Subgroups — C₂×#20276587 #27537390

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₁₂	C₂	1	2	yes	maximal
3	#20276587	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810451	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810428	C₂	1	2	yes	maximal
6	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₆	—	2	2	no	—
9	C₂×Dic₄₂₄	C₄	1	8	yes	complemented
10	C₂×Dic₄₂₄	C₂²	1	4	yes	—
11	C₄×Dic₂₁₂	—	2	4	no	—
12	C₄×Dic₂₁₂	C₂²	1	8	yes	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133142	—	2	2	no	—
15	#16810451	C₂²	1	4	yes	—
16	#16810428	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#346	C₂²	1	4	yes	—
21	C₂²×Dic₁₀₆	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	4	no	—
24	C₂×Dic₂₁₂	C₂×C₄	1	8	yes	—
25	C₂×Dic₂₁₂	C₂³	1	4	yes	—
26	C₄×Dic₁₀₆	—	2	8	no	—
27	C₄×Dic₁₀₆	—	4	4	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×#11719919	—	4	2	no	—
30	C₂×C₄×Dic₅₃	—	4	2	no	—
31	Dic₄₂₄	C₂×C₄	1	16	yes	complemented
32	#13133151	—	2	2	no	—
33	#13133151	C₂³	1	4	yes	—
34	#13133142	—	2	8	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
42	C₄×C₈×C₅₃	D₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₅₃	D₈	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	—	4	2	no	—
47	C₂²×Dic₅₃	—	4	2	no	—
48	C₂×Dic₁₀₆	—	2	8	no	—
49	C₂×Dic₁₀₆	—	2	4	no	—
50	C₂×Dic₁₀₆	—	4	2	no	—
51	C₂×Dic₁₀₆	—	4	4	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	2	yes	—
57	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₄²×C₅₃	D₈	1	2	yes	—
59	C₄²×C₅₃	C₂×D₄	1	2	yes	—
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	#11719919	—	4	8	no	—
63	C₄×Dic₅₃	—	4	8	no	—
64	Dic₂₁₂	—	2	8	no	—
65	Dic₂₁₂	—	2	4	no	—
66	Dic₂₁₂	C₂²×C₄	1	8	yes	—
67	C₁₆×C₅₃	—	2	2	no	—
68	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
73	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#135	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×Dic₅₃	—	4	4	no	—
77	C₂×Dic₅₃	—	4	8	no	—
78	C₂×Dic₅₃	—	4	4	no	—
79	C₂×Dic₅₃	—	8	4	no	—
80	Dic₁₀₆	—	2	8	no	—
81	Dic₁₀₆	—	4	4	no	—
82	Dic₁₀₆	—	4	8	no	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	C₄×D₄	1	2	yes	—
85	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
86	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
87	C₂×#2176	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₄×D₈	1	2	yes	—
89	C₂²×C₅₃	#510	1	4	yes	—
90	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	C₄×D₈	1	2	yes	—
93	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
94	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
95	Dic₅₃	—	4	8	no	—
96	Dic₅₃	—	8	4	no	—
97	C₂²×Q₃₂	—	53	1	no	—
98	C₂×C₄×Q₁₆	—	53	2	no	—
99	#2176	—	53	8	no	—
100	C₂×#367	—	53	1	no	—
101	C₂×#358	—	53	2	no	—
102	C₂×#346	D₅₃	1	1	yes	characteristic
103	C₂×C₅₃	#2176	1	4	yes	complemented
104	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
105	C₂×C₅₃	C₂×#510	1	2	yes	—
106	C₂²×Q₁₆	—	53	2	no	—
107	C₂×C₄×Q₈	—	106	2	no	—
108	C₂×Q₃₂	—	53	8	no	—
109	C₂×Q₃₂	—	53	4	no	—
110	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
111	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
112	C₄×Q₁₆	—	53	8	no	—
113	C₄×Q₁₆	—	106	4	no	—
114	C₂×#107	—	53	2	no	—
115	C₂×#103	—	106	2	no	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	#367	—	53	4	no	—
118	#358	—	53	8	no	—
119	#346	C₂×D₅₃	1	4	yes	—
120	C₅₃	C₂×#2176	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	106	2	no	—
122	C₂×Q₁₆	—	53	8	no	—
123	C₂×Q₁₆	—	53	4	no	—
124	C₂×Q₁₆	—	106	2	no	—
125	C₂×Q₁₆	—	106	4	no	—
126	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₄×Q₈	—	106	8	no	—
129	C₄×Q₈	—	212	4	no	—
130	C₂×C₄⋊C₄	—	106	2	no	—
131	C₂×C₄⋊C₄	—	212	2	no	—
132	C₂×C₄²	—	212	2	no	—
133	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
134	Q₃₂	—	53	16	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
137	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
138	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
139	#107	—	53	4	no	—
140	#107	—	106	2	no	—
141	#103	—	106	8	no	—
142	C₄×C₈	D₂₁₂	1	2	yes	—
143	C₄×C₈	C₂²×D₅₃	1	2	yes	—
144	C₂×Q₈	—	106	8	no	—
145	C₂×Q₈	—	106	4	no	—
146	C₂×Q₈	—	212	2	no	—
147	C₂×Q₈	—	212	4	no	—
148	C₂²×C₄	—	212	2	no	—
149	C₂²×C₄	—	212	2	no	—
150	C₂²×C₄	D₄₂₄	1	2	yes	—
151	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
152	Q₁₆	—	53	8	no	—
153	Q₁₆	—	106	8	no	—
154	Q₁₆	—	106	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
157	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
158	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
159	C₂×C₈	#11719945	1	2	yes	—
160	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
161	C₄⋊C₄	—	106	4	no	—
162	C₄⋊C₄	—	212	2	no	—
163	C₄⋊C₄	—	212	8	no	—
164	C₄²	—	212	8	no	—
165	C₄²	D₄₂₄	1	2	yes	—
166	C₄²	C₂×D₂₁₂	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
169	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
170	Q₈	—	106	8	no	—
171	Q₈	—	212	4	no	—
172	Q₈	—	212	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	424	4	no	—
175	C₂×C₄	—	212	4	no	—
176	C₂×C₄	—	212	4	no	—
177	C₂×C₄	—	212	8	no	—
178	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
179	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
180	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
181	C₂×C₄	#13133239	1	2	yes	—
182	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	C₄×D₂₁₂	1	2	yes	—
185	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
186	C₈	C₂×#11719945	1	1	yes	characteristic
187	C₂²	C₄×D₄₂₄	1	2	yes	—
188	C₂²	#16811066	1	4	yes	—
189	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	424	4	no	—
192	C₄	—	212	8	no	—
193	C₄	C₄×D₄₂₄	1	2	yes	—
194	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
195	C₄	C₂×#13133239	1	1	yes	characteristic
196	C₂	#20276587	1	4	yes	direct product factor
197	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic
198	C₂	C₂×#16811066	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#2176
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #20276587	C₂	direct (N × H)
9. C₂×Dic₄₂₄	C₄	semidirect
31. Dic₄₂₄	C₂×C₄	semidirect
103. C₂×C₅₃	#2176	semidirect
120. C₅₃	C₂×#2176	semidirect
196. C₂	#20276587	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×Dic₄₂₄	2
C₂×C₄×Dic₂₁₂	2
#20276587	2
C₂×#16810451	2
C₂×#16810428	2
C₂×C₅₃×#346	2
C₂×#2176	53

Subgroups — C2×#20276587 #27537390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276587 #27537390