Subgroups — #27295636

Order 13568.

270 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272778	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₄⋊#11719908	—	2	1	no	—
9	C₄⋊#11719908	C₂²	1	2	yes	—
10	C₈×#88322	C₄	1	2	yes	—
11	C₈×#88322	C₂²	1	2	yes	—
12	C₂×M₃₂×C₅₃	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	—	2	1	no	—
15	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×#88322	—	2	1	no	—
17	C₂²×#88322	C₈	1	2	yes	—
18	#13133157	—	4	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₄×#88322	—	2	1	no	—
21	C₄×#88322	C₈	1	1	yes	characteristic
22	C₄×#88322	C₈	1	1	yes	characteristic
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₄×#88322	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₅₃	—	4	2	no	—
29	C₂×C₁₆×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×#88322	—	2	4	no	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	2	4	no	—
51	C₂×#88322	C₂×C₈	1	2	yes	—
52	C₂×#88322	C₂×C₈	1	2	yes	—
53	C₁₆×C₅₃	—	4	2	no	—
54	#11719908	—	4	2	no	—
55	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#106	1	2	yes	—
62	C₂×C₄×C₅₃	#108	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	#106	1	2	yes	—
69	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	#88322	—	2	4	no	—
72	#88322	—	2	2	no	—
73	#88322	M₃₂	1	4	yes	complemented
74	#11996	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	#334	1	2	yes	—
78	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	#334	1	2	yes	—
82	C₄×C₅₃	#364	1	2	yes	—
83	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
84	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
85	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
86	#2150	—	53	1	no	—
87	#2150	D₅₃	1	1	yes	characteristic
88	C₂×C₈²	—	53	1	no	—
89	#1367	—	53	4	no	—
90	C₂×C₅₃	—	2	2	no	—
91	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
92	C₂×C₅₃	#2150	1	1	yes	characteristic
93	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
94	C₂×M₃₂	—	2	1	no	—
95	C₂×M₃₂	—	106	1	no	—
96	C₂×C₄×C₈	—	53	1	no	—
97	C₂×C₄×C₈	—	53	1	no	—
98	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₁₆	—	2	1	no	—
100	C₄⋊C₁₆	—	53	2	no	—
101	C₄⋊C₁₆	—	106	1	no	—
102	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
103	C₈²	—	53	2	no	—
104	C₈²	—	53	2	no	—
105	C₅₃	#11996	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	53	2	no	—
107	C₂²×C₈	—	106	1	no	—
108	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
109	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
111	M₃₂	—	4	2	no	—
112	M₃₂	—	212	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	106	2	no	—
115	C₄×C₈	—	53	2	no	—
116	C₄×C₈	—	53	2	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	106	1	no	—
120	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
121	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
125	C₂²×C₄	#11719912	1	1	yes	characteristic
126	C₂²×C₄	#11719919	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	53	2	no	—
129	C₂×C₈	—	53	2	no	—
130	C₂×C₈	—	106	4	no	—
131	C₂×C₈	—	106	4	no	—
132	C₂×C₈	—	106	2	no	—
133	C₂×C₈	#11719919	1	1	yes	characteristic
134	C₂×C₈	#11719919	1	1	yes	characteristic
135	C₂×C₈	#11719923	1	2	yes	—
136	C₂×C₈	#11719924	1	2	yes	—
137	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₄²	C₈×D₅₃	1	1	yes	characteristic
140	C₄²	#11719912	1	1	yes	characteristic
141	C₄²	#11719919	1	1	yes	characteristic
142	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₁₆	—	4	2	no	—
144	C₁₆	—	212	2	no	—
145	C₂³	#13133146	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#13133146	1	1	yes	characteristic
149	C₂×C₄	#13133146	1	1	yes	characteristic
150	C₂×C₄	#13133146	1	1	yes	characteristic
151	C₂×C₄	#13133148	1	2	yes	—
152	C₂×C₄	#13133152	1	2	yes	—
153	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
154	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	53	4	no	—
159	C₈	—	106	4	no	—
160	C₈	—	106	2	no	—
161	C₈	#13133148	1	2	yes	—
162	C₈	C₂×#11719924	1	1	yes	characteristic
163	C₈	C₂×#11719923	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16810373	1	2	yes	—
167	C₂²	C₂×#13133146	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#16810373	1	2	yes	—
171	C₄	#16810443	1	2	yes	—
172	C₄	C₂×#13133148	1	1	yes	characteristic
173	C₄	C₂×#13133152	1	1	yes	characteristic
174	C₄	M₃₂×D₅₃	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	C₂×#16810373	1	1	yes	characteristic
177	C₂	#20276470	1	1	yes	characteristic
178	C₂	C₂×#16810443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#11996
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #20272778	C₂	semidirect
73. #88322	M₃₂	semidirect
105. C₅₃	#11996	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×C₈×#88322	2
#20272778	2
C₅₃×#2150	2
#11996	53