Subgroups — C₂×#20276862 #27538701

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 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	#20276862	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810450	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810424	C₂	1	2	yes	maximal
6	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	—	2	1	no	—
8	C₂²×#11719908	C₂²	1	2	yes	—
9	#16811028	—	2	2	no	—
10	#16811028	C₂²	1	4	yes	complemented
11	C₂×#13133151	—	2	1	no	—
12	C₂×#13133151	C₂²	1	2	yes	—
13	#16810450	C₂²	1	8	yes	complemented
14	C₄⋊#11719908	C₂²	1	4	yes	complemented
15	#16810424	C₂²	1	8	yes	—
16	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₅₃	—	2	1	no	—
18	C₄×D₈×C₅₃	—	2	2	no	—
19	C₄×D₈×C₅₃	C₂²	1	4	yes	—
20	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₅₃	—	2	1	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₀₆	—	4	1	no	—
24	#13133217	—	2	4	no	—
25	#13133217	—	4	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	4	1	no	—
28	C₂×#11719908	—	2	4	no	—
29	C₂×#11719908	C₂³	1	4	yes	—
30	#13133151	—	2	2	no	—
31	#13133151	—	2	4	no	—
32	#13133151	C₂³	1	4	yes	—
33	C₂²×D₄×C₅₃	—	2	1	no	—
34	C₂³×C₄×C₅₃	—	4	1	no	—
35	C₂×D₈×C₅₃	—	2	1	no	—
36	C₂×D₈×C₅₃	—	2	2	no	—
37	C₂×D₈×C₅₃	Q₈	1	4	yes	complemented
38	C₂×D₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄×D₄×C₅₃	—	2	4	no	—
42	C₄×D₄×C₅₃	—	4	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	#107×C₅₃	—	2	1	no	—
47	#107×C₅₃	C₂³	1	2	yes	—
48	#102×C₅₃	—	2	4	no	—
49	C₄×C₈×C₅₃	D₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₂⁴×C₅₃	—	4	1	no	—
52	C₂×D₄×C₅₃	—	2	4	no	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	4	1	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	1	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	—	8	4	no	—
59	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	4	2	no	—
63	C₂×Dic₁₀₆	—	4	4	no	—
64	C₂×Dic₁₀₆	—	8	4	no	—
65	D₈×C₅₃	—	2	4	no	—
66	D₈×C₅₃	—	2	2	no	—
67	D₈×C₅₃	C₂×Q₈	1	4	yes	complemented
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₅₃	#41	1	2	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	4	1	no	—
76	C₂²⋊C₄×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	D₈	1	2	yes	—
78	C₄²×C₅₃	C₂×D₄	1	2	yes	—
79	C₄⋊Dic₅₃	—	2	4	no	—
80	C₄⋊Dic₅₃	—	4	2	no	—
81	C₄⋊Dic₅₃	—	4	4	no	—
82	#11719908	—	2	4	no	—
83	C₂³×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	—	4	4	no	—
85	C₂³×C₅₃	—	8	2	no	—
86	C₂³×C₅₃	—	8	2	no	—
87	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
88	D₄×C₅₃	—	2	4	no	—
89	D₄×C₅₃	—	4	2	no	—
90	D₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	—	8	8	no	—
95	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
96	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₅₃	#137	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₅₃	—	4	4	no	—
103	C₂×Dic₅₃	—	8	4	no	—
104	Dic₁₀₆	—	8	8	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂×#2226	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	4	4	no	—
111	C₂²×C₅₃	—	8	2	no	—
112	C₂²×C₅₃	—	8	2	no	—
113	C₂²×C₅₃	—	8	8	no	—
114	C₂²×C₅₃	#474	1	2	yes	—
115	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
116	C₂²×C₅₃	#509	1	2	yes	—
117	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	8	2	no	—
120	C₄×C₅₃	#474	1	2	yes	—
121	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
122	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
124	Dic₅₃	—	8	4	no	—
125	C₂×#500	—	53	1	no	—
126	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
127	#2226	—	53	8	no	—
128	C₂×#367	—	53	2	no	—
129	C₂×C₄⋊C₁₆	—	53	1	no	—
130	C₂×#357	—	53	2	no	—
131	C₂×C₅₃	—	8	4	no	—
132	C₂×C₅₃	#2226	1	4	yes	complemented
133	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
134	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
136	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
137	C₂×C₄⋊Q₈	—	106	1	no	—
138	C₂×C₄×D₄	—	2	1	no	—
139	C₂²×C₁₆	—	53	2	no	—
