Subgroups — #27384557

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	4	yes	maximal
2	C₂×#16810647	C₂	1	2	yes	maximal
3	#20274890	C₂	1	4	yes	maximal, complemented
4	#20274887	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	4	yes	—
7	#16810672	—	2	16	no	—
8	C₂×#13133153	C₂²	1	16	yes	—
9	#16810647	—	2	4	no	—
10	#16810647	C₂²	1	4	yes	—
11	#16810647	C₂²	1	4	yes	—
12	C₂×C₄⋊#88322	—	2	4	no	—
13	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
16	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	16	no	—
21	C₂×#11719917	C₂³	1	8	yes	—
22	C₂²×#88322	—	2	4	no	—
23	#13133153	—	2	32	no	—
24	C₄⋊#88322	—	2	8	no	—
25	C₄⋊#88322	—	2	8	no	—
26	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
29	C₂×M₁₆×C₅₃	D₄	1	8	yes	—
30	C₂×M₁₆×C₅₃	Q₈	1	8	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	4	yes	—
35	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#97×C₅₃	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	4	4	no	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₅₃	C₄⋊C₄	1	32	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
56	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
57	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
58	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₅₃	C₂×D₄	1	2	yes	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	#11719917	—	2	16	no	—
65	#11719917	—	4	8	no	—
66	C₂×#88322	—	2	8	no	—
67	C₂×#88322	—	4	8	no	—
68	C₂³×C₅₃	—	4	2	no	—
69	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
71	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
79	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₅₃	#127	1	4	yes	—
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
83	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
88	C₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
89	#88322	—	4	8	no	—
90	#20806	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	4	no	—
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	—	4	4	no	—
94	C₂²×C₅₃	#384	1	4	yes	—
95	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
97	C₄×C₅₃	—	2	4	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	#384	1	4	yes	—
100	C₄×C₅₃	#430	1	4	yes	complemented
101	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
102	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
103	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₂×#430	—	53	4	no	—
105	C₂×#423	—	53	2	no	—
106	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
107	#1849	—	53	4	no	—
108	#1848	—	53	4	no	—
109	C₂×C₅₃	—	2	2	no	—
110	C₂×C₅₃	—	2	4	no	—
111	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#430	1	2	yes	—
113	C₂²×M₁₆	—	53	4	no	—
114	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
115	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
116	#430	—	106	16	no	—
117	C₂×#108	—	53	16	no	—
118	#423	—	53	4	no	—
119	#423	—	53	4	no	—
120	#423	—	106	4	no	—
121	C₂×C₄⋊C₈	—	106	4	no	—
122	C₄×M₁₆	Dic₅₃	1	4	yes	—
123	C₄×M₁₆	Dic₅₃	1	4	yes	—
124	C₂×#97	C₂×D₅₃	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
126	C₅₃	#20806	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₄	Dic₁₀₆	1	2	yes	—
128	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
129	C₂×M₁₆	—	53	8	no	—
130	C₂×M₁₆	—	106	4	no	—
131	C₂×M₁₆	—	106	16	no	—
132	C₂×M₁₆	Dic₁₀₆	1	8	yes	—
133	C₂×M₁₆	D₂₁₂	1	8	yes	—
134	C₂×M₁₆	C₂×Dic₅₃	1	8	yes	—
135	C₂²×C₈	—	106	4	no	—
136	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
137	C₂×C₄²	Dic₁₀₆	1	2	yes	—
138	C₂×C₄²	D₂₁₂	1	4	yes	—
139	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
140	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
141	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
143	#108	—	106	32	no	—
144	C₄⋊C₈	—	106	8	no	—
145	C₄⋊C₈	—	106	8	no	—
146	#97	C₂×Dic₅₃	1	8	yes	—
147	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
148	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
149	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
152	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
153	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
154	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
155	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
157	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
158	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
159	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
160	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
161	M₁₆	—	106	16	no	—
162	M₁₆	—	212	8	no	—
163	M₁₆	C₄⋊Dic₅₃	1	32	yes	—
164	C₂×C₈	—	106	8	no	—
165	C₂×C₈	—	212	8	no	—
166	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
167	C₂×C₈	C₂×D₂₁₂	1	8	yes	—
168	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
169	C₄²	C₄⋊Dic₅₃	1	4	yes	—
170	C₄²	C₄⋊Dic₅₃	1	4	yes	—
171	C₄²	C₂×D₂₁₂	1	2	yes	—
172	C₄²	C₂×D₂₁₂	1	2	yes	—
173	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
174	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
175	C₄²	C₂²×Dic₅₃	1	2	yes	—
176	C₂³	—	4	2	no	—
177	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
178	C₂³	#13133217	1	4	yes	—
179	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
185	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
186	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
187	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
189	C₂×C₄	#13133214	1	4	yes	—
190	C₂×C₄	#13133217	1	4	yes	—
191	C₂×C₄	#13133217	1	4	yes	—
192	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
193	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
194	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₁₀₆	1	2	yes	—
196	C₈	—	212	8	no	—
197	C₈	C₂×C₄⋊Dic₅₃	1	16	yes	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	#16810493	1	4	yes	—
202	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
203	C₂²	C₂×#13133217	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	#16810493	1	4	yes	—
207	C₄	#16810672	1	4	yes	complemented
208	C₄	C₂²×C₄⋊Dic₅₃	1	2	yes	—
209	C₄	C₂×#13133214	1	1	yes	characteristic
210	C₄	C₂×#13133217	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	2	4	no	—
213	C₂	C₂×#16810493	1	1	yes	characteristic
214	C₂	C₂×#16810672	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#20806
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20274890	C₂	semidirect
4. #20274887	C₂	semidirect
100. C₄×C₅₃	#430	semidirect
126. C₅₃	#20806	semidirect
207. C₄	#16810672	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
C₂×#16810647	2
#20274890	2
#20274887	2
C₂×C₄×M₁₆×C₅₃	2
#20806	53