Subgroups — #28310781

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811510	C₂	1	2	yes	maximal
2	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
4	(C₂×#6157)⋊C₁₆	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	D₅₃⋊(C₄⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₈×#6157	C₂²	1	2	yes	—
10	C₄×C₄⋊#6157	C₄	1	1	yes	characteristic
11	#16811510	—	2	4	no	—
12	#16811509	C₄	1	2	yes	complemented
13	C₄×#11719910	C₄	1	2	yes	complemented
14	(C₂×#6157)⋊C₈	C₄	1	2	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊#11719908	C₄	1	1	yes	characteristic, complemented
18	#16810441	C₄	1	2	yes	complemented
19	C₅₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
20	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊#6157	C₈	1	2	yes	—
22	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
23	D₅₃⋊C₁₆	—	2	1	no	—
24	D₅₃⋊C₁₆	—	2	1	no	—
25	D₅₃⋊C₁₆	D₄	1	2	yes	—
26	D₅₃⋊C₁₆	Q₈	1	2	yes	—
27	C₂×#11719910	C₂×C₄	1	2	yes	—
28	C₂×#11719910	C₂×C₄	1	2	yes	—
29	#13133284	C₈	1	2	yes	—
30	C₄⋊#88324	C₈	1	2	yes	—
31	C₈×#6157	—	2	2	no	—
32	C₈×#6157	—	2	2	no	—
33	(C₂×#6157)⋊C₄	C₈	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
36	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719908	C₂×C₄	1	2	yes	—
38	C₁₆×D₅₃	—	2	2	no	—
39	C₁₆×D₅₃	—	2	2	no	—
40	C₄×#88322	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×#6157	C₂×C₈	1	4	yes	—
46	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×#88324	C₂×C₈	1	4	yes	—
50	C₄⋊#6157	—	2	4	no	—
51	C₄×#6157	—	2	2	no	—
52	C₄×#6157	—	2	2	no	—
53	D₅₃⋊C₈	—	2	2	no	—
54	D₅₃⋊C₈	—	2	2	no	—
55	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₄²	1	1	yes	characteristic
58	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂×#88322	C₄²	1	1	yes	characteristic
62	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₅₃	—	2	1	no	—
64	C₈×D₅₃	—	2	1	no	—
65	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
66	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₈×D₅₃	#41	1	1	yes	characteristic
68	C₈×D₅₃	#41	1	1	yes	characteristic
69	#11719910	—	2	2	no	—
70	#11719910	C₄⋊C₄	1	4	yes	complemented
71	C₁₆×C₅₃	—	2	2	no	—
72	#11719908	—	2	2	no	—
73	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×#6157	—	2	4	no	—
75	C₂×#6157	—	2	4	no	—
76	C₂×#6157	—	2	4	no	—
77	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
81	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₄×D₅₃	—	2	2	no	—
84	C₄×D₅₃	M₃₂	1	4	yes	—
85	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
86	C₄×D₅₃	C₄×Q₈	1	1	yes	characteristic
87	C₄×D₅₃	#131	1	1	yes	characteristic
88	C₄×D₅₃	#131	1	1	yes	characteristic
89	#88324	—	2	4	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
93	#88322	—	2	1	no	—
94	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
95	#88322	#117	1	1	yes	characteristic
96	#12662	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
99	C₂×D₅₃	C₈×Q₈	1	1	yes	characteristic
100	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
101	C₂×D₅₃	#504	1	2	yes	—
102	#6157	—	4	4	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	#346	1	2	yes	complemented
105	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₅₃	#405	1	1	yes	characteristic
107	Dic₅₃	—	2	1	no	—
108	Dic₅₃	#346	1	2	yes	complemented
109	Dic₅₃	C₈×D₄	1	1	yes	characteristic
110	Dic₅₃	C₈×Q₈	1	1	yes	characteristic
111	C₂×C₄⋊C₁₆	—	53	1	no	—
112	C₂×#346	—	53	2	no	—
113	C₂×C₄×C₁₆	—	53	1	no	—
114	C₈×C₄⋊C₄	—	53	1	no	—
115	#1374	—	53	2	no	—
