Subgroups — #28307825

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291072	C₂	1	2	yes	maximal
2	#20289628	C₂	1	2	yes	maximal, complemented
3	C₈×#11719910	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719910	C₄	1	4	yes	—
6	C₂²⋊#11719910	C₄	1	4	yes	—
7	C₄×#11719910	C₂²	1	4	yes	—
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₈×#88322	C₄	1	1	yes	characteristic
11	C₈×#88322	C₄	1	1	yes	characteristic
12	C₈×#88322	C₂²	1	1	yes	characteristic
13	C₈×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#11719910	C₂×C₄	1	8	yes	—
16	C₂²×#88322	C₈	1	2	yes	—
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₄×#88322	C₈	1	2	yes	—
19	C₄×#88322	C₂×C₄	1	1	yes	characteristic
20	C₄×#88322	C₂×C₄	1	2	yes	—
21	C₄×#88322	C₂×C₄	1	1	yes	characteristic
22	C₄×#88322	C₂×C₄	1	1	yes	characteristic
23	C₄×#88322	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₈	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₈	1	2	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₁₆	1	8	yes	complemented
33	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
35	C₄²×C₅₃	C₄²	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×#88322	—	2	4	no	—
40	C₂×#88322	—	2	4	no	—
41	C₂×#88322	C₄²	1	2	yes	—
42	C₂×#88322	C₂×C₈	1	2	yes	—
43	C₂×#88322	C₂×C₈	1	2	yes	—
44	C₂×#88322	C₂²×C₄	1	2	yes	—
45	#11719910	—	2	8	no	—
46	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
57	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
58	#88322	—	2	2	no	—
59	#88322	—	2	2	no	—
60	#88322	M₃₂	1	4	yes	—
61	#88322	#131	1	4	yes	—
62	#11410	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
66	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
70	C₄×C₅₃	#405	1	2	yes	—
71	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
72	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
73	#2162	—	53	2	no	—
74	C₂×C₈²	—	53	1	no	—
75	#1312	—	53	2	no	—
76	C₈×C₁₆	—	53	2	no	—
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#1450	1	1	yes	characteristic
79	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
80	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
81	C₂×C₄×C₈	—	53	1	no	—
82	C₂×C₄×C₈	—	53	1	no	—
83	C₂×C₄×C₈	#6157	1	1	yes	characteristic
84	C₄⋊C₁₆	—	53	4	no	—
85	C₂²⋊C₁₆	—	53	4	no	—
86	C₄×C₁₆	—	53	4	no	—
87	C₈²	—	53	1	no	—
88	C₈²	—	53	1	no	—
89	C₈²	—	53	1	no	—
90	C₈²	—	53	1	no	—
91	C₅₃	#11410	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	53	2	no	—
93	C₂²×C₈	—	53	2	no	—
94	C₂²×C₈	#88324	1	2	yes	—
95	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
96	C₂×C₁₆	—	53	8	no	—
97	C₄×C₈	—	53	2	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	—	53	2	no	—
103	C₄×C₈	#88324	1	2	yes	—
104	C₄×C₈	C₂×#6157	1	1	yes	characteristic
105	C₄×C₈	C₂×#6157	1	1	yes	characteristic
106	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
107	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
108	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
109	C₂×C₈	—	53	2	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	106	4	no	—
114	C₂×C₈	—	106	4	no	—
115	C₂×C₈	#11719910	1	8	yes	complemented
116	C₂×C₈	C₂×#88324	1	2	yes	—
117	C₂×C₈	C₂×#88324	1	2	yes	—
118	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
119	C₄²	C₄×#6157	1	1	yes	characteristic
120	C₄²	C₂×#88324	1	1	yes	characteristic
121	C₄²	C₂²×#6157	1	1	yes	characteristic
122	C₁₆	—	106	8	no	—
123	C₂³	C₄×#88324	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₄×#88324	1	1	yes	characteristic
127	C₂×C₄	C₄×#88324	1	1	yes	characteristic
128	C₂×C₄	C₄×#88324	1	1	yes	characteristic
129	C₂×C₄	C₂×#11719910	1	4	yes	—
130	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
131	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
132	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
133	C₈	—	53	4	no	—
134	C₈	—	53	4	no	—
135	C₈	—	106	2	no	—
136	C₈	—	106	2	no	—
137	C₈	C₂×#11719910	1	4	yes	—
138	C₈	C₂×#11719910	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₄×#11719910	1	2	yes	—
142	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	C₄×#11719910	1	2	yes	—
146	C₄	#16811586	1	2	yes	—
147	C₄	C₂²×#11719910	1	2	yes	—
148	C₄	D₅₃⋊M₃₂	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#20289918	1	1	yes	characteristic
151	C₂	C₂×C₄×#11719910	1	1	yes	characteristic
152	C₂	#20290921	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₄²	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#11410
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289628	C₂	semidirect
3. C₈×#11719910	C₂	semidirect
32. C₂×C₈×C₅₃	C₁₆	semidirect
91. C₅₃	#11410	semidirect
115. C₂×C₈	#11719910	semidirect

Maximal subgroups

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

subgroup	index
#20291072	2
#20289628	2
C₈×#11719910	2
C₂×C₈×#88322	2
#11410	53