Subgroups — C₂×#20276991 #27539571

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 188 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	#20276991	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×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133209	—	2	1	no	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	#16811025	—	2	2	no	—
11	#16811025	C₂²	1	4	yes	—
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133149	—	2	1	no	—
14	#16810450	C₂²	1	8	yes	—
15	#16810448	C₂²	1	8	yes	—
16	C₄×#11719908	C₂²	1	4	yes	—
17	C₂×#125×C₅₃	—	2	1	no	—
18	C₅₃×#499	—	2	2	no	—
19	C₅₃×#499	C₂²	1	4	yes	—
20	C₂×#107×C₅₃	C₂²	1	2	yes	—
21	C₂×#106×C₅₃	—	2	1	no	—
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	2	no	—
26	C₂×#11719919	—	4	2	no	—
27	C₂×#11719908	Q₈	1	4	yes	—
28	C₂×#11719908	Q₈	1	4	yes	—
29	C₂×#11719908	C₂³	1	2	yes	—
30	C₂×#11719908	C₂³	1	2	yes	—
31	#13133151	—	2	2	no	—
32	#13133151	C₂³	1	4	yes	—
33	#13133149	—	2	4	no	—
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	#125×C₅₃	—	2	4	no	—
37	#125×C₅₃	—	4	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	#107×C₅₃	—	2	2	no	—
42	#107×C₅₃	C₂³	1	4	yes	—
43	#106×C₅₃	—	2	4	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	D₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	8	no	—
59	C₄²×C₅₃	D₈	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₅₃	—	2	4	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	#11719919	—	4	8	no	—
64	#11719908	C₂×Q₈	1	4	yes	—
65	#11719908	C₂×Q₈	1	4	yes	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₅₃	—	4	4	no	—
76	C₂×Dic₅₃	—	8	4	no	—
77	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
78	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
79	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂×#2253	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	#500	1	2	yes	—
83	C₂²×C₅₃	#508	1	4	yes	—
84	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
85	C₄×C₅₃	—	2	4	no	—
86	C₄×C₅₃	—	8	4	no	—
87	C₄×C₅₃	#500	1	2	yes	—
88	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
90	Dic₅₃	—	8	4	no	—
91	C₂×#499	—	53	1	no	—
92	C₂×#499	D₅₃	1	1	yes	characteristic
93	#2253	—	53	8	no	—
94	C₂×#367	—	53	2	no	—
95	C₂×#366	—	53	2	no	—
96	C₂×C₄×C₁₆	—	53	1	no	—
97	C₂×C₅₃	#2253	1	4	yes	complemented
98	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#508	1	2	yes	—
100	C₂×#125	—	2	1	no	—
101	C₂×#125	—	106	1	no	—
102	C₂²×C₁₆	—	53	1	no	—
103	C₂²×C₁₆	—	53	1	no	—
104	#499	—	2	2	no	—
105	#499	—	53	4	no	—
106	#499	—	106	2	no	—
107	#499	C₂×D₅₃	1	4	yes	—
108	C₂×#107	—	53	2	no	—
109	C₂×#107	C₂×D₅₃	1	2	yes	—
110	C₂×#106	—	2	1	no	—
111	C₂×#106	—	106	1	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	#367	—	53	8	no	—
114	#366	—	53	8	no	—
115	C₄×C₁₆	—	53	4	no	—
116	C₅₃	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	#125	—	2	4	no	—
120	#125	—	4	2	no	—
121	#125	—	106	4	no	—
122	#125	—	212	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	2	no	—
125	C₂×C₄⋊C₄	—	106	2	no	—
126	C₂×C₄⋊C₄	—	212	2	no	—
127	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
128	C₂×C₁₆	—	53	4	no	—
129	C₂×C₁₆	—	53	4	no	—
130	C₂×C₁₆	—	53	2	no	—
131	C₂×C₁₆	—	53	2	no	—
132	#107	—	2	2	no	—
133	#107	—	53	4	no	—
134	#107	—	106	2	no	—
135	#107	C₂²×D₅₃	1	4	yes	—
136	#106	—	2	4	no	—
137	#106	—	106	4	no	—
138	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
139	C₄×C₈	C₂²×D₅₃	1	2	yes	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	212	2	no	—
142	C₂²×C₄	#11719922	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
144	C₂×C₈	#11719925	1	4	yes	—
145	C₂×C₈	Q₈×D₅₃	1	2	yes	—
146	C₂×C₈	Q₈×D₅₃	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
149	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	106	4	no	—
154	C₄⋊C₄	—	212	2	no	—
155	C₄⋊C₄	—	212	8	no	—
156	C₄²	#11719922	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
158	C₁₆	—	53	4	no	—
159	C₁₆	—	53	4	no	—
160	C₂³	C₂×#11719922	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	424	4	no	—
165	C₂×C₄	—	212	4	no	—
166	C₂×C₄	#13133215	1	2	yes	—
167	C₂×C₄	C₂×#11719922	1	4	yes	—
168	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
169	C₂×C₄	C₂×#11719925	1	2	yes	—
170	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
171	C₈	#13133215	1	2	yes	—
172	C₈	C₂×#11719925	1	4	yes	—
173	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
174	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
175	C₂²	#16811026	1	2	yes	—
176	C₂²	#16811053	1	4	yes	—
177	C₂²	C₂²×#11719922	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	424	4	no	—
181	C₄	#16811026	1	2	yes	—
182	C₄	C₂×#13133215	1	1	yes	characteristic
183	C₄	C₂²×#11719925	1	1	yes	characteristic
184	C₂	#20276991	1	4	yes	direct product factor
185	C₂	C₂×#16811026	1	1	yes	characteristic
186	C₂	C₂×#16811053	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	C₂×#2253
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #20276991	C₂	direct (N × H)
97. C₂×C₅₃	#2253	semidirect
116. C₅₃	C₂×#2253	semidirect
184. C₂	#20276991	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
#20276991	2
C₂×#16810450	2
C₂×#16810448	2
C₂×C₄×#11719908	2
C₂×C₅₃×#499	2
C₂×#2253	53

Subgroups — C2×#20276991 #27539571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276991 #27539571