Subgroups — #27293803

Order 13568.

402 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂²×#11719908	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	2	yes	—
9	C₄×#11719908	—	2	1	no	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₂×#131×C₅₃	—	2	1	no	—
12	C₅₃×#532	C₄	1	1	yes	characteristic
13	C₅₃×#436	—	2	1	no	—
14	C₅₃×#436	C₄	1	2	yes	—
15	C₈×D₄×C₅₃	—	2	1	no	—
16	C₈×D₄×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#423	C₄	1	1	yes	characteristic
18	C₅₃×#408	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	—	4	2	no	—
21	C₂×#11719908	—	2	4	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#41×C₅₃	—	2	1	no	—
25	#131×C₅₃	—	4	2	no	—
26	#131×C₅₃	—	4	2	no	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
33	#127×C₅₃	C₈	1	1	yes	characteristic
34	#124×C₅₃	—	2	1	no	—
35	#121×C₅₃	—	2	1	no	—
36	#121×C₅₃	C₈	1	2	yes	—
37	C₄×D₄×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	1	no	—
41	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₂²⋊C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	#41×C₅₃	—	4	2	no	—
49	#41×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	2	1	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	1	no	—
53	C₂×D₄×C₅₃	—	2	1	no	—
54	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
59	M₁₆×C₅₃	—	4	2	no	—
60	M₁₆×C₅₃	—	4	4	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	D₈	1	2	yes	—
68	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
73	C₂²⋊C₄×C₅₃	—	2	2	no	—
74	C₂²⋊C₄×C₅₃	—	2	2	no	—
75	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	#11719908	—	2	4	no	—
80	#11719908	—	4	2	no	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
83	Q₈×C₅₃	—	4	2	no	—
84	D₄×C₅₃	—	2	4	no	—
85	D₄×C₅₃	—	4	4	no	—
86	D₄×C₅₃	—	4	2	no	—
87	D₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#102	1	2	yes	—
97	C₂×C₄×C₅₃	#104	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₅₃	—	2	1	no	—
102	C₈×C₅₃	—	2	1	no	—
103	C₈×C₅₃	—	4	2	no	—
104	C₈×C₅₃	#102	1	2	yes	—
105	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
106	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
107	#11584	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	4	2	no	—
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
114	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	4	2	no	—
118	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
119	C₄×C₅₃	#350	1	2	yes	—
120	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
121	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
122	C₄×C₅₃	#504	1	2	yes	—
123	#2931	D₅₃	1	1	yes	characteristic
124	#2150	—	53	1	no	—
125	C₂×C₄×C₁₆	—	53	1	no	—
126	D₄⋊C₁₆	—	53	4	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	—	8	2	no	—
129	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
130	C₂×C₅₃	#2113	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
132	C₂×#131	—	2	1	no	—
133	#532	Dic₅₃	1	1	yes	characteristic
134	C₂×M₃₂	—	106	1	no	—
135	C₂²×C₁₆	—	106	1	no	—
136	#436	—	2	1	no	—
137	#436	Dic₅₃	1	2	yes	—
138	C₈×D₄	—	2	1	no	—
139	C₈×D₄	C₂×D₅₃	1	2	yes	—
140	#423	Dic₅₃	1	1	yes	characteristic
141	#408	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₄⋊C₁₆	—	53	2	no	—
144	C₄⋊C₁₆	—	106	1	no	—
145	C₄×C₁₆	—	53	2	no	—
146	C₄×C₁₆	—	106	1	no	—
147	C₅₃	#11584	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	2	1	no	—
149	#131	—	4	2	no	—
150	#131	—	4	2	no	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₄⋊Q₈	#88322	1	1	yes	characteristic
157	#127	#88322	1	1	yes	characteristic
158	#124	—	2	1	no	—
159	#121	—	2	1	no	—
160	#121	#88322	1	2	yes	—
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
163	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
164	M₃₂	—	212	2	no	—
165	C₂×C₁₆	—	106	4	no	—
166	C₂×C₁₆	—	106	2	no	—
167	C₂×C₁₆	—	106	2	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	C₂×#88322	1	2	yes	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
185	C₂²×C₄	#11719917	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
187	M₁₆	—	4	2	no	—
188	M₁₆	—	4	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	#11719922	1	2	yes	—
194	C₂×C₈	#11719923	1	2	yes	—
195	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂×#88322	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₄²	C₂×#88322	1	1	yes	characteristic
204	C₄²	#11719917	1	1	yes	characteristic
205	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
207	C₁₆	—	106	4	no	—
208	C₁₆	—	212	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂²⋊#88322	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	D₄	—	2	4	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	2	no	—
215	D₄	—	4	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
224	C₂×C₄	#13133138	1	2	yes	—
225	C₂×C₄	#13133144	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
228	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	4	2	no	—
232	C₈	#13133138	1	2	yes	—
233	C₈	C₂×#11719922	1	1	yes	characteristic
234	C₈	C₂×#11719923	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	D₄⋊#88322	1	2	yes	—
241	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	2	no	—
245	C₄	D₄⋊#88322	1	2	yes	—
246	C₄	#16810407	1	2	yes	—
247	C₄	C₂×#13133138	1	1	yes	characteristic
248	C₄	C₂×#13133144	1	1	yes	characteristic
249	C₄	#16811040	1	2	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	8	2	no	—
252	C₂	C₂×D₄⋊#88322	1	1	yes	characteristic
253	C₂	#20276329	1	1	yes	characteristic
254	C₂	C₂×#16810407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2931	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₅₃×#532	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2931	D₅₃
hypocenter	C₅₃	#11584
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. C₂×C₄×#11719908	C₂	semidirect
3. D₄⋊#11719908	C₂	semidirect
147. C₅₃	#11584	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×C₄×#11719908	2
D₄⋊#11719908	2
C₅₃×#2931	2
#11584	53