Subgroups — C₈×#13133305 #28313765

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291521	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133305	C₂	1	1	yes	maximal, characteristic
3	#20289921	C₂	1	3	yes	maximal, complemented
4	#20289917	C₂	1	3	yes	maximal, complemented
5	C₈×#11719936	C₂	1	3	yes	maximal, complemented
6	C₈×D₅₃⋊C₈	C₂	1	3	yes	maximal, complemented
7	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133305	C₄	1	2	yes	—
9	Q₈×#88324	C₄	1	2	yes	—
10	#16811651	C₄	1	6	yes	—
11	#16811642	C₂²	1	3	yes	—
12	#16811588	C₄	1	6	yes	—
13	C₄×#11719936	C₂²	1	3	yes	—
14	C₄×D₅₃⋊C₈	C₂²	1	3	yes	—
15	C₄×D₅₃⋊C₈	C₂²	1	3	yes	—
16	#88324⋊C₈	C₂²	1	6	yes	—
17	C₈×#88324	C₂²	1	1	yes	characteristic
18	C₈×#88324	C₂²	1	3	yes	—
19	C₈×#88324	C₂²	1	3	yes	—
20	C₈×#88324	C₂²	1	3	yes	—
21	C₄×#11719949	C₂²	1	1	yes	characteristic
22	Q₈×#88322	C₄	1	1	yes	characteristic
23	C₈×D₂₁₂	C₄	1	3	yes	—
24	#16810688	C₄	1	3	yes	—
25	#16810675	C₂²	1	3	yes	—
26	C₄×C₈×D₅₃	C₂²	1	3	yes	—
27	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
28	#13133305	C₈	1	16	yes	direct product factor
29	C₂×#11719936	C₂×C₄	1	6	yes	—
30	C₂×D₅₃⋊C₈	C₂×C₄	1	6	yes	—
31	C₂×D₅₃⋊C₈	C₂×C₄	1	6	yes	—
32	C₄⋊#88324	C₂×C₄	1	6	yes	—
33	#13133274	C₂×C₄	1	6	yes	—
34	#13133272	C₂×C₄	1	6	yes	—
35	C₄×#88324	C₂×C₄	1	6	yes	—
36	C₄×#88324	C₂×C₄	1	6	yes	—
37	C₄×#88324	C₂³	1	1	yes	characteristic
38	C₄×#88324	C₂³	1	1	yes	characteristic
39	C₄×#88324	C₂³	1	3	yes	—
40	C₄×#88324	C₂³	1	3	yes	—
41	C₂×#11719949	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
43	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
44	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×D₂₁₂	C₂×C₄	1	3	yes	—
46	#13133179	C₂×C₄	1	3	yes	—
47	#13133175	C₂×C₄	1	3	yes	—
48	C₄²×D₅₃	C₂³	1	3	yes	—
49	C₄⋊#88322	C₂×C₄	1	3	yes	—
50	#13133129	C₂×C₄	1	6	yes	—
51	C₄×#88322	C₂×C₄	1	3	yes	—
52	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
53	C₈×Dic₅₃	C₂³	1	3	yes	—
54	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
57	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
58	#11719949	C₂×C₈	1	8	yes	—
59	C₂×D₂₁₂	C₄²	1	3	yes	—
60	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
61	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
62	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
63	C₂×#88324	C₂²×C₄	1	6	yes	—
64	C₂×#88324	C₂²×C₄	1	2	yes	—
65	C₂×#88324	C₂²×C₄	1	6	yes	—
66	C₂×#88324	C₂²×C₄	1	2	yes	—
67	#11719936	C₂×C₈	1	24	yes	complemented
68	D₅₃⋊C₈	—	2	12	no	—
69	D₅₃⋊C₈	C₂×C₈	1	24	yes	complemented
70	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	C₄²	1	3	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
74	#11719921	C₄²	1	6	yes	—
75	C₄⋊Dic₅₃	C₄²	1	3	yes	—
76	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
