Subgroups — #97×#88324 #28300034

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290429	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×#16811464	C₂	1	2	yes	maximal
4	C₂×C₈×#88324	C₂	1	2	yes	maximal
5	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88324	C₄	1	4	yes	—
7	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88324	C₂²	1	2	yes	—
10	#16811489	C₄	1	4	yes	—
11	#16811464	C₄	1	8	yes	complemented
12	C₈×#88324	C₄	1	8	yes	complemented
13	C₂×#13133128	C₄	1	1	yes	characteristic
14	C₂×#13133127	C₄	1	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×#88324	C₂×C₄	1	2	yes	—
19	C₂²×#88324	C₂×C₄	1	4	yes	—
20	C₂²×#88324	C₂×C₄	1	2	yes	—
21	C₄×#88324	—	2	8	no	—
22	C₄×#88324	C₂×C₄	1	4	yes	—
23	C₄×#88324	C₂×C₄	1	4	yes	—
24	C₄×#88324	C₂×C₄	1	4	yes	—
25	C₄×#88324	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	#13133128	C₈	1	4	yes	complemented
32	#13133127	C₈	1	8	yes	complemented
33	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
34	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#97×C₅₃	C₈	1	4	yes	complemented
38	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₄²	1	2	yes	—
41	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×#88324	—	2	4	no	—
44	C₂×#88324	—	2	8	no	—
45	C₂×#88324	C₄²	1	4	yes	—
46	C₂×#88324	C₄²	1	8	yes	—
47	C₂×#88324	C₄²	1	4	yes	—
48	C₂×#88324	M₁₆	1	8	yes	—
49	C₂×C₈×C₅₃	C₄²	1	4	yes	—
50	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
52	C₄²×C₅₃	C₂×C₈	1	2	yes	—
53	C₄²×C₅₃	C₂×C₈	1	2	yes	—
54	C₄×Dic₅₃	—	2	4	no	—
55	C₄×Dic₅₃	—	2	4	no	—
56	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
57	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
58	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
59	C₂×#88322	C₄²	1	4	yes	—
60	C₂×#88322	C₂×C₈	1	4	yes	—
61	C₂×#88322	C₂×C₈	1	4	yes	—
62	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
73	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
76	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
77	C₂×Dic₅₃	#131	1	4	yes	—
78	#88324	—	2	4	no	—
79	#88324	—	2	8	no	—
80	#88324	#97	1	8	yes	direct product factor
81	C₈×C₅₃	C₄×C₈	1	8	yes	complemented
82	#88322	C₄×C₈	1	8	yes	complemented
83	C₈×#97	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	C₄³	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
87	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
88	C₂²×C₅₃	#405	1	2	yes	—
89	C₄×C₅₃	—	2	4	no	—
90	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
91	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
92	Dic₅₃	—	2	4	no	—
93	Dic₅₃	C₂×#97	1	2	yes	—
94	Dic₅₃	C₄×M₁₆	1	2	yes	—
95	Dic₅₃	#405	1	4	yes	—
96	#1744	—	53	1	no	—
97	C₄×#97	—	53	1	no	—
98	C₄²×C₈	—	53	1	no	—
99	C₂×#322	—	53	2	no	—
100	C₂×C₈²	—	53	2	no	—
101	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
102	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
103	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
104	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
105	C₂×C₅₃	#1726	1	1	yes	characteristic
106	C₂×#97	—	53	2	no	—
107	C₂×#97	—	53	1	no	—
108	C₂×#97	#6157	1	1	yes	characteristic
