Subgroups — #347×#6157 #28310194

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290932	C₂	1	1	yes	maximal, characteristic
2	#20290930	C₂	1	2	yes	maximal
3	M₃₂×#6157	C₂	1	2	yes	maximal
4	#97×#6157	C₂	1	1	yes	maximal, characteristic
5	#347×D₅₃	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊M₃₂	C₂²	1	2	yes	—
7	#16811582	C₄	1	1	yes	characteristic
8	C₂×#13133271	C₄	1	1	yes	characteristic
9	C₂×#13133271	C₂²	1	1	yes	characteristic
10	C₂×C₈×#6157	C₄	1	1	yes	characteristic
11	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
12	C₄²×#6157	C₄	1	1	yes	characteristic
13	#16811516	C₄	1	2	yes	complemented
14	#16811515	C₄	1	2	yes	complemented
15	#16811514	C₄	1	4	yes	complemented
16	#16811513	C₄	1	4	yes	complemented
17	#16811510	C₄	1	4	yes	complemented
18	C₁₆×#6157	C₄	1	4	yes	complemented
19	M₃₂×D₅₃	C₂²	1	2	yes	—
20	#97×D₅₃	C₂²	1	1	yes	characteristic
21	#16810401	C₄	1	1	yes	characteristic, complemented
22	#16810400	C₄	1	2	yes	complemented
23	C₅₃×#347	C₄	1	1	yes	characteristic, complemented
24	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#6157	—	2	2	no	—
27	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
29	#13133292	C₂×C₄	1	4	yes	—
30	#13133291	C₂×C₄	1	4	yes	—
31	D₅₃⋊C₁₆	C₂×C₄	1	4	yes	—
32	#13133272	—	2	2	no	—
33	#13133271	—	2	2	no	—
34	#13133271	C₂×C₄	1	2	yes	—
35	#13133271	C₂×C₄	1	2	yes	—
36	C₈×#6157	—	2	2	no	—
37	C₈×#6157	C₂×C₄	1	2	yes	—
38	C₈×#6157	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
41	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
42	#13133157	C₂×C₄	1	2	yes	—
43	#13133156	C₂×C₄	1	4	yes	—
44	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
45	#13133128	C₂×C₄	1	1	yes	characteristic
46	#13133127	C₂×C₄	1	1	yes	characteristic
47	#13133127	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
49	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#6157	—	2	1	no	—
51	C₂²×#6157	M₁₆	1	2	yes	—
52	C₂×C₄×D₅₃	—	2	1	no	—
53	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×#88324	—	2	1	no	—
55	C₂×#88324	M₁₆	1	2	yes	—
56	C₄×#6157	—	2	1	no	—
57	C₄×#6157	—	4	4	no	—
58	C₄×#6157	C₄²	1	2	yes	—
59	C₄×#6157	C₄²	1	1	yes	characteristic
60	C₄×#6157	C₄²	1	1	yes	characteristic
61	C₄×#6157	M₁₆	1	2	yes	—
62	D₅₃⋊C₈	—	2	1	no	—
63	D₅₃⋊C₈	C₄²	1	1	yes	characteristic
64	D₅₃⋊C₈	C₄²	1	1	yes	characteristic
65	D₅₃⋊C₈	C₄²	1	2	yes	—
66	D₅₃⋊C₈	M₁₆	1	2	yes	—
67	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₄²	1	1	yes	characteristic
70	C₄×Dic₅₃	—	2	1	no	—
71	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
72	C₂×#88322	C₄²	1	1	yes	characteristic
73	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
74	C₈×D₅₃	—	2	1	no	—
75	C₈×D₅₃	—	2	1	no	—
