Subgroups — C₂×#20274304 #27371732

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 272 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, complemented
2	C₂×#16810995	C₂	1	2	yes	maximal, complemented
3	C₂×#16810969	C₂	1	2	yes	maximal, complemented
4	#20274304	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719923	—	2	4	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	#16811028	—	2	2	no	—
10	#16811028	C₂²	1	4	yes	complemented
11	#16810995	—	2	4	no	—
12	#16810995	C₂²	1	8	yes	complemented
13	#16810969	—	2	4	no	—
14	#16810969	C₂²	1	8	yes	complemented
15	C₂×#13133151	—	2	2	no	—
16	C₂×#13133138	—	2	4	no	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₈×#88322	C₂²	1	4	yes	complemented
20	C₂²×D₈×C₅₃	—	2	2	no	—
21	C₂×#127×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#493	—	2	2	no	—
23	C₅₃×#493	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₀₆	—	2	2	no	—
26	C₂×#11719923	—	2	16	no	—
27	C₂×#11719923	—	4	16	no	—
28	C₂²×#88322	D₄	1	2	yes	—
29	C₂²×#88322	D₄	1	2	yes	—
30	#13133217	—	2	2	no	—
31	#13133217	—	2	2	no	—
32	#13133217	—	2	2	no	—
33	#13133217	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	#13133151	—	2	8	no	—
37	#13133138	—	2	16	no	—
38	C₄×#88322	D₄	1	2	yes	—
39	C₄×#88322	D₄	1	2	yes	—
40	C₄×#88322	C₂³	1	2	yes	—
41	C₄×#88322	C₂³	1	2	yes	—
42	C₂²×D₄×C₅₃	—	2	2	no	—
43	C₂²×D₄×C₅₃	—	2	2	no	—
44	C₂²×D₄×C₅₃	—	2	2	no	—
45	C₂×D₈×C₅₃	—	2	8	no	—
46	C₂×D₈×C₅₃	—	4	8	no	—
47	C₂²×C₈×C₅₃	D₄	1	2	yes	—
48	#127×C₅₃	—	2	2	no	—
49	#127×C₅₃	—	2	2	no	—
50	#127×C₅₃	—	2	2	no	—
51	#127×C₅₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂⁴×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	2	4	no	—
57	C₂×D₄×C₅₃	—	2	4	no	—
58	C₂×D₄×C₅₃	—	2	4	no	—
59	C₂×D₄×C₅₃	—	4	2	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂×D₄×C₅₃	—	4	2	no	—
62	C₂×D₄×C₅₃	—	4	8	no	—
63	C₂×D₄×C₅₃	—	4	8	no	—
64	C₂×D₄×C₅₃	—	4	8	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	4	2	no	—
69	C₂×Dic₁₀₆	—	2	4	no	—
70	C₂×Dic₁₀₆	—	4	8	no	—
71	C₂×Dic₁₀₆	—	4	2	no	—
72	D₈×C₅₃	—	4	16	no	—
73	C₂×C₈×C₅₃	SD₁₆	1	8	yes	complemented
74	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
79	#11719923	—	4	32	no	—
80	C₄⋊Dic₅₃	—	2	4	no	—
81	C₄⋊Dic₅₃	—	2	4	no	—
82	C₄⋊Dic₅₃	—	4	2	no	—
83	C₄⋊Dic₅₃	—	4	2	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	D₈	1	8	yes	complemented
86	C₂×#88322	C₂×D₄	1	4	yes	—
87	C₂×#88322	C₂×D₄	1	4	yes	—
88	C₂³×C₅₃	—	4	4	no	—
89	C₂³×C₅₃	—	8	4	no	—
90	C₂³×C₅₃	—	8	4	no	—
91	C₂³×C₅₃	—	8	4	no	—
92	C₂³×C₅₃	#127	1	1	yes	characteristic
93	D₄×C₅₃	—	4	8	no	—
94	D₄×C₅₃	—	4	8	no	—
95	D₄×C₅₃	—	4	8	no	—
96	D₄×C₅₃	—	8	4	no	—
97	D₄×C₅₃	—	8	4	no	—
98	D₄×C₅₃	—	8	4	no	—
99	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
