Subgroups — #27275304

Order 13568.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	2	yes	maximal
4	#20272550	C₂	1	2	yes	maximal
5	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133128	—	2	1	no	—
9	C₂×C₄×#88322	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	#16810376	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	C₄	1	4	yes	—
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	C₂×C₄	1	4	yes	—
22	C₄⋊#88322	C₂×C₄	1	4	yes	—
23	#13133128	—	4	2	no	—
24	C₄×#88322	—	4	2	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	C₄²	1	4	yes	—
43	C₄²×C₅₃	—	2	4	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	4	2	no	—
46	C₄²×C₅₃	D₈	1	2	yes	—
47	C₄²×C₅₃	Q₁₆	1	2	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
49	C₄²×C₅₃	#41	1	2	yes	—
50	C₂×#88322	—	2	4	no	—
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	#107	1	4	yes	—
64	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₅₃	#131	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#135	1	2	yes	—
72	C₈×C₅₃	—	2	4	no	—
73	C₈×C₅₃	—	4	2	no	—
74	#88322	—	4	4	no	—
75	#88322	—	4	4	no	—
76	#8278	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₅₃	#424	1	2	yes	—
79	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₅₃	#427	1	1	yes	characteristic
81	C₂²×C₅₃	#430	1	2	yes	—
82	C₄×C₅₃	—	2	4	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	#405	1	2	yes	—
86	C₄×C₅₃	#433	1	2	yes	—
87	C₄×C₅₃	C₄×D₈	1	2	yes	—
88	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
89	#1767	—	53	1	no	—
90	C₄×C₄⋊C₈	—	53	1	no	—
91	#1744	D₅₃	1	1	yes	characteristic
92	C₂×#335	—	53	2	no	—
93	#1277	—	53	2	no	—
94	C₂×C₅₃	#1571	1	2	yes	—
95	C₂×C₅₃	#1572	1	2	yes	—
96	C₂×C₅₃	#1768	1	1	yes	characteristic
97	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
98	C₂×C₅₃	#1778	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	53	2	no	—
100	C₂×C₄⋊C₈	—	53	2	no	—
101	C₂×#97	—	106	1	no	—
102	C₂×C₄×C₈	—	106	1	no	—
103	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	C₄³	C₂×D₅₃	1	1	yes	characteristic
106	#336	—	106	1	no	—
107	#336	—	106	1	no	—
108	#336	Dic₅₃	1	4	yes	—
109	#335	—	53	8	no	—
110	C₅₃	#8278	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	106	2	no	—
112	C₂²×C₈	—	106	2	no	—
113	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
114	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
115	C₂×C₄²	Dic₁₀₆	1	2	yes	—
116	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
117	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
119	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
120	C₄⋊C₈	—	53	4	no	—
121	C₄⋊C₈	—	53	4	no	—
122	C₄⋊C₈	—	106	2	no	—
123	C₄⋊C₈	—	106	2	no	—
124	#97	—	212	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	212	2	no	—
127	C₄×C₈	C₄×D₅₃	1	4	yes	—
128	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
129	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
130	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
132	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	106	4	no	—
137	C₂×C₈	—	106	4	no	—
138	C₂×C₈	—	212	2	no	—
139	C₂×C₈	—	212	2	no	—
140	C₂×C₈	—	212	2	no	—
141	C₂×C₈	—	212	2	no	—
142	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	D₄₂₄	1	2	yes	—
147	C₄²	Dic₂₁₂	1	2	yes	—
148	C₄²	C₂×C₄×D₅₃	1	2	yes	—
149	C₄²	#11719945	1	2	yes	—
150	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#13133151	1	4	yes	—
158	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	#13133176	1	1	yes	characteristic
160	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
161	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
162	C₂×C₄	#13133224	1	4	yes	—
163	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
164	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
165	C₂×C₄	#13133239	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	212	4	no	—
169	C₈	—	212	4	no	—
170	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂²	#16810650	1	2	yes	—
172	C₂²	C₂×#13133151	1	1	yes	characteristic
173	C₂²	#16810661	1	1	yes	characteristic
174	C₂²	#16810672	1	2	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	#16810579	1	2	yes	—
179	C₄	#16810684	1	2	yes	—
180	C₄	C₄×D₄₂₄	1	2	yes	—
181	C₄	C₄×Dic₂₁₂	1	2	yes	—
182	C₂	#20273490	1	2	yes	—
183	C₂	#20273495	1	2	yes	—
184	C₂	#20274572	1	1	yes	characteristic
185	C₂	C₄×#13133151	1	1	yes	characteristic
186	C₂	#20274621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8278
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
12. #16810376	C₄	semidirect
110. C₅₃	#8278	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₄×C₄⋊#88322	2
C₂×#16810376	2
#20272550	2
C₅₃×#1744	2
#8278	53