Subgroups — C₂×#20289775 #28310276

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133288	C₂	1	2	yes	maximal
2	#20289775	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719936	—	2	2	no	—
5	C₂×#13133288	—	2	4	no	—
6	C₂×#13133288	—	2	2	no	—
7	C₂×#13133288	C₄	1	8	yes	—
8	C₂×#13133288	C₂²	1	4	yes	—
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₅₃	—	2	4	no	—
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#11719936	—	2	8	no	—
16	C₂×#11719936	—	2	4	no	—
17	C₂×#11719936	—	4	2	no	—
18	C₂×#11719936	—	4	4	no	—
19	C₂²×#88324	—	4	2	no	—
20	#13133288	—	2	8	no	—
21	#13133288	—	2	4	no	—
22	#13133288	C₂×C₄	1	8	yes	—
23	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂²×#88322	Q₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₃	D₄	1	4	yes	—
28	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	C₂×C₄	1	2	yes	—
33	C₁₆×D₅₃	—	2	4	no	—
34	C₁₆×D₅₃	—	2	4	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
41	C₂×C₄×D₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
44	C₂×C₄×D₅₃	SD₁₆	1	4	yes	—
45	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×#88324	—	4	8	no	—
48	C₂×#88324	—	4	4	no	—
49	#11719936	—	2	8	no	—
50	#11719936	—	4	8	no	—
51	#11719936	—	4	4	no	—
52	C₂×C₈×C₅₃	C₄²	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#88322	C₄⋊C₄	1	4	yes	—
58	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
59	C₈×D₅₃	C₄²	1	2	yes	—
60	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
61	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
62	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
63	C₈×D₅₃	C₂²×C₄	1	2	yes	—
64	C₈×D₅₃	C₂×D₄	1	2	yes	—
65	C₁₆×C₅₃	—	2	2	no	—
66	#11719908	—	2	2	no	—
67	C₂³×C₅₃	#107	1	1	yes	characteristic
68	C₂²×D₅₃	—	2	2	no	—
69	C₂²×D₅₃	—	2	2	no	—
70	C₂²×D₅₃	#103	1	4	yes	—
71	C₂²×D₅₃	#107	1	1	yes	characteristic
72	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#95	1	2	yes	—
74	C₂×C₄×C₅₃	#106	1	2	yes	—
75	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	#102	1	4	yes	—
78	C₂×Dic₅₃	#107	1	1	yes	characteristic
79	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
80	C₄×D₅₃	#95	1	2	yes	—
81	C₄×D₅₃	#102	1	4	yes	—
82	C₄×D₅₃	#103	1	4	yes	—
83	C₄×D₅₃	#106	1	2	yes	—
84	C₄×D₅₃	C₂×C₂²⋊C₄	1	2	yes	—
85	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
86	#88324	—	4	8	no	—
87	C₈×C₅₃	#95	1	2	yes	—
88	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
89	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	#88322	#95	1	2	yes	—
91	#88322	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×#1381	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	#340	1	2	yes	—
94	C₂²×C₅₃	#368	1	2	yes	—
95	C₂²×C₅₃	#368	1	2	yes	—
96	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
97	C₂×D₅₃	—	2	8	no	—
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	—	2	1	no	—
100	C₂×D₅₃	—	2	1	no	—
101	C₂×D₅₃	—	2	1	no	—
102	C₂×D₅₃	#340	1	2	yes	—
103	C₂×D₅₃	C₂×#103	1	2	yes	—
104	C₄×C₅₃	#340	1	2	yes	—
105	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
107	Dic₅₃	#340	1	2	yes	—
108	Dic₅₃	C₂×#102	1	2	yes	—
109	C₂³×C₁₆	—	53	1	no	—
110	C₂²×#108	—	53	2	no	—
111	#1381	—	53	4	no	—
112	C₂×C₅₃	#1381	1	4	yes	complemented
113	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
116	D₅₃	—	2	2	no	—
117	D₅₃	—	2	2	no	—
118	C₂²×M₁₆	—	106	2	no	—
119	C₂³×C₈	—	53	1	no	—
