Subgroups — #28328317

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811682	C₂	1	2	yes	maximal, complemented
2	#20291101	C₂	1	4	yes	maximal, complemented
3	#20291094	C₂	1	4	yes	maximal, complemented
4	#20291047	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290947	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
7	#20283426	C₂	1	1	yes	maximal, characteristic
8	#16811682	—	2	8	no	—
9	C₂×#13133292	C₂²	1	4	yes	—
10	C₂×#13133291	C₂²	1	2	yes	—
11	C₂×D₅₃⋊C₁₆	C₂²	1	4	yes	—
12	C₂²×#11719910	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	1	yes	characteristic
14	C₄⋊#11719910	C₂²	1	4	yes	—
15	#16811518	C₂²	1	8	yes	—
16	C₄×#11719910	C₂²	1	2	yes	—
17	C₄×#11719910	C₂²	1	2	yes	—
18	C₂×#13133225	C₂²	1	2	yes	—
19	#16811173	C₄	1	1	yes	characteristic
20	#16810700	C₄	1	4	yes	—
21	#16810688	C₂²	1	4	yes	—
22	#16810591	C₄	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	C₅₃×#423	C₄	1	1	yes	characteristic
25	#13133292	—	2	8	no	—
26	#13133291	—	2	4	no	—
27	D₅₃⋊C₁₆	—	2	8	no	—
28	C₂×#11719910	—	2	2	no	—
29	C₂×#11719910	D₄	1	4	yes	complemented
30	C₂×#11719910	C₂³	1	4	yes	—
31	C₂×#11719910	C₂³	1	2	yes	—
32	C₂×#11719910	C₂³	1	1	yes	characteristic
33	C₂×#11719910	C₂³	1	1	yes	characteristic
34	C₂×#11719945	C₂×C₄	1	2	yes	—
35	#13133225	—	2	4	no	—
36	#13133225	—	2	4	no	—
37	C₂×#11719912	C₂×C₄	1	4	yes	—
38	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
39	C₂²×#88322	C₂³	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂³	1	4	yes	—
41	#13133217	C₈	1	1	yes	characteristic
42	#13133214	C₈	1	1	yes	characteristic
43	#13133205	C₈	1	2	yes	—
44	#13133191	C₈	1	4	yes	—
45	C₄×D₂₁₂	C₂×C₄	1	4	yes	—
46	#13133129	C₂×C₄	1	8	yes	—
47	C₄×#88322	C₂×C₄	1	2	yes	—
48	C₄×#88322	C₂³	1	2	yes	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₅₃	C₂×C₈	1	4	yes	—
55	#11719945	—	2	4	no	—
56	#11719945	—	2	4	no	—
57	C₂×D₂₁₂	C₂×C₈	1	4	yes	—
58	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
60	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
61	M₁₆×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
63	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
66	#11719921	C₂×C₈	1	8	yes	—
67	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
68	C₂×#88322	—	2	2	no	—
69	C₂×#88322	C₂²×C₄	1	2	yes	—
70	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
71	C₂×#88322	C₂×D₄	1	2	yes	—
72	C₂×#88322	#41	1	2	yes	—
73	C₂×#88322	C₂⁴	1	1	yes	characteristic
74	#11719912	—	2	8	no	—
75	C₈×D₅₃	—	2	4	no	—
76	C₈×D₅₃	—	2	4	no	—
77	#11719910	—	2	4	no	—
78	#11719910	—	2	2	no	—
79	#11719910	C₂×D₄	1	2	yes	—
80	#11719910	C₂×D₄	1	2	yes	—
81	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	#131	1	2	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₅₃	—	2	8	no	—
91	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
92	D₂₁₂	—	2	4	no	—
93	D₂₁₂	—	2	8	no	—
94	C₄×D₅₃	—	2	4	no	—
95	C₄×D₅₃	—	2	4	no	—
96	Dic₁₀₆	—	2	4	no	—
97	C₈×C₅₃	—	2	4	no	—
98	#88322	—	2	2	no	—
99	#88322	C₄×D₄	1	2	yes	—
100	#88322	C₂²×D₄	1	1	yes	characteristic
101	#88322	C₂×#41	1	1	yes	characteristic
102	#33434	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	C₈×D₄	1	2	yes	—
105	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
106	C₂×D₅₃	—	2	8	no	—
107	C₂×D₅₃	—	2	4	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	C₈×D₄	1	2	yes	—
110	C₄×C₅₃	#504	1	4	yes	complemented
111	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
113	Dic₅₃	—	2	4	no	—
114	C₂×#504	—	53	2	no	—
115	#2931	—	53	1	no	—
116	#2169	—	53	4	no	—
117	D₄×C₁₆	—	53	4	no	—
118	#2150	—	53	1	no	—
119	#2113	—	53	2	no	—
120	C₂×C₄×C₁₆	—	53	1	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
123	C₂×C₅₃	C₂×#504	1	2	yes	—
124	D₅₃	—	4	4	no	—
