yes	maximal, characteristic
2	#20276883	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810445	C₂	1	2	yes	maximal, complemented
4	C₂×#16810444	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810432	C₂	1	2	yes	maximal
6	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133239	C₂²	1	1	yes	characteristic
8	C₂×#13133223	—	2	1	no	—
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810741	—	2	2	no	—
11	#16810741	C₂²	1	4	yes	—
12	C₂×#13133153	C₂²	1	1	yes	characteristic
13	C₂×#13133145	—	2	1	no	—
14	#16810445	C₂²	1	8	yes	complemented
15	#16810444	C₂²	1	4	yes	complemented
16	#16810432	C₂²	1	8	yes	—
17	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
18	C₅₃×#498	—	2	2	no	—
19	C₅₃×#498	C₂²	1	4	yes	complemented
20	C₂×#106×C₅₃	—	2	1	no	—
21	C₂×#106×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719945	—	2	1	no	—
24	#13133239	—	2	1	no	—
25	#13133239	—	2	1	no	—
26	#13133239	Q₈	1	2	yes	complemented
27	#13133239	C₂³	1	2	yes	—
28	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic, complemented
29	C₂×#11719913	—	2	1	no	—
30	C₂×D₄₂₄	Q₈	1	1	yes	characteristic, complemented
31	#13133223	—	2	4	no	—
32	#13133223	—	4	2	no	—
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719912	—	4	1	no	—
35	C₂×C₈×D₅₃	—	4	1	no	—
36	#13133157	—	2	2	no	—
37	#13133157	C₂³	1	4	yes	—
38	C₂×#11719908	—	2	2	no	—
39	#13133153	—	2	1	no	—
40	#13133153	C₂³	1	2	yes	—
41	#13133145	—	2	4	no	—
42	C₂²×Q₈×C₅₃	—	4	1	no	—
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₅₃	—	2	4	no	—
46	C₄⋊Q₈×C₅₃	—	4	2	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	#106×C₅₃	—	2	2	no	—
51	#106×C₅₃	—	2	4	no	—
52	#106×C₅₃	C₂³	1	4	yes	—
53	C₄×C₈×C₅₃	D₄	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂×Q₈×C₅₃	—	4	4	no	—
56	C₂×Q₈×C₅₃	—	8	4	no	—
57	C₂²×C₄×C₅₃	—	2	1	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	—	4	1	no	—
61	#11719945	—	2	2	no	—
62	#11719945	—	2	2	no	—
63	#11719945	—	4	1	no	—
64	#11719945	—	4	1	no	—
65	C₂×D₂₁₂	—	2	1	no	—
66	C₂×C₄×D₅₃	—	4	1	no	—
67	C₂×Dic₁₀₆	—	2	1	no	—
68	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
69	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
70	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	—	2	1	no	—
82	C₄²×C₅₃	C₂×D₄	1	2	yes	—
83	#11719917	—	2	2	no	—
84	#11719917	—	4	1	no	—
85	C₂×#88322	—	4	1	no	—
86	Dic₂₁₂	—	2	1	no	—
87	Dic₂₁₂	C₂×Q₈	1	2	yes	complemented
88	D₄₂₄	—	2	1	no	—
89	D₄₂₄	C₂×Q₈	1	2	yes	complemented
90	#11719913	—	2	2	no	—
91	#11719913	—	2	2	no	—
92	#11719912	—	4	4	no	—
93	C₈×D₅₃	—	4	4	no	—
94	#11719908	—	2	4	no	—
95	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂²×D₅₃	—	4	1	no	—
97	Q₈×C₅₃	—	8	8	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	8	4	no	—
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂²⋊D₅₃	—	4	2	no	—
109	C₂²⋊D₅₃	—	4	2	no	—
110	C₂×Dic₅₃	—	4	1	no	—
111	D₂₁₂	—	2	2	no	—
112	D₂₁₂	—	4	1	no	—
113	C₄×D₅₃	—	4	2	no	—
114	C₄×D₅₃	—	4	2	no	—
115	Dic₁₀₆	—	2	2	no	—
116	Dic₁₀₆	—	4	1	no	—
117	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
120	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
121	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
123	#88322	—	4	2	no	—
124	C₂×#2230	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	#476	1	1	yes	characteristic
127	C₂²×C₅₃	#476	1	1	yes	characteristic
128	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×D₅₃	—	4	2	no	—
130	C₂×D₅₃	—	8	2	no	—
131	C₄×C₅₃	—	2	4	no	—
132	C₄×C₅₃	—	8	4	no	—
133	C₄×C₅₃	#476	1	1	yes	characteristic
134	C₄×C₅₃	#476	1	1	yes	characteristic
135	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
137	Dic₅₃	—	4	2	no	—
138	C₂×#498	D₅₃	1	1	yes	characteristic, complemented
139	C₂×#443	—	53	1	no	—
140	#2230	—	53	8	no	—
141	C₂×#365	—	53	2	no	—
142	C₂×#364	—	53	1	no	—
143	C₂×#360	—	53	2	no	—
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	#2230	1	2	yes	complemented
147	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
148	D₅₃	—	8	2	no	—
149	C₂×#135	—	53	1	no	—
150	C₂×#131	—	106	1	no	—
151	C₂×C₄⋊Q₈	—	2	1	no	—
152	C₂×M₃₂	—	53	2	no	—
153	#498	—	2	2	no	—
154	#498	C₂×D₅₃	1	4	yes	complemented
155	#443	—	53	4	no	—
156	#443	—	106	2	no	—
157	C₂×#108	—	53	1	no	—
158	C₂×#106	—	2	1	no	—
159	C₂×#106	C₂×D₅₃	1	2	yes	—
160	C₂×#104	—	106	1	no	—
