Subgroups — C₂×#20274072 #27366417

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810820	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274072	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810367	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719923	—	2	2	no	—
8	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	#16810820	—	2	4	no	—
10	#16810820	C₂²	1	8	yes	complemented
11	C₈×Dic₁₀₆	—	2	2	no	—
12	C₈×Dic₁₀₆	C₂²	1	4	yes	complemented
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×#13133146	—	2	1	no	—
16	C₂×#13133138	—	2	2	no	—
17	C₂×C₈×Dic₅₃	—	2	1	no	—
18	#16810376	C₂²	1	4	yes	complemented
19	#16810367	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₅₃	—	2	1	no	—
21	C₂²×D₈×C₅₃	—	2	1	no	—
22	C₂×#127×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#493	—	2	2	no	—
24	C₅₃×#493	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
27	C₂×#11719923	—	2	8	no	—
28	C₂×#11719923	—	4	8	no	—
29	C₂²×#88322	—	2	1	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₄×Dic₁₀₆	—	2	1	no	—
32	C₄×Dic₁₀₆	—	2	1	no	—
33	C₄×Dic₁₀₆	D₄	1	2	yes	—
34	C₄×Dic₁₀₆	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
36	C₂×#11719919	—	2	1	no	—
37	C₂×C₄×Dic₅₃	—	2	1	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	—	2	1	no	—
40	C₄⋊#88322	C₂³	1	4	yes	—
41	C₄⋊#88322	C₂³	1	2	yes	—
42	#13133146	—	2	4	no	—
43	#13133138	—	2	8	no	—
44	C₈×Dic₅₃	—	2	4	no	—
45	C₂²×D₄×C₅₃	—	2	2	no	—
46	C₂²×D₄×C₅₃	—	2	2	no	—
47	C₂²×D₄×C₅₃	—	2	2	no	—
48	C₂×D₈×C₅₃	—	2	4	no	—
49	C₂×D₈×C₅₃	—	2	4	no	—
50	C₂×D₈×C₅₃	—	4	4	no	—
51	C₂×D₈×C₅₃	—	4	4	no	—
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	#127×C₅₃	—	2	2	no	—
55	#127×C₅₃	—	2	2	no	—
56	#127×C₅₃	—	2	2	no	—
57	#127×C₅₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₃	D₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂⁴×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	2	4	no	—
63	C₂×D₄×C₅₃	—	2	4	no	—
64	C₂×D₄×C₅₃	—	2	4	no	—
65	C₂×D₄×C₅₃	—	4	2	no	—
66	C₂×D₄×C₅₃	—	4	2	no	—
67	C₂×D₄×C₅₃	—	4	2	no	—
68	C₂×D₄×C₅₃	—	4	4	no	—
69	C₂×D₄×C₅₃	—	4	4	no	—
70	C₂×D₄×C₅₃	—	4	4	no	—
71	C₂×D₄×C₅₃	—	4	4	no	—
72	C₂×D₄×C₅₃	—	8	4	no	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
76	C₂²×Dic₅₃	—	2	1	no	—
77	C₂²×Dic₅₃	—	2	1	no	—
78	C₂×Dic₁₀₆	—	2	1	no	—
79	C₂×Dic₁₀₆	—	2	2	no	—
80	C₂×Dic₁₀₆	D₈	1	4	yes	complemented
81	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
82	D₈×C₅₃	—	4	8	no	—
83	D₈×C₅₃	—	4	8	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	#41	1	1	yes	characteristic
91	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
92	#11719923	—	4	16	no	—
93	C₄⋊Dic₅₃	—	2	1	no	—
94	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
95	#11719919	—	2	2	no	—
96	#11719919	—	2	2	no	—
97	C₄×Dic₅₃	—	2	2	no	—
98	C₄×Dic₅₃	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	2	4	no	—
101	C₂×#88322	—	4	4	no	—
102	C₂×#88322	—	4	2	no	—
103	C₂³×C₅₃	—	4	4	no	—
104	C₂³×C₅₃	—	8	4	no	—
105	C₂³×C₅₃	—	8	4	no	—
106	C₂³×C₅₃	—	8	4	no	—
107	C₂³×C₅₃	#121	1	1	yes	characteristic
108	D₄×C₅₃	—	4	4	no	—
