Subgroups — #27295351

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	#20272765	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	—	2	1	no	—
6	C₂²×Dic₂₁₂	C₄	1	2	yes	—
7	C₂×#13133217	—	2	1	no	—
8	#16811003	—	2	1	no	—
9	#16811003	—	2	1	no	—
10	#16811003	C₄	1	2	yes	—
11	#16811003	C₂²	1	2	yes	—
12	#16810676	—	2	1	no	—
13	#16810376	C₂²	1	2	yes	—
14	#16810374	—	2	1	no	—
15	C₂×M₃₂×C₅₃	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	—	2	1	no	—
18	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	4	2	no	—
24	C₂×Dic₂₁₂	—	4	2	no	—
25	C₂×Dic₂₁₂	—	4	4	no	—
26	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
27	#13133217	—	2	2	no	—
28	#13133217	—	2	2	no	—
29	#13133217	—	4	1	no	—
30	#13133217	—	4	1	no	—
31	C₂×C₄⋊Dic₅₃	—	4	1	no	—
32	C₂×C₄⋊Dic₅₃	—	4	1	no	—
33	C₄⋊#88322	—	2	2	no	—
34	C₂²⋊#88322	—	4	1	no	—
35	C₄×#88322	—	4	1	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	M₃₂×C₅₃	—	4	2	no	—
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	4	2	no	—
49	C₂×Dic₁₀₆	—	2	4	no	—
50	C₂×Dic₁₀₆	—	4	4	no	—
51	C₂×Dic₁₀₆	—	4	4	no	—
52	C₂×Dic₁₀₆	—	4	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	D₈	1	2	yes	—
57	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	D₈	1	1	yes	characteristic
62	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	4	2	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	Dic₂₁₂	—	4	4	no	—
70	Dic₂₁₂	—	4	4	no	—
71	Dic₂₁₂	—	4	4	no	—
72	Dic₂₁₂	—	4	2	no	—
73	Dic₂₁₂	—	4	2	no	—
74	C₁₆×C₅₃	—	4	2	no	—
75	C₂³×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	#100	1	2	yes	—
79	C₂×C₄×C₅₃	#102	1	2	yes	—
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₅₃	—	4	4	no	—
87	C₂×Dic₅₃	—	8	4	no	—
88	Dic₁₀₆	—	4	4	no	—
89	Dic₁₀₆	—	4	4	no	—
90	Dic₁₀₆	—	8	4	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	#102	1	2	yes	—
94	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
95	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	#88322	—	8	2	no	—
97	#11932	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	#331	1	2	yes	—
101	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	#331	1	2	yes	—
105	C₄×C₅₃	#362	1	2	yes	—
106	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
107	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
108	C₄×C₅₃	#510	1	2	yes	—
109	Dic₅₃	—	8	4	no	—
110	C₂×#494	—	53	1	no	—
111	#2150	D₅₃	1	1	yes	characteristic
112	#1564	—	53	1	no	—
113	#1362	—	53	4	no	—
114	C₂×C₅₃	—	2	2	no	—
115	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
116	C₂×C₅₃	#2141	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
118	C₂²×Q₁₆	—	53	2	no	—
119	C₂²×Q₁₆	—	106	1	no	—
120	C₂×C₄⋊Q₈	—	106	1	no	—
121	C₂×M₃₂	—	2	1	no	—
122	#494	—	53	2	no	—
123	#494	—	53	2	no	—
124	#494	—	106	1	no	—
125	#494	—	106	1	no	—
126	#431	—	106	1	no	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	1	no	—
129	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
130	#335	—	53	2	no	—
131	#334	—	106	1	no	—
132	C₅₃	#11932	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	106	2	no	—
134	C₂×Q₁₆	—	53	4	no	—
135	C₂×Q₁₆	—	106	2	no	—
136	C₂×Q₁₆	—	106	2	no	—
137	C₂×Q₁₆	—	106	2	no	—
138	C₂×Q₁₆	—	212	4	no	—
139	C₂×Q₁₆	—	212	2	no	—
140	C₂×Q₁₆	—	212	2	no	—
141	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	106	2	no	—
144	C₄⋊Q₈	—	106	2	no	—
145	C₄⋊Q₈	—	212	1	no	—
146	C₄⋊Q₈	—	212	1	no	—
147	C₂×C₄⋊C₄	—	212	1	no	—
148	C₂×C₄⋊C₄	—	212	1	no	—
149	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
150	M₃₂	—	4	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₄⋊C₈	—	106	2	no	—
153	C₂²⋊C₈	—	212	1	no	—
154	C₄×C₈	—	212	1	no	—
155	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
156	C₄×C₈	D₂₁₂	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×Q₈	—	106	4	no	—
160	C₂×Q₈	—	212	4	no	—
161	C₂×Q₈	—	212	4	no	—
162	C₂×Q₈	—	212	2	no	—
163	C₂²×C₄	—	212	2	no	—
164	C₂²×C₄	#11719913	1	1	yes	characteristic
165	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
166	C₂²×C₄	#11719921	1	1	yes	characteristic
167	Q₁₆	—	212	4	no	—
168	Q₁₆	—	212	4	no	—
169	Q₁₆	—	212	4	no	—
170	Q₁₆	—	212	2	no	—
171	Q₁₆	—	212	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	#11719921	1	1	yes	characteristic
175	C₂×C₈	#11719921	1	1	yes	characteristic
176	C₂×C₈	#11719922	1	2	yes	—
177	C₂×C₈	#11719924	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄⋊C₄	—	212	2	no	—
181	C₄⋊C₄	—	212	2	no	—
182	C₄⋊C₄	—	212	2	no	—
183	C₄⋊C₄	—	212	2	no	—
184	C₄²	#11719913	1	1	yes	characteristic
185	C₄²	D₄₂₄	1	1	yes	characteristic
186	C₄²	#11719921	1	1	yes	characteristic
187	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
188	C₁₆	—	4	2	no	—
189	C₂³	#13133139	1	1	yes	characteristic
190	Q₈	—	424	4	no	—
191	Q₈	—	212	4	no	—
192	Q₈	—	212	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	424	4	no	—
196	C₂×C₄	—	212	4	no	—
197	C₂×C₄	#13133133	1	2	yes	—
198	C₂×C₄	#13133137	1	2	yes	—
199	C₂×C₄	#13133139	1	1	yes	characteristic
200	C₂×C₄	#13133139	1	1	yes	characteristic
201	C₂×C₄	#13133139	1	1	yes	characteristic
202	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	424	2	no	—
208	C₈	#13133137	1	2	yes	—
209	C₈	C₂×#11719922	1	1	yes	characteristic
210	C₈	C₂×#11719924	1	1	yes	characteristic
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#16810366	1	2	yes	—
214	C₂²	C₂×#13133139	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	424	4	no	—
218	C₄	#16810366	1	2	yes	—
219	C₄	#16810440	1	2	yes	—
220	C₄	C₂×#13133133	1	1	yes	characteristic
221	C₄	C₂×#13133137	1	1	yes	characteristic
222	C₄	#16811066	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₂×#16810366	1	1	yes	characteristic
225	C₂	#20276436	1	1	yes	characteristic
226	C₂	C₂×#16810440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#11932
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #20272765	C₂	semidirect
132. C₅₃	#11932	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811003	2
#20273463	2
#20272765	2
C₅₃×#2150	2
#11932	53