Subgroups — #28316097

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291630	C₂	1	2	yes	maximal
2	#20290268	C₂	1	6	yes	maximal, complemented
3	#20290264	C₂	1	3	yes	maximal, complemented
4	C₂×#16811487	C₂	1	3	yes	maximal, complemented
5	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88324	—	2	2	no	—
7	D₄×#88324	—	2	6	no	—
8	#16811639	C₂²	1	6	yes	—
9	C₂×C₄⋊#88324	C₂²	1	6	yes	—
10	#16811487	C₂²	1	3	yes	—
11	#16811487	C₂²	1	6	yes	—
12	#16811487	C₂²	1	1	yes	characteristic
13	#16811485	C₂²	1	6	yes	—
14	C₂×#13133224	C₄	1	1	yes	characteristic
15	#41×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810581	C₄	1	3	yes	—
17	#16810578	C₄	1	3	yes	—
18	M₁₆×Dic₅₃	C₂²	1	3	yes	—
19	C₂×C₈×Dic₅₃	C₂²	1	3	yes	—
20	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
21	C₂²×#88324	—	2	6	no	—
22	C₄⋊#88324	—	2	6	no	—
23	C₄⋊#88324	C₂³	1	6	yes	—
24	C₄⋊#88324	C₂³	1	2	yes	—
25	C₂²⋊#88324	—	2	6	no	—
26	C₄×#88324	—	2	6	no	—
27	#13133224	C₈	1	8	yes	complemented
28	C₂×#11719917	C₂×C₄	1	3	yes	—
29	C₂²×#88322	C₂×C₄	1	3	yes	—
30	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
31	Q₈×Dic₅₃	Q₈	1	1	yes	characteristic
32	D₄×Dic₅₃	D₄	1	3	yes	—
33	D₄×Dic₅₃	Q₈	1	3	yes	—
34	#13133176	C₂×C₄	1	3	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
36	#13133127	C₂×C₄	1	6	yes	—
37	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
39	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₈×Dic₅₃	C₂³	1	3	yes	—
41	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#131×C₅₃	—	2	4	no	—
43	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
45	#41×C₅₃	C₂×C₈	1	4	yes	—
46	#41×C₅₃	M₁₆	1	4	yes	—
47	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₅₃	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₅₃	C₂×D₄	1	3	yes	—
51	C₂²×Dic₅₃	C₂×Q₈	1	3	yes	—
52	C₂×#88324	—	2	2	no	—
53	C₂×#88324	—	2	6	no	—
54	C₂×#88324	—	8	6	no	—
55	M₁₆×C₅₃	—	2	6	no	—
56	C₂×C₈×C₅₃	—	2	6	no	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
59	C₂²⋊Dic₅₃	C₄⋊C₄	1	6	yes	—
60	C₄⋊Dic₅₃	C₄⋊C₄	1	3	yes	—
61	C₄⋊Dic₅₃	C₄⋊C₄	1	3	yes	—
62	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
63	C₄×Dic₅₃	C₂×D₄	1	3	yes	—
64	C₄×Dic₅₃	C₂×Q₈	1	3	yes	—
65	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
66	#11719917	C₂×C₈	1	12	yes	complemented
67	C₂×#88322	C₂×C₈	1	12	yes	complemented
68	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
69	C₂×#88322	C₂²×C₄	1	3	yes	—
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
71	Q₈×C₅₃	C₄⋊C₈	1	4	yes	—
72	D₄×C₅₃	C₄⋊C₈	1	12	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	—	2	3	no	—
78	C₂×Dic₅₃	—	2	3	no	—
79	C₂×Dic₅₃	—	4	3	no	—
80	C₂×Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×Dic₅₃	C₂×C₄⋊C₄	1	3	yes	—
82	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
84	#88324	—	4	6	no	—
85	#88324	—	4	2	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	2	3	no	—
88	#88322	C₂²×C₈	1	2	yes	—
89	#88322	C₂²×C₈	1	6	yes	—
90	#17789	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	3	no	—
92	C₂²×C₅₃	—	2	3	no	—
93	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
94	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
96	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
98	Dic₅₃	—	2	3	no	—
99	Dic₅₃	—	2	1	no	—
100	Dic₅₃	—	2	3	no	—
101	Dic₅₃	#576	1	1	yes	characteristic
102	Dic₅₃	#578	1	1	yes	characteristic
103	#2988	—	53	2	no	—
104	C₄×#131	—	53	1	no	—
105	#1565	—	53	6	no	—
106	#1564	—	53	3	no	—
