Subgroups — #27294181

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272676	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	—	2	1	no	—
6	#16810647	—	2	1	no	—
7	C₄×#11719908	—	2	1	no	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	#16810376	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	C₅₃×#532	—	2	1	no	—
12	C₅₃×#500	C₄	1	1	yes	characteristic
13	C₅₃×#493	C₄	1	1	yes	characteristic
14	C₅₃×#486	—	2	1	no	—
15	C₅₃×#466	—	2	1	no	—
16	C₅₃×#466	C₄	1	2	yes	—
17	C₄×D₈×C₅₃	—	2	1	no	—
18	C₄×D₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	4	1	no	—
21	C₂×#11719908	—	2	4	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	1	no	—
25	C₂×#41×C₅₃	—	4	1	no	—
26	C₂×D₈×C₅₃	—	2	1	no	—
27	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	—	2	1	no	—
31	#127×C₅₃	—	2	1	no	—
32	#124×C₅₃	—	4	1	no	—
33	#121×C₅₃	—	2	2	no	—
34	#121×C₅₃	—	4	1	no	—
35	C₄×D₄×C₅₃	—	2	2	no	—
36	C₄×D₄×C₅₃	—	4	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	#107×C₅₃	—	2	1	no	—
39	#107×C₅₃	C₂×C₄	1	2	yes	—
40	#102×C₅₃	—	2	2	no	—
41	#102×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	#41×C₅₃	—	8	2	no	—
47	#41×C₅₃	—	8	2	no	—
48	C₂×Q₈×C₅₃	—	4	1	no	—
49	C₂×D₄×C₅₃	—	2	2	no	—
50	C₂×D₄×C₅₃	—	4	2	no	—
51	C₂×D₄×C₅₃	—	4	1	no	—
52	C₂×D₄×C₅₃	—	4	1	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
57	D₈×C₅₃	—	2	4	no	—
58	D₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₂²⋊C₄×C₅₃	—	4	2	no	—
68	C₂²⋊C₄×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	D₈	1	1	yes	characteristic
71	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	#11719917	—	8	2	no	—
74	C₂×#88322	—	4	2	no	—
75	#11719908	—	2	4	no	—
76	C₂³×C₅₃	—	4	2	no	—
77	C₂³×C₅₃	#102	1	1	yes	characteristic
78	Q₈×C₅₃	—	8	2	no	—
79	D₄×C₅₃	—	4	2	no	—
80	D₄×C₅₃	—	4	2	no	—
81	D₄×C₅₃	—	8	4	no	—
82	D₄×C₅₃	—	8	2	no	—
83	D₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	8	2	no	—
89	C₂×C₄×C₅₃	—	8	2	no	—
90	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
94	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	2	1	no	—
100	C₈×C₅₃	#104	1	2	yes	—
101	C₈×C₅₃	#131	1	2	yes	—
102	#88322	—	8	2	no	—
103	#11671	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	8	2	no	—
107	C₂²×C₅₃	—	8	2	no	—
108	C₂²×C₅₃	—	8	2	no	—
109	C₂²×C₅₃	#357	1	2	yes	—
110	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	#357	1	2	yes	—
115	C₄×C₅₃	#359	1	2	yes	—
116	C₄×C₅₃	#408	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
118	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
119	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
120	#3169	D₅₃	1	1	yes	characteristic
121	C₂×C₄×C₁₆	—	53	1	no	—
122	#1567	—	53	1	no	—
123	#1331	—	53	4	no	—
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	—	16	2	no	—
126	C₂×C₅₃	#1492	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
129	#532	—	2	1	no	—
130	C₂²×C₁₆	—	106	1	no	—
131	#500	Dic₅₃	1	1	yes	characteristic
132	#493	Dic₅₃	1	1	yes	characteristic
133	#486	—	2	1	no	—
134	#466	—	2	1	no	—
135	#466	Dic₅₃	1	2	yes	—
136	C₄×D₈	—	2	1	no	—
137	C₄×D₈	C₂×D₅₃	1	2	yes	—
138	#423	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₄×C₁₆	—	53	2	no	—
141	C₄×C₁₆	—	106	1	no	—
142	#335	—	53	2	no	—
143	#335	—	106	1	no	—
144	C₅₃	#11671	1	1	yes	characteristic, complemented, Sylow
145	C₂×#41	—	4	1	no	—
146	C₂×D₈	—	2	1	no	—
147	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
148	C₂×M₁₆	—	212	1	no	—
149	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₄⋊Q₈	—	2	1	no	—
152	#127	—	2	1	no	—
153	#124	—	4	1	no	—
154	#121	—	2	2	no	—
155	#121	—	4	1	no	—
156	C₄×D₄	—	2	2	no	—
157	C₄×D₄	—	4	1	no	—
158	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₁₆	—	106	4	no	—
160	C₂×C₁₆	—	106	2	no	—
161	#107	—	2	1	no	—
162	#107	C₂×Dic₅₃	1	2	yes	—
163	C₄⋊C₈	—	106	2	no	—
164	C₄⋊C₈	—	212	1	no	—
165	#102	—	2	2	no	—
166	#102	—	2	2	no	—
167	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	#41	—	8	2	no	—
172	#41	—	8	2	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	#11719922	1	1	yes	characteristic
180	C₂²×C₄	#11719923	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	D₈	—	2	4	no	—
183	D₈	—	4	2	no	—
184	M₁₆	—	424	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	#11719922	1	1	yes	characteristic
197	C₄²	#11719923	1	1	yes	characteristic
198	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
200	C₁₆	—	106	4	no	—
201	C₂³	—	4	2	no	—
202	C₂³	#13133138	1	1	yes	characteristic
203	Q₈	—	8	2	no	—
204	D₄	—	4	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	4	no	—
207	D₄	—	8	2	no	—
208	D₄	—	8	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	#13133138	1	1	yes	characteristic
216	C₂×C₄	#13133138	1	1	yes	characteristic
217	C₂×C₄	#13133138	1	1	yes	characteristic
218	C₂×C₄	#13133158	1	2	yes	—
219	C₂×C₄	#13133161	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
222	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	424	2	no	—
226	C₈	#13133144	1	2	yes	—
227	C₈	#13133224	1	2	yes	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	#16810424	1	2	yes	—
234	C₂²	C₂×#13133138	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	8	2	no	—
238	C₄	#16810424	1	2	yes	—
239	C₄	#16810430	1	2	yes	—
240	C₄	#16810590	1	1	yes	characteristic
241	C₄	C₂×#13133144	1	1	yes	characteristic
242	C₄	C₂×#13133158	1	1	yes	characteristic
243	C₄	C₂×#13133161	1	1	yes	characteristic
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	#20273166	1	1	yes	characteristic
247	C₂	C₂×#16810424	1	1	yes	characteristic
248	C₂	C₂×#16810430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#11671
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×#11719908	C₂	semidirect
2. #20273474	C₂	semidirect
3. #20272676	C₂	semidirect
144. C₅₃	#11671	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
#20273474	2
#20272676	2
C₅₃×#3169	2
#11671	53