103	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
104	C₂×C₄×C₅₃	#135	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	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₅₃	—	4	4	no	—
109	C₂×Dic₅₃	—	8	4	no	—
110	Dic₁₀₆	—	4	8	no	—
111	Dic₁₀₆	—	8	4	no	—
112	C₈×C₅₃	C₂×SD₁₆	1	8	yes	complemented
113	#88322	—	2	4	no	—
114	#88322	C₂×D₈	1	8	yes	complemented
115	C₂×#1706	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	8	4	no	—
117	C₂²×C₅₃	—	8	4	no	—
118	C₂²×C₅₃	—	8	4	no	—
119	C₂²×C₅₃	—	16	8	no	—
120	C₂²×C₅₃	#492	1	2	yes	—
121	C₂²×C₅₃	#493	1	2	yes	—
122	C₂²×C₅₃	#495	1	2	yes	—
123	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
124	C₄×C₅₃	#492	1	2	yes	—
125	C₄×C₅₃	#493	1	2	yes	—
126	C₄×C₅₃	#495	1	2	yes	—
127	C₄×C₅₃	C₂²×D₈	1	2	yes	—
128	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
129	C₄×C₅₃	C₂×#135	1	2	yes	—
130	Dic₅₃	—	8	4	no	—
131	C₂×#500	—	53	1	no	—
132	C₂×#493	D₅₃	1	1	yes	characteristic
133	C₂×#492	—	53	2	no	—
134	C₂×#486	—	53	2	no	—
135	#1706	—	53	8	no	—
136	C₂×C₈²	—	53	1	no	—
137	C₂×C₅₃	—	16	4	no	—
138	C₂×C₅₃	#1706	1	4	yes	complemented
139	C₂×C₅₃	C₂×#492	1	1	yes	characteristic
140	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
142	C₂²×SD₁₆	—	106	4	no	—
143	C₂²×D₈	—	2	2	no	—
144	C₂×C₄⋊Q₈	—	53	2	no	—
145	C₂×#127	C₂×D₅₃	1	2	yes	—
146	#500	—	53	4	no	—
147	#500	—	106	2	no	—
148	#493	—	2	2	no	—
149	#493	C₂×D₅₃	1	4	yes	—
150	#492	—	53	8	no	—
151	#492	—	106	4	no	—
152	#486	—	53	8	no	—
153	#486	—	106	4	no	—
154	C₂×#107	—	106	2	no	—
155	C₂×#102	—	106	4	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₈²	—	53	4	no	—
160	C₅₃	C₂×#1706	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	106	2	no	—
162	C₂²×D₄	—	2	2	no	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	—	2	2	no	—
165	C₂×SD₁₆	—	106	16	no	—
166	C₂×SD₁₆	—	212	16	no	—
167	C₂×D₈	—	2	8	no	—
168	C₂×D₈	—	4	8	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	2	no	—
171	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
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	#127	—	2	2	no	—
177	#127	—	2	2	no	—
178	#127	—	2	2	no	—
179	#127	C₂²×D₅₃	1	4	yes	—
180	C₂×C₄⋊C₄	—	106	2	no	—
181	C₂×C₄⋊C₄	—	106	2	no	—
182	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
183	#107	—	106	8	no	—
184	#102	—	106	16	no	—
185	C₄×C₈	—	53	2	no	—
186	C₄×C₈	—	53	2	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	53	2	no	—
189	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
190	C₄×C₈	C₂²×D₅₃	1	2	yes	—
191	C₂⁴	—	4	2	no	—
192	C₂×Q₈	—	106	4	no	—
193	C₂×Q₈	—	212	8	no	—
194	C₂×Q₈	—	212	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	8	no	—
202	C₂×D₄	—	4	8	no	—
203	C₂×D₄	—	4	8	no	—
204	C₂²×C₄	—	212	2	no	—
205	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	SD₁₆	—	212	32	no	—
