Subgroups — C₂×#20289918 #28313749

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811639	C₂	1	2	yes	maximal
2	#20289918	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16811464	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#88324	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₄	1	8	yes	—
7	#16811639	C₂²	1	8	yes	—
8	C₂×C₄⋊#88324	C₄	1	4	yes	—
9	C₂×C₂²⋊#88324	C₄	1	4	yes	—
10	C₂×C₄×#88324	C₂²	1	4	yes	—
11	#16811464	C₂²	1	8	yes	complemented
12	C₈×#88324	C₂²	1	8	yes	complemented
13	C₂³×#88322	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₄	1	4	yes	—
17	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
21	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
22	C₂²×#88324	C₂×C₄	1	8	yes	—
23	C₄⋊#88324	C₂×C₄	1	16	yes	—
24	C₂²⋊#88324	C₂×C₄	1	16	yes	—
25	C₄×#88324	C₂×C₄	1	8	yes	—
26	C₄×#88324	C₂³	1	8	yes	—
27	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₈	1	8	yes	complemented
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂²×#88322	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
39	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
40	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
41	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
42	C₈×Dic₅₃	C₂³	1	2	yes	—
43	C₈×Dic₅₃	C₂³	1	2	yes	—
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₈	1	8	yes	complemented
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂²×Dic₅₃	—	2	8	no	—
57	C₂²×Dic₅₃	C₄²	1	2	yes	—
58	C₂²×Dic₅₃	C₄²	1	4	yes	—
59	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
60	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
61	C₂×#88324	—	2	16	no	—
62	C₂×#88324	C₂²×C₄	1	16	yes	—
63	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	complemented
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₄×Dic₅₃	C₄²	1	2	yes	—
69	C₄×Dic₅₃	C₄²	1	4	yes	—
70	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
71	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
73	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
74	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
75	C₂×#88322	C₂×C₈	1	4	yes	—
76	C₂×#88322	C₂×C₈	1	4	yes	—
77	C₂×#88322	C₂×C₈	1	16	yes	complemented
78	C₂×#88322	C₂²×C₄	1	2	yes	—
79	C₂×#88322	C₂²×C₄	1	2	yes	—
80	C₂³×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	C₄×C₈	1	4	yes	—
82	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
83	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
85	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
92	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×Dic₅₃	—	2	4	no	—
94	C₂×Dic₅₃	—	2	8	no	—
95	C₂×Dic₅₃	—	2	8	no	—
96	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
97	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
98	C₂×Dic₅₃	#131	1	8	yes	—
99	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
100	#88324	—	2	16	no	—
101	C₈×C₅₃	C₂²×C₈	1	4	yes	—
102	C₈×C₅₃	C₂²×C₈	1	4	yes	—
103	#88322	C₂²×C₈	1	4	yes	—
104	#88322	C₂²×C₈	1	4	yes	—
105	C₂×#1450	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	2	8	no	—
108	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
109	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
110	C₂²×C₅₃	#405	1	4	yes	—
111	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
113	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
114	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
116	Dic₅₃	—	2	4	no	—
117	Dic₅₃	#405	1	4	yes	—
118	Dic₅₃	C₂×#131	1	4	yes	—
119	C₂×#431	—	53	2	no	—
120	C₂²×C₄×C₈	—	53	1	no	—
121	#1450	—	53	8	no	—
122	C₂×#322	—	53	2	no	—
123	C₂×C₈²	—	53	2	no	—
124	C₂×C₅₃	—	2	4	no	—
125	C₂×C₅₃	#1450	1	4	yes	complemented
126	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#405	1	2	yes	—
128	C₂³×C₈	—	53	1	no	—
129	C₂³×C₈	#6157	1	1	yes	characteristic
130	C₂²×C₄²	—	53	1	no	—
131	#431	—	53	8	no	—
132	#431	—	53	8	no	—
133	C₂×C₄⋊C₈	—	53	4	no	—
134	C₂×C₂²⋊C₈	—	53	4	no	—
135	C₂×C₄×C₈	—	53	4	no	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	4	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	—	53	4	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	—	53	1	no	—
