Subgroups — #27294043

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	#20276518	C₂	1	1	yes	maximal, characteristic
2	#20276514	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Q₈×C₅₃	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	C₄×#11719908	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₅₃	—	2	1	no	—
13	C₈×Q₈×C₅₃	—	2	1	no	—
14	C₈×Q₈×C₅₃	C₄	1	2	yes	—
15	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×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	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×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	4	no	—
45	C₂×Q₈×C₅₃	—	2	1	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	—	2	4	no	—
74	Q₈×C₅₃	—	2	2	no	—
75	Q₈×C₅₃	M₃₂	1	4	yes	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	4	2	no	—
94	C₈×C₅₃	#137	1	2	yes	—
95	#11639	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	#350	1	2	yes	—
99	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₄×C₅₃	—	2	4	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	#419	1	1	yes	characteristic
107	C₄×C₅₃	#421	1	1	yes	characteristic
108	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
109	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
110	#2163	—	53	1	no	—
111	#2162	—	53	1	no	—
112	Q₈⋊C₁₆	—	53	4	no	—
113	C₂×C₅₃	—	2	2	no	—
114	C₂×C₅₃	#1479	1	1	yes	characteristic
115	C₂×C₅₃	#2112	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
117	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
118	C₈×Q₈	—	2	1	no	—
119	C₈×Q₈	—	2	1	no	—
120	C₈×Q₈	Dic₅₃	1	2	yes	—
121	C₈×Q₈	C₂×D₅₃	1	2	yes	—
122	C₂×C₄⋊C₈	—	2	1	no	—
123	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
124	C₂×C₄×C₈	—	2	1	no	—
125	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
126	C₄⋊C₁₆	—	53	2	no	—
127	C₄⋊C₁₆	—	106	1	no	—
128	C₂²⋊C₁₆	—	106	1	no	—
129	C₂²⋊C₁₆	—	106	1	no	—
130	#346	—	106	1	no	—
131	C₄×C₁₆	—	53	2	no	—
132	C₅₃	#11639	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	#88322	1	1	yes	characteristic
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	#88322	1	2	yes	—
140	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂×C₁₆	—	106	2	no	—
146	C₂×C₁₆	—	106	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	1	no	—
150	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	C₂×#88322	1	2	yes	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
163	C₂²×C₄	#11719917	1	1	yes	characteristic
164	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	2	no	—
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	2	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₂×#88322	1	2	yes	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₂×#88322	1	1	yes	characteristic
180	C₄²	#11719917	1	1	yes	characteristic
181	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
183	C₁₆	—	212	2	no	—
184	C₁₆	—	212	2	no	—
185	C₂³	C₂²⋊#88322	1	1	yes	characteristic
186	Q₈	—	2	4	no	—
187	Q₈	—	2	2	no	—
188	Q₈	#13133157	1	4	yes	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
198	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
199	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
202	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	#13133248	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#16810407	1	2	yes	—
211	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	#16810632	1	1	yes	characteristic
219	C₄	#16810640	1	1	yes	characteristic
220	C₄	C₂×#13133157	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	#20273110	1	1	yes	characteristic
223	C₂	#20276326	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	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#11639
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#11719908	C₂	semidirect
132. C₅₃	#11639	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
#20276514	2
Q₈⋊#11719908	2
C₂×C₈×Q₈×C₅₃	2
#11639	53