Subgroups — C₂×#20276620 #27537576

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20276620	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810441	C₂	1	2	yes	maximal
4	C₂×#16810398	C₂	1	2	yes	maximal
5	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₀₆	C₄	1	4	yes	—
11	C₈×Dic₁₀₆	C₂²	1	4	yes	—
12	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
13	C₂×#13133146	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	#16810441	C₂²	1	8	yes	—
16	#16810398	C₂²	1	8	yes	—
17	C₄×#11719908	C₂²	1	4	yes	—
18	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	—
21	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
22	C₂²×#88322	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₀₆	C₈	1	4	yes	—
24	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
26	C₂×#11719919	C₈	1	2	yes	—
27	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	Q₈	1	4	yes	complemented
30	C₂×#11719908	C₂³	1	2	yes	—
31	C₂×#11719908	C₂³	1	2	yes	—
32	C₄⋊#88322	C₂×C₄	1	4	yes	—
33	#13133146	C₂×C₄	1	8	yes	—
34	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
35	C₈×Dic₅₃	C₂³	1	4	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	—	2	4	no	—
40	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
47	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
48	C₂×Dic₁₀₆	—	2	4	no	—
49	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₃	#41	1	2	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×C₈	1	2	yes	—
57	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
58	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
59	#11719919	C₂×C₈	1	8	yes	—
60	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
61	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	C₂²×C₄	1	4	yes	—
64	C₁₆×C₅₃	—	2	4	no	—
65	#11719908	—	2	2	no	—
66	#11719908	C₂×Q₈	1	4	yes	complemented
67	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
70	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₅₃	#131	1	2	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	4	no	—
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
79	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
80	Dic₁₀₆	—	2	8	no	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	C₄×Q₈	1	2	yes	—
83	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
85	#88322	—	2	4	no	—
86	C₂×#2183	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
88	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
89	C₂²×C₅₃	#504	1	2	yes	—
90	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	C₈×Q₈	1	2	yes	—
93	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
94	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
96	Dic₅₃	—	2	4	no	—
97	Dic₅₃	—	2	4	no	—
98	C₂×C₈×Q₈	—	53	1	no	—
99	#2183	—	53	8	no	—
100	C₂×C₄⋊C₁₆	—	53	2	no	—
101	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
102	C₂×#346	—	53	2	no	—
103	C₂×C₄×C₁₆	—	53	1	no	—
104	C₂×C₅₃	#2183	1	4	yes	complemented
105	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
106	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
108	C₂×C₄×Q₈	—	53	1	no	—
109	C₂²×C₁₆	—	53	1	no	—
110	C₂²×C₁₆	—	53	1	no	—
111	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
112	C₈×Q₈	—	53	4	no	—
113	C₈×Q₈	—	53	4	no	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₂×C₄⋊C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	2	no	—
117	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₄⋊C₁₆	—	53	8	no	—
119	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	—
120	#346	—	53	8	no	—
121	C₄×C₁₆	—	53	4	no	—
122	C₅₃	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	53	1	no	—
124	C₂²×C₈	—	53	2	no	—
125	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	C₄×Q₈	—	53	4	no	—
128	C₄×Q₈	—	53	4	no	—
129	C₂×C₄⋊C₄	—	53	2	no	—
130	C₂×C₄⋊C₄	—	53	1	no	—
131	C₂×C₄²	—	53	2	no	—
132	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₁₆	—	2	4	no	—
134	C₂×C₁₆	—	53	4	no	—
135	C₂×C₁₆	—	53	2	no	—
136	C₂×C₁₆	—	53	2	no	—
137	C₂×C₁₆	—	106	2	no	—
138	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
139	C₄⋊C₈	—	53	8	no	—
140	C₄⋊C₈	—	53	4	no	—
141	C₄×C₈	—	53	4	no	—
142	C₄×C₈	—	53	4	no	—
143	C₄×C₈	C₄×D₅₃	1	2	yes	—
144	C₄×C₈	C₂²×D₅₃	1	2	yes	—
145	C₂×Q₈	—	53	4	no	—
146	C₂×Q₈	—	106	4	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	—	53	2	no	—
149	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	53	4	no	—
154	C₂×C₈	—	106	4	no	—
155	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₂×C₈	#11719947	1	2	yes	—
158	C₂×C₈	Q₈×D₅₃	1	2	yes	—
159	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
160	C₄⋊C₄	—	53	8	no	—
161	C₄⋊C₄	—	53	4	no	—
162	C₄²	—	53	4	no	—
163	C₄²	—	53	4	no	—
164	C₄²	C₈×D₅₃	1	2	yes	—
165	C₄²	C₂×C₄×D₅₃	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₁₆	—	53	4	no	—
168	C₁₆	—	106	2	no	—
169	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
170	Q₈	—	106	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	53	4	no	—
173	C₂×C₄	—	53	4	no	—
174	C₂×C₄	—	106	4	no	—
175	C₂×C₄	—	106	4	no	—
176	C₂×C₄	#13133156	1	4	yes	—
177	C₂×C₄	#13133182	1	2	yes	—
178	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
180	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
181	C₂×C₄	#13133225	1	2	yes	—
182	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	106	4	no	—
185	C₈	#13133182	1	2	yes	—
186	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
187	C₈	C₂×#11719947	1	1	yes	characteristic
188	C₂²	#16810747	1	2	yes	—
189	C₂²	C₂×#13133156	1	2	yes	—
190	C₂²	#16811041	1	2	yes	—
191	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	106	4	no	—
194	C₄	—	106	4	no	—
195	C₄	#16810747	1	2	yes	—
196	C₄	C₂×#13133156	1	4	yes	—
197	C₄	C₂×#13133182	1	1	yes	characteristic
198	C₄	C₂×#13133225	1	1	yes	characteristic
199	C₂	#20276620	1	4	yes	direct product factor
200	C₂	C₂×#16810747	1	1	yes	characteristic
201	C₂	C₂²×#13133156	1	1	yes	characteristic
202	C₂	C₂×#16811041	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2183
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276620	C₂	direct (N × H)
29. C₂×#11719908	Q₈	semidirect
66. #11719908	C₂×Q₈	semidirect
104. C₂×C₅₃	#2183	semidirect
122. C₅₃	C₂×#2183	semidirect
199. C₂	#20276620	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×C₈×Dic₁₀₆	2
#20276620	2
C₂×#16810441	2
C₂×#16810398	2
C₂×C₄×#11719908	2
C₂×C₅₃×C₄⋊C₁₆	2
C₂×#2183	53

Subgroups — C2×#20276620 #27537576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276620 #27537576