142	#322	—	53	8	no	—
143	C₈²	—	53	8	no	—
144	C₅₃	C₂×#1450	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	—	53	2	no	—
146	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
147	C₂²×C₈	—	53	1	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	53	4	no	—
150	C₂²×C₈	—	53	8	no	—
151	C₂²×C₈	—	53	8	no	—
152	C₂²×C₈	#88324	1	8	yes	complemented
153	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
154	C₂²×C₈	C₂×#6157	1	4	yes	—
155	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
156	C₂×C₄²	—	53	2	no	—
157	C₂×C₄²	—	53	4	no	—
158	C₂×C₄²	—	53	2	no	—
159	C₂×C₄²	—	53	1	no	—
160	C₂×C₄²	—	53	1	no	—
161	C₂×C₄²	—	53	2	no	—
162	C₄⋊C₈	—	53	16	no	—
163	C₂²⋊C₈	—	53	16	no	—
164	C₄×C₈	—	53	8	no	—
165	C₄×C₈	—	53	8	no	—
166	C₄×C₈	—	53	2	no	—
167	C₄×C₈	—	53	2	no	—
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	—	53	4	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	—	53	4	no	—
172	C₂⁴	C₄×#6157	1	1	yes	characteristic
173	C₂²×C₄	—	53	2	no	—
174	C₂²×C₄	—	53	4	no	—
175	C₂²×C₄	—	53	2	no	—
176	C₂²×C₄	—	53	4	no	—
177	C₂²×C₄	—	106	8	no	—
178	C₂²×C₄	D₅₃⋊C₈	1	4	yes	—
179	C₂²×C₄	C₄×#6157	1	2	yes	—
180	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
181	C₂²×C₄	C₂×#88324	1	4	yes	—
182	C₂²×C₄	C₂²×#6157	1	2	yes	—
183	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
184	C₂×C₈	—	53	4	no	—
185	C₂×C₈	—	53	4	no	—
186	C₂×C₈	—	53	16	no	—
187	C₂×C₈	—	53	2	no	—
188	C₂×C₈	—	53	2	no	—
189	C₂×C₈	—	53	16	no	—
190	C₂×C₈	—	106	16	no	—
191	C₂×C₈	C₂×#88324	1	4	yes	—
192	C₂×C₈	C₂×#88324	1	4	yes	—
193	C₂×C₈	C₂×#88324	1	16	yes	complemented
194	C₂×C₈	C₂²×#6157	1	2	yes	—
195	C₂×C₈	C₂²×#6157	1	2	yes	—
196	C₄²	—	53	2	no	—
197	C₄²	—	53	4	no	—
198	C₄²	—	53	4	no	—
199	C₄²	—	53	2	no	—
200	C₄²	—	53	1	no	—
201	C₄²	—	53	1	no	—
202	C₄²	—	53	2	no	—
203	C₂³	—	2	4	no	—
204	C₂³	C₄×#88324	1	4	yes	—
205	C₂³	C₂×C₄×#6157	1	2	yes	—
206	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
207	C₂×C₄	—	53	8	no	—
208	C₂×C₄	—	53	2	no	—
209	C₂×C₄	—	53	2	no	—
210	C₂×C₄	—	53	4	no	—
211	C₂×C₄	—	106	8	no	—
212	C₂×C₄	—	106	8	no	—
213	C₂×C₄	—	106	4	no	—
214	C₂×C₄	C₄×#88324	1	8	yes	—
215	C₂×C₄	C₄×#88324	1	4	yes	—
216	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
218	C₂×C₄	C₂²×#88324	1	2	yes	—
219	C₂×C₄	C₂²×#88324	1	4	yes	—
220	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
221	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
222	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
223	C₈	—	53	4	no	—
224	C₈	—	53	4	no	—
225	C₈	—	106	16	no	—
226	C₈	C₂²×#88324	1	4	yes	—
227	C₈	C₂²×#88324	1	4	yes	—
228	C₂²	—	2	8	no	—
229	C₂²	—	2	4	no	—
230	C₂²	C₂×C₄×#88324	1	4	yes	—
231	C₂²	C₂×C₄×#88324	1	2	yes	—
232	C₂²	#16811586	1	4	yes	—
233	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
234	C₄	—	53	4	no	—
235	C₄	—	53	4	no	—
236	C₄	—	106	4	no	—
237	C₄	C₂×C₄×#88324	1	2	yes	—
238	C₄	C₂×C₄×#88324	1	4	yes	—
239	C₄	C₂³×#88324	1	1	yes	characteristic
240	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
241	C₂	—	2	4	no	—
242	C₂	#20289918	1	4	yes	direct product factor
243	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
244	C₂	C₂×#16811586	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂³×C₈	#6157
hypocenter	C₅₃	C₂×#1450
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #20289918	C₂	direct (N × H)
3. C₂×#16811464	C₂	semidirect
4. C₂×C₈×#88324	C₂	semidirect
11. #16811464	C₂²	semidirect
12. C₈×#88324	C₂²	semidirect
28. C₂²×#88322	C₈	semidirect
45. C₂²×C₈×C₅₃	C₈	semidirect
65. C₂×C₈×C₅₃	C₂×C₈	semidirect
77. C₂×#88322	C₂×C₈	semidirect
125. C₂×C₅₃	#1450	semidirect
144. C₅₃	C₂×#1450	semidirect
152. C₂²×C₈	#88324	semidirect
193. C₂×C₈	C₂×#88324	semidirect
242. C₂	#20289918	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×#16811639	2
#20289918	2
C₂×#16811464	2
C₂×C₈×#88324	2
C₂²×C₈×Dic₅₃	2
C₂×#1450	53

Subgroups — C2×#20289918 #28313749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289918 #28313749