Subgroups — #27293845

Order 13568.

386 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
2	#20273463	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	C₂×#13133157	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	#16810399	—	2	1	no	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	#16810376	C₂²	1	2	yes	—
10	#16810374	—	2	1	no	—
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	#13133157	—	2	4	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₂²⋊#88322	—	4	1	no	—
24	C₄×#88322	—	4	1	no	—
25	C₂²×Q₈×C₅₃	—	2	1	no	—
26	C₂×Q₁₆×C₅₃	—	2	1	no	—
27	C₂×Q₁₆×C₅₃	—	2	2	no	—
28	C₂×Q₁₆×C₅₃	C₈	1	4	yes	complemented
29	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	—	4	1	no	—
35	C₄×Q₈×C₅₃	—	4	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
38	C₂×C₄²×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	1	no	—
41	#107×C₅₃	C₂×C₄	1	2	yes	—
42	#103×C₅₃	—	2	2	no	—
43	#103×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	C₂×Q₈×C₅₃	—	2	4	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	4	1	no	—
51	C₂×Q₈×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	4	1	no	—
53	C₂²×C₄×C₅₃	—	4	1	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	Q₁₆×C₅₃	—	2	4	no	—
58	Q₁₆×C₅₃	—	2	2	no	—
59	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
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	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	—	4	2	no	—
72	C₄²×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₅₃	D₈	1	1	yes	characteristic
75	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×#88322	—	4	2	no	—
78	#11719908	—	2	4	no	—
79	C₂³×C₅₃	#102	1	1	yes	characteristic
80	Q₈×C₅₃	—	2	4	no	—
81	Q₈×C₅₃	—	4	4	no	—
82	Q₈×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	8	2	no	—
89	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
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	#104	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
100	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	#88322	—	8	2	no	—
103	#11594	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
107	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	4	4	no	—
112	C₄×C₅₃	—	8	2	no	—
113	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
114	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
116	C₄×C₅₃	#510	1	2	yes	—
117	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
118	C₄×M₃₂	—	53	1	no	—
119	#1564	—	53	1	no	—
120	#1332	—	53	4	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
123	C₂×C₅₃	#2141	1	1	yes	characteristic
124	C₂×C₅₃	#2143	1	1	yes	characteristic
125	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	—	2	1	no	—
127	C₂×M₃₂	—	106	1	no	—
128	C₄×Q₁₆	—	2	1	no	—
129	C₄×Q₁₆	—	2	1	no	—
130	C₄×Q₁₆	Dic₅₃	1	2	yes	—
131	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
132	#431	—	106	1	no	—
133	C₂×#107	Dic₅₃	1	1	yes	characteristic
134	C₂×#103	—	2	1	no	—
135	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
136	#346	—	106	1	no	—
137	C₄×C₁₆	—	53	2	no	—
138	#335	—	53	2	no	—
139	#334	—	106	1	no	—
140	C₅₃	#11594	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	2	1	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	—	2	1	no	—
144	C₂×Q₁₆	#88322	1	4	yes	complemented
145	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
146	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₄×Q₈	—	4	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	4	1	no	—
154	C₂×C₄²	—	4	1	no	—
155	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
156	M₃₂	—	106	4	no	—
157	C₂×C₁₆	—	106	2	no	—
158	#107	—	2	1	no	—
159	#107	C₂×Dic₅₃	1	2	yes	—
160	C₄⋊C₈	—	106	2	no	—
161	#103	—	2	2	no	—
162	#103	—	2	2	no	—
163	C₂²⋊C₈	—	212	1	no	—
164	C₄×C₈	—	212	1	no	—
165	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	—	4	1	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	#11719922	1	1	yes	characteristic
176	C₂²×C₄	#11719923	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
178	Q₁₆	—	2	4	no	—
179	Q₁₆	—	2	2	no	—
180	Q₁₆	C₂×#88322	1	4	yes	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	C₂×#88322	1	2	yes	—
184	C₂×C₈	#11719917	1	2	yes	—
185	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	#11719922	1	1	yes	characteristic
196	C₄²	#11719923	1	1	yes	characteristic
197	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₁₆	—	106	4	no	—
200	C₂³	#13133138	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂²⋊#88322	1	2	yes	—
211	C₂×C₄	#13133138	1	1	yes	characteristic
212	C₂×C₄	#13133138	1	1	yes	characteristic
213	C₂×C₄	#13133138	1	1	yes	characteristic
214	C₂×C₄	#13133144	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
217	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	424	2	no	—
221	C₈	C₂²⋊#88322	1	2	yes	—
222	C₈	C₂²×#88322	1	1	yes	characteristic
223	C₈	C₂×#11719917	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	D₄⋊#88322	1	2	yes	—
227	C₂²	C₂×#13133138	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	8	2	no	—
233	C₄	D₄⋊#88322	1	2	yes	—
234	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
235	C₄	C₂×#13133144	1	1	yes	characteristic
236	C₄	#16811064	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	C₂×D₄⋊#88322	1	1	yes	characteristic
239	C₂	#20276435	1	1	yes	characteristic
240	C₂	#20276443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₈
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	#11594
LCS	G ▷ 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
28. C₂×Q₁₆×C₅₃	C₈	semidirect
140. C₅₃	#11594	semidirect
144. C₂×Q₁₆	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133157	2
#20273463	2
#20272679	2
C₂×C₄×Q₁₆×C₅₃	2
#11594	53