Subgroups — C₄×#16810424 #27299296

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810424	C₂	1	2	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₅₃	—	2	1	no	—
9	#16810424	C₄	1	8	yes	direct product factor
10	C₄×#11719908	—	2	2	no	—
11	#16810386	—	2	1	no	—
12	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₅₃	—	2	1	no	—
14	C₄×D₈×C₅₃	—	2	2	no	—
15	C₄×D₈×C₅₃	C₄	1	4	yes	complemented
16	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#102×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄×Dic₅₃	—	4	1	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	C₂×C₄	1	4	yes	complemented
24	#13133151	—	2	2	no	—
25	#13133151	C₂×C₄	1	4	yes	complemented
26	#13133124	—	4	1	no	—
27	C₂²×D₄×C₅₃	—	2	1	no	—
28	C₂³×C₄×C₅₃	—	4	1	no	—
29	C₂×D₈×C₅₃	—	2	1	no	—
30	C₂×D₈×C₅₃	—	2	2	no	—
31	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₅₃	—	2	4	no	—
36	C₄×D₄×C₅₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	1	no	—
41	#107×C₅₃	C₂×C₄	1	2	yes	—
42	#102×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂⁴×C₅₃	—	4	1	no	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	2	4	no	—
48	C₂×D₄×C₅₃	—	4	1	no	—
49	C₂×D₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	4	4	no	—
52	C₂²×C₄×C₅₃	—	8	2	no	—
53	C₂²×C₄×C₅₃	—	8	2	no	—
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	4	2	no	—
58	D₈×C₅₃	—	2	2	no	—
59	D₈×C₅₃	—	2	4	no	—
60	D₈×C₅₃	C₄²	1	4	yes	complemented
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	2	yes	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₂²⋊C₄×C₅₃	—	4	4	no	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅₃	SD₁₆	1	2	yes	—
72	C₄⋊Dic₅₃	—	2	4	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄×Dic₅₃	—	8	2	no	—
75	#11719908	—	2	4	no	—
76	C₂³×C₅₃	—	4	2	no	—
77	C₂³×C₅₃	—	4	4	no	—
78	C₂³×C₅₃	—	8	1	no	—
79	C₂³×C₅₃	—	8	1	no	—
80	C₂³×C₅₃	—	8	2	no	—
81	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
82	D₄×C₅₃	—	2	4	no	—
83	D₄×C₅₃	—	4	2	no	—
84	D₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	8	1	no	—
88	C₂×C₄×C₅₃	—	8	1	no	—
89	C₂×C₄×C₅₃	—	8	8	no	—
90	C₂×C₄×C₅₃	#102	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#135	1	2	yes	—
98	C₂×Dic₅₃	—	4	4	no	—
99	C₂×Dic₅₃	—	8	2	no	—
100	C₂×Dic₅₃	—	8	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
103	C₈×C₅₃	#117	1	1	yes	characteristic
104	C₈×C₅₃	C₄×D₄	1	2	yes	—
105	C₄×#357	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	4	4	no	—
107	C₂²×C₅₃	—	8	2	no	—
108	C₂²×C₅₃	—	8	2	no	—
109	C₂²×C₅₃	—	8	4	no	—
110	C₂²×C₅₃	—	8	4	no	—
111	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
112	C₂²×C₅₃	C₄×D₈	1	2	yes	—
113	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
115	C₂²×C₅₃	#508	1	2	yes	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	8	2	no	—
118	C₄×C₅₃	#357	1	4	yes	complemented
119	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₅₃	#416	1	1	yes	characteristic
121	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
122	Dic₅₃	—	8	4	no	—
123	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
124	C₂×#357	—	53	2	no	—
125	C₂×C₄×C₁₆	—	53	1	no	—
126	C₄×#107	—	53	1	no	—
127	#1380	—	53	2	no	—
128	C₂×C₅₃	—	8	4	no	—
129	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
131	C₂×C₅₃	#2138	1	1	yes	characteristic
132	C₂×C₅₃	C₄×D₁₆	1	2	yes	—
133	C₂×C₅₃	C₄×SD₃₂	1	2	yes	—
134	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×C₄×D₄	—	2	1	no	—
136	C₂²×C₁₆	—	53	2	no	—
137	C₄×D₈	—	2	2	no	—
138	C₄×D₈	Dic₅₃	1	4	yes	complemented
139	C₂×#107	—	53	2	no	—
140	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
141	C₂×#102	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	106	1	no	—
144	#357	—	53	8	no	—
