Subgroups — C₂×#20291083 #28328227

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291083	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11719910	C₂	1	2	yes	maximal, complemented
3	C₂×#16811518	C₂	1	2	yes	maximal
4	C₂×#16811512	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16811509	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810580	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719910	C₂²	1	2	yes	—
8	C₂²×#11719910	C₂²	1	2	yes	—
9	C₄⋊#11719910	C₂²	1	8	yes	complemented
10	#16811518	C₂²	1	8	yes	—
11	#16811512	C₂²	1	4	yes	complemented
12	#16811509	C₂²	1	4	yes	complemented
13	C₂²×#11719912	C₄	1	1	yes	characteristic
14	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×#13133174	C₄	1	1	yes	characteristic
16	#16810580	C₄	1	4	yes	—
17	#16810580	C₂²	1	4	yes	—
18	C₂×#13133127	C₄	1	1	yes	characteristic
19	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#11719910	—	2	4	no	—
23	C₂×#11719910	—	2	4	no	—
24	C₂×#11719910	C₂³	1	4	yes	—
25	C₂×#11719910	C₂³	1	4	yes	—
26	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#11719912	—	2	4	no	—
28	C₂×#11719912	C₂×C₄	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	4	no	—
32	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₃	C₂³	1	2	yes	—
34	#13133174	C₈	1	4	yes	—
35	#13133174	C₂×C₄	1	4	yes	—
36	C₂×#11719919	C₈	1	2	yes	—
37	C₂×#11719921	C₈	1	2	yes	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	#13133127	C₂×C₄	1	4	yes	—
40	C₄×#88322	C₂×C₄	1	2	yes	—
41	C₄×#88322	C₂³	1	2	yes	—
42	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
43	C₈×Dic₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	#97×C₅₃	C₂×C₄	1	4	yes	—
48	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
52	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₅₃	—	2	2	no	—
54	C₂×C₄×D₅₃	—	2	2	no	—
55	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₅₃	C₂×C₈	1	2	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
63	#11719921	C₂×C₈	1	8	yes	—
64	#11719919	C₂×C₈	1	8	yes	—
65	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
66	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	C₂²×C₄	1	2	yes	—
69	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
70	C₂×#88322	#41	1	4	yes	—
71	C₂×#88322	C₂⁴	1	1	yes	characteristic
72	#11719912	—	2	8	no	—
73	C₈×D₅₃	—	2	4	no	—
74	C₈×D₅₃	—	2	4	no	—
75	#11719910	—	2	4	no	—
76	#11719910	—	2	4	no	—
77	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₅₃	—	2	2	no	—
79	C₂²×D₅₃	—	2	4	no	—
80	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	#117	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	2	4	no	—
89	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
90	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
91	C₄×D₅₃	—	2	2	no	—
92	C₄×D₅₃	—	2	2	no	—
93	C₄×D₅₃	—	2	4	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
96	#88322	—	2	2	no	—
97	#88322	#117	1	2	yes	—
98	#88322	C₂×#41	1	2	yes	—
99	C₂×#2164	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	#431	1	2	yes	—
101	C₂²×C₅₃	#504	1	4	yes	—
102	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
103	C₂×D₅₃	—	2	4	no	—
104	C₂×D₅₃	—	2	2	no	—
105	C₂×D₅₃	—	4	4	no	—
106	C₄×C₅₃	—	2	4	no	—
107	C₄×C₅₃	#431	1	2	yes	—
108	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
109	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
110	Dic₅₃	—	2	4	no	—
111	Dic₅₃	—	2	2	no	—
112	C₂×#405	—	53	1	no	—
113	#2164	—	53	8	no	—
114	C₂×C₄⋊C₁₆	—	53	2	no	—
115	C₂×C₂²⋊C₁₆	—	53	2	no	—
116	C₂×#346	—	53	1	no	—
117	C₂×C₄×C₁₆	—	53	1	no	—
118	C₂×C₅₃	#2164	1	4	yes	complemented
119	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#504	1	2	yes	—
121	D₅₃	—	4	2	no	—
122	C₂²×M₁₆	—	53	1	no	—
