Subgroups — C₂×#20276592 #27537396

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₂₄	C₂	1	2	yes	maximal, complemented
3	#20276592	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810425	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₂₄	C₂²	1	2	yes	—
8	C₂×C₄×D₂₁₂	—	2	2	no	—
9	C₂×D₈₄₈	C₄	1	8	yes	complemented
10	C₂×D₈₄₈	C₂²	1	4	yes	—
11	C₄×D₄₂₄	—	2	4	no	—
12	C₄×D₄₂₄	C₂²	1	8	yes	complemented
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133139	—	2	2	no	—
15	#16810451	C₂²	1	4	yes	complemented
16	#16810425	C₂²	1	8	yes	complemented
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	complemented
21	C₂²×D₂₁₂	—	2	2	no	—
22	C₂²×C₄×D₅₃	—	4	2	no	—
23	C₂×D₄₂₄	—	2	2	no	—
24	C₂×D₄₂₄	—	2	4	no	—
25	C₂×D₄₂₄	C₂×C₄	1	8	yes	complemented
26	C₂×D₄₂₄	C₂³	1	4	yes	—
27	C₄×D₂₁₂	—	2	8	no	—
28	C₄×D₂₁₂	—	4	4	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×#11719921	—	4	2	no	—
31	D₈₄₈	C₂×C₄	1	16	yes	complemented
32	#13133151	—	2	2	no	—
33	#13133151	C₂³	1	4	yes	—
34	#13133139	—	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	complemented
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	C₂³×D₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	D₈	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	4	2	no	—
48	C₂×D₂₁₂	—	2	8	no	—
49	C₂×D₂₁₂	—	2	4	no	—
50	C₂×D₂₁₂	—	4	2	no	—
51	C₂×D₂₁₂	—	4	4	no	—
52	C₂×C₄×D₅₃	—	4	8	no	—
53	C₂×C₄×D₅₃	—	8	8	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	2	yes	—
59	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₄²×C₅₃	D₈	1	2	yes	—
61	C₄²×C₅₃	C₂×D₄	1	2	yes	—
62	#11719921	—	4	8	no	—
63	C₄⋊Dic₅₃	—	2	4	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	D₄₂₄	—	2	8	no	—
66	D₄₂₄	—	2	4	no	—
67	D₄₂₄	C₂²×C₄	1	8	yes	complemented
68	C₁₆×C₅₃	—	2	2	no	—
69	C₁₆×C₅₃	C₂²×C₄	1	4	yes	complemented
70	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₅₃	—	4	4	no	—
72	C₂²×D₅₃	—	4	8	no	—
73	C₂²×D₅₃	—	8	4	no	—
74	C₂²×D₅₃	—	8	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#135	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	8	4	no	—
83	D₂₁₂	—	2	8	no	—
84	D₂₁₂	—	4	4	no	—
85	D₂₁₂	—	4	8	no	—
86	C₄×D₅₃	—	8	16	no	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	C₄×D₄	1	2	yes	—
89	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
91	C₂×#2177	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	C₄×D₈	1	2	yes	—
93	C₂²×C₅₃	#509	1	4	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
95	C₂×D₅₃	—	4	8	no	—
96	C₂×D₅₃	—	8	4	no	—
97	C₂×D₅₃	—	8	4	no	—
98	C₂×D₅₃	—	8	16	no	—
99	C₄×C₅₃	—	2	4	no	—
100	C₄×C₅₃	C₄×D₈	1	2	yes	—
101	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
103	Dic₅₃	—	8	4	no	—
104	C₂²×D₁₆	—	53	1	no	—
105	C₂×C₄×D₈	—	53	2	no	—
106	#2177	—	53	8	no	—
107	C₂×#367	—	53	1	no	—
108	C₂×#357	—	53	2	no	—
109	C₂×#346	D₅₃	1	1	yes	characteristic, complemented
110	C₂×C₅₃	#2177	1	4	yes	complemented
111	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#509	1	2	yes	—
113	D₅₃	—	8	8	no	—
114	C₂²×D₈	—	53	2	no	—
115	C₂×C₄×D₄	—	106	2	no	—
116	C₂×D₁₆	—	53	8	no	—
117	C₂×D₁₆	—	53	4	no	—
118	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
119	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
120	C₄×D₈	—	53	8	no	—
121	C₄×D₈	—	106	4	no	—
122	C₂×#107	—	53	2	no	—
123	C₂×#102	—	106	2	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	#367	—	53	4	no	—
126	#357	—	53	8	no	—
127	#346	C₂×D₅₃	1	4	yes	complemented
128	C₅₃	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
129	C₂²×D₄	—	106	2	no	—
130	C₂³×C₄	—	212	2	no	—
131	C₂×D₈	—	53	8	no	—
132	C₂×D₈	—	53	4	no	—
133	C₂×D₈	—	106	2	no	—
134	C₂×D₈	—	106	4	no	—
