Subgroups — #27293903

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 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
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2960	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	C₅₃×#533	C₄	1	1	yes	characteristic
12	C₅₃×#446	—	2	1	no	—
13	C₅₃×#446	C₄	1	2	yes	—
14	C₈×Q₈×C₅₃	—	2	1	no	—
15	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#424	—	2	1	no	—
17	C₅₃×#423	C₄	1	1	yes	characteristic
18	C₅₃×#405	—	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×M₁₆×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
29	#125×C₅₃	—	2	1	no	—
30	C₂²⋊Q₈×C₅₃	—	2	1	no	—
31	C₂²⋊Q₈×C₅₃	C₈	1	2	yes	—
32	C₄×Q₈×C₅₃	—	2	1	no	—
33	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
34	#117×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	1	no	—
39	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
40	#97×C₅₃	—	2	2	no	—
41	C₄×C₈×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	1	no	—
47	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	M₁₆×C₅₃	—	4	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
57	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
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	1	no	—
61	C₄⋊C₄×C₅₃	—	2	1	no	—
62	C₄⋊C₄×C₅₃	—	2	1	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	2	2	no	—
65	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
66	C₂²⋊C₄×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	#11719908	—	2	4	no	—
73	#11719908	—	4	2	no	—
74	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
75	Q₈×C₅₃	—	2	4	no	—
76	Q₈×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#103	1	2	yes	—
85	C₂×C₄×C₅₃	#104	1	2	yes	—
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	—	4	2	no	—
92	C₈×C₅₃	#103	1	2	yes	—
93	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
95	#11608	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
99	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	4	2	no	—
103	C₄×C₅₃	—	4	2	no	—
104	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
105	C₄×C₅₃	#350	1	2	yes	—
106	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
107	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
108	C₄×C₅₃	#504	1	2	yes	—
109	#2960	D₅₃	1	1	yes	characteristic
110	#2150	—	53	1	no	—
111	C₂×C₄×C₁₆	—	53	1	no	—
112	Q₈⋊C₁₆	—	53	4	no	—
113	C₂×C₅₃	—	2	2	no	—
114	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
115	C₂×C₅₃	#2113	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
117	#533	Dic₅₃	1	1	yes	characteristic
118	C₂×M₃₂	—	106	1	no	—
119	C₂²×C₁₆	—	106	1	no	—
120	#446	—	2	1	no	—
121	#446	Dic₅₃	1	2	yes	—
122	C₈×Q₈	—	2	1	no	—
123	C₈×Q₈	C₂×D₅₃	1	2	yes	—
124	#424	—	2	1	no	—
125	#423	Dic₅₃	1	1	yes	characteristic
126	#405	—	2	1	no	—
127	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
128	C₄⋊C₁₆	—	53	2	no	—
129	C₄⋊C₁₆	—	106	1	no	—
130	C₄×C₁₆	—	53	2	no	—
131	C₄×C₁₆	—	106	1	no	—
132	C₅₃	#11608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	2	1	no	—
134	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₄⋊Q₈	#88322	1	1	yes	characteristic
137	C₄⋊Q₈	#88322	1	1	yes	characteristic
138	#125	—	2	1	no	—
139	C₂²⋊Q₈	—	2	1	no	—
140	C₂²⋊Q₈	#88322	1	2	yes	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
143	#117	—	2	1	no	—
144	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
145	M₃₂	—	212	2	no	—
146	C₂×C₁₆	—	106	4	no	—
147	C₂×C₁₆	—	106	2	no	—
148	C₂×C₁₆	—	106	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
153	#97	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	C₂×#88322	1	2	yes	—
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	M₁₆	—	4	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	#11719924	1	2	yes	—
168	C₂×C₈	#11719925	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	C₂×#88322	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	C₂×#88322	1	1	yes	characteristic
182	C₄²	#11719917	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
185	C₁₆	—	106	4	no	—
186	C₁₆	—	212	2	no	—
187	C₂³	C₂²⋊#88322	1	1	yes	characteristic
188	Q₈	—	2	4	no	—
189	Q₈	—	4	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
197	C₂×C₄	#13133141	1	2	yes	—
198	C₂×C₄	#13133144	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
200	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
201	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	#13133141	1	2	yes	—
206	C₈	C₂×#11719924	1	1	yes	characteristic
207	C₈	C₂×#11719925	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	Q₈⋊#88322	1	2	yes	—
211	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	Q₈⋊#88322	1	2	yes	—
217	C₄	#16810407	1	2	yes	—
218	C₄	C₂×#13133141	1	1	yes	characteristic
219	C₄	C₂×#13133144	1	1	yes	characteristic
220	C₄	#16811040	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₂×Q₈⋊#88322	1	1	yes	characteristic
223	C₂	#20276329	1	1	yes	characteristic
224	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×#2960	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#11608
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#11719908	C₂	semidirect
132. C₅₃	#11608	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
Q₈⋊#11719908	2
C₅₃×#2960	2
#11608	53