123	C₂³×C₈	—	53	1	no	—
124	C₂×#117	—	53	1	no	—
125	C₂²×C₁₆	—	53	2	no	—
126	C₂²×C₁₆	—	53	2	no	—
127	#405	—	53	4	no	—
128	#405	—	53	4	no	—
129	C₂×#97	—	53	1	no	—
130	C₂×#97	#6157	1	1	yes	characteristic
131	C₂×C₄×C₈	—	53	1	no	—
132	C₂×C₄×C₈	—	53	1	no	—
133	C₄⋊C₁₆	—	53	8	no	—
134	C₂²⋊C₁₆	—	53	8	no	—
135	#346	—	53	4	no	—
136	C₄×C₁₆	—	53	4	no	—
137	C₅₃	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	—	53	1	no	—
139	C₂×M₁₆	—	53	4	no	—
140	C₂×M₁₆	—	106	4	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	53	2	no	—
144	C₂²×C₈	—	53	2	no	—
145	C₂²×C₈	—	106	4	no	—
146	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
147	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
148	#117	—	53	4	no	—
149	#117	—	53	4	no	—
150	C₂×C₄⋊C₄	—	53	2	no	—
151	C₂×C₂²⋊C₄	—	53	2	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
154	C₂×C₁₆	—	53	4	no	—
155	C₂×C₁₆	—	53	4	no	—
156	C₂×C₁₆	—	106	4	no	—
157	C₂×C₁₆	—	106	4	no	—
158	#97	—	53	4	no	—
159	#97	C₂×#6157	1	4	yes	—
160	C₄×C₈	—	53	2	no	—
161	C₄×C₈	—	53	2	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	—	53	2	no	—
164	C₂⁴	—	53	1	no	—
165	C₂²×C₄	—	53	2	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	2	no	—
168	C₂²×C₄	—	53	2	no	—
169	C₂²×C₄	—	106	2	no	—
170	C₂²×C₄	—	106	2	no	—
171	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
172	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
173	M₁₆	—	106	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	53	1	no	—
177	C₂×C₈	—	53	1	no	—
178	C₂×C₈	—	53	2	no	—
179	C₂×C₈	—	106	4	no	—
180	C₂×C₈	—	106	4	no	—
181	C₂×C₈	—	106	2	no	—
182	C₂×C₈	#11719936	1	4	yes	—
183	C₂×C₈	C₂²×#6157	1	2	yes	—
184	C₂×C₈	C₂²×#6157	1	2	yes	—
185	C₄⋊C₄	—	53	8	no	—
186	C₂²⋊C₄	—	53	8	no	—
187	C₄²	—	53	2	no	—
188	C₄²	—	53	2	no	—
189	C₄²	D₅₃⋊C₈	1	2	yes	—
190	C₄²	C₂²×#6157	1	2	yes	—
191	C₁₆	—	106	4	no	—
192	C₁₆	—	106	4	no	—
193	C₂³	—	53	2	no	—
194	C₂³	—	106	2	no	—
195	C₂³	—	106	4	no	—
196	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	53	4	no	—
199	C₂×C₄	—	53	2	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	—	106	4	no	—
202	C₂×C₄	—	106	4	no	—
203	C₂×C₄	—	106	2	no	—
204	C₂×C₄	—	106	2	no	—
205	C₂×C₄	#13133296	1	2	yes	—
206	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
207	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
208	C₂×C₄	C₂×#11719936	1	2	yes	—
209	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	53	2	no	—
212	C₈	—	53	2	no	—
213	C₈	—	106	2	no	—
214	C₈	C₂×#11719936	1	4	yes	—
215	C₂²	—	106	2	no	—
216	C₂²	—	106	4	no	—
217	C₂²	—	212	4	no	—
218	C₂²	#16811642	1	2	yes	—
219	C₂²	#16811682	1	4	yes	—
220	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	106	2	no	—
223	C₄	—	106	4	no	—
224	C₄	#16811642	1	2	yes	—
225	C₄	C₂×#13133296	1	1	yes	characteristic
226	C₄	C₂²×#11719936	1	1	yes	characteristic
227	C₂	—	212	2	no	—
228	C₂	#20291083	1	4	yes	direct product factor
229	C₂	C₂×#16811642	1	1	yes	characteristic
230	C₂	C₂×#16811682	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#97×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	C₂×#2164
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
1. #20291083	C₂	direct (N × H)
2. C₂×C₄⋊#11719910	C₂	semidirect
4. C₂×#16811512	C₂	semidirect
5. C₂×#16811509	C₂	semidirect
9. C₄⋊#11719910	C₂²	semidirect
11. #16811512	C₂²	semidirect
12. #16811509	C₂²	semidirect
118. C₂×C₅₃	#2164	semidirect
137. C₅₃	C₂×#2164	semidirect
228. C₂	#20291083	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291083	2
C₂×C₄⋊#11719910	2
C₂×#16811518	2
C₂×#16811512	2
C₂×#16811509	2
C₂×#16810580	2
C₂×#2164	53

Subgroups — C2×#20291083 #28328227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291083 #28328227