120	C₂²×C₁₆	—	53	1	no	—
121	C₂²×C₁₆	—	53	2	no	—
122	C₂²×C₁₆	—	53	2	no	—
123	C₂²×C₁₆	—	106	4	no	—
124	C₂²×C₁₆	#6157	1	1	yes	characteristic
125	C₂×#108	—	53	4	no	—
126	C₂×#108	—	53	8	no	—
127	C₂×#108	—	106	2	no	—
128	C₂×#108	—	106	4	no	—
129	C₅₃	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	—	53	1	no	—
131	C₂×M₁₆	—	106	8	no	—
132	C₂×M₁₆	—	106	4	no	—
133	C₂×M₁₆	—	212	4	no	—
134	C₂×M₁₆	—	212	2	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	4	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	53	4	no	—
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	—	212	2	no	—
143	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	106	4	no	—
147	C₂×C₁₆	—	106	4	no	—
148	C₂×C₁₆	—	106	2	no	—
149	C₂×C₁₆	C₂×#6157	1	2	yes	—
150	#108	—	53	8	no	—
151	#108	—	106	8	no	—
152	#108	—	106	4	no	—
153	C₂⁴	—	53	1	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	4	no	—
156	C₂²×C₄	—	53	4	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
162	M₁₆	—	106	8	no	—
163	M₁₆	—	212	4	no	—
164	M₁₆	—	212	8	no	—
165	C₂×C₈	—	53	4	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	4	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	212	4	no	—
175	C₂×C₈	—	212	8	no	—
176	C₂×C₈	C₄×#6157	1	2	yes	—
177	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
178	C₂×C₈	C₂²⋊#6157	1	2	yes	—
179	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	106	2	no	—
182	C₂³	—	53	1	no	—
183	C₂³	—	53	1	no	—
184	C₂³	—	53	4	no	—
185	C₂³	—	106	2	no	—
186	C₂³	—	106	2	no	—
187	C₂³	#13133286	1	1	yes	characteristic
188	C₂×C₄	—	53	1	no	—
189	C₂×C₄	—	53	1	no	—
190	C₂×C₄	—	53	4	no	—
191	C₂×C₄	—	53	2	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	4	no	—
196	C₂×C₄	—	53	4	no	—
197	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
198	C₂×C₄	#13133285	1	2	yes	—
199	C₂×C₄	#13133286	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
201	C₈	—	53	2	no	—
202	C₈	—	53	2	no	—
203	C₈	—	212	8	no	—
204	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
205	C₈	C₂×C₄×#6157	1	1	yes	characteristic
206	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
207	C₂²	—	53	2	no	—
208	C₂²	—	53	2	no	—
209	C₂²	—	106	8	no	—
210	C₂²	—	106	1	no	—
211	C₂²	—	106	1	no	—
212	C₂²	—	106	1	no	—
213	C₂²	—	106	1	no	—
214	C₂²	#16811497	1	2	yes	—
215	C₂²	#16811551	1	2	yes	—
216	C₂²	#16811552	1	2	yes	—
217	C₂²	C₂×#13133286	1	1	yes	characteristic
218	C₄	—	53	2	no	—
219	C₄	—	53	2	no	—
220	C₄	#16811497	1	2	yes	—
221	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
222	C₄	C₂×#13133285	1	1	yes	characteristic
223	C₂	—	106	2	no	—
224	C₂	—	106	2	no	—
225	C₂	#20289775	1	4	yes	direct product factor
226	C₂	C₂×#16811497	1	1	yes	characteristic
227	C₂	C₂×#16811551	1	1	yes	characteristic
228	C₂	C₂×#16811552	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133286
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂³×C₅₃	#107
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#1381
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20289775	C₂	direct (N × H)
112. C₂×C₅₃	#1381	semidirect
129. C₅₃	C₂×#1381	semidirect
225. C₂	#20289775	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×#13133288	2
#20289775	2
C₂²×C₁₆×D₅₃	2
C₂×#1381	53

Subgroups — C2×#20289775 #28310276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289775 #28310276