107	C₂×#335	—	53	3	no	—
108	C₂×C₅₃	—	2	6	no	—
109	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₅₃	#2913	1	1	yes	characteristic
111	C₂×C₅₃	#2916	1	1	yes	characteristic
112	C₂×#131	—	53	1	no	—
113	C₂×#131	#6157	1	1	yes	characteristic
114	C₄×#41	—	53	1	no	—
115	C₈×Q₈	—	106	2	no	—
116	C₈×D₄	—	106	6	no	—
117	#431	—	53	6	no	—
118	C₂×C₄⋊C₈	—	53	6	no	—
119	#405	—	53	3	no	—
120	#405	—	53	3	no	—
121	C₄×M₁₆	—	53	3	no	—
122	C₂×C₄×C₈	—	53	3	no	—
123	#335	—	53	3	no	—
124	#335	—	53	6	no	—
125	#335	—	53	1	no	—
126	#334	—	53	6	no	—
127	C₅₃	#17789	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₂×#6157	1	1	yes	characteristic
129	#131	—	2	4	no	—
130	#131	—	53	8	no	—
131	C₂×M₁₆	—	53	3	no	—
132	C₂×M₁₆	C₂×#6157	1	3	yes	—
133	C₂²×C₈	—	53	3	no	—
134	C₂²×C₈	—	106	6	no	—
135	C₂²×C₈	C₂×#6157	1	3	yes	—
136	C₄×Q₈	—	53	1	no	—
137	C₄×Q₈	—	53	1	no	—
138	C₄×D₄	—	53	3	no	—
139	C₄×D₄	—	53	3	no	—
140	#117	—	53	3	no	—
141	C₂×C₄²	—	53	3	no	—
142	C₄⋊C₈	—	53	2	no	—
143	C₄⋊C₈	—	53	6	no	—
144	C₄⋊C₈	—	106	6	no	—
145	C₂²⋊C₈	—	106	6	no	—
146	#97	—	53	6	no	—
147	C₄×C₈	—	53	3	no	—
148	C₄×C₈	—	53	3	no	—
149	C₄×C₈	—	53	3	no	—
150	C₄×C₈	—	53	1	no	—
151	C₄×C₈	—	106	6	no	—
152	#41	D₅₃⋊C₈	1	4	yes	—
153	#41	#11719940	1	4	yes	—
154	C₂×Q₈	C₄⋊#6157	1	1	yes	characteristic
155	C₂×D₄	C₄⋊#6157	1	3	yes	—
156	C₂²×C₄	—	53	3	no	—
157	C₂²×C₄	—	53	3	no	—
158	C₂²×C₄	C₂²×#6157	1	3	yes	—
159	M₁₆	—	2	6	no	—
160	M₁₆	—	53	12	no	—
161	C₂×C₈	—	2	6	no	—
162	C₂×C₈	—	53	12	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	3	no	—
165	C₂×C₈	—	106	2	no	—
166	C₂×C₈	—	106	6	no	—
167	C₂×C₈	—	424	6	no	—
168	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
169	C₂×C₈	C₂²×#6157	1	3	yes	—
170	C₄⋊C₄	—	53	3	no	—
171	C₄⋊C₄	—	53	3	no	—
172	C₂²⋊C₄	—	53	6	no	—
173	C₄²	—	53	3	no	—
174	C₄²	—	53	3	no	—
175	C₄²	—	53	1	no	—
176	C₄²	—	53	3	no	—
177	C₂³	C₂×C₄⋊#6157	1	3	yes	—
178	Q₈	#13133284	1	4	yes	—
179	D₄	#13133284	1	12	yes	—
180	C₂×C₄	—	53	1	no	—
181	C₂×C₄	—	53	3	no	—
182	C₂×C₄	—	53	3	no	—
183	C₂×C₄	—	53	1	no	—
184	C₂×C₄	—	106	3	no	—
185	C₂×C₄	—	106	3	no	—
186	C₂×C₄	—	212	3	no	—
187	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
188	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
189	C₂×C₄	C₂×#11719940	1	6	yes	—
190	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	3	no	—
193	C₈	—	53	2	no	—
194	C₈	—	53	6	no	—
195	C₈	—	212	2	no	—
196	C₈	—	212	6	no	—
197	C₂²	—	2	3	no	—
198	C₂²	—	2	3	no	—
199	C₂²	C₂×#13133284	1	6	yes	—
200	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
201	C₄	—	53	1	no	—
202	C₄	—	53	1	no	—
203	C₄	—	106	3	no	—
204	C₄	—	106	1	no	—
205	C₄	—	106	3	no	—
206	C₄	C₂×#13133284	1	6	yes	—
207	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
208	C₄	C₂²×#11719940	1	1	yes	characteristic
209	C₂	—	2	6	no	—
210	C₂	C₂²×#13133284	1	1	yes	characteristic
211	C₂	#20291370	1	1	yes	characteristic
212	C₂	#20291376	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#131×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂×#131	#6157
hypocenter	C₅₃	#17789
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #20290268	C₂	semidirect
3. #20290264	C₂	semidirect
4. C₂×#16811487	C₂	semidirect
27. #13133224	C₈	semidirect
66. #11719917	C₂×C₈	semidirect
67. C₂×#88322	C₂×C₈	semidirect
127. C₅₃	#17789	semidirect

Maximal subgroups

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

subgroup	index
#20291630	2
#20290268	2
#20290264	2
C₂×#16811487	2
#131×Dic₅₃	2
#17789	53