161	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
162	#365	—	53	8	no	—
163	#364	—	53	4	no	—
164	#360	—	53	8	no	—
165	C₅₃	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
166	C₂×#41	—	106	1	no	—
167	C₂²×Q₈	—	4	1	no	—
168	#135	—	53	2	no	—
169	#135	—	53	2	no	—
170	#135	—	106	1	no	—
171	#135	—	106	1	no	—
172	C₂×Q₁₆	—	53	1	no	—
173	C₂×SD₁₆	—	106	1	no	—
174	C₂×D₈	—	53	1	no	—
175	#131	—	106	4	no	—
176	#131	—	212	2	no	—
177	C₂×M₁₆	—	106	1	no	—
178	C₂×M₁₆	—	212	1	no	—
179	C₂²×C₈	—	212	1	no	—
180	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	C₄⋊Q₈	—	2	4	no	—
183	C₄⋊Q₈	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
187	M₃₂	—	53	4	no	—
188	M₃₂	—	106	2	no	—
189	C₂×C₁₆	—	106	2	no	—
190	#108	—	53	2	no	—
191	#108	—	106	1	no	—
192	#106	—	2	4	no	—
193	#106	—	2	2	no	—
194	#106	C₂²×D₅₃	1	4	yes	—
195	#104	—	106	4	no	—
196	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
197	C₄×C₈	C₂²×D₅₃	1	2	yes	—
198	#41	—	106	2	no	—
199	#41	—	106	2	no	—
200	#41	—	212	1	no	—
201	#41	—	212	1	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	8	4	no	—
204	C₂×Q₈	—	106	1	no	—
205	C₂×D₄	—	106	1	no	—
206	C₂×D₄	—	212	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	212	1	no	—
210	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
211	Q₁₆	—	53	2	no	—
212	Q₁₆	—	106	1	no	—
213	SD₁₆	—	106	2	no	—
214	SD₁₆	—	106	2	no	—
215	D₈	—	53	2	no	—
216	D₈	—	106	1	no	—
217	M₁₆	—	106	2	no	—
218	M₁₆	—	212	1	no	—
219	M₁₆	—	212	4	no	—
220	C₂×C₈	—	212	1	no	—
221	C₂×C₈	—	212	4	no	—
222	C₂×C₈	#11719924	1	1	yes	characteristic
223	C₂×C₈	#11719924	1	1	yes	characteristic
224	C₂×C₈	#11719924	1	2	yes	—
225	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
227	C₂×C₈	#11719949	1	1	yes	characteristic
228	C₂×C₈	#11719949	1	1	yes	characteristic
229	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
231	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
237	C₁₆	—	106	4	no	—
238	C₂³	—	212	1	no	—
239	C₂³	#13133243	1	1	yes	characteristic
240	Q₈	—	8	8	no	—
241	Q₈	—	106	2	no	—
242	Q₈	—	212	1	no	—
243	D₄	—	106	2	no	—
244	D₄	—	212	2	no	—
245	D₄	—	212	2	no	—
246	D₄	—	212	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	212	1	no	—
252	C₂×C₄	—	212	2	no	—
253	C₂×C₄	—	212	2	no	—
254	C₂×C₄	#13133193	1	1	yes	characteristic
255	C₂×C₄	#13133193	1	1	yes	characteristic
256	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
258	C₂×C₄	#13133243	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₈	—	212	2	no	—
261	C₈	#13133193	1	1	yes	characteristic
262	C₈	#13133193	1	1	yes	characteristic
263	C₈	C₂×#11719924	1	2	yes	—
264	C₈	C₂×#11719924	1	2	yes	—
265	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
266	C₈	C₂×#11719949	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	424	2	no	—
269	C₂²	—	212	2	no	—
270	C₂²	#16810910	1	1	yes	characteristic
271	C₂²	#16810910	1	1	yes	characteristic
272	C₂²	C₂×#13133243	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	8	4	no	—
275	C₄	—	212	2	no	—
276	C₄	#16810910	1	1	yes	characteristic
277	C₄	#16810910	1	1	yes	characteristic
278	C₄	C₂×#13133193	1	1	yes	characteristic
279	C₄	C₂²×#11719924	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	424	2	no	—
283	C₂	#20276883	1	2	yes	direct product factor
284	C₂	C₂×#16810910	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810910
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#498	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂×#13133239	C₂²
hypercenter	C₂×#498	D₅₃
hypocenter	C₅₃	C₂×#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. #20276883	C₂	direct (N × H)
3. C₂×#16810445	C₂	semidirect
4. C₂×#16810444	C₂	semidirect
6. C₂×C₅₃×#498	C₂	semidirect
14. #16810445	C₂²	semidirect
15. #16810444	C₂²	semidirect
19. C₅₃×#498	C₂²	semidirect
26. #13133239	Q₈	semidirect
28. C₂×Dic₂₁₂	Q₈	semidirect
30. C₂×D₄₂₄	Q₈	semidirect
87. Dic₂₁₂	C₂×Q₈	semidirect
89. D₄₂₄	C₂×Q₈	semidirect
138. C₂×#498	D₅₃	semidirect
146. C₂×C₅₃	#2230	semidirect
154. #498	C₂×D₅₃	semidirect
165. C₅₃	C₂×#2230	semidirect
283. C₂	#20276883	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×#16810741	2
#20276883	2
C₂×#16810445	2
C₂×#16810444	2
C₂×#16810432	2
C₂×C₅₃×#498	2
C₂×#2230	53

Subgroups — C2×#20276883 #27538819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276883 #27538819