76	C₈×D₅₃	C₂²×C₄	1	2	yes	—
77	C₈×D₅₃	C₂²×C₄	1	2	yes	—
78	#11719910	C₄²	1	8	yes	complemented
79	C₁₆×C₅₃	C₄²	1	4	yes	complemented
80	#11719908	C₄²	1	4	yes	complemented
81	C₂²×D₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×#6157	—	2	4	no	—
83	C₂×#6157	—	2	2	no	—
84	C₂×#6157	—	4	2	no	—
85	C₂×#6157	#97	1	2	yes	—
86	C₂×#6157	#97	1	2	yes	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×Dic₅₃	—	2	1	no	—
90	C₂×Dic₅₃	C₂×M₁₆	1	1	yes	characteristic
91	C₄×D₅₃	—	4	2	no	—
92	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
93	C₄×D₅₃	C₂×C₄²	1	1	yes	characteristic
94	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
96	#88324	—	2	1	no	—
97	#88324	—	2	1	no	—
98	#88324	#97	1	4	yes	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	C₂×C₄²	1	2	yes	—
101	#88322	—	2	1	no	—
102	#88322	C₂×C₄²	1	2	yes	—
103	C₄×#347	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
105	C₂×D₅₃	—	2	2	no	—
106	C₂×D₅₃	C₂×#97	1	1	yes	characteristic
107	C₂×D₅₃	C₄×M₁₆	1	1	yes	characteristic
108	#6157	—	2	2	no	—
109	#6157	—	4	2	no	—
110	#6157	#347	1	4	yes	direct product factor
111	C₄×C₅₃	—	4	1	no	—
112	C₄×C₅₃	C₄³	1	1	yes	characteristic
113	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
114	Dic₅₃	—	4	1	no	—
115	Dic₅₃	C₂×#97	1	1	yes	characteristic
116	Dic₅₃	C₄×M₁₆	1	1	yes	characteristic
117	C₂×#347	—	53	2	no	—
118	C₂×#347	—	53	1	no	—
119	C₂×#347	—	53	1	no	—
120	C₄×M₃₂	—	53	2	no	—
121	C₄×#97	—	53	1	no	—
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
124	D₅₃	—	2	1	no	—
125	D₅₃	C₂×#347	1	2	yes	—
126	C₂×M₃₂	—	53	2	no	—
127	C₂×M₃₂	—	53	2	no	—
128	C₂×#97	—	53	1	no	—
129	C₂×#97	—	53	1	no	—
130	C₂×#97	—	53	1	no	—
131	C₂×#97	—	53	1	no	—
132	C₂×C₄×C₈	—	53	1	no	—
133	C₂×C₄×C₈	—	53	1	no	—
134	C₄³	—	53	1	no	—
135	#347	—	53	4	no	—
136	#347	—	53	4	no	—
137	#347	—	53	2	no	—
138	#347	—	53	2	no	—
139	#347	—	53	2	no	—
140	#347	—	53	1	no	—
141	#347	#6157	1	1	yes	characteristic, direct product factor
142	#346	—	53	4	no	—
143	C₄×C₁₆	—	53	4	no	—
144	C₅₃	C₄×#347	1	1	yes	characteristic, complemented, Sylow
145	C₂²×C₈	—	53	1	no	—
146	C₂²×C₈	—	53	1	no	—
147	C₂²×C₈	—	53	1	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂×C₄²	—	53	1	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	1	no	—
152	C₂×C₄²	—	106	2	no	—
153	M₃₂	—	53	4	no	—
154	M₃₂	—	53	4	no	—
155	M₃₂	—	53	2	no	—
156	M₃₂	—	53	4	no	—
157	M₃₂	C₂×#6157	1	2	yes	—
158	C₂×C₁₆	—	53	4	no	—
159	C₂×C₁₆	—	53	4	no	—
160	#97	—	53	2	no	—
161	#97	—	53	2	no	—
162	#97	—	53	1	no	—
163	#97	—	53	1	no	—
164	#97	—	53	1	no	—
165	#97	—	106	2	no	—
166	#97	—	106	2	no	—
167	#97	C₂×#6157	1	1	yes	characteristic
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	—	53	2	no	—