109	D₄×C₅₃	—	4	4	no	—
110	D₄×C₅₃	—	4	4	no	—
111	D₄×C₅₃	—	4	4	no	—
112	D₄×C₅₃	—	8	4	no	—
113	D₄×C₅₃	—	8	4	no	—
114	D₄×C₅₃	—	8	4	no	—
115	D₄×C₅₃	—	8	4	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₅₃	#135	1	2	yes	—
123	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	2	4	no	—
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	4	2	no	—
131	Dic₁₀₆	—	2	2	no	—
132	Dic₁₀₆	—	2	4	no	—
133	Dic₁₀₆	C₂×D₈	1	4	yes	complemented
134	C₈×C₅₃	—	2	2	no	—
135	C₈×C₅₃	C₂×SD₁₆	1	4	yes	complemented
136	#88322	—	4	4	no	—
137	#88322	—	4	2	no	—
138	C₂×#1668	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	—	8	4	no	—
140	C₂²×C₅₃	—	8	4	no	—
141	C₂²×C₅₃	—	8	4	no	—
142	C₂²×C₅₃	—	16	8	no	—
143	C₂²×C₅₃	#460	1	2	yes	—
144	C₂²×C₅₃	#466	1	2	yes	—
145	C₂²×C₅₃	#472	1	2	yes	—
146	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	#460	1	2	yes	—
150	C₄×C₅₃	#466	1	2	yes	—
151	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
154	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	Dic₅₃	—	2	4	no	—
156	Dic₅₃	—	4	2	no	—
157	C₂×#493	D₅₃	1	1	yes	characteristic
158	C₂×#460	—	53	2	no	—
159	C₂×C₈×Q₈	—	53	1	no	—
160	#1668	—	53	8	no	—
161	C₂×#335	—	53	1	no	—
162	C₂×#331	—	53	2	no	—
163	C₂×C₅₃	—	16	4	no	—
164	C₂×C₅₃	#1668	1	4	yes	complemented
165	C₂×C₅₃	C₂×#460	1	1	yes	characteristic
166	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
168	C₂²×SD₁₆	—	106	2	no	—
169	C₂²×D₈	—	2	1	no	—
170	C₂²×D₈	—	2	1	no	—
171	C₂×#127	C₂×D₅₃	1	2	yes	—
172	C₂×C₄×Q₈	—	53	1	no	—
173	#493	—	2	2	no	—
174	#493	C₂×D₅₃	1	4	yes	—
175	#460	—	53	8	no	—
176	#460	—	106	4	no	—
177	C₈×Q₈	—	53	4	no	—
178	C₈×Q₈	—	106	2	no	—
179	C₂×C₄⋊C₈	—	53	1	no	—
180	C₂×C₄⋊C₈	—	53	2	no	—
181	C₂×C₄⋊C₈	—	106	1	no	—
182	C₂×#102	—	106	2	no	—
183	C₂×C₄×C₈	—	106	1	no	—
184	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
185	#335	—	53	4	no	—
186	#331	—	53	8	no	—
187	C₅₃	C₂×#1668	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	53	1	no	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂×SD₁₆	—	106	8	no	—
193	C₂×SD₁₆	—	212	8	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	4	4	no	—
197	C₂×D₈	—	4	4	no	—
198	C₂²×C₈	—	106	2	no	—
199	C₂²×C₈	—	106	1	no	—
200	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
202	#127	—	2	2	no	—
203	#127	—	2	2	no	—
204	#127	—	2	2	no	—
205	#127	C₂²×D₅₃	1	4	yes	—
206	C₄×Q₈	—	53	2	no	—
207	C₄×Q₈	—	53	2	no	—
208	C₄×Q₈	—	106	1	no	—
209	C₄×Q₈	—	106	1	no	—
210	C₂×C₄⋊C₄	—	53	1	no	—
211	C₂×C₄⋊C₄	—	106	1	no	—
212	C₂×C₄²	—	106	1	no	—
213	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
214	C₄⋊C₈	—	53	2	no	—
215	C₄⋊C₈	—	53	4	no	—
216	C₄⋊C₈	—	106	4	no	—
217	C₄⋊C₈	—	106	2	no	—
218	C₄⋊C₈	—	106	1	no	—
219	#102	—	106	8	no	—
220	C₄×C₈	—	106	4	no	—
221	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
222	C₄×C₈	C₂²×D₅₃	1	2	yes	—
223	C₂⁴	—	4	2	no	—
224	C₂×Q₈	—	53	4	no	—
225	C₂×Q₈	—	53	2	no	—
226	C₂×Q₈	—	106	1	no	—
227	C₂×Q₈	—	106	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	8	4	no	—
239	C₂²×C₄	—	106	1	no	—
240	C₂²×C₄	—	106	1	no	—
