Subgroups — #27271008

Order 13568.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	—	2	1	no	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₄×#11719908	—	2	1	no	—
7	C₄×#11719908	C₄	1	2	yes	complemented
8	#16810376	—	2	1	no	—
9	#16810376	C₄	1	2	yes	complemented
10	C₂×#107×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
13	C₅₃×#340	—	2	1	no	—
14	C₂²×#88322	—	4	1	no	—
15	C₂×#11719908	—	2	4	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	4	1	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄²×C₅₃	—	4	1	no	—
23	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
24	#107×C₅₃	—	2	2	no	—
25	#107×C₅₃	C₈	1	4	yes	complemented
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
30	#95×C₅₃	—	4	1	no	—
31	C₂²×C₄×C₅₃	—	4	2	no	—
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
42	C₄⋊C₄×C₅₃	—	2	4	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄²×C₅₃	—	8	1	no	—
45	C₄²×C₅₃	—	8	1	no	—
46	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
50	C₂×#88322	—	4	2	no	—
51	C₂×#88322	—	8	1	no	—
52	C₂×#88322	—	8	1	no	—
53	#11719908	—	2	4	no	—
54	C₂³×C₅₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	4	4	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#104	1	2	yes	—
65	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
67	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
69	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
74	C₈×C₅₃	#97	1	1	yes	characteristic
75	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
76	#88322	—	8	2	no	—
77	#7171	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
79	C₂²×C₅₃	#340	1	1	yes	characteristic
80	C₂²×C₅₃	#357	1	1	yes	characteristic
81	C₂²×C₅₃	#358	1	1	yes	characteristic
82	C₂²×C₅₃	#359	1	2	yes	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
87	C₄×C₅₃	#336	1	1	yes	characteristic
88	C₄×C₅₃	#339	1	1	yes	characteristic
89	C₄×C₅₃	#357	1	1	yes	characteristic
90	C₄×C₅₃	#358	1	1	yes	characteristic
91	C₄×C₅₃	#366	1	1	yes	characteristic
92	C₄×C₅₃	#367	1	1	yes	characteristic
93	C₂×C₄×C₁₆	—	53	1	no	—
94	C₄×#107	D₅₃	1	1	yes	characteristic
95	C₂×#335	—	53	1	no	—
96	C₂×C₅₃	#1279	1	1	yes	characteristic
97	C₂×C₅₃	#1331	1	2	yes	—
98	C₂×C₅₃	#1332	1	2	yes	—
99	C₂×C₅₃	#1380	1	1	yes	characteristic
100	C₂×C₅₃	#1382	1	1	yes	characteristic
101	C₂²×C₁₆	—	106	1	no	—
102	C₂×#107	Dic₅₃	1	2	yes	—
103	C₂×C₄⋊C₈	—	106	1	no	—
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄×C₁₆	—	53	2	no	—
107	C₄×C₁₆	—	106	1	no	—
108	#340	—	2	1	no	—
109	#335	—	53	2	no	—
110	#335	—	106	1	no	—
111	C₅₃	#7171	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	212	1	no	—
113	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
118	C₂×C₁₆	—	106	4	no	—
119	C₂×C₁₆	—	106	2	no	—
120	#107	—	2	2	no	—
121	#107	#88322	1	4	yes	complemented
122	C₄⋊C₈	—	106	2	no	—
123	C₄⋊C₈	—	212	1	no	—
124	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
125	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
126	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
127	C₄×C₈	D₂₁₂	1	1	yes	characteristic
128	#95	—	4	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	#11719922	1	1	yes	characteristic
131	C₂²×C₄	#11719923	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	424	1	no	—
135	C₂×C₈	—	424	1	no	—
136	C₂×C₈	—	212	2	no	—
137	C₂×C₈	C₂×#88322	1	2	yes	—
138	C₂×C₈	#11719917	1	2	yes	—
139	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
140	C₂×C₈	#11719919	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
142	C₂×C₈	#11719921	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	#11719919	1	1	yes	characteristic
148	C₄²	#11719921	1	1	yes	characteristic
149	C₄²	#11719924	1	1	yes	characteristic
150	C₄²	#11719925	1	1	yes	characteristic
151	C₁₆	—	106	4	no	—
152	C₂³	#13133138	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	#13133124	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊#88322	1	2	yes	—
159	C₂×C₄	#13133137	1	1	yes	characteristic
160	C₂×C₄	#13133140	1	1	yes	characteristic
161	C₂×C₄	#13133141	1	1	yes	characteristic
162	C₂×C₄	#13133144	1	2	yes	—
163	C₂×C₄	#13133148	1	1	yes	characteristic
164	C₂×C₄	#13133150	1	1	yes	characteristic
165	C₂×C₄	#13133158	1	1	yes	characteristic
166	C₂×C₄	#13133160	1	1	yes	characteristic
167	C₂×C₄	#13133161	1	1	yes	characteristic
168	C₂×C₄	#13133163	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	424	2	no	—
172	C₈	C₄×#88322	1	1	yes	characteristic
173	C₈	#13133128	1	1	yes	characteristic
174	C₈	C₄⋊#88322	1	2	yes	—
175	C₂²	D₄⋊#88322	1	2	yes	—
176	C₂²	#16810385	1	1	yes	characteristic
177	C₂²	#16810424	1	1	yes	characteristic
178	C₂²	#16810427	1	1	yes	characteristic
179	C₂²	#16810430	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	Q₈⋊#88322	1	2	yes	—
184	C₄	#16810378	1	1	yes	characteristic
185	C₄	#16810382	1	1	yes	characteristic
186	C₄	#16810423	1	1	yes	characteristic
187	C₄	#16810426	1	1	yes	characteristic
188	C₄	#16810448	1	1	yes	characteristic
189	C₄	#16810450	1	1	yes	characteristic
190	C₂	#20272556	1	1	yes	characteristic
191	C₂	#20272676	1	2	yes	—
192	C₂	#20272679	1	2	yes	—
193	C₂	#20272806	1	1	yes	characteristic
194	C₂	#20272810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	#7171
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. C₄×#11719908	C₄	semidirect
9. #16810376	C₄	semidirect
25. #107×C₅₃	C₈	semidirect
111. C₅₃	#7171	semidirect
121. #107	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
C₂×#16810376	2
C₄×#107×C₅₃	2
#7171	53