77	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
78	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
79	C₂×#88322	C₂²×C₄	1	3	yes	—
80	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
81	C₈×D₅₃	—	2	6	no	—
82	C₈×D₅₃	—	2	6	no	—
83	C₂²×D₅₃	C₂×C₄²	1	3	yes	—
84	Q₈×C₅₃	C₄×C₈	1	4	yes	—
85	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
86	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
87	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
89	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
91	D₂₁₂	C₄×C₈	1	12	yes	—
92	C₄×D₅₃	—	2	3	no	—
93	C₄×D₅₃	—	2	3	no	—
94	C₄×D₅₃	—	2	6	no	—
95	C₄×D₅₃	C₂²×C₈	1	12	yes	—
96	#88324	—	2	6	no	—
97	#88324	C₂²×C₈	1	12	yes	—
98	#88324	C₂²×C₈	1	4	yes	—
99	#88324	#131	1	4	yes	complemented
100	C₈×C₅₃	—	2	3	no	—
101	C₈×C₅₃	#131	1	2	yes	complemented
102	#88322	—	2	3	no	—
103	#88322	#131	1	2	yes	complemented
104	C₈×#131	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
106	C₂×D₅₃	—	2	3	no	—
107	C₂×D₅₃	—	2	3	no	—
108	C₂×D₅₃	C₂×C₄×C₈	1	6	yes	—
109	C₄×C₅₃	—	2	3	no	—
110	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
111	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
112	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
113	Dic₅₃	—	2	3	no	—
114	Dic₅₃	C₂³×C₈	1	2	yes	—
115	Dic₅₃	C₂×#131	1	2	yes	—
116	C₈×#41	—	53	1	no	—
117	C₈×#41	—	53	1	no	—
118	C₄×#131	—	53	1	no	—
119	#1450	—	53	3	no	—
120	C₈×M₁₆	—	53	3	no	—
121	C₈×M₁₆	—	53	3	no	—
122	C₂×C₈²	—	53	3	no	—
123	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
124	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
125	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
126	D₅₃	—	2	6	no	—
127	C₂×#131	—	53	2	no	—
128	C₄×#41	—	53	1	no	—
129	C₈×Q₈	—	53	2	no	—
130	C₈×Q₈	—	53	1	no	—
131	C₈×Q₈	#6157	1	1	yes	characteristic
132	C₈×D₄	—	53	6	no	—
133	C₈×D₄	—	53	3	no	—
134	C₈×D₄	—	53	3	no	—
135	#431	—	53	3	no	—
136	#431	—	53	3	no	—
137	#405	—	53	6	no	—
138	C₄×M₁₆	—	53	3	no	—
139	C₂×C₄×C₈	—	53	3	no	—
140	C₂×C₄×C₈	—	53	3	no	—
141	C₂×C₄×C₈	—	53	3	no	—
142	#322	—	53	6	no	—
143	C₈²	—	53	1	no	—
144	C₈²	—	53	3	no	—
145	C₈²	—	53	3	no	—
146	C₈²	—	53	3	no	—
147	C₅₃	C₈×#131	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	53	1	no	—
149	#131	—	53	16	no	—
150	C₂×M₁₆	—	53	6	no	—
151	C₂²×C₈	—	53	6	no	—
152	C₂²×C₈	—	53	6	no	—
153	C₂²×C₈	—	53	3	no	—
154	C₂²×C₈	—	53	3	no	—
155	C₄×Q₈	—	53	1	no	—
156	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
157	C₄×D₄	—	53	3	no	—
158	C₄×D₄	—	53	3	no	—
159	#117	—	53	3	no	—
160	C₂×C₄²	—	53	3	no	—
161	C₄⋊C₈	—	53	6	no	—
162	C₄⋊C₈	—	53	3	no	—
163	C₄⋊C₈	C₂×#6157	1	3	yes	—
164	C₂²⋊C₈	—	53	6	no	—
165	C₂²⋊C₈	—	53	6	no	—
166	#97	—	53	6	no	—
167	C₄×C₈	—	53	1	no	—
168	C₄×C₈	—	53	1	no	—
