Subgroups — #27295692

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	2	yes	maximal
2	#20272785	C₂	1	2	yes	maximal, complemented
3	#20272779	C₂	1	2	yes	maximal, complemented
4	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719908	C₂²	1	4	yes	—
6	C₂²⋊#11719908	—	2	2	no	—
7	C₄×#11719908	—	2	2	no	—
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	C₈²×C₅₃	C₄	1	1	yes	characteristic
13	C₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719908	—	2	4	no	—
16	C₂²×C₈×C₅₃	—	2	1	no	—
17	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	—	2	1	no	—
23	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	4	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	C₁₆	1	4	yes	complemented
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
54	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	#11719908	—	4	4	no	—
56	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	—	2	1	no	—
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	#108	1	2	yes	—
75	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
76	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
77	C₈×C₅₃	#135	1	2	yes	—
78	#12012	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
82	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
86	C₄×C₅₃	#364	1	2	yes	—
87	C₄×C₅₃	#427	1	1	yes	characteristic
88	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
90	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
91	#2162	—	53	2	no	—
92	C₂×C₈²	D₅₃	1	1	yes	characteristic
93	#1370	—	53	2	no	—
94	#1367	—	53	2	no	—
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	#1564	1	1	yes	characteristic
97	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
99	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
100	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₄⋊C₁₆	—	53	4	no	—
103	C₂²⋊C₁₆	—	106	2	no	—
104	C₄×C₁₆	—	106	2	no	—
105	C₈²	Dic₅₃	1	1	yes	characteristic
106	C₈²	Dic₅₃	1	1	yes	characteristic
107	C₈²	C₂×D₅₃	1	1	yes	characteristic
108	C₈²	C₂×D₅₃	1	1	yes	characteristic
109	C₅₃	#12012	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	#88322	1	1	yes	characteristic
112	C₂²×C₈	#88322	1	1	yes	characteristic
113	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
114	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
115	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂×C₁₆	—	106	4	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	#88322	1	1	yes	characteristic
119	C₄×C₈	#88322	1	1	yes	characteristic
120	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
121	C₄×C₈	D₂₁₂	1	1	yes	characteristic
122	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
123	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
124	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
125	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
129	C₂²×C₄	#11719917	1	1	yes	characteristic
130	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	#11719908	1	4	yes	complemented
138	C₂×C₈	C₂×#88322	1	1	yes	characteristic
139	C₂×C₈	C₂×#88322	1	1	yes	characteristic
140	C₂×C₈	C₂×#88322	1	1	yes	characteristic
141	C₂×C₈	C₂×#88322	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
145	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
146	C₄²	C₂×#88322	1	1	yes	characteristic
147	C₄²	#11719917	1	1	yes	characteristic
148	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
149	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₁₆	—	212	4	no	—
151	C₂³	C₄⋊#88322	1	1	yes	characteristic
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
156	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
157	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719908	1	2	yes	—
159	C₂×C₄	#13133157	1	2	yes	—
160	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
162	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	#13133153	1	2	yes	—
170	C₈	C₂×#11719908	1	2	yes	—
171	C₈	C₂×#11719908	1	2	yes	—
172	C₈	#13133239	1	2	yes	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#11719908	1	2	yes	—
176	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₄⋊#11719908	1	2	yes	—
180	C₄	#16810444	1	2	yes	—
181	C₄	#16810661	1	1	yes	characteristic
182	C₄	C₂×#13133153	1	1	yes	characteristic
183	C₄	C₂²×#11719908	1	1	yes	characteristic
184	C₄	C₂×#13133157	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#20273463	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#11719908	1	1	yes	characteristic
188	C₂	C₂×#16810444	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#12012
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #20272785	C₂	semidirect
3. #20272779	C₂	semidirect
42. C₂×C₈×C₅₃	C₁₆	semidirect
109. C₅₃	#12012	semidirect
137. C₂×C₈	#11719908	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
#20272785	2
#20272779	2
C₂×C₈²×C₅₃	2
#12012	53