145	C₄×C₁₆	—	106	2	no	—
146	#340	—	106	1	no	—
147	C₅₃	C₄×#357	1	1	yes	characteristic, complemented, Sylow
148	C₂²×D₄	—	2	1	no	—
149	C₂³×C₄	—	4	1	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	C₄×D₅₃	1	4	yes	—
153	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
154	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₄×D₄	—	2	4	no	—
157	C₄×D₄	—	4	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	106	2	no	—
160	C₂×C₂²⋊C₄	—	4	1	no	—
161	C₂×C₄²	—	212	1	no	—
162	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂×C₁₆	—	53	4	no	—
164	C₂×C₁₆	—	106	2	no	—
165	C₂×C₁₆	—	106	2	no	—
166	#107	—	2	1	no	—
167	#107	—	53	4	no	—
168	#107	—	106	2	no	—
169	#107	C₂×Dic₅₃	1	2	yes	—
170	#102	—	2	4	no	—
171	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
172	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
173	#95	—	212	1	no	—
174	C₂⁴	—	4	1	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	—	8	2	no	—
182	C₂²×C₄	—	8	2	no	—
183	C₂²×C₄	—	212	2	no	—
184	C₂²×C₄	#11719922	1	1	yes	characteristic
185	C₂²×C₄	#11719922	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	D₈	—	2	2	no	—
188	D₈	—	2	4	no	—
189	D₈	C₄×Dic₅₃	1	4	yes	complemented
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
193	C₂×C₈	#11719945	1	2	yes	—
194	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	106	4	no	—
199	C₄⋊C₄	—	212	2	no	—
200	C₂²⋊C₄	—	4	4	no	—
201	C₄²	—	424	2	no	—
202	C₄²	#11719923	1	2	yes	—
203	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
204	C₁₆	—	106	4	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	8	1	no	—
208	C₂³	—	8	1	no	—
209	C₂³	—	8	2	no	—
210	C₂³	C₂×#11719922	1	1	yes	characteristic
211	D₄	—	2	4	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	8	8	no	—
219	C₂×C₄	—	424	2	no	—
220	C₂×C₄	—	424	2	no	—
221	C₂×C₄	—	212	4	no	—
222	C₂×C₄	#13133138	1	2	yes	—
223	C₂×C₄	#13133138	1	2	yes	—
224	C₂×C₄	#13133158	1	2	yes	—
225	C₂×C₄	#13133161	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
228	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
229	C₂×C₄	#13133237	1	2	yes	—
230	C₈	—	2	2	no	—
231	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
232	C₈	#13133176	1	1	yes	characteristic
233	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	C₂×#13133138	1	1	yes	characteristic
240	C₂²	C₄×#11719922	1	2	yes	—
241	C₂²	C₂×#13133158	1	1	yes	characteristic
242	C₂²	C₂×#13133161	1	1	yes	characteristic
243	C₂²	#16811053	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	8	2	no	—
246	C₄	—	424	4	no	—
247	C₄	#16810424	1	4	yes	direct product factor
248	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₄	#16810619	1	1	yes	characteristic
250	C₄	C₄×#11719923	1	2	yes	—
251	C₂	—	8	4	no	—
252	C₂	C₄×#13133138	1	1	yes	characteristic
253	C₂	C₂×#16810424	1	1	yes	characteristic
254	C₂	#20276422	1	1	yes	characteristic
255	C₂	C₄×#13133158	1	2	yes	—
256	C₂	C₄×#13133161	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₈×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	C₄×#357
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11719908	C₂	semidirect
3. C₄×#13133151	C₂	semidirect
4. #20272806	C₂	semidirect
9. #16810424	C₄	direct (N × H)
15. C₄×D₈×C₅₃	C₄	semidirect
23. C₂×#11719908	C₂×C₄	semidirect
25. #13133151	C₂×C₄	semidirect
60. D₈×C₅₃	C₄²	semidirect
118. C₄×C₅₃	#357	semidirect
138. C₄×D₈	Dic₅₃	semidirect
147. C₅₃	C₄×#357	semidirect
189. D₈	C₄×Dic₅₃	semidirect
247. C₄	#16810424	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×#16810424	2
C₂×C₄×#11719908	2
C₄×#13133151	2
#20272806	2
C₂×C₄×D₈×C₅₃	2
C₄×#357	53

Subgroups — C4×#16810424 #27299296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810424 #27299296