Subgroups — #97×#88322 #27273443

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274461	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810344	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	2	yes	maximal
5	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	C₄	1	4	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810378	C₄	1	4	yes	—
11	#16810344	C₄	1	8	yes	complemented
12	C₈×#88322	C₄	1	8	yes	complemented
13	C₂×#97×C₅₃	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂²×#88322	C₂×C₄	1	4	yes	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₄×#88322	—	2	8	no	—
20	C₄×#88322	C₂×C₄	1	4	yes	—
21	C₄×#88322	C₂×C₄	1	4	yes	—
22	C₄×#88322	C₂×C₄	1	4	yes	—
23	C₄×#88322	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#97×C₅₃	C₈	1	16	yes	complemented
30	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₄²	1	8	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	—
38	C₄²×C₅₃	—	2	4	no	—
39	C₄²×C₅₃	—	2	4	no	—
40	C₄²×C₅₃	C₂×C₈	1	8	yes	—
41	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
42	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×#88322	—	2	4	no	—
44	C₂×#88322	—	2	8	no	—
45	C₂×#88322	C₄²	1	4	yes	—
46	C₂×#88322	C₄²	1	8	yes	—
47	C₂×#88322	C₄²	1	4	yes	—
48	C₂×#88322	M₁₆	1	8	yes	—
49	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	4	no	—
51	C₂×C₄×C₅₃	—	2	4	no	—
52	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₅₃	#131	1	4	yes	—
60	C₈×C₅₃	C₄×C₈	1	16	yes	complemented
61	#88322	—	2	4	no	—
62	#88322	—	2	8	no	—
63	#88322	#97	1	8	yes	direct product factor
64	C₈×#97	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₄³	1	1	yes	characteristic
66	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
67	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
68	C₂²×C₅₃	#405	1	2	yes	—
69	C₄×C₅₃	—	2	8	no	—
70	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
71	C₄×C₅₃	C₂×#97	1	2	yes	—
72	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
73	C₄×C₅₃	#405	1	4	yes	—
74	#1744	—	53	1	no	—
75	C₄×#97	D₅₃	1	1	yes	characteristic
76	C₄²×C₈	—	53	1	no	—
77	C₂×#322	—	53	2	no	—
78	C₂×C₈²	—	53	2	no	—
79	C₂×C₅₃	C₈×M₁₆	1	4	yes	—
80	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
81	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
82	C₂×C₅₃	#1726	1	1	yes	characteristic
83	C₂×#97	Dic₅₃	1	4	yes	—
84	C₂×C₄×C₈	—	53	1	no	—
85	C₂×C₄×C₈	—	53	2	no	—
86	C₂×C₄×C₈	—	53	1	no	—
87	C₂×C₄×C₈	—	53	4	no	—
88	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
89	C₄³	C₂×D₅₃	1	1	yes	characteristic
90	#336	—	53	4	no	—
91	#322	—	53	8	no	—
92	C₈²	—	53	8	no	—
93	C₅₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	53	2	no	—
95	C₂²×C₈	—	53	4	no	—
96	C₂²×C₈	—	53	2	no	—
97	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
98	C₂×C₄²	C₄×D₅₃	1	2	yes	—
99	C₂×C₄²	C₄×D₅₃	1	2	yes	—
100	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
101	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
102	#97	#88322	1	16	yes	direct product factor
103	C₄×C₈	—	53	4	no	—
104	C₄×C₈	—	53	4	no	—
105	C₄×C₈	—	53	4	no	—
106	C₄×C₈	—	53	4	no	—
107	C₄×C₈	—	106	8	no	—
108	C₄×C₈	C₄×D₅₃	1	4	yes	—
109	C₄×C₈	C₄×D₅₃	1	4	yes	—
110	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
111	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₂×C₈	—	53	4	no	—
115	C₂×C₈	—	53	8	no	—
116	C₂×C₈	—	53	4	no	—
117	C₂×C₈	—	53	8	no	—
118	C₂×C₈	—	106	4	no	—
119	C₂×C₈	—	106	8	no	—
120	C₂×C₈	C₂×#88322	1	16	yes	—
121	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	C₂×#88322	1	8	yes	—
125	C₄²	C₂×C₄×D₅₃	1	2	yes	—
126	C₄²	C₂×C₄×D₅₃	1	2	yes	—
127	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	C₄×#88322	1	8	yes	—
131	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
132	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
133	C₂×C₄	C₄²×D₅₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
135	C₂×C₄	C₂²×#88322	1	4	yes	—
136	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
137	C₂×C₄	#13133225	1	4	yes	—
138	C₈	—	53	8	no	—
139	C₈	—	106	4	no	—
140	C₈	—	106	8	no	—
141	C₈	C₄×#88322	1	16	yes	complemented
142	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
143	C₂²	C₂×C₄×#88322	1	2	yes	—
144	C₂²	M₁₆×Dic₅₃	1	2	yes	—
145	C₂²	#16810581	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₂×C₄×#88322	1	4	yes	—
148	C₄	#97×D₅₃	1	2	yes	—
149	C₄	#16810575	1	2	yes	—
150	C₄	#16810580	1	4	yes	—
151	C₂	M₁₆×#88322	1	4	yes	—
152	C₂	C₄²×#88322	1	1	yes	characteristic
153	C₂	#97×Dic₅₃	1	1	yes	characteristic
154	C₂	#20274403	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. #16810344	C₄	semidirect
12. C₈×#88322	C₄	semidirect
29. #97×C₅₃	C₈	semidirect
60. C₈×C₅₃	C₄×C₈	semidirect
63. #88322	#97	direct (N × H)
93. C₅₃	C₈×#97	semidirect
102. #97	#88322	direct (N × H)
141. C₈	C₄×#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274461	2
C₄²×#88322	2
C₂×#16810344	2
C₂×C₈×#88322	2
C₄×#97×C₅₃	2
C₈×#97	53