Subgroups — #28311423

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811544	C₂	1	4	yes	maximal
2	C₂×C₄⋊#11719910	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719910	C₂²	1	4	yes	—
5	#16811544	—	2	8	no	—
6	C₄⋊#11719910	—	2	4	no	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
10	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719910	—	2	4	no	—
13	C₂×#11719910	—	2	8	no	—
14	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
15	C₂²×#88322	C₂³	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
18	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₄×#88322	D₄	1	2	yes	—
21	C₄×#88322	Q₈	1	2	yes	—
22	C₈×Dic₅₃	—	2	4	no	—
23	C₈×Dic₅₃	D₄	1	4	yes	—
24	C₈×Dic₅₃	Q₈	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
32	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₄²×C₅₃	M₁₆	1	2	yes	—
38	C₄×Dic₅₃	—	2	2	no	—
39	C₄×Dic₅₃	—	2	4	no	—
40	C₄×Dic₅₃	C₁₆	1	4	yes	complemented
41	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
42	C₄×Dic₅₃	M₁₆	1	4	yes	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	C₄⋊C₄	1	2	yes	—
45	C₂×#88322	C₂×D₄	1	2	yes	—
46	C₂×#88322	C₂×D₄	1	1	yes	characteristic
47	C₂×#88322	C₂×Q₈	1	2	yes	—
48	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
49	#11719910	—	4	4	no	—
50	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	2	2	no	—
62	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
63	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
64	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
65	C₂×Dic₅₃	M₃₂	1	4	yes	—
66	C₈×C₅₃	—	2	4	no	—
67	#88322	—	2	2	no	—
68	#88322	#127	1	1	yes	characteristic
69	#88322	C₄⋊Q₈	1	2	yes	—
70	#88322	C₄⋊Q₈	1	1	yes	characteristic
71	#12898	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
73	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
75	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
76	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	C₄⋊C₁₆	1	4	yes	complemented
79	C₄×C₅₃	#384	1	1	yes	characteristic
80	C₄×C₅₃	#423	1	2	yes	—
81	C₄×C₅₃	#423	1	1	yes	characteristic
82	Dic₅₃	—	2	4	no	—
83	Dic₅₃	C₄⋊C₁₆	1	8	yes	complemented
84	C₂×C₄⋊C₁₆	—	53	2	no	—
85	C₂×C₄⋊C₁₆	—	53	4	no	—
86	C₄²×C₈	—	53	1	no	—
87	C₂×C₅₃	#1842	1	1	yes	characteristic
88	C₂×C₅₃	C₂×C₄⋊C₁₆	1	2	yes	—
89	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
90	C₂×C₅₃	#2150	1	2	yes	—
91	C₂×C₅₃	#2150	1	1	yes	characteristic
92	C₂²×C₁₆	—	53	4	no	—
93	C₂×C₄×C₈	—	53	2	no	—
94	C₂×C₄×C₈	—	53	1	no	—
95	C₂×C₄×C₈	—	53	2	no	—
96	C₂×C₄×C₈	#6157	1	1	yes	characteristic
97	C₄³	—	53	1	no	—
98	C₄⋊C₁₆	—	106	4	no	—
99	C₄⋊C₁₆	—	106	8	no	—
100	C₅₃	#12898	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	—	53	1	no	—
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	C₂×#6157	1	2	yes	—
104	C₂×C₄²	—	53	2	no	—
105	C₂×C₄²	—	53	1	no	—
106	C₂×C₄²	—	53	2	no	—
107	C₂×C₄²	—	53	1	no	—
108	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
109	C₂×C₁₆	—	106	4	no	—
110	C₂×C₁₆	—	106	8	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	53	4	no	—
113	C₄×C₈	—	53	4	no	—
114	C₄×C₈	—	53	2	no	—
115	C₄×C₈	—	53	2	no	—
116	C₄×C₈	—	106	4	no	—
117	C₂²×C₄	—	53	2	no	—
118	C₂²×C₄	—	53	2	no	—
119	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
121	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	2	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	53	2	no	—
129	C₂×C₈	—	106	2	no	—
130	C₂×C₈	C₄⋊#6157	1	4	yes	—
131	C₄²	—	53	4	no	—
132	C₄²	—	53	4	no	—
133	C₄²	—	53	4	no	—
134	C₄²	—	106	4	no	—
135	C₄²	—	106	2	no	—
136	C₄²	C₄⋊#6157	1	2	yes	—
137	C₄²	#11719940	1	2	yes	—
138	C₁₆	—	212	4	no	—
139	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	53	4	no	—
142	C₂×C₄	—	53	4	no	—
143	C₂×C₄	—	53	4	no	—
144	C₂×C₄	—	53	4	no	—
145	C₂×C₄	—	106	2	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	#13133284	1	2	yes	—
148	C₂×C₄	#13133284	1	2	yes	—
149	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
150	C₂×C₄	#13133292	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
153	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
154	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	53	1	no	—
157	C₈	—	53	2	no	—
158	C₈	—	53	1	no	—
159	C₈	—	106	2	no	—
160	C₂²	D₅₃⋊(C₄⋊C₈)	1	2	yes	—
161	C₂²	C₂×#13133284	1	1	yes	characteristic
162	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
163	C₂²	D₅₃⋊M₃₂	1	2	yes	—
164	C₂²	C₂×#13133292	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	53	8	no	—
167	C₄	—	106	4	no	—
168	C₄	#16811544	1	4	yes	complemented
169	C₄	#16811569	1	1	yes	characteristic
170	C₄	#16811626	1	2	yes	—
171	C₄	#16811628	1	1	yes	characteristic
172	C₂	#20290648	1	1	yes	characteristic
173	C₂	D₅₃⋊(C₄⋊C₁₆)	1	2	yes	—
174	C₂	C₂×#16811544	1	1	yes	characteristic
175	C₂	#20291048	1	2	yes	—
176	C₂	#20291049	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12898
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
40. C₄×Dic₅₃	C₁₆	semidirect
78. C₄×C₅₃	C₄⋊C₁₆	semidirect
83. Dic₅₃	C₄⋊C₁₆	semidirect
100. C₅₃	#12898	semidirect
168. C₄	#16811544	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811544	2
C₂×C₄⋊#11719910	2
C₄×C₈×Dic₅₃	2
#12898	53