Subgroups — #27537084

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811040	C₂	1	2	yes	maximal
2	#20276474	C₂	1	6	yes	maximal, complemented
3	#20276471	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊#11719908	C₂	1	3	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811040	—	2	8	no	—
7	C₂×#13133157	C₂²	1	6	yes	—
8	C₂²×#11719908	C₂²	1	6	yes	—
9	C₄⋊#11719908	C₂²	1	1	yes	characteristic
10	C₄⋊#11719908	C₂²	1	3	yes	—
11	C₄⋊#11719908	C₂²	1	6	yes	—
12	C₄⋊#11719908	C₂²	1	6	yes	—
13	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
16	C₅₃×#405	C₄	1	3	yes	—
17	C₅₃×#405	C₂²	1	3	yes	—
18	C₄×M₁₆×C₅₃	C₄	1	3	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
20	#13133157	—	2	12	no	—
21	C₂×#11719908	—	2	12	no	—
22	C₂×#11719908	C₂³	1	2	yes	—
23	C₂×#11719908	C₂³	1	6	yes	—
24	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#131×C₅₃	—	2	4	no	—
26	#131×C₅₃	D₄	1	4	yes	—
27	#131×C₅₃	Q₈	1	4	yes	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
29	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
32	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
34	C₄×D₄×C₅₃	C₈	1	3	yes	—
35	C₄×D₄×C₅₃	C₈	1	3	yes	—
36	#117×C₅₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
38	#97×C₅₃	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂³	1	3	yes	—
43	#41×C₅₃	C₄⋊C₄	1	4	yes	—
44	#41×C₅₃	M₁₆	1	4	yes	—
45	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₅₃	C₂×C₈	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
50	M₁₆×C₅₃	—	2	6	no	—
51	M₁₆×C₅₃	C₄⋊C₄	1	12	yes	—
52	C₂×C₈×C₅₃	—	2	6	no	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
58	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
60	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
61	C₂²⋊C₄×C₅₃	C₂×C₈	1	6	yes	—
62	C₄²×C₅₃	C₂×C₈	1	3	yes	—
63	C₄²×C₅₃	C₂×C₈	1	3	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
66	#11719908	—	2	2	no	—
67	#11719908	—	2	6	no	—
68	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
69	Q₈×C₅₃	C₄⋊C₈	1	4	yes	—
70	D₄×C₅₃	C₄⋊C₈	1	12	yes	—
71	C₂×C₄×C₅₃	—	2	6	no	—
72	C₂×C₄×C₅₃	—	4	3	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₅₃	—	2	1	no	—
82	C₈×C₅₃	—	2	3	no	—
83	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
84	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
85	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
86	#33398	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	3	no	—
88	C₂²×C₅₃	—	2	3	no	—
89	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
90	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₅₃	—	2	3	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	3	no	—
94	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
95	C₄×C₅₃	#504	1	2	yes	complemented
96	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
98	C₂×#504	—	53	2	no	—
99	C₄×#131	D₅₃	1	1	yes	characteristic
100	#2151	—	53	6	no	—
101	#2150	—	53	3	no	—
102	C₂×C₄⋊C₁₆	—	53	3	no	—
103	C₂×C₅₃	—	2	6	no	—
104	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
106	C₂×C₅₃	#3407	1	1	yes	characteristic
107	C₂×#131	Dic₅₃	1	1	yes	characteristic
108	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
109	C₄×#41	Dic₅₃	1	1	yes	characteristic
110	#504	—	106	8	no	—
111	C₂×M₃₂	—	53	6	no	—
112	C₂²×C₁₆	—	53	6	no	—
113	#405	Dic₅₃	1	3	yes	—
114	#405	C₂×D₅₃	1	3	yes	—
115	C₄×M₁₆	Dic₅₃	1	3	yes	—
116	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
117	C₄⋊C₁₆	—	53	1	no	—
118	C₄⋊C₁₆	—	53	3	no	—
119	C₄⋊C₁₆	—	53	6	no	—
120	C₄⋊C₁₆	—	53	6	no	—
121	C₅₃	#33398	1	1	yes	characteristic, complemented, Sylow
122	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
123	#131	—	2	4	no	—
124	#131	Dic₁₀₆	1	4	yes	—
125	#131	D₂₁₂	1	4	yes	—
126	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
127	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
128	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
129	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
130	C₄×Q₈	#88322	1	1	yes	characteristic
131	C₄×Q₈	#88322	1	1	yes	characteristic
132	C₄×D₄	#88322	1	3	yes	—
133	C₄×D₄	#88322	1	3	yes	—
134	#117	C₂×Dic₅₃	1	3	yes	—
135	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
136	M₃₂	—	106	12	no	—
137	C₂×C₁₆	—	53	2	no	—
138	C₂×C₁₆	—	53	6	no	—
139	C₂×C₁₆	—	106	12	no	—
140	#97	C₂×Dic₅₃	1	6	yes	—
141	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
142	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
143	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	C₄×C₈	C₂²×D₅₃	1	3	yes	—
145	#41	#11719917	1	4	yes	—
146	#41	C₄⋊Dic₅₃	1	4	yes	—
147	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
148	C₂×D₄	C₂×#88322	1	3	yes	—
149	C₂²×C₄	C₂×#88322	1	3	yes	—
150	C₂²×C₄	C₂×#88322	1	3	yes	—
151	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
152	M₁₆	—	2	6	no	—
153	M₁₆	C₄⋊Dic₅₃	1	12	yes	—
154	C₂×C₈	—	2	6	no	—
155	C₂×C₈	C₂×Dic₁₀₆	1	6	yes	—
156	C₂×C₈	C₂×D₂₁₂	1	6	yes	—
157	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
158	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
159	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
160	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
161	C₄⋊C₄	C₂×#88322	1	3	yes	—
162	C₄⋊C₄	C₂×#88322	1	3	yes	—
163	C₂²⋊C₄	C₂×#88322	1	6	yes	—
164	C₄²	C₂×#88322	1	3	yes	—
165	C₄²	C₂×#88322	1	3	yes	—
166	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
167	C₄²	C₂²×Dic₅₃	1	3	yes	—
168	C₁₆	—	106	2	no	—
169	C₁₆	—	106	6	no	—
170	C₂³	C₂²×#88322	1	3	yes	—
171	Q₈	C₄⋊#88322	1	4	yes	—
172	D₄	C₄⋊#88322	1	12	yes	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
176	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
177	C₂×C₄	C₂²×#88322	1	3	yes	—
178	C₂×C₄	C₂²×#88322	1	3	yes	—
179	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
180	C₂×C₄	C₂²×#88322	1	3	yes	—
181	C₂×C₄	C₂×#11719917	1	6	yes	—
182	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	C₂×C₄⋊Dic₅₃	1	6	yes	—
186	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
187	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₄⋊#88322	1	6	yes	—
191	C₂²	C₂³×#88322	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₄⋊#88322	1	6	yes	—
196	C₄	#16811040	1	2	yes	complemented
197	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
198	C₄	C₂²×#11719917	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
201	C₂	C₂×#16811040	1	1	yes	characteristic
202	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×#131×C₅₃	C₂
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#33398
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20276474	C₂	semidirect
3. #20276471	C₂	semidirect
4. C₂×C₄⋊#11719908	C₂	semidirect
95. C₄×C₅₃	#504	semidirect
121. C₅₃	#33398	semidirect
196. C₄	#16811040	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811040	2
#20276474	2
#20276471	2
C₂×C₄⋊#11719908	2
C₄×#131×C₅₃	2
#33398	53