Subgroups — C₂×#20276497 #27537119

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	#20276497	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810450	C₂	1	2	yes	maximal, complemented
5	C₂×#16810448	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133176	—	2	1	no	—
8	C₂²×#11719908	C₄	1	8	yes	complemented
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	C₂²×#11719908	C₂²	1	4	yes	—
12	#16810661	—	2	2	no	—
13	#16810661	C₂²	1	4	yes	—
14	C₂×#13133151	C₂²	1	2	yes	—
15	C₂×#13133149	—	2	1	no	—
16	#16810450	C₂²	1	8	yes	complemented
17	#16810448	C₂²	1	8	yes	complemented
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#117×C₅₃	—	2	1	no	—
20	C₅₃×#427	—	2	2	no	—
21	C₅₃×#427	C₂²	1	4	yes	—
22	C₂×#107×C₅₃	C₂²	1	2	yes	—
23	C₂×#106×C₅₃	—	2	1	no	—
24	#13133176	—	2	4	no	—
25	#13133176	—	4	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
28	C₂×C₄×Dic₅₃	—	4	1	no	—
29	C₂×#11719908	C₂×C₄	1	4	yes	—
30	C₂×#11719908	C₂×C₄	1	4	yes	—
31	C₂×#11719908	C₂×C₄	1	16	yes	complemented
32	C₂×#11719908	C₂³	1	2	yes	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	#13133151	—	2	2	no	—
35	#13133151	C₂³	1	4	yes	—
36	#13133149	—	2	4	no	—
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	#117×C₅₃	—	2	4	no	—
45	#117×C₅₃	—	4	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
48	C₂×C₄²×C₅₃	—	4	1	no	—
49	#107×C₅₃	—	2	2	no	—
50	#107×C₅₃	C₂³	1	4	yes	—
51	#106×C₅₃	—	2	4	no	—
52	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	D₈	1	2	yes	—
56	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×Dic₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
62	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	—	4	2	no	—
72	C₂²⋊C₄×C₅₃	—	4	4	no	—
73	C₄²×C₅₃	—	4	4	no	—
74	C₂²⋊Dic₅₃	—	4	4	no	—
75	C₄⋊Dic₅₃	—	2	4	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₄×Dic₅₃	—	4	4	no	—
78	#11719908	C₂²×C₄	1	4	yes	—
79	#11719908	C₂²×C₄	1	4	yes	—
80	C₂³×C₅₃	—	2	4	no	—
81	C₂³×C₅₃	#107	1	4	yes	—
82	C₂³×C₅₃	C₂×D₈	1	2	yes	—
83	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	8	4	no	—
86	C₂×C₄×C₅₃	#107	1	8	yes	—
87	C₂×C₄×C₅₃	#107	1	4	yes	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×Dic₅₃	—	4	4	no	—
96	C₂×Dic₅₃	—	8	4	no	—
97	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×#2158	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	4	no	—
103	C₂²×C₅₃	—	2	8	no	—
104	C₂²×C₅₃	C₂×#107	1	4	yes	—
105	C₂²×C₅₃	C₂×#107	1	2	yes	—
106	C₂²×C₅₃	#508	1	4	yes	—
107	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
108	C₄×C₅₃	—	8	4	no	—
109	C₄×C₅₃	C₂×#107	1	4	yes	—
110	C₄×C₅₃	C₂×#107	1	2	yes	—
111	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
113	Dic₅₃	—	8	4	no	—
114	C₂³×C₁₆	—	53	1	no	—
115	C₂×#427	—	53	1	no	—
116	C₂×#427	D₅₃	1	1	yes	characteristic
117	#2158	—	53	8	no	—
118	C₂×#367	—	53	2	no	—
119	C₂×#366	—	53	2	no	—
120	C₂×C₅₃	—	2	4	no	—
121	C₂×C₅₃	#2158	1	4	yes	complemented
122	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#508	1	2	yes	—
124	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×#117	—	2	1	no	—
126	C₂×#117	—	106	1	no	—
127	C₂²×C₁₆	—	53	8	no	—
128	C₂²×C₁₆	—	53	1	no	—
129	C₂²×C₁₆	—	53	1	no	—
130	C₂²×C₁₆	—	53	4	no	—
131	#427	—	2	2	no	—
132	#427	—	53	4	no	—
133	#427	—	106	2	no	—
134	#427	C₂×D₅₃	1	4	yes	—
135	C₂×#107	—	53	2	no	—
136	C₂×#107	C₂×D₅₃	1	2	yes	—
137	C₂×#106	—	2	1	no	—
138	C₂×#106	—	106	1	no	—
139	#367	—	53	8	no	—
140	#366	—	53	8	no	—
141	C₅₃	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂²×C₈	Dic₁₀₆	1	4	yes	—