140	#500	—	53	4	no	—
141	#500	—	106	2	no	—
142	C₄×D₈	—	2	2	no	—
143	C₄×D₈	C₂×D₅₃	1	4	yes	—
144	C₂×#107	—	53	2	no	—
145	C₂×#107	—	106	1	no	—
146	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
147	C₂×#102	—	2	1	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#367	—	53	8	no	—
150	C₄⋊C₁₆	—	53	4	no	—
151	#357	—	53	8	no	—
152	C₅₃	C₂×#2226	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	212	1	no	—
154	C₂²×D₄	—	2	1	no	—
155	C₂³×C₄	—	4	1	no	—
156	C₂×D₈	—	2	2	no	—
157	C₂×D₈	—	2	1	no	—
158	C₂×D₈	Dic₁₀₆	1	4	yes	complemented
159	C₂×D₈	C₂²×D₅₃	1	2	yes	—
160	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	106	4	no	—
163	C₄⋊Q₈	—	212	2	no	—
164	C₄×D₄	—	2	4	no	—
165	C₄×D₄	—	4	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	106	2	no	—
168	C₂×C₄⋊C₄	—	212	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₁₆	—	53	4	no	—
172	C₂×C₁₆	—	106	4	no	—
173	#107	—	2	1	no	—
174	#107	—	53	4	no	—
175	#107	—	106	2	no	—
176	#107	—	106	4	no	—
177	#107	C₂²×D₅₃	1	2	yes	—
178	#102	—	2	4	no	—
179	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
180	C₄×C₈	C₂²×D₅₃	1	2	yes	—
181	C₂⁴	—	4	1	no	—
182	C₂×Q₈	—	424	4	no	—
183	C₂×Q₈	—	212	4	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	8	4	no	—
191	C₂²×C₄	—	212	2	no	—
192	C₂²×C₄	#11719922	1	1	yes	characteristic
193	C₂²×C₄	#11719922	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	D₈	—	2	4	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂×Dic₁₀₆	1	4	yes	complemented
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
200	C₂×C₈	#11719945	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	106	4	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄⋊C₄	—	212	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	#11719922	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
212	C₁₆	—	106	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	2	no	—
216	C₂³	—	8	2	no	—
217	C₂³	C₂×#11719922	1	1	yes	characteristic
218	Q₈	—	424	8	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	8	no	—
226	C₂×C₄	—	424	4	no	—
227	C₂×C₄	—	212	4	no	—
228	C₂×C₄	#13133161	1	4	yes	—
229	C₂×C₄	#13133196	1	2	yes	—
230	C₂×C₄	C₂×#11719922	1	2	yes	—
231	C₂×C₄	C₂×#11719922	1	2	yes	—
232	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
233	C₂×C₄	#13133247	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#13133196	1	2	yes	—
237	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
238	C₈	C₂×#11719945	1	1	yes	characteristic
239	C₂²	—	4	4	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	8	no	—
243	C₂²	#16810903	1	2	yes	—
244	C₂²	C₂×#13133161	1	2	yes	—
245	C₂²	#16811058	1	2	yes	—
246	C₂²	C₂²×#11719922	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	424	4	no	—
250	C₄	#16810903	1	2	yes	—
251	C₄	C₂×#13133161	1	4	yes	—
252	C₄	C₂×#13133196	1	1	yes	characteristic
253	C₄	C₂×#13133247	1	1	yes	characteristic
254	C₂	—	8	4	no	—
255	C₂	#20276862	1	4	yes	direct product factor
256	C₂	C₂×#16810903	1	1	yes	characteristic
257	C₂	C₂²×#13133161	1	1	yes	characteristic
258	C₂	C₂×#16811058	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₈×C₅₃	C₂²
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	C₂×#2226
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16811028	C₂	semidirect
2. #20276862	C₂	direct (N × H)
3. C₂×#16810450	C₂	semidirect
4. C₂×C₄⋊#11719908	C₂	semidirect
10. #16811028	C₂²	semidirect
13. #16810450	C₂²	semidirect
14. C₄⋊#11719908	C₂²	semidirect
37. C₂×D₈×C₅₃	Q₈	semidirect
67. D₈×C₅₃	C₂×Q₈	semidirect
132. C₂×C₅₃	#2226	semidirect
152. C₅₃	C₂×#2226	semidirect
158. C₂×D₈	Dic₁₀₆	semidirect
197. D₈	C₂×Dic₁₀₆	semidirect
255. C₂	#20276862	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
#20276862	2
C₂×#16810450	2
C₂×C₄⋊#11719908	2
C₂×#16810424	2
C₂×C₄×D₈×C₅₃	2
C₂×#2226	53

Subgroups — C2×#20276862 #27538701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276862 #27538701