Subgroups — #27353329

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283283	C₂	1	2	yes	maximal
2	#20273490	C₂	1	6	yes	maximal, complemented
3	#20273474	C₂	1	3	yes	maximal, complemented
4	C₂×#16810376	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133224	—	2	2	no	—
7	#16810650	—	2	6	no	—
8	#16810647	C₂²	1	6	yes	—
9	C₂×C₄⋊#88322	C₂²	1	6	yes	—
10	#16810376	C₂²	1	1	yes	characteristic
11	#16810376	C₂²	1	3	yes	—
12	#16810376	C₂²	1	6	yes	—
13	#16810376	C₂²	1	6	yes	—
14	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
17	C₈×D₄×C₅₃	C₄	1	3	yes	—
18	C₈×D₄×C₅₃	C₄	1	3	yes	—
19	C₅₃×#431	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
21	#13133224	—	4	8	no	—
22	C₂×#11719917	—	2	6	no	—
23	C₂²×#88322	—	2	6	no	—
24	C₄⋊#88322	—	2	6	no	—
25	C₄⋊#88322	—	2	6	no	—
26	C₄⋊#88322	C₂³	1	2	yes	—
27	C₄⋊#88322	C₂³	1	6	yes	—
28	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
31	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₄×D₄×C₅₃	C₂×C₄	1	3	yes	—
34	C₄×D₄×C₅₃	Q₈	1	3	yes	—
35	#117×C₅₃	D₄	1	3	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
37	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
38	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
39	C₂²⋊C₈×C₅₃	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂³	1	3	yes	—
44	#41×C₅₃	D₈	1	4	yes	—
45	#41×C₅₃	Q₁₆	1	4	yes	—
46	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×D₄×C₅₃	C₄⋊C₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
51	C₂×C₈×C₅₃	—	2	6	no	—
52	C₂×C₈×C₅₃	—	4	3	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	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
58	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
59	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
60	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
61	C₄²×C₅₃	C₂×D₄	1	3	yes	—
62	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
63	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
64	#11719917	—	4	12	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	6	no	—
67	C₂×#88322	—	4	12	no	—
68	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₅₃	#107	1	4	yes	—
70	D₄×C₅₃	#107	1	12	yes	—
71	C₂×C₄×C₅₃	—	2	3	no	—
72	C₂×C₄×C₅₃	—	2	3	no	—
73	C₂×C₄×C₅₃	—	4	3	no	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	6	yes	—
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
79	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₅₃	—	2	3	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	3	no	—
85	C₈×C₅₃	#131	1	2	yes	complemented
86	#88322	—	4	2	no	—
87	#88322	—	4	6	no	—
88	#17828	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	—	2	3	no	—
90	C₂²×C₅₃	—	2	3	no	—
91	C₂²×C₅₃	C₂×#107	1	6	yes	—
92	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₅₃	—	2	3	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	3	no	—
96	C₄×C₅₃	C₂×#107	1	6	yes	—
97	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
98	C₄×C₅₃	#568	1	1	yes	characteristic
99	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
101	C₈×#41	D₅₃	1	1	yes	characteristic
102	#2906	—	53	2	no	—
103	#1571	—	53	6	no	—
104	#1567	—	53	3	no	—
105	C₂×#335	—	53	3	no	—
106	C₂×C₅₃	—	2	6	no	—
107	C₂×C₅₃	#2906	1	1	yes	characteristic
108	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
109	C₂×C₅₃	#2916	1	1	yes	characteristic
110	C₂×#131	—	106	2	no	—
111	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
112	C₈×Q₈	Dic₅₃	1	1	yes	characteristic
113	C₈×Q₈	Dic₅₃	1	1	yes	characteristic