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	2	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	2	no	—
113	C₂×C₄×C₈	—	53	4	no	—
114	C₄³	—	53	1	no	—
115	#336	—	53	4	no	—
116	#322	—	53	8	no	—
117	C₈²	—	53	8	no	—
118	C₅₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	53	2	no	—
120	C₂²×C₈	—	53	2	no	—
121	C₂²×C₈	—	53	4	no	—
122	C₂²×C₈	—	53	2	no	—
123	C₂²×C₈	C₂×#6157	1	2	yes	—
124	C₂×C₄²	—	53	1	no	—
125	C₂×C₄²	—	53	1	no	—
126	C₂×C₄²	—	53	2	no	—
127	C₂×C₄²	—	53	2	no	—
128	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
129	#97	—	53	8	no	—
130	#97	—	53	4	no	—
131	#97	#88324	1	4	yes	direct product factor
132	C₄×C₈	—	53	4	no	—
133	C₄×C₈	—	53	4	no	—
134	C₄×C₈	—	53	4	no	—
135	C₄×C₈	—	53	4	no	—
136	C₄×C₈	—	53	4	no	—
137	C₄×C₈	—	53	4	no	—
138	C₄×C₈	—	106	8	no	—
139	C₂²×C₄	—	53	1	no	—
140	C₂²×C₄	—	53	1	no	—
141	C₂²×C₄	—	53	2	no	—
142	C₂²×C₄	C₄×#6157	1	2	yes	—
143	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
144	C₂×C₈	—	53	4	no	—
145	C₂×C₈	—	53	8	no	—
146	C₂×C₈	—	53	4	no	—
147	C₂×C₈	—	53	4	no	—
148	C₂×C₈	—	53	4	no	—
149	C₂×C₈	—	53	8	no	—
150	C₂×C₈	—	53	4	no	—
151	C₂×C₈	—	106	4	no	—
152	C₂×C₈	—	106	8	no	—
153	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
154	C₂×C₈	C₄×#6157	1	4	yes	—
155	C₂×C₈	C₂×#88324	1	4	yes	—
156	C₄²	—	53	2	no	—
157	C₄²	—	53	2	no	—
158	C₄²	—	53	4	no	—
159	C₄²	—	106	4	no	—
160	C₄²	—	106	4	no	—
161	C₄²	D₅₃⋊C₈	1	2	yes	—
162	C₄²	C₂×#88324	1	2	yes	—
163	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	53	1	no	—
167	C₂×C₄	—	53	1	no	—
168	C₂×C₄	—	53	2	no	—
169	C₂×C₄	—	53	4	no	—
170	C₂×C₄	—	53	4	no	—
171	C₂×C₄	—	53	4	no	—
172	C₂×C₄	—	106	2	no	—
173	C₂×C₄	—	106	2	no	—
174	C₂×C₄	C₄×#88324	1	4	yes	—
175	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
176	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
177	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
178	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
179	C₈	—	53	8	no	—
180	C₈	—	53	8	no	—
181	C₈	—	106	4	no	—
182	C₈	—	106	8	no	—
183	C₈	C₄×#88324	1	8	yes	complemented
184	C₂²	C₄²×#6157	1	1	yes	characteristic
185	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
186	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
187	C₂²	M₁₆×#6157	1	2	yes	—
188	C₂²	#16811589	1	2	yes	—
189	C₄	—	2	4	no	—
190	C₄	—	53	2	no	—
191	C₄	—	53	2	no	—
192	C₄	—	53	4	no	—
193	C₄	—	106	4	no	—
194	C₄	C₂×C₄×#88324	1	2	yes	—
195	C₄	C₄×D₅₃⋊C₈	1	2	yes	—
196	C₂	M₁₆×#88324	1	2	yes	—
197	C₂	#20289913	1	2	yes	—
198	C₂	C₄²×#88324	1	1	yes	characteristic
199	C₂	#97×#6157	1	1	yes	characteristic
200	C₂	#20290373	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₂×#97×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	C₈×#97
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
11. #16811464	C₄	semidirect
12. C₈×#88324	C₄	semidirect
31. #13133128	C₈	semidirect
32. #13133127	C₈	semidirect
37. #97×C₅₃	C₈	semidirect
80. #88324	#97	direct (N × H)
81. C₈×C₅₃	C₄×C₈	semidirect
82. #88322	C₄×C₈	semidirect
118. C₅₃	C₈×#97	semidirect
131. #97	#88324	direct (N × H)
183. C₈	C₄×#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290429	2
C₄²×#88324	2
C₂×#16811464	2
C₂×C₈×#88324	2
#97×Dic₅₃	2
C₈×#97	53