Subgroups — #346×#6157 #28310185

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290916	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811510	C₂	1	2	yes	maximal
3	C₂×C₁₆×#6157	C₂	1	2	yes	maximal
4	C₄×C₈×#6157	C₂	1	1	yes	maximal, characteristic
5	#346×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
7	C₄×D₅₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×#6157	C₄	1	1	yes	characteristic
9	C₂×C₈×#6157	C₄	1	1	yes	characteristic
10	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
11	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
12	C₄²×#6157	C₄	1	1	yes	characteristic
13	#16811512	C₄	1	2	yes	complemented
14	#16811511	C₄	1	2	yes	complemented
15	#16811510	C₄	1	8	yes	complemented
16	C₁₆×#6157	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
18	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	#16810399	C₄	1	1	yes	characteristic, complemented
20	#16810398	C₄	1	2	yes	complemented
21	C₅₃×#346	C₄	1	1	yes	characteristic, complemented
22	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#6157	C₈	1	4	yes	—
25	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
27	D₅₃⋊C₁₆	C₂×C₄	1	4	yes	—
28	D₅₃⋊C₁₆	C₂×C₄	1	4	yes	—
29	C₂×#11719910	C₂×C₄	1	4	yes	—
30	C₄×#88324	C₈	1	4	yes	—
31	C₈×#6157	C₈	1	4	yes	—
32	C₈×#6157	C₈	1	4	yes	—
33	C₈×#6157	C₂×C₄	1	4	yes	—
34	C₈×#6157	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
37	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×#11719908	C₂×C₄	1	2	yes	—
39	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
40	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
41	C₄×#88322	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×#6157	C₂×C₈	1	2	yes	—
47	C₂²×#6157	C₂×C₈	1	2	yes	—
48	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
49	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×#88324	C₂×C₈	1	2	yes	—
51	C₂×#88324	C₂×C₈	1	2	yes	—
52	C₄×#6157	—	2	8	no	—
53	C₄×#6157	C₄²	1	2	yes	—
54	C₄×#6157	C₄²	1	2	yes	—
55	C₄×#6157	C₂×C₈	1	2	yes	—
56	C₄×#6157	C₂×C₈	1	2	yes	—
57	D₅₃⋊C₈	C₄²	1	2	yes	—
58	D₅₃⋊C₈	C₄²	1	2	yes	—
59	D₅₃⋊C₈	C₂×C₈	1	2	yes	—
60	D₅₃⋊C₈	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₄²	1	1	yes	characteristic
64	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
66	C₂×#88322	C₄²	1	1	yes	characteristic
67	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
68	C₈×D₅₃	C₂×C₈	1	2	yes	—
69	C₈×D₅₃	C₂×C₈	1	2	yes	—
70	C₈×D₅₃	C₂²×C₄	1	2	yes	—
71	C₈×D₅₃	C₂²×C₄	1	2	yes	—
72	#11719910	C₄²	1	8	yes	complemented
73	C₁₆×C₅₃	C₄²	1	4	yes	complemented
74	#11719908	C₄²	1	4	yes	complemented
75	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×#6157	—	2	4	no	—
77	C₂×#6157	C₄×C₈	1	4	yes	—
78	C₂×#6157	C₄×C₈	1	4	yes	—
79	C₂×#6157	M₃₂	1	8	yes	—
80	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
83	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₄×D₅₃	—	2	4	no	—
85	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
86	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
87	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
89	#88324	C₄×C₈	1	4	yes	—
90	#88324	C₄×C₈	1	4	yes	—
91	C₈×C₅₃	C₄×C₈	1	2	yes	—
92	C₈×C₅₃	C₂×C₄²	1	2	yes	—
93	#88322	C₄×C₈	1	2	yes	—
94	#88322	C₂×C₄²	1	2	yes	—
95	C₄×#346	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂×D₅₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₂×D₅₃	C₂×C₄×C₈	1	1	yes	characteristic
99	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
100	#6157	—	2	4	no	—
101	#6157	#346	1	8	yes	direct product factor
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	C₄³	1	1	yes	characteristic