125	C₂×#131	—	53	2	no	—
126	#532	—	53	1	no	—
127	#504	—	106	8	no	—
128	C₂×M₃₂	—	53	2	no	—
129	C₂×M₃₂	—	53	4	no	—
130	C₂²×C₁₆	—	53	1	no	—
131	C₂²×C₁₆	—	53	1	no	—
132	C₂²×C₁₆	—	53	4	no	—
133	#436	—	53	4	no	—
134	C₈×D₄	—	53	4	no	—
135	#423	#6157	1	1	yes	characteristic
136	#408	—	53	2	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₄⋊C₁₆	—	53	4	no	—
139	C₂²⋊C₁₆	—	53	8	no	—
140	C₄×C₁₆	—	53	2	no	—
141	C₄×C₁₆	—	53	2	no	—
142	C₅₃	#33434	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	53	2	no	—
144	#131	—	106	4	no	—
145	#131	—	106	4	no	—
146	C₂×M₁₆	—	53	4	no	—
147	C₂×M₁₆	C₂×#6157	1	2	yes	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	53	1	no	—
150	C₂²×C₈	—	53	4	no	—
151	C₄⋊Q₈	—	53	1	no	—
152	#127	—	53	1	no	—
153	#124	—	53	2	no	—
154	#121	—	53	4	no	—
155	C₄×D₄	—	53	4	no	—
156	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
157	M₃₂	—	106	8	no	—
158	M₃₂	—	106	4	no	—
159	C₂×C₁₆	—	53	1	no	—
160	C₂×C₁₆	—	53	1	no	—
161	C₂×C₁₆	—	53	4	no	—
162	C₂×C₁₆	—	53	2	no	—
163	C₂×C₁₆	—	53	4	no	—
164	C₂×C₁₆	—	106	8	no	—
165	C₂×C₁₆	—	106	2	no	—
166	C₄⋊C₈	C₂×#6157	1	4	yes	—
167	C₂²⋊C₈	—	53	8	no	—
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	—	53	2	no	—
170	#41	—	106	4	no	—
171	#41	—	106	4	no	—
172	C₂×Q₈	—	53	2	no	—
173	C₂×D₄	—	53	4	no	—
174	C₂×D₄	—	53	4	no	—
175	C₂×D₄	—	53	2	no	—
176	C₂²×C₄	—	53	4	no	—
177	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
178	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	106	8	no	—
181	C₂×C₈	—	53	2	no	—
182	C₂×C₈	—	53	2	no	—
183	C₂×C₈	—	53	1	no	—
184	C₂×C₈	—	53	1	no	—
185	C₂×C₈	—	53	2	no	—
186	C₂×C₈	—	106	4	no	—
187	C₂×C₈	—	106	4	no	—
188	C₂×C₈	—	106	2	no	—
189	C₂×C₈	C₂²×#6157	1	4	yes	—
190	C₄⋊C₄	—	53	4	no	—
191	C₂²⋊C₄	—	53	8	no	—
192	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
193	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
194	C₄²	C₂²×#6157	1	2	yes	—
195	C₁₆	—	53	2	no	—
196	C₁₆	—	53	2	no	—
197	C₁₆	—	106	4	no	—
198	C₁₆	—	106	2	no	—
199	C₂³	—	53	4	no	—
200	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
201	Q₈	—	106	4	no	—
202	D₄	—	106	8	no	—
203	D₄	—	106	4	no	—
204	D₄	—	106	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	53	4	no	—
207	C₂×C₄	—	106	4	no	—
208	C₂×C₄	—	106	4	no	—
209	C₂×C₄	D₄×#6157	1	2	yes	—
210	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
211	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
212	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
213	C₂×C₄	#13133305	1	2	yes	—
214	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	53	2	no	—
217	C₈	—	53	1	no	—
218	C₈	—	53	1	no	—
219	C₈	—	106	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	106	4	no	—
222	C₂²	—	106	8	no	—
223	C₂²	#16811651	1	2	yes	—
224	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	106	4	no	—
227	C₄	#16811651	1	2	yes	—
228	C₄	#16811682	1	4	yes	complemented
229	C₄	C₂×D₄×#6157	1	1	yes	characteristic
230	C₄	C₂×#13133305	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	212	4	no	—
233	C₂	C₂×#16811651	1	1	yes	characteristic
234	C₂	C₂×#16811682	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#423	C₄
prime radical	#16811173	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#423	#6157
hypocenter	C₅₃	#33434
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×#16811682	C₂	semidirect
2. #20291101	C₂	semidirect
3. #20291094	C₂	semidirect
4. #20291047	C₂	semidirect
5. #20290947	C₂	semidirect
6. C₂×C₄×#11719910	C₂	semidirect
29. C₂×#11719910	D₄	semidirect
110. C₄×C₅₃	#504	semidirect
142. C₅₃	#33434	semidirect
228. C₄	#16811682	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811682	2
#20291101	2
#20291094	2
#20291047	2
#20290947	2
C₂×C₄×#11719910	2
#20283426	2
#33434	53