114	C₈×D₄	Dic₅₃	1	3	yes	—
115	C₈×D₄	Dic₅₃	1	3	yes	—
116	#431	C₂×D₅₃	1	3	yes	—
117	#424	—	106	6	no	—
118	#423	—	53	6	no	—
119	C₂×C₄⋊C₈	—	53	6	no	—
120	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
121	#335	—	53	1	no	—
122	#335	—	53	3	no	—
123	#335	—	53	6	no	—
124	#335	—	53	6	no	—
125	C₅₃	#17828	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	D₂₁₂	1	1	yes	characteristic
127	#131	—	212	8	no	—
128	C₂×M₁₆	—	106	6	no	—
129	C₂²×C₈	—	106	6	no	—
130	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
131	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
132	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
133	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
134	C₄×D₄	Dic₁₀₆	1	3	yes	—
135	C₄×D₄	C₂×Dic₅₃	1	3	yes	—
136	#117	D₂₁₂	1	3	yes	—
137	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
138	C₄⋊C₈	—	53	2	no	—
139	C₄⋊C₈	—	53	6	no	—
140	C₄⋊C₈	—	106	6	no	—
141	C₄⋊C₈	—	106	6	no	—
142	C₄⋊C₈	C₂×Dic₅₃	1	3	yes	—
143	C₄⋊C₈	C₂×Dic₅₃	1	3	yes	—
144	C₂²⋊C₈	C₂×Dic₅₃	1	6	yes	—
145	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
146	C₄×C₈	C₂×Dic₅₃	1	3	yes	—
147	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
148	C₄×C₈	C₂²×D₅₃	1	3	yes	—
149	#41	D₄₂₄	1	4	yes	—
150	#41	Dic₂₁₂	1	4	yes	—
151	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
152	C₂×D₄	C₄⋊Dic₅₃	1	3	yes	—
153	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
154	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
155	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
156	M₁₆	—	212	12	no	—
157	C₂×C₈	—	2	6	no	—
158	C₂×C₈	—	4	3	no	—
159	C₂×C₈	—	106	2	no	—
160	C₂×C₈	—	106	6	no	—
161	C₂×C₈	—	212	12	no	—
162	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
163	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
164	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
165	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₄⋊C₄	C₄⋊Dic₅₃	1	3	yes	—
167	C₄⋊C₄	C₄⋊Dic₅₃	1	3	yes	—
168	C₂²⋊C₄	C₄⋊Dic₅₃	1	6	yes	—
169	C₄²	C₂×Dic₁₀₆	1	3	yes	—
170	C₄²	C₂×D₂₁₂	1	3	yes	—
171	C₄²	C₂²×Dic₅₃	1	3	yes	—
172	C₄²	C₂³×D₅₃	1	1	yes	characteristic
173	C₂³	C₂×C₄⋊Dic₅₃	1	3	yes	—
174	Q₈	#13133151	1	4	yes	—
175	D₄	#13133151	1	12	yes	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	2	3	no	—
178	C₂×C₄	—	4	3	no	—
179	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
182	C₂×C₄	C₂×D₄₂₄	1	6	yes	—
183	C₂×C₄	C₂×Dic₂₁₂	1	6	yes	—
184	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
186	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	212	2	no	—
191	C₈	—	212	6	no	—
192	C₈	#13133224	1	2	yes	complemented
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×#13133151	1	6	yes	—
196	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×#13133151	1	6	yes	—
201	C₄	C₂×#13133224	1	1	yes	characteristic
202	C₄	#16811390	1	1	yes	characteristic
203	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
204	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	#20283283	1	1	yes	characteristic
207	C₂	C₂²×#13133151	1	1	yes	characteristic
208	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#17828
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #20273490	C₂	semidirect
3. #20273474	C₂	semidirect
4. C₂×#16810376	C₂	semidirect
85. C₈×C₅₃	#131	semidirect
125. C₅₃	#17828	semidirect
192. C₈	#13133224	semidirect

Maximal subgroups

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

subgroup	index
#20283283	2
#20273490	2
#20273474	2
C₂×#16810376	2
C₈×#41×C₅₃	2
#17828	53