170	C₄×C₈	—	106	2	no	—
171	C₂²×C₄	—	53	1	no	—
172	C₂²×C₄	—	53	2	no	—
173	C₂²×C₄	—	106	1	no	—
174	C₂²×C₄	—	106	1	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	53	1	no	—
179	C₂×C₈	—	53	2	no	—
180	C₂×C₈	—	53	2	no	—
181	C₂×C₈	—	53	1	no	—
182	C₂×C₈	—	53	1	no	—
183	C₂×C₈	—	53	1	no	—
184	C₂×C₈	—	106	1	no	—
185	C₂×C₈	—	106	1	no	—
186	C₂×C₈	—	106	1	no	—
187	C₂×C₈	—	106	1	no	—
188	C₂×C₈	C₄×#6157	1	1	yes	characteristic
189	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
190	C₄²	—	53	1	no	—
191	C₄²	—	53	1	no	—
192	C₄²	—	53	2	no	—
193	C₄²	—	53	2	no	—
194	C₄²	—	53	1	no	—
195	C₄²	—	106	1	no	—
196	C₄²	—	106	1	no	—
197	C₄²	—	212	4	no	—
198	C₄²	C₄×#6157	1	1	yes	characteristic
199	C₁₆	—	53	8	no	—
200	C₁₆	—	53	4	no	—
201	C₁₆	C₄×#6157	1	4	yes	complemented
202	C₂³	—	53	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	53	2	no	—
205	C₂×C₄	—	53	1	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	1	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	2	no	—
211	C₂×C₄	—	106	4	no	—
212	C₂×C₄	—	106	2	no	—
213	C₂×C₄	—	106	1	no	—
214	C₂×C₄	—	212	2	no	—
215	C₂×C₄	—	212	2	no	—
216	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	53	4	no	—
219	C₈	—	53	2	no	—
220	C₈	—	106	1	no	—
221	C₈	—	106	1	no	—
222	C₈	—	106	1	no	—
223	C₈	C₂×C₄×#6157	1	2	yes	—
224	C₂²	—	53	1	no	—
225	C₂²	—	53	1	no	—
226	C₂²	—	106	2	no	—
227	C₂²	M₁₆×#6157	1	1	yes	characteristic
228	C₄	—	4	1	no	—
229	C₄	—	53	4	no	—
230	C₄	—	53	1	no	—
231	C₄	—	53	1	no	—
232	C₄	—	106	2	no	—
233	C₄	—	212	1	no	—
234	C₄	—	212	2	no	—
235	C₄	C₄²×#6157	1	1	yes	characteristic
236	C₄	M₁₆×#6157	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	53	2	no	—
239	C₂	—	106	1	no	—
240	C₂	#97×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#6157
commutator	C₄×C₅₃	C₄³
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#347	C₄
socle	C₂×C₅₃	C₄×#97
prime radical	C₂²×D₅₃	C₂×M₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	#347	#6157
hypocenter	C₅₃	C₄×#347
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #347

Complements

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

N	H ≅ G/N	type
13. #16811516	C₄	semidirect
14. #16811515	C₄	semidirect
15. #16811514	C₄	semidirect
16. #16811513	C₄	semidirect
17. #16811510	C₄	semidirect
18. C₁₆×#6157	C₄	semidirect
21. #16810401	C₄	semidirect
22. #16810400	C₄	semidirect
23. C₅₃×#347	C₄	semidirect
78. #11719910	C₄²	semidirect
79. C₁₆×C₅₃	C₄²	semidirect
80. #11719908	C₄²	semidirect
110. #6157	#347	direct (N × H)
141. #347	#6157	direct (N × H)
144. C₅₃	C₄×#347	semidirect
201. 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
#20290932	2
#20290930	2
M₃₂×#6157	2
#97×#6157	2
#347×D₅₃	2
C₄×#347	53