Subgroups — C₂×#20272997 #27349669

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810674	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272997	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810345	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	C₄	1	2	yes	—
8	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	#16810674	C₄	1	8	yes	—
10	#16810674	C₂²	1	8	yes	—
11	C₂×#13133146	C₄	1	4	yes	—
12	C₂×#13133129	C₄	1	4	yes	—
13	C₂×C₄×#88322	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
16	C₄×C₈×D₅₃	C₄	1	4	yes	—
17	C₄×C₈×D₅₃	C₂²	1	4	yes	—
18	#16810345	C₂²	1	8	yes	complemented
19	C₈×#88322	C₂²	1	4	yes	complemented
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₈²×C₅₃	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₅₃	C₈	1	16	yes	complemented
28	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
31	C₄²×D₅₃	C₂×C₄	1	2	yes	—
32	C₄²×D₅₃	C₂×C₄	1	4	yes	—
33	C₄²×D₅₃	C₂³	1	2	yes	—
34	#13133146	C₂×C₄	1	16	yes	—
35	#13133129	C₂×C₄	1	16	yes	—
36	C₄×#88322	C₂×C₄	1	2	yes	—
37	C₄×#88322	C₂×C₄	1	4	yes	—
38	C₄×#88322	C₂³	1	2	yes	—
39	C₄×#88322	C₂³	1	4	yes	—
40	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₅₃	C₂³	1	2	yes	—
47	C₄×C₈×C₅₃	C₂³	1	4	yes	—
48	C₂³×D₅₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂²×Dic₅₃	C₄²	1	2	yes	—
52	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	8	no	—
54	C₂×C₄×D₅₃	C₄²	1	4	yes	—
55	C₂×C₄×D₅₃	C₄²	1	2	yes	—
56	C₂×C₄×D₅₃	C₂×C₈	1	8	yes	—
57	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	—	2	8	no	—
60	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
63	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₄×Dic₅₃	C₄²	1	2	yes	—
67	C₄×Dic₅₃	C₄²	1	4	yes	—
68	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
69	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
70	C₂×#88322	—	2	8	no	—
71	C₂×#88322	C₂×C₈	1	8	yes	—
72	C₂×#88322	C₂²×C₄	1	8	yes	—
73	C₂×#88322	C₂²×C₄	1	4	yes	—
74	C₈×D₅₃	C₂×C₈	1	32	yes	complemented
75	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂²×D₅₃	—	2	4	no	—
77	C₂²×D₅₃	C₄×C₈	1	4	yes	—
78	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
82	C₂×C₄×C₅₃	#131	1	8	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
85	C₂×Dic₅₃	—	2	4	no	—
86	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
87	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
88	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
89	C₄×D₅₃	—	2	8	no	—
90	C₄×D₅₃	—	2	8	no	—
91	C₄×D₅₃	C₄×C₈	1	8	yes	—
92	C₄×D₅₃	C₂²×C₈	1	8	yes	—
93	C₈×C₅₃	—	2	8	no	—
94	C₈×C₅₃	C₂²×C₈	1	8	yes	—
95	#88322	—	2	8	no	—
96	#88322	C₂²×C₈	1	8	yes	—
97	C₂×#1450	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
99	C₂²×C₅₃	#405	1	4	yes	—
100	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
101	C₂×D₅₃	—	2	4	no	—
102	C₂×D₅₃	—	2	8	no	—
103	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
104	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
105	C₄×C₅₃	#405	1	4	yes	—
106	C₄×C₅₃	C₂³×C₈	1	2	yes	—
107	C₄×C₅₃	C₂×#131	1	4	yes	—
108	Dic₅₃	—	2	4	no	—
109	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
110	C₂×#431	—	53	2	no	—
111	C₂²×C₄×C₈	—	53	1	no	—
112	#1450	—	53	8	no	—
113	C₂×#322	—	53	2	no	—
114	C₂×C₈²	—	53	1	no	—
115	C₂×C₈²	D₅₃	1	1	yes	characteristic, complemented
116	C₂×C₅₃	#1450	1	4	yes	complemented
117	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#405	1	2	yes	—
119	D₅₃	—	2	4	no	—
120	C₂³×C₈	—	53	2	no	—
121	C₂²×C₄²	—	53	1	no	—
122	#431	—	53	8	no	—
123	#431	—	53	8	no	—
124	C₂×C₄⋊C₈	—	53	4	no	—
125	C₂×C₂²⋊C₈	—	53	4	no	—
126	C₂×C₄×C₈	—	53	2	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₂×C₄×C₈	—	53	2	no	—