241	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₄	#11719947	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
244	SD₁₆	—	212	16	no	—
245	D₈	—	4	8	no	—
246	D₈	—	4	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	106	4	no	—
249	C₂×C₈	—	106	2	no	—
250	C₂×C₈	—	212	4	no	—
251	C₂×C₈	—	212	2	no	—
252	C₂×C₈	#11719923	1	4	yes	complemented
253	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
255	C₄⋊C₄	—	53	2	no	—
256	C₄⋊C₄	—	106	1	no	—
257	C₄⋊C₄	—	106	2	no	—
258	C₄⋊C₄	—	106	2	no	—
259	C₄²	—	106	2	no	—
260	C₄²	—	106	2	no	—
261	C₄²	D₄×D₅₃	1	1	yes	characteristic
262	C₄²	#11719947	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
264	C₄²	C₂³×D₅₃	1	1	yes	characteristic
265	C₂³	—	4	4	no	—
266	C₂³	—	8	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	#13133188	1	1	yes	characteristic
270	Q₈	—	53	4	no	—
271	Q₈	—	106	4	no	—
272	Q₈	—	106	2	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	106	4	no	—
283	C₂×C₄	—	106	2	no	—
284	C₂×C₄	—	106	2	no	—
285	C₂×C₄	—	212	2	no	—
286	C₂×C₄	#13133188	1	1	yes	characteristic
287	C₂×C₄	#13133188	1	1	yes	characteristic
288	C₂×C₄	#13133188	1	1	yes	characteristic
289	C₂×C₄	D₈×D₅₃	1	2	yes	—
290	C₂×C₄	C₂×#11719923	1	2	yes	—
291	C₂×C₄	#13133236	1	2	yes	—
292	C₂×C₄	#13133243	1	2	yes	—
293	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	212	4	no	—
298	C₈	—	212	2	no	—
299	C₈	C₂×#11719923	1	4	yes	complemented
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	8	4	no	—
303	C₂²	—	16	8	no	—
304	C₂²	#16810820	1	2	yes	—
305	C₂²	#16810856	1	2	yes	—
306	C₂²	#16810890	1	2	yes	—
307	C₂²	C₂×#13133188	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	106	4	no	—
311	C₄	—	212	2	no	—
312	C₄	#16810820	1	2	yes	—
313	C₄	#16810856	1	2	yes	—
314	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
315	C₄	C₂²×#11719923	1	1	yes	characteristic
316	C₄	C₂×#13133236	1	1	yes	characteristic
317	C₄	C₂×#13133243	1	1	yes	characteristic
318	C₂	—	16	4	no	—
319	C₂	#20274072	1	4	yes	direct product factor
320	C₂	C₂×#16810820	1	1	yes	characteristic
321	C₂	C₂×#16810856	1	1	yes	characteristic
322	C₂	C₂×#16810890	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂³×C₅₃	#121
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	C₂×#1668
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16810820	C₂	semidirect
2. C₂×C₈×Dic₁₀₆	C₂	semidirect
3. #20274072	C₂	direct (N × H)
4. C₂×#16810376	C₂	semidirect
5. C₂×#16810367	C₂	semidirect
10. #16810820	C₂²	semidirect
12. C₈×Dic₁₀₆	C₂²	semidirect
18. #16810376	C₂²	semidirect
19. #16810367	C₂²	semidirect
80. C₂×Dic₁₀₆	D₈	semidirect
85. C₂×C₈×C₅₃	SD₁₆	semidirect
133. Dic₁₀₆	C₂×D₈	semidirect
135. C₈×C₅₃	C₂×SD₁₆	semidirect
164. C₂×C₅₃	#1668	semidirect
187. C₅₃	C₂×#1668	semidirect
252. C₂×C₈	#11719923	semidirect
299. C₈	C₂×#11719923	semidirect
319. C₂	#20274072	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×#16810820	2
C₂×C₈×Dic₁₀₆	2
#20274072	2
C₂×#16810376	2
C₂×#16810367	2
C₂×C₅₃×#493	2
C₂×#1668	53

Subgroups — C2×#20274072 #27366417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274072 #27366417