116	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
118	C₂×C₅₃	#2108	1	1	yes	characteristic
119	D₅₃	#2169	1	2	yes	—
120	D₅₃	#2183	1	2	yes	—
121	C₂²×C₁₆	—	53	1	no	—
122	C₂²×C₁₆	—	53	1	no	—
123	C₂²×C₁₆	—	53	2	no	—
124	C₂×C₄⋊C₈	—	53	1	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	2	no	—
127	C₄×C₄⋊C₄	—	53	1	no	—
128	C₄⋊C₁₆	—	53	2	no	—
129	C₄⋊C₁₆	—	53	1	no	—
130	C₄⋊C₁₆	#6157	1	1	yes	characteristic, complemented
131	#346	—	106	4	no	—
132	C₄×C₁₆	—	53	2	no	—
133	C₄×C₁₆	—	53	2	no	—
134	#336	—	53	2	no	—
135	C₅₃	#12662	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	2	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂×C₄⋊C₄	—	53	2	no	—
140	C₂×C₄²	—	53	1	no	—
141	C₂×C₄²	—	53	2	no	—
142	C₂×C₁₆	—	53	2	no	—
143	C₂×C₁₆	—	53	2	no	—
144	C₂×C₁₆	—	53	2	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	53	2	no	—
147	C₂×C₁₆	—	106	2	no	—
148	C₂×C₁₆	—	106	2	no	—
149	C₂×C₁₆	—	106	1	no	—
150	C₂×C₁₆	—	106	1	no	—
151	C₂×C₁₆	C₂×#6157	1	2	yes	—
152	C₄⋊C₈	—	53	2	no	—
153	C₄⋊C₈	—	53	2	no	—
154	C₄×C₈	—	53	1	no	—
155	C₄×C₈	—	53	1	no	—
156	C₄×C₈	—	53	1	no	—
157	C₄×C₈	—	106	2	no	—
158	C₄×C₈	—	106	2	no	—
159	C₄×C₈	C₂×#6157	1	1	yes	characteristic
160	#95	—	53	2	no	—
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	4	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	53	1	no	—
169	C₂×C₈	—	53	1	no	—
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	1	no	—
174	C₂×C₈	—	106	2	no	—
175	C₂×C₈	—	106	2	no	—
176	C₂×C₈	C₄×#6157	1	1	yes	characteristic
177	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
178	C₄⋊C₄	—	106	4	no	—
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	106	2	no	—
183	C₄²	—	106	2	no	—
184	C₄²	C₄×#6157	1	1	yes	characteristic
185	C₁₆	—	2	2	no	—
186	C₁₆	—	53	4	no	—
187	C₁₆	—	106	2	no	—
188	C₁₆	—	106	2	no	—
189	C₂³	—	53	1	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	4	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	106	2	no	—
199	C₂×C₄	—	106	4	no	—
200	C₂×C₄	—	106	4	no	—
201	C₂×C₄	—	106	4	no	—
202	C₂×C₄	C₈×#6157	1	1	yes	characteristic
203	C₂×C₄	C₈×#6157	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	53	1	no	—
207	C₈	—	53	1	no	—
208	C₈	—	106	4	no	—
209	C₈	—	106	1	no	—
210	C₈	D₄×#6157	1	1	yes	characteristic
211	C₈	Q₈×#6157	1	1	yes	characteristic
212	C₂²	—	53	2	no	—
213	C₂²	—	53	2	no	—
214	C₂²	—	53	1	no	—
215	C₂²	—	53	1	no	—
216	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	53	2	no	—
219	C₄	—	53	1	no	—
220	C₄	—	53	1	no	—
221	C₄	—	106	1	no	—
222	C₄	—	212	4	no	—
223	C₄	#16811510	1	2	yes	complemented
224	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
225	C₄	#16811589	1	1	yes	characteristic
226	C₂	—	53	2	no	—
227	C₂	—	53	2	no	—
228	C₂	#20290508	1	1	yes	characteristic
229	C₂	C₂×#16811510	1	1	yes	characteristic
230	C₂	#20290927	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×C₄⋊C₁₆	C₄
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₄⋊C₁₆	#6157
hypocenter	C₅₃	#12662
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
12. #16811509	C₄	semidirect
13. C₄×#11719910	C₄	semidirect
17. C₄⋊#11719908	C₄	semidirect
18. #16810441	C₄	semidirect
19. C₅₃×C₄⋊C₁₆	C₄	semidirect
70. #11719910	C₄⋊C₄	semidirect
104. C₄×C₅₃	#346	semidirect
108. Dic₅₃	#346	semidirect
130. C₄⋊C₁₆	#6157	semidirect
135. C₅₃	#12662	semidirect
223. C₄	#16811510	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811510	2
C₄×D₅₃⋊C₁₆	2
C₈×C₄⋊#6157	2
(C₂×#6157)⋊C₁₆	2
C₄⋊C₁₆×D₅₃	2
#12662	53