129	C₂×C₄×C₈	—	53	4	no	—
130	C₂×C₄×C₈	—	53	4	no	—
131	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	#322	—	53	8	no	—
134	C₈²	—	53	4	no	—
135	C₈²	C₂×D₅₃	1	4	yes	complemented
136	C₅₃	C₂×#1450	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	53	1	no	—
138	C₂³×C₄	—	53	2	no	—
139	C₂²×C₈	—	53	2	no	—
140	C₂²×C₈	—	53	8	no	—
141	C₂²×C₈	—	53	16	no	—
142	C₂²×C₈	—	53	4	no	—
143	C₂²×C₈	C₄×D₅₃	1	4	yes	—
144	C₂²×C₈	C₄×D₅₃	1	2	yes	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	53	2	no	—
147	C₂×C₄²	—	53	2	no	—
148	C₂×C₄²	—	53	4	no	—
149	C₂×C₄²	—	53	2	no	—
150	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
151	C₄⋊C₈	—	53	16	no	—
152	C₂²⋊C₈	—	53	16	no	—
153	C₄×C₈	—	53	8	no	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	53	4	no	—
156	C₄×C₈	—	53	4	no	—
157	C₄×C₈	—	53	2	no	—
158	C₄×C₈	C₄×D₅₃	1	2	yes	—
159	C₄×C₈	C₄×D₅₃	1	4	yes	—
160	C₄×C₈	C₂²×D₅₃	1	2	yes	—
161	C₄×C₈	C₂²×D₅₃	1	4	yes	—
162	C₂⁴	—	53	1	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	2	no	—
165	C₂²×C₄	—	53	8	no	—
166	C₂²×C₄	—	53	2	no	—
167	C₂²×C₄	—	53	2	no	—
168	C₂²×C₄	—	53	4	no	—
169	C₂²×C₄	—	53	4	no	—
170	C₂²×C₄	—	106	8	no	—
171	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
173	C₂×C₈	—	2	8	no	—
174	C₂×C₈	—	53	8	no	—
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	53	32	no	—
177	C₂×C₈	—	53	8	no	—
178	C₂×C₈	—	106	8	no	—
179	C₂×C₈	C₈×D₅₃	1	8	yes	—
180	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
181	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
182	C₄²	—	53	4	no	—
183	C₄²	—	53	2	no	—
184	C₄²	—	53	2	no	—
185	C₄²	—	53	4	no	—
186	C₄²	C₂×C₄×D₅₃	1	2	yes	—
187	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₄²	C₂³×D₅₃	1	1	yes	characteristic
189	C₂³	—	53	2	no	—
190	C₂³	—	53	4	no	—
191	C₂³	—	106	4	no	—
192	C₂³	C₄²×D₅₃	1	1	yes	characteristic
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	4	no	—
195	C₂×C₄	—	53	8	no	—
196	C₂×C₄	—	53	8	no	—
197	C₂×C₄	—	53	4	no	—
198	C₂×C₄	—	106	4	no	—
199	C₂×C₄	—	106	8	no	—
200	C₂×C₄	—	106	8	no	—
201	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
202	C₂×C₄	C₄²×D₅₃	1	2	yes	—
203	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
204	C₂×C₄	#13133223	1	8	yes	—
205	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
207	C₈	—	2	8	no	—
208	C₈	—	53	8	no	—
209	C₈	—	106	8	no	—
210	C₈	C₂×C₈×D₅₃	1	8	yes	—
211	C₂²	—	53	4	no	—
212	C₂²	—	106	8	no	—
213	C₂²	—	106	4	no	—
214	C₂²	C₄×C₈×D₅₃	1	2	yes	—
215	C₂²	#16810577	1	4	yes	—
216	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
217	C₄	—	53	4	no	—
218	C₄	—	106	4	no	—
219	C₄	C₄×C₈×D₅₃	1	2	yes	—
220	C₄	#16810577	1	4	yes	—
221	C₄	C₂²×C₈×D₅₃	1	2	yes	—
222	C₄	C₂×#13133223	1	4	yes	—
223	C₂	—	106	4	no	—
224	C₂	#20272997	1	4	yes	direct product factor
225	C₂	C₂×C₄×C₈×D₅₃	1	1	yes	characteristic
226	C₂	C₂×#16810577	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	C₂×#1450
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #20272997	C₂	direct (N × H)
4. C₂×#16810345	C₂	semidirect
5. C₂×C₈×#88322	C₂	semidirect
6. C₂×C₈²×C₅₃	C₂	semidirect
18. #16810345	C₂²	semidirect
19. C₈×#88322	C₂²	semidirect
22. C₈²×C₅₃	C₂²	semidirect
27. C₂×C₈×D₅₃	C₈	semidirect
74. C₈×D₅₃	C₂×C₈	semidirect
115. C₂×C₈²	D₅₃	semidirect
116. C₂×C₅₃	#1450	semidirect
135. C₈²	C₂×D₅₃	semidirect
136. C₅₃	C₂×#1450	semidirect
224. C₂	#20272997	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×#16810674	2
C₂×C₄×C₈×D₅₃	2
#20272997	2
C₂×#16810345	2
C₂×C₈×#88322	2
C₂×C₈²×C₅₃	2
C₂×#1450	53

Subgroups — C2×#20272997 #27349669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272997 #27349669