104	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
105	Dic₅₃	—	2	2	no	—
106	Dic₅₃	C₂×C₄×C₈	1	1	yes	characteristic
107	Dic₅₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂×#346	—	53	2	no	—
109	C₂×#346	—	53	1	no	—
110	C₂×#346	—	53	1	no	—
111	C₂×C₄×C₁₆	—	53	2	no	—
112	C₄²×C₈	—	53	1	no	—
113	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
114	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
115	D₅₃	C₂×#346	1	2	yes	—
116	D₅₃	C₄×M₃₂	1	2	yes	—
117	C₂²×C₁₆	—	53	2	no	—
118	C₂²×C₁₆	—	53	2	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	—	53	1	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₄³	—	53	1	no	—
126	#346	—	53	8	no	—
127	#346	—	53	2	no	—
128	#346	—	53	2	no	—
129	#346	—	53	2	no	—
130	#346	—	53	1	no	—
131	#346	#6157	1	1	yes	characteristic, direct product factor
132	C₄×C₁₆	—	53	8	no	—
133	C₅₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	53	1	no	—
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₄²	—	53	4	no	—
142	C₂×C₁₆	—	53	4	no	—
143	C₂×C₁₆	—	53	4	no	—
144	C₂×C₁₆	—	53	4	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	53	4	no	—
147	C₂×C₁₆	—	53	4	no	—
148	C₂×C₁₆	C₂×#6157	1	2	yes	—
149	C₄×C₈	—	53	4	no	—
150	C₄×C₈	—	53	4	no	—
151	C₄×C₈	—	53	4	no	—
152	C₄×C₈	—	53	4	no	—
153	C₄×C₈	—	53	4	no	—
154	C₄×C₈	—	53	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	53	1	no	—
157	C₄×C₈	C₂×#6157	1	1	yes	characteristic
158	C₂²×C₄	—	53	2	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	2	no	—
161	C₂²×C₄	—	53	2	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	53	1	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	C₄×#6157	1	1	yes	characteristic
175	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
176	C₄²	—	53	2	no	—
177	C₄²	—	53	2	no	—
178	C₄²	—	53	2	no	—
179	C₄²	—	53	2	no	—
180	C₄²	—	53	2	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	106	8	no	—
183	C₄²	C₄×#6157	1	1	yes	characteristic
184	C₁₆	—	53	8	no	—
185	C₁₆	—	53	4	no	—
186	C₁₆	C₄×#6157	1	4	yes	complemented
187	C₂³	—	53	1	no	—
188	C₂×C₄	—	53	4	no	—
189	C₂×C₄	—	53	8	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	4	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	106	4	no	—
198	C₂×C₄	—	106	4	no	—
199	C₂×C₄	C₈×#6157	1	2	yes	—
200	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
201	C₈	—	53	4	no	—
202	C₈	—	53	4	no	—
203	C₈	—	53	2	no	—
204	C₈	—	53	2	no	—
205	C₈	C₈×#6157	1	2	yes	—
206	C₈	C₂×C₄×#6157	1	2	yes	—
207	C₂²	—	53	1	no	—
208	C₂²	—	53	1	no	—
209	C₂²	—	53	4	no	—
210	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	53	8	no	—
213	C₄	—	53	1	no	—
214	C₄	—	53	1	no	—
215	C₄	—	106	4	no	—
216	C₄	—	106	2	no	—
217	C₄	C₄²×#6157	1	1	yes	characteristic
218	C₄	C₂×C₈×#6157	1	1	yes	characteristic
219	C₂	—	53	2	no	—
220	C₂	—	53	2	no	—
221	C₂	C₄×C₈×#6157	1	1	yes	characteristic
222	C₂	M₃₂×#6157	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₅₃×#346	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	C₄×#346
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. #16811512	C₄	semidirect
14. #16811511	C₄	semidirect
15. #16811510	C₄	semidirect
16. C₁₆×#6157	C₄	semidirect
19. #16810399	C₄	semidirect
20. #16810398	C₄	semidirect
21. C₅₃×#346	C₄	semidirect
72. #11719910	C₄²	semidirect
73. C₁₆×C₅₃	C₄²	semidirect
74. #11719908	C₄²	semidirect
101. #6157	#346	direct (N × H)
131. #346	#6157	direct (N × H)
133. C₅₃	C₄×#346	semidirect
186. C₁₆	C₄×#6157	semidirect

Maximal subgroups

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

subgroup	index
#20290916	2
C₂×#16811510	2
C₂×C₁₆×#6157	2
C₄×C₈×#6157	2
#346×D₅₃	2
C₄×#346	53