Subgroups — #27274764

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273097	C₂	1	2	yes	maximal
2	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272553	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	Q₈⋊#88322	C₄	1	4	yes	—
9	D₄⋊#88322	C₄	1	4	yes	—
10	#16810344	—	2	1	no	—
11	C₈×#88322	—	2	2	no	—
12	C₈×#88322	—	2	1	no	—
13	C₂×#131×C₅₃	C₄	1	2	yes	—
14	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#405	—	2	2	no	—
16	C₅₃×#405	C₄	1	2	yes	—
17	C₄×M₁₆×C₅₃	—	2	1	no	—
18	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	C₂×C₄	1	4	yes	—
23	C₂²⋊#88322	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	C₂×C₄	1	4	yes	—
27	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#131×C₅₃	—	2	4	no	—
29	#131×C₅₃	C₈	1	8	yes	complemented
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×D₄×C₅₃	—	2	2	no	—
36	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
37	#117×C₅₃	—	2	1	no	—
38	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	—	2	1	no	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#97×C₅₃	—	2	2	no	—
42	#97×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	2	1	no	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	#41×C₅₃	—	2	2	no	—
49	#41×C₅₃	C₂×C₈	1	4	yes	—
50	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
51	C₂×D₄×C₅₃	—	2	1	no	—
52	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	2	yes	—
57	M₁₆×C₅₃	—	2	2	no	—
58	M₁₆×C₅₃	—	2	8	no	—
59	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
68	C₂²⋊C₄×C₅₃	—	2	1	no	—
69	C₂²⋊C₄×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×#88322	—	2	8	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	#117	1	1	yes	characteristic
80	Q₈×C₅₃	—	2	1	no	—
81	Q₈×C₅₃	C₄×C₈	1	2	yes	—
82	D₄×C₅₃	—	2	4	no	—
83	D₄×C₅₃	—	2	1	no	—
84	D₄×C₅₃	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
96	C₈×C₅₃	—	2	4	no	—
97	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
98	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
99	#88322	—	2	8	no	—
100	#88322	—	4	4	no	—
101	#8184	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	4	2	no	—
105	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₅₃	#431	1	2	yes	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
111	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
112	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
113	C₄×C₅₃	#443	1	4	yes	—
114	C₄×#131	D₅₃	1	1	yes	characteristic
115	#1476	—	53	2	no	—
116	#1450	—	53	1	no	—
117	C₂×C₈²	—	53	1	no	—
118	#1278	—	53	2	no	—
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₅₃	#1764	1	2	yes	—
123	C₂×#131	Dic₅₃	1	2	yes	—
124	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
125	#431	—	53	2	no	—
126	#405	—	2	2	no	—
127	#405	Dic₅₃	1	2	yes	—
128	C₄×M₁₆	—	2	1	no	—
129	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	1	no	—
131	C₂×C₄×C₈	—	53	2	no	—
132	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
133	Q₈⋊C₈	—	53	4	no	—
134	D₄⋊C₈	—	53	4	no	—
135	#322	—	106	1	no	—
136	C₈²	—	106	2	no	—
137	C₈²	—	106	1	no	—
138	C₅₃	#8184	1	1	yes	characteristic, complemented, Sylow
139	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
140	#131	—	2	4	no	—
141	#131	#88322	1	8	yes	complemented
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	106	2	no	—
146	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
147	C₄×Q₈	C₄×D₅₃	1	2	yes	—
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	C₄×D₅₃	1	2	yes	—
150	#117	—	2	1	no	—
151	#117	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
154	C₄⋊C₈	—	53	4	no	—
155	C₂²⋊C₈	—	106	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	1	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	—	53	4	no	—
161	C₄×C₈	—	106	2	no	—
162	C₄×C₈	—	106	2	no	—
163	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
164	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
166	#41	—	2	2	no	—
167	#41	C₂×#88322	1	4	yes	—
168	C₂×Q₈	C₄×Dic₅₃	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₄×Dic₅₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	#11719945	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
175	M₁₆	—	2	2	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	C₂×#88322	1	4	yes	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	106	8	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	C₂×#88322	1	4	yes	—
184	C₂×C₈	#11719917	1	4	yes	—
185	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
186	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	C₂×C₄×D₅₃	1	2	yes	—
194	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	C₂³	#13133176	1	1	yes	characteristic
198	Q₈	—	2	1	no	—
199	Q₈	C₄×#88322	1	2	yes	—
200	D₄	—	2	4	no	—
201	D₄	—	2	1	no	—
202	D₄	C₄×#88322	1	2	yes	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
211	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
212	C₂×C₄	C₂²×#88322	1	2	yes	—
213	C₂×C₄	C₂×#11719917	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	—	106	8	no	—
216	C₈	—	212	4	no	—
217	C₈	C₂²⋊#88322	1	4	yes	—
218	C₈	C₂²⋊#88322	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₂²	#16810676	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×C₄×#88322	1	1	yes	characteristic
228	C₄	C₄×#11719917	1	1	yes	characteristic
229	C₄	C₂×C₂²⋊#88322	1	2	yes	—
230	C₄	#16810738	1	4	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
234	C₂	#20274550	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#8184
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20272998	C₂	semidirect
3. C₂×C₈×#88322	C₂	semidirect
4. #20272553	C₂	semidirect
29. #131×C₅₃	C₈	semidirect
138. C₅₃	#8184	semidirect
141. #131	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273097	2
#20272998	2
C₂×C₈×#88322	2
#20272553	2
C₄×#131×C₅₃	2
#8184	53