Subgroups — #27295803

Order 13568.

244 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276547	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272785	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272632	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133157	—	2	1	no	—
9	C₂²×#11719908	—	2	1	no	—
10	C₄⋊#11719908	C₂²	1	1	yes	characteristic
11	C₄⋊#11719908	C₂²	1	1	yes	characteristic
12	C₂²⋊#11719908	—	2	1	no	—
13	C₂²⋊#11719908	—	2	1	no	—
14	C₄×#11719908	C₂²	1	1	yes	characteristic
15	C₄×#11719908	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
17	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#335	C₄	1	1	yes	characteristic
20	C₅₃×D₄⋊C₈	C₄	1	1	yes	characteristic
21	C₅₃×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₅₃	C₂²	1	1	yes	characteristic
23	#13133157	—	4	2	no	—
24	C₂×#11719908	—	2	1	no	—
25	C₂×#11719908	—	2	1	no	—
26	C₂×#11719908	—	2	1	no	—
27	C₂×#11719908	—	4	2	no	—
28	C₂×#11719908	D₄	1	1	yes	characteristic
29	C₂×#11719908	D₄	1	1	yes	characteristic
30	C₂×D₈×C₅₃	C₈	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	—	2	1	no	—
33	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#107×C₅₃	C₈	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#102×C₅₃	C₈	1	1	yes	characteristic
39	#102×C₅₃	C₈	1	1	yes	characteristic
40	C₂²⋊C₈×C₅₃	—	2	1	no	—
41	C₂²⋊C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	D₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	—	4	1	no	—
57	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₅₃	—	2	1	no	—
64	C₂²⋊C₄×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	#11719908	—	2	2	no	—
67	#11719908	—	2	1	no	—
68	#11719908	—	4	1	no	—
69	#11719908	—	4	1	no	—
70	#11719908	D₈	1	2	yes	complemented
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	D₄×C₅₃	—	2	1	no	—
74	D₄×C₅₃	—	2	1	no	—
75	D₄×C₅₃	—	2	1	no	—
76	D₄×C₅₃	M₃₂	1	2	yes	complemented
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	1	no	—
81	C₂×C₄×C₅₃	—	4	1	no	—
82	C₂×C₄×C₅₃	—	4	1	no	—
83	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	—	2	1	no	—
89	C₈×C₅₃	—	4	1	no	—
90	C₈×C₅₃	—	4	1	no	—
91	C₈×C₅₃	—	4	1	no	—
92	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
93	C₈×C₅₃	#135	1	1	yes	characteristic
94	#12038	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	4	1	no	—
97	C₂²×C₅₃	—	4	1	no	—
98	C₂²×C₅₃	—	4	1	no	—
99	C₂²×C₅₃	—	4	1	no	—
100	C₂²×C₅₃	—	4	1	no	—
101	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	4	1	no	—
104	C₄×C₅₃	—	4	1	no	—
105	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
106	C₄×C₅₃	#443	1	1	yes	characteristic
107	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
108	C₄×C₅₃	#504	1	1	yes	characteristic
109	#2169	—	53	1	no	—
110	D₄×C₁₆	—	53	1	no	—
111	C₈×D₈	D₅₃	1	1	yes	characteristic
112	#1370	—	53	1	no	—
113	D₄⋊C₁₆	—	53	1	no	—
114	D₄⋊C₁₆	—	53	1	no	—
115	#1311	—	53	1	no	—
116	C₂×C₅₃	—	4	2	no	—
117	C₂×C₅₃	—	8	1	no	—
118	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
119	C₂×C₅₃	#2168	1	1	yes	characteristic
120	C₂×C₅₃	#2171	1	1	yes	characteristic
121	C₂×M₃₂	—	106	1	no	—
122	C₂²×C₁₆	—	106	1	no	—
123	C₄×D₈	Dic₅₃	1	1	yes	characteristic
124	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
125	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
126	C₄⋊C₁₆	—	53	1	no	—
127	C₄⋊C₁₆	—	53	1	no	—
128	C₂²⋊C₁₆	—	106	1	no	—
129	C₂²⋊C₁₆	—	106	1	no	—
130	C₄×C₁₆	—	53	1	no	—
131	C₄×C₁₆	—	53	1	no	—
132	#335	Dic₅₃	1	1	yes	characteristic
133	D₄⋊C₈	Dic₅₃	1	1	yes	characteristic
134	D₄⋊C₈	Dic₅₃	1	1	yes	characteristic
135	C₈²	C₂×D₅₃	1	1	yes	characteristic
136	C₅₃	#12038	1	1	yes	characteristic, complemented, Sylow
137	C₂×D₈	#88322	1	1	yes	characteristic
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
141	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
142	M₃₂	—	212	2	no	—
143	C₂×C₁₆	—	53	1	no	—
144	C₂×C₁₆	—	53	1	no	—
145	C₂×C₁₆	—	106	1	no	—
146	C₂×C₁₆	—	106	1	no	—
147	C₂×C₁₆	—	106	1	no	—
148	C₂×C₁₆	—	212	2	no	—
149	#107	#88322	1	1	yes	characteristic
150	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
152	#102	#88322	1	1	yes	characteristic
153	#102	#88322	1	1	yes	characteristic
154	C₂²⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	1	no	—
156	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×D₄	C₂×#88322	1	1	yes	characteristic
160	C₂×D₄	C₂×#88322	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	D₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂×#88322	1	1	yes	characteristic
172	C₂×C₈	C₂×#88322	1	1	yes	characteristic
173	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
174	C₂×C₈	#11719947	1	1	yes	characteristic
175	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
176	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
180	C₁₆	—	53	2	no	—
181	C₁₆	—	106	2	no	—
182	C₁₆	—	106	1	no	—
183	C₁₆	—	212	1	no	—
184	C₁₆	—	212	1	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	2	1	no	—
189	D₄	—	2	1	no	—
190	D₄	#13133157	1	2	yes	complemented
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
199	C₂×C₄	#13133224	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	D₈×D₅₃	1	1	yes	characteristic
207	C₈	#13133236	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	D₄×#88322	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	D₈×Dic₅₃	1	1	yes	characteristic
219	C₄	#16810740	1	1	yes	characteristic
220	C₄	C₂×#13133157	1	1	yes	characteristic
221	C₄	#16811040	1	1	yes	characteristic
222	C₂	—	4	2	no	—
223	C₂	—	8	1	no	—
224	C₂	D₈×#88322	1	1	yes	characteristic
225	C₂	#20276541	1	1	yes	characteristic
226	C₂	#20276557	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×D₈×C₅₃	C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×D₈	D₅₃
hypocenter	C₅₃	#12038
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #20276547	C₂	semidirect
2. D₄×#11719908	C₂	semidirect
3. #20272785	C₂	semidirect
4. D₄⋊#11719908	C₂	semidirect
5. D₄⋊#11719908	C₂	semidirect
6. #20272632	C₂	semidirect
70. #11719908	D₈	semidirect
76. D₄×C₅₃	M₃₂	semidirect
136. C₅₃	#12038	semidirect
190. D₄	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276547	2
D₄×#11719908	2
#20272785	2
D₄⋊#11719908	2
#20272632	2
C₈×D₈×C₅₃	2
#12038	53