Subgroups — C₂×#20276998 #27539586

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	#20276998	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810450	C₂	1	2	yes	maximal
4	C₂×#16810448	C₂	1	2	yes	maximal
5	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133209	—	2	1	no	—
8	C₂²×#11719908	C₂²	1	2	yes	—
9	#16811025	—	2	2	no	—
10	#16811025	C₂²	1	4	yes	—
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133151	C₂²	1	1	yes	characteristic
13	C₂×#13133149	—	2	1	no	—
14	#16810450	C₂²	1	8	yes	—
15	#16810448	C₂²	1	8	yes	—
16	#16810399	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
18	C₅₃×#500	—	2	2	no	—
19	C₅₃×#500	C₂²	1	4	yes	—
20	C₂×#107×C₅₃	—	2	1	no	—
21	C₂×#107×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	#13133209	—	2	4	no	—
24	#13133209	—	4	2	no	—
25	C₂×C₄⋊Dic₅₃	—	2	1	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×#11719919	—	4	1	no	—
28	C₂×#11719919	—	4	1	no	—
29	C₂×#11719908	Q₈	1	8	yes	complemented
30	C₂×#11719908	C₂³	1	4	yes	—
31	#13133151	—	2	1	no	—
32	#13133151	—	2	1	no	—
33	#13133151	C₂³	1	2	yes	—
34	#13133151	C₂³	1	2	yes	—
35	#13133149	—	2	4	no	—
36	C₂²×Q₈×C₅₃	—	4	1	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₅₃	—	2	4	no	—
40	C₄⋊Q₈×C₅₃	—	4	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
43	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
44	#107×C₅₃	—	2	2	no	—
45	#107×C₅₃	—	2	4	no	—
46	#107×C₅₃	C₂³	1	4	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂×Q₈×C₅₃	—	4	4	no	—
50	C₂×Q₈×C₅₃	—	8	4	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	4	1	no	—
56	C₂²×Dic₅₃	—	4	1	no	—
57	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	4	no	—
65	C₄²×C₅₃	D₈	1	2	yes	—
66	C₄²×C₅₃	C₂×D₄	1	2	yes	—
67	C₄⋊Dic₅₃	—	2	2	no	—
68	C₄⋊Dic₅₃	—	2	2	no	—
69	C₄⋊Dic₅₃	—	4	1	no	—
70	C₄⋊Dic₅₃	—	4	1	no	—
71	#11719919	—	4	4	no	—
72	#11719919	—	4	4	no	—
73	#11719908	C₂×Q₈	1	8	yes	complemented
74	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₅₃	—	8	8	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	—	8	4	no	—
80	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₅₃	—	4	2	no	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	—	8	2	no	—
89	C₂×Dic₅₃	—	8	2	no	—
90	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
91	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
92	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
93	C₂×#2255	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#500	1	2	yes	—
95	C₂²×C₅₃	#509	1	2	yes	—
96	C₂²×C₅₃	#510	1	2	yes	—
97	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	—	8	4	no	—
101	C₄×C₅₃	#500	1	2	yes	—
102	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₅₃	—	8	2	no	—
105	Dic₅₃	—	8	2	no	—
106	C₂×#500	D₅₃	1	1	yes	characteristic
107	C₂×#499	—	53	1	no	—
108	#2255	—	53	8	no	—
109	C₂×#367	—	53	2	no	—
110	C₂×#366	—	53	2	no	—
111	C₂×#346	—	53	1	no	—
112	C₂×C₅₃	#2255	1	4	yes	complemented
113	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×#125	—	106	1	no	—
118	C₂²×C₁₆	—	53	2	no	—
119	#500	—	2	2	no	—
120	#500	C₂×D₅₃	1	4	yes	—
121	#499	—	53	4	no	—
122	#499	—	106	2	no	—
123	C₂×#107	—	2	1	no	—
124	C₂×#107	—	53	1	no	—
125	C₂×#107	—	53	1	no	—
126	C₂×#107	C₂×D₅₃	1	2	yes	—
127	C₂×#106	—	106	1	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	#367	—	53	8	no	—
130	#366	—	53	8	no	—
131	#346	—	53	4	no	—
132	C₅₃	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	4	1	no	—
134	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	—	4	2	no	—
138	#125	—	106	4	no	—
139	#125	—	212	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	106	1	no	—
143	C₂×C₄⋊C₄	—	106	1	no	—
144	C₂×C₄⋊C₄	—	212	1	no	—
145	C₂×C₄⋊C₄	—	212	1	no	—
146	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₁₆	—	53	8	no	—
148	C₂×C₁₆	—	53	4	no	—
149	#107	—	2	4	no	—
150	#107	—	2	2	no	—
151	#107	—	53	2	no	—
152	#107	—	53	2	no	—
153	#107	—	106	1	no	—
154	#107	—	106	1	no	—
155	#107	C₂²×D₅₃	1	4	yes	—
156	#106	—	106	4	no	—
157	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
158	C₄×C₈	C₂²×D₅₃	1	2	yes	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	8	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	—	212	1	no	—
163	C₂²×C₄	—	212	1	no	—
164	C₂²×C₄	#11719922	1	1	yes	characteristic
165	C₂²×C₄	#11719922	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₈	#11719925	1	4	yes	—
168	C₂×C₈	Q₈×D₅₃	1	4	yes	—
169	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
171	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	106	2	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	—	212	1	no	—
178	C₄⋊C₄	—	212	1	no	—
179	C₄⋊C₄	—	212	4	no	—
180	C₄⋊C₄	—	212	4	no	—
181	C₄²	#11719922	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
183	C₁₆	—	53	8	no	—
184	C₂³	C₂×#11719922	1	1	yes	characteristic
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	—	424	2	no	—
191	C₂×C₄	—	424	2	no	—
192	C₂×C₄	—	212	2	no	—
193	C₂×C₄	—	212	2	no	—
194	C₂×C₄	#13133215	1	2	yes	—
195	C₂×C₄	C₂×#11719922	1	2	yes	—
196	C₂×C₄	C₂×#11719922	1	2	yes	—
197	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
198	C₂×C₄	C₂×#11719925	1	2	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
200	C₈	#13133215	1	2	yes	—
201	C₈	C₂×#11719925	1	4	yes	—
202	C₈	C₂×Q₈×D₅₃	1	2	yes	—
203	C₂²	#16811026	1	2	yes	—
204	C₂²	#16811059	1	2	yes	—
205	C₂²	#16811064	1	2	yes	—
206	C₂²	C₂²×#11719922	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₄	—	424	2	no	—
211	C₄	—	424	2	no	—
212	C₄	#16811026	1	2	yes	—
213	C₄	C₂×#13133215	1	1	yes	characteristic
214	C₄	C₂²×#11719925	1	1	yes	characteristic
215	C₂	#20276998	1	4	yes	direct product factor
216	C₂	C₂×#16811026	1	1	yes	characteristic
217	C₂	C₂×#16811059	1	1	yes	characteristic
218	C₂	C₂×#16811064	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	C₂×#2255
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #20276998	C₂	direct (N × H)
29. C₂×#11719908	Q₈	semidirect
73. #11719908	C₂×Q₈	semidirect
112. C₂×C₅₃	#2255	semidirect
132. C₅₃	C₂×#2255	semidirect
215. C₂	#20276998	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×#16811025	2
#20276998	2
C₂×#16810450	2
C₂×#16810448	2
C₂×#16810399	2
C₂×C₅₃×#500	2
C₂×#2255	53

Subgroups — C2×#20276998 #27539586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276998 #27539586