169	C₄×C₈	—	53	3	no	—
170	C₄×C₈	—	53	3	no	—
171	C₄×C₈	—	53	6	no	—
172	C₄×C₈	—	53	6	no	—
173	C₄×C₈	—	53	3	no	—
174	C₄×C₈	—	53	1	no	—
175	C₄×C₈	—	53	3	no	—
176	C₄×C₈	C₂×#6157	1	3	yes	—
177	#41	—	53	8	no	—
178	C₂×Q₈	C₄×#6157	1	1	yes	characteristic
179	C₂×D₄	—	53	3	no	—
180	C₂²×C₄	—	53	3	no	—
181	C₂²×C₄	—	53	3	no	—
182	C₂²×C₄	—	53	3	no	—
183	M₁₆	—	53	24	no	—
184	C₂×C₈	—	53	2	no	—
185	C₂×C₈	—	53	2	no	—
186	C₂×C₈	—	53	6	no	—
187	C₂×C₈	—	53	24	no	—
188	C₂×C₈	—	53	1	no	—
189	C₂×C₈	—	53	6	no	—
190	C₂×C₈	—	53	3	no	—
191	C₂×C₈	—	106	6	no	—
192	C₂×C₈	—	106	12	no	—
193	C₂×C₈	—	106	6	no	—
194	C₂×C₈	C₂²×#6157	1	3	yes	—
195	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
196	C₄⋊C₄	—	53	3	no	—
197	C₄⋊C₄	C₄×#6157	1	3	yes	—
198	C₂²⋊C₄	—	53	6	no	—
199	C₄²	—	53	3	no	—
200	C₄²	—	53	3	no	—
201	C₄²	—	53	1	no	—
202	C₄²	C₂²×#6157	1	3	yes	—
203	C₂³	—	53	3	no	—
204	Q₈	C₈×#6157	1	4	yes	—
205	D₄	—	53	12	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	1	no	—
208	C₂×C₄	—	53	12	no	—
209	C₂×C₄	—	53	3	no	—
210	C₂×C₄	—	106	6	no	—
211	C₂×C₄	—	106	3	no	—
212	C₂×C₄	—	106	3	no	—
213	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
214	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
215	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
216	C₈	—	2	3	no	—
217	C₈	—	53	4	no	—
218	C₈	—	53	12	no	—
219	C₈	—	53	4	no	—
220	C₈	—	53	2	no	—
221	C₈	—	106	6	no	—
222	C₈	—	106	3	no	—
223	C₈	#13133305	1	2	yes	direct product factor
224	C₂²	—	53	6	no	—
225	C₂²	—	106	3	no	—
226	C₂²	—	106	3	no	—
227	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
228	C₄	—	2	3	no	—
229	C₄	—	53	2	no	—
230	C₄	—	53	2	no	—
231	C₄	—	106	3	no	—
232	C₄	C₂×C₈×#6157	1	6	yes	—
233	C₄	D₅₃⋊#131	1	1	yes	characteristic
234	C₄	C₂×#13133305	1	1	yes	characteristic
235	C₂	—	106	6	no	—
236	C₂	C₂²×C₈×#6157	1	1	yes	characteristic
237	C₂	#131×#6157	1	1	yes	characteristic
238	C₂	C₄×#13133305	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂×#11719949	C₂×C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	C₈×#131
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20289921	C₂	semidirect
4. #20289917	C₂	semidirect
5. C₈×#11719936	C₂	semidirect
6. C₈×D₅₃⋊C₈	C₂	semidirect
28. #13133305	C₈	direct (N × H)
67. #11719936	C₂×C₈	semidirect
69. D₅₃⋊C₈	C₂×C₈	semidirect
99. #88324	#131	semidirect
101. C₈×C₅₃	#131	semidirect
103. #88322	#131	semidirect
147. C₅₃	C₈×#131	semidirect
223. C₈	#13133305	direct (N × H)

Maximal subgroups

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

subgroup	index
#20291521	2
C₄×#13133305	2
#20289921	2
#20289917	2
C₈×#11719936	2
C₈×D₅₃⋊C₈	2
C₈×#11719949	2
C₈×#131	53

Subgroups — C8×#13133305 #28313765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133305 #28313765