144	C₂²×C₈	C₄×D₅₃	1	4	yes	—
145	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
147	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
149	#117	—	2	4	no	—
150	#117	—	4	2	no	—
151	#117	—	106	4	no	—
152	#117	—	212	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	106	2	no	—
155	C₂×C₂²⋊C₄	—	4	1	no	—
156	C₂×C₂²⋊C₄	—	212	1	no	—
157	C₂×C₄²	—	4	1	no	—
158	C₂×C₄²	—	212	1	no	—
159	C₂×C₁₆	—	53	4	no	—
160	C₂×C₁₆	—	53	4	no	—
161	C₂×C₁₆	—	53	16	no	—
162	C₂×C₁₆	—	53	2	no	—
163	C₂×C₁₆	—	53	2	no	—
164	#107	—	2	2	no	—
165	#107	—	53	4	no	—
166	#107	—	106	2	no	—
167	#107	C₂²×D₅₃	1	4	yes	—
168	#106	—	2	4	no	—
169	#106	—	106	4	no	—
170	C₂⁴	#11719922	1	1	yes	characteristic
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	212	2	no	—
173	C₂²×C₄	#11719919	1	4	yes	—
174	C₂²×C₄	#11719922	1	2	yes	—
175	C₂²×C₄	#11719922	1	1	yes	characteristic
176	C₂²×C₄	#11719925	1	4	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
179	C₂×C₈	#11719919	1	4	yes	—
180	C₂×C₈	#11719919	1	8	yes	—
181	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
182	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
183	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
184	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
186	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	106	4	no	—
191	C₄⋊C₄	—	212	2	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	212	4	no	—
194	C₄²	—	4	4	no	—
195	C₄²	—	212	4	no	—
196	C₁₆	—	53	4	no	—
197	C₁₆	—	53	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	#13133150	1	4	yes	—
200	C₂³	C₂×#11719922	1	2	yes	—
201	C₂³	C₂×#11719922	1	1	yes	characteristic
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	4	no	—
204	C₂×C₄	—	424	4	no	—
205	C₂×C₄	—	212	4	no	—
206	C₂×C₄	#13133150	1	8	yes	—
207	C₂×C₄	#13133150	1	4	yes	—
208	C₂×C₄	C₂×#11719919	1	4	yes	—
209	C₂×C₄	C₂×#11719919	1	2	yes	—
210	C₂×C₄	C₂×#11719922	1	2	yes	—
211	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
212	C₂×C₄	C₂×#11719925	1	4	yes	—
213	C₂×C₄	C₂×#11719925	1	2	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₈	C₂×#11719919	1	4	yes	—
216	C₈	C₂×#11719919	1	2	yes	—
217	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
218	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
219	C₂²	—	2	8	no	—
220	C₂²	—	2	4	no	—
221	C₂²	C₂×#13133150	1	4	yes	—
222	C₂²	C₂×#13133150	1	2	yes	—
223	C₂²	#16811053	1	4	yes	—
224	C₂²	C₂²×#11719922	1	1	yes	characteristic
225	C₄	—	8	4	no	—
226	C₄	—	424	4	no	—
227	C₄	C₂×#13133150	1	2	yes	—
228	C₄	C₂×#13133150	1	4	yes	—
229	C₄	C₂²×#11719919	1	1	yes	characteristic
230	C₄	C₂²×#11719925	1	1	yes	characteristic
231	C₂	—	2	4	no	—
232	C₂	#20276497	1	4	yes	direct product factor
233	C₂	C₂²×#13133150	1	1	yes	characteristic
234	C₂	C₂×#16811053	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	C₂×#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20276497	C₂	direct (N × H)
4. C₂×#16810450	C₂	semidirect
5. C₂×#16810448	C₂	semidirect
8. C₂²×#11719908	C₄	semidirect
16. #16810450	C₂²	semidirect
17. #16810448	C₂²	semidirect
31. C₂×#11719908	C₂×C₄	semidirect
121. C₂×C₅₃	#2158	semidirect
141. C₅₃	C₂×#2158	semidirect
232. C₂	#20276497	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂×#16810661	2
#20276497	2
C₂×#16810450	2
C₂×#16810448	2
C₂×C₅₃×#427	2
C₂×#2158	53

Subgroups — C2×#20276497 #27537119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276497 #27537119