209	D₈	—	4	16	no	—
210	C₂×C₈	—	53	8	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	53	4	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	#11719923	1	8	yes	complemented
215	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
216	C₄⋊C₄	—	106	4	no	—
217	C₄⋊C₄	—	106	4	no	—
218	C₄⋊C₄	—	212	2	no	—
219	C₄⋊C₄	—	212	2	no	—
220	C₄²	D₄×D₅₃	1	1	yes	characteristic
221	C₄²	D₄×D₅₃	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
223	C₄²	C₂³×D₅₃	1	1	yes	characteristic
224	C₂³	—	4	4	no	—
225	C₂³	—	8	4	no	—
226	C₂³	—	8	4	no	—
227	C₂³	—	8	4	no	—
228	C₂³	#13133213	1	1	yes	characteristic
229	Q₈	—	424	4	no	—
230	Q₈	—	212	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	D₄	—	4	8	no	—
234	D₄	—	8	4	no	—
235	D₄	—	8	4	no	—
236	D₄	—	8	4	no	—
237	C₂×C₄	—	424	4	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	#13133213	1	1	yes	characteristic
240	C₂×C₄	#13133213	1	1	yes	characteristic
241	C₂×C₄	#13133213	1	1	yes	characteristic
242	C₂×C₄	D₈×D₅₃	1	4	yes	—
243	C₂×C₄	C₂×#11719923	1	4	yes	—
244	C₂×C₄	#13133236	1	4	yes	—
245	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
248	C₈	—	53	8	no	—
249	C₈	—	106	4	no	—
250	C₈	C₂×#11719923	1	8	yes	complemented
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	16	8	no	—
255	C₂²	#16810995	1	2	yes	—
256	C₂²	#16810998	1	2	yes	—
257	C₂²	#16811006	1	2	yes	—
258	C₂²	C₂×#13133213	1	1	yes	characteristic
259	C₄	—	424	4	no	—
260	C₄	#16810995	1	2	yes	—
261	C₄	#16810998	1	2	yes	—
262	C₄	#16811006	1	2	yes	—
263	C₄	C₂×D₈×D₅₃	1	2	yes	—
264	C₄	C₂²×#11719923	1	2	yes	—
265	C₄	C₂×#13133236	1	2	yes	—
266	C₂	—	16	4	no	—
267	C₂	#20274304	1	4	yes	direct product factor
268	C₂	C₂×#16810995	1	1	yes	characteristic
269	C₂	C₂×#16810998	1	1	yes	characteristic
270	C₂	C₂×#16811006	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂³×C₅₃	#127
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	C₂×#1706
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×#16811028	C₂	semidirect
2. C₂×#16810995	C₂	semidirect
3. C₂×#16810969	C₂	semidirect
4. #20274304	C₂	direct (N × H)
5. C₂×C₈×#88322	C₂	semidirect
10. #16811028	C₂²	semidirect
12. #16810995	C₂²	semidirect
14. #16810969	C₂²	semidirect
19. C₈×#88322	C₂²	semidirect
73. C₂×C₈×C₅₃	SD₁₆	semidirect
85. C₂×#88322	D₈	semidirect
112. C₈×C₅₃	C₂×SD₁₆	semidirect
114. #88322	C₂×D₈	semidirect
138. C₂×C₅₃	#1706	semidirect
160. C₅₃	C₂×#1706	semidirect
214. C₂×C₈	#11719923	semidirect
250. C₈	C₂×#11719923	semidirect
267. C₂	#20274304	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×#16810995	2
C₂×#16810969	2
#20274304	2
C₂×C₈×#88322	2
C₂×C₅₃×#493	2
C₂×#1706	53

Subgroups — C2×#20274304 #27371732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274304 #27371732