135	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₄×D₄	—	106	8	no	—
138	C₄×D₄	—	212	4	no	—
139	C₂×C₄⋊C₄	—	106	2	no	—
140	C₂×C₂²⋊C₄	—	212	2	no	—
141	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
142	D₁₆	—	53	16	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
146	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
147	#107	—	53	4	no	—
148	#107	—	106	2	no	—
149	#102	—	106	8	no	—
150	C₄×C₈	D₂₁₂	1	2	yes	—
151	C₄×C₈	C₂²×D₅₃	1	2	yes	—
152	C₂⁴	—	212	2	no	—
153	C₂×D₄	—	106	8	no	—
154	C₂×D₄	—	106	4	no	—
155	C₂×D₄	—	212	2	no	—
156	C₂×D₄	—	212	4	no	—
157	C₂²×C₄	—	424	8	no	—
158	C₂²×C₄	—	212	2	no	—
159	C₂²×C₄	—	212	8	no	—
160	C₂²×C₄	D₄₂₄	1	2	yes	—
161	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
162	D₈	—	53	8	no	—
163	D₈	—	106	8	no	—
164	D₈	—	106	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
167	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
168	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
169	C₂×C₈	#11719945	1	2	yes	—
170	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
171	C₄⋊C₄	—	106	4	no	—
172	C₄⋊C₄	—	212	2	no	—
173	C₂²⋊C₄	—	212	8	no	—
174	C₄²	D₄₂₄	1	2	yes	—
175	C₄²	C₂×D₂₁₂	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂×C₄×D₅₃	1	4	yes	complemented
178	C₂³	—	424	4	no	—
179	C₂³	—	424	4	no	—
180	C₂³	—	212	4	no	—
181	C₂³	—	212	8	no	—
182	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
183	D₄	—	106	8	no	—
184	D₄	—	212	4	no	—
185	D₄	—	212	8	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	424	16	no	—
188	C₂×C₄	—	424	4	no	—
189	C₂×C₄	—	212	4	no	—
190	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
191	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
192	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
193	C₂×C₄	#13133239	1	2	yes	—
194	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	C₄×D₂₁₂	1	2	yes	—
197	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
198	C₈	C₂×#11719945	1	1	yes	characteristic
199	C₂²	—	424	4	no	—
200	C₂²	—	424	4	no	—
201	C₂²	—	424	16	no	—
202	C₂²	—	212	8	no	—
203	C₂²	C₄×D₄₂₄	1	2	yes	—
204	C₂²	#16811061	1	4	yes	—
205	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	424	4	no	—
208	C₄	C₄×D₄₂₄	1	2	yes	—
209	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
210	C₄	C₂×#13133239	1	1	yes	characteristic
211	C₂	—	424	8	no	—
212	C₂	#20276592	1	4	yes	direct product factor
213	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic
214	C₂	C₂×#16811061	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	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₈₄₈	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#2177
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
2. C₂×C₄×D₄₂₄	C₂	semidirect
3. #20276592	C₂	direct (N × H)
4. C₂×#16810451	C₂	semidirect
5. C₂×#16810425	C₂	semidirect
6. C₂×C₅₃×#346	C₂	semidirect
9. C₂×D₈₄₈	C₄	semidirect
12. C₄×D₄₂₄	C₂²	semidirect
15. #16810451	C₂²	semidirect
16. #16810425	C₂²	semidirect
20. C₅₃×#346	C₂²	semidirect
25. C₂×D₄₂₄	C₂×C₄	semidirect
31. D₈₄₈	C₂×C₄	semidirect
39. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
67. D₄₂₄	C₂²×C₄	semidirect
69. C₁₆×C₅₃	C₂²×C₄	semidirect
109. C₂×#346	D₅₃	semidirect
110. C₂×C₅₃	#2177	semidirect
127. #346	C₂×D₅₃	semidirect
128. C₅₃	C₂×#2177	semidirect
144. C₂×C₁₆	C₄×D₅₃	semidirect
177. C₁₆	C₂×C₄×D₅₃	semidirect
212. C₂	#20276592	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×D₈₄₈	2
C₂×C₄×D₄₂₄	2
#20276592	2
C₂×#16810451	2
C₂×#16810425	2
C₂×C₅₃×#346	2
C₂×#2177	53

Subgroups — C2×#20276592 #27537396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276592 #27537396