Subgroups — #27294063

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	1	yes	maximal, characteristic
2	#20276518	C₂	1	1	yes	maximal, characteristic
3	#20272697	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719908	—	2	1	no	—
6	C₄⋊#11719908	C₂²	1	2	yes	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	—	2	1	no	—
9	#16810399	—	2	1	no	—
10	#16810399	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₅₃×#446	—	2	1	no	—
13	C₅₃×#446	—	2	1	no	—
14	C₅₃×#446	C₄	1	2	yes	—
15	C₅₃×#446	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#97×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	—	2	1	no	—
27	C₄×Q₈×C₅₃	—	2	1	no	—
28	C₄×Q₈×C₅₃	C₈	1	2	yes	—
29	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	—	2	1	no	—
37	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
38	#97×C₅₃	—	2	2	no	—
39	#97×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₅₃	—	2	1	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	#11719908	—	4	2	no	—
71	#11719908	—	4	2	no	—
72	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
73	Q₈×C₅₃	—	4	2	no	—
74	Q₈×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	4	2	no	—
93	C₈×C₅₃	#137	1	2	yes	—
94	#11644	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	#349	1	2	yes	—
98	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	4	2	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	#419	1	1	yes	characteristic
107	C₄×C₅₃	#421	1	1	yes	characteristic
108	C₂×#446	D₅₃	1	1	yes	characteristic
109	#2163	—	53	1	no	—
110	#2163	—	53	1	no	—
111	#1338	—	53	4	no	—
112	C₂×C₅₃	—	2	2	no	—
113	C₂×C₅₃	#1479	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
115	C₂×C₅₃	#2117	1	1	yes	characteristic
116	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
117	#446	—	2	1	no	—
118	#446	—	2	1	no	—
119	#446	Dic₅₃	1	2	yes	—
120	#446	C₂×D₅₃	1	2	yes	—
121	C₂×C₄⋊C₈	—	2	1	no	—
122	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
123	C₂×#97	—	2	1	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₄⋊C₁₆	—	53	2	no	—
126	C₄⋊C₁₆	—	106	1	no	—
127	C₂²⋊C₁₆	—	106	1	no	—
128	C₂²⋊C₁₆	—	106	1	no	—
129	#346	—	53	2	no	—
130	#346	—	106	1	no	—
131	C₅₃	#11644	1	1	yes	characteristic, complemented, Sylow
132	C₂²×Q₈	#88322	1	1	yes	characteristic
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	#88322	1	2	yes	—
139	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂×C₁₆	—	106	2	no	—
145	C₂×C₁₆	—	106	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	1	no	—
149	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
150	#97	—	2	2	no	—
151	#97	—	2	2	no	—
152	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	C₂×#88322	1	2	yes	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
162	C₂²×C₄	#11719917	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	C₂×#88322	1	2	yes	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₂×#88322	1	1	yes	characteristic
179	C₄²	#11719917	1	1	yes	characteristic
180	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
181	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
182	C₁₆	—	212	2	no	—
183	C₁₆	—	212	2	no	—
184	C₂³	C₂²⋊#88322	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
200	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	#13133248	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16810405	1	2	yes	—
209	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	#16810632	1	1	yes	characteristic
218	C₄	#16810640	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#20273110	1	1	yes	characteristic
221	C₂	C₂×#16810405	1	1	yes	characteristic
222	C₂	#20276343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#446	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#11644
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #20272697	C₂	semidirect
131. C₅₃	#11644	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
#20272697	2
C₂×C₅₃×#446	2
#11644	53