Subgroups — C₂×#20272634 #27293079

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272634	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₄	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₄×#11719908	C₄	1	2	yes	—
9	C₄×#11719908	C₂²	1	2	yes	—
10	C₈×#88322	C₄	1	2	yes	—
11	C₈×#88322	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
15	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×#88322	C₈	1	2	yes	—
17	C₂²×#88322	C₈	1	2	yes	—
18	C₂×#11719908	—	2	4	no	—
19	C₂×#11719908	C₂×C₄	1	4	yes	—
20	C₄×#88322	C₈	1	2	yes	—
21	C₄×#88322	C₈	1	2	yes	—
22	C₄×#88322	C₂×C₄	1	2	yes	—
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	—	2	4	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
39	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₄²	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×#88322	—	2	4	no	—
47	C₂×#88322	C₁₆	1	8	yes	complemented
48	C₂×#88322	C₂×C₈	1	4	yes	—
49	C₂×#88322	C₂×C₈	1	4	yes	—
50	C₁₆×C₅₃	—	2	4	no	—
51	#11719908	—	2	4	no	—
52	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#97	1	2	yes	—
57	C₂×C₄×C₅₃	#97	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
59	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
60	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₅₃	#97	1	2	yes	—
64	C₈×C₅₃	#97	1	2	yes	—
65	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
66	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
67	#88322	—	2	4	no	—
68	#88322	C₂×C₁₆	1	8	yes	complemented
69	C₂×#1312	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	#322	1	2	yes	—
71	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
72	C₂²×C₅₃	#346	1	2	yes	—
73	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₅₃	#322	1	2	yes	—
75	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
76	C₄×C₅₃	#346	1	2	yes	—
77	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
78	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
79	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
80	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
81	C₂×C₄×C₁₆	—	53	1	no	—
82	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
83	C₂×C₈²	—	53	1	no	—
84	#1312	—	53	4	no	—
85	C₂×C₅₃	#1312	1	4	yes	complemented
86	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
87	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
89	C₂²×C₁₆	—	53	2	no	—
90	C₂²×C₁₆	Dic₅₃	1	2	yes	—
91	C₂×C₄×C₈	—	53	1	no	—
92	C₂×C₄×C₈	—	53	1	no	—
93	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
94	C₄×C₁₆	—	53	2	no	—
95	C₄×C₁₆	—	53	2	no	—
96	C₄×C₁₆	Dic₅₃	1	2	yes	—
97	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
98	C₈²	—	53	2	no	—
99	C₈²	—	53	2	no	—
100	C₅₃	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	—	53	2	no	—
102	C₂²×C₈	—	53	2	no	—
103	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
105	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
106	C₂×C₁₆	—	2	4	no	—
107	C₂×C₁₆	—	53	4	no	—
108	C₂×C₁₆	—	106	4	no	—
109	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
110	C₄×C₈	—	53	2	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	—	53	2	no	—
113	C₄×C₈	—	53	2	no	—
114	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
119	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
120	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
121	C₂×C₈	—	53	8	no	—
122	C₂×C₈	—	53	4	no	—
123	C₂×C₈	—	53	4	no	—
124	C₂×C₈	—	106	4	no	—
125	C₂×C₈	#11719912	1	4	yes	—
126	C₂×C₈	#11719917	1	4	yes	—
127	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
128	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
131	C₄²	C₈×D₅₃	1	1	yes	characteristic
132	C₄²	C₈×D₅₃	1	1	yes	characteristic
133	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
134	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
135	C₁₆	—	2	4	no	—
136	C₁₆	—	106	4	no	—
137	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
138	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
139	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
141	C₂×C₄	#13133127	1	2	yes	—
142	C₂×C₄	#13133128	1	2	yes	—
143	C₂×C₄	C₁₆×D₅₃	1	4	yes	—
144	C₂×C₄	#13133156	1	4	yes	—
145	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
148	C₈	—	53	8	no	—
149	C₈	—	106	4	no	—
150	C₈	#13133127	1	2	yes	—
151	C₈	#13133128	1	2	yes	—
152	C₈	C₂×#11719912	1	2	yes	—
153	C₈	C₂×#11719917	1	2	yes	—
154	C₂²	#16810345	1	2	yes	—
155	C₂²	C₁₆×Dic₅₃	1	2	yes	—
156	C₂²	#16810398	1	2	yes	—
157	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
158	C₄	#16810345	1	2	yes	—
159	C₄	C₁₆×Dic₅₃	1	2	yes	—
160	C₄	#16810398	1	2	yes	—
161	C₄	C₂×#13133127	1	1	yes	characteristic
162	C₄	C₂×#13133128	1	1	yes	characteristic
163	C₄	C₂×C₁₆×D₅₃	1	2	yes	—
164	C₄	C₂×#13133156	1	2	yes	—
165	C₂	#20272634	1	4	yes	direct product factor
166	C₂	C₂×#16810345	1	1	yes	characteristic
167	C₂	C₂×C₁₆×Dic₅₃	1	1	yes	characteristic
168	C₂	C₂×#16810398	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1312
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20272634	C₂	direct (N × H)
47. C₂×#88322	C₁₆	semidirect
68. #88322	C₂×C₁₆	semidirect
85. C₂×C₅₃	#1312	semidirect
100. C₅₃	C₂×#1312	semidirect
165. C₂	#20272634	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×C₄×#11719908	2
C₂×C₈×#88322	2
#20272634	2
C₂×C₄×C₁₆×C₅₃	2
C₂×#1312	53

Subgroups — C2×#20272634 #27293079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272634 #27293079