Subgroups — #27294123

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	1	yes	maximal, characteristic
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20272679	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810647	—	2	1	no	—
6	C₄⋊#11719908	—	2	1	no	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	#16810376	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
13	C₄×Q₁₆×C₅₃	—	2	1	no	—
14	C₄×Q₁₆×C₅₃	—	2	1	no	—
15	C₄×Q₁₆×C₅₃	C₄	1	2	yes	—
16	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#103×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	4	1	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	4	1	no	—
24	C₂²×Q₈×C₅₃	—	2	1	no	—
25	C₂×Q₁₆×C₅₃	—	2	4	no	—
26	C₂×Q₁₆×C₅₃	—	2	1	no	—
27	C₂×Q₁₆×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	2	no	—
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₄×Q₈×C₅₃	—	4	1	no	—
33	C₄×Q₈×C₅₃	—	4	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
36	C₂×C₄²×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	#103×C₅₃	—	2	2	no	—
41	#103×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	C₂×Q₈×C₅₃	—	2	4	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂²×C₄×C₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
55	Q₁₆×C₅₃	—	2	4	no	—
56	Q₁₆×C₅₃	—	2	2	no	—
57	Q₁₆×C₅₃	M₁₆	1	4	yes	complemented
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	4	1	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	—	4	1	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₃	D₈	1	1	yes	characteristic
72	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	#11719917	—	8	2	no	—
75	C₂×#88322	—	4	2	no	—
76	#11719908	—	4	2	no	—
77	C₂³×C₅₃	#102	1	1	yes	characteristic
78	Q₈×C₅₃	—	2	4	no	—
79	Q₈×C₅₃	—	4	4	no	—
80	Q₈×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	—	4	4	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	8	2	no	—
88	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
92	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
100	#88322	—	8	2	no	—
101	#11658	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	#358	1	2	yes	—
105	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	4	4	no	—
110	C₄×C₅₃	—	8	2	no	—
111	C₄×C₅₃	#358	1	2	yes	—
112	C₄×C₅₃	#407	1	1	yes	characteristic
113	C₄×C₅₃	#421	1	1	yes	characteristic
114	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
115	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
116	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
117	#2162	—	53	1	no	—
118	#1567	—	53	1	no	—
119	#1332	—	53	4	no	—
120	C₂×C₅₃	—	2	2	no	—
121	C₂×C₅₃	#1486	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
123	C₂×C₅₃	#2143	1	1	yes	characteristic
124	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
125	C₂×C₄×Q₈	—	2	1	no	—
126	C₄×Q₁₆	—	2	1	no	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	Dic₅₃	1	2	yes	—
129	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
130	C₂×#107	Dic₅₃	1	1	yes	characteristic
131	#423	—	106	1	no	—
132	C₂×#103	—	2	1	no	—
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₄⋊C₁₆	—	106	1	no	—
135	C₂²⋊C₁₆	—	106	1	no	—
136	C₄×C₁₆	—	53	2	no	—
137	#335	—	53	2	no	—
138	#335	—	106	1	no	—
139	C₅₃	#11658	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	—	2	1	no	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
144	C₂×M₁₆	—	212	1	no	—
145	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	4	1	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄²	—	4	1	no	—
154	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂×C₁₆	—	106	2	no	—
156	#107	—	2	1	no	—
157	#107	C₂×Dic₅₃	1	2	yes	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄⋊C₈	—	212	1	no	—
160	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	#11719922	1	1	yes	characteristic
173	C₂²×C₄	#11719923	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	2	no	—
177	Q₁₆	#11719917	1	4	yes	complemented
178	M₁₆	—	424	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	212	2	no	—
182	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
185	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	#11719922	1	1	yes	characteristic
193	C₄²	#11719923	1	1	yes	characteristic
194	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₁₆	—	212	2	no	—
197	C₂³	#13133138	1	1	yes	characteristic
198	Q₈	—	2	4	no	—
199	Q₈	—	4	4	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	#13133138	1	1	yes	characteristic
209	C₂×C₄	#13133138	1	1	yes	characteristic
210	C₂×C₄	#13133138	1	1	yes	characteristic
211	C₂×C₄	#13133160	1	2	yes	—
212	C₂×C₄	#13133163	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
215	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	424	2	no	—
220	C₈	C₂×#11719917	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#16810427	1	2	yes	—
224	C₂²	C₂×#13133138	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	4	no	—
229	C₄	—	8	2	no	—
230	C₄	#16810427	1	2	yes	—
231	C₄	#16810587	1	1	yes	characteristic
232	C₄	#16810640	1	1	yes	characteristic
233	C₄	C₂×#13133160	1	1	yes	characteristic
234	C₄	C₂×#13133163	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	#20273140	1	1	yes	characteristic
237	C₂	C₂×#16810427	1	1	yes	characteristic
238	C₂	#20276443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#11658
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20272679	C₂	semidirect
57. Q₁₆×C₅₃	M₁₆	semidirect
139. C₅₃	#11658	semidirect
177. Q₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
#20273474	2
#20272679	2
C₂×C₄×Q₁₆×C₅₃	2
#11658	53