Subgroups — #27354816

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20283649	C₂	1	2	yes	maximal
3	#20273567	C₂	1	8	yes	maximal, complemented
4	#20273474	C₂	1	1	yes	maximal, characteristic
5	#20273171	C₂	1	2	yes	maximal
6	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
7	#16811180	C₂²	1	2	yes	—
8	#16811028	C₄	1	1	yes	characteristic
9	#16811003	C₄	1	1	yes	characteristic
10	#16810974	C₄	1	2	yes	—
11	#16810864	C₄	1	4	yes	—
12	#16810753	—	2	2	no	—
13	#16810753	C₂²	1	4	yes	—
14	C₈×Dic₁₀₆	—	2	2	no	—
15	C₈×Dic₁₀₆	C₂²	1	4	yes	—
16	C₄×Dic₂₁₂	C₂²	1	4	yes	—
17	#16810649	—	2	2	no	—
18	#16810647	C₂²	1	2	yes	—
19	#16810578	—	2	2	no	—
20	#16810376	C₂²	1	4	yes	—
21	#16810355	C₂²	1	8	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#322	C₂²	1	4	yes	—
26	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
27	C₂×#11719917	—	2	2	no	—
28	#13133217	C₂×C₄	1	2	yes	—
29	#13133217	C₂×C₄	1	2	yes	—
30	#13133209	—	2	2	no	—
31	#13133196	—	2	2	no	—
32	#13133196	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₀₆	—	2	2	no	—
34	C₄×Dic₁₀₆	C₂³	1	4	yes	—
35	#13133176	—	2	2	no	—
36	#13133151	C₂×C₄	1	4	yes	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	C₂³	1	4	yes	—
39	#13133146	—	2	4	no	—
40	#13133146	—	2	4	no	—
41	#13133142	C₂×C₄	1	8	yes	—
42	#13133127	—	2	4	no	—
43	C₈×Dic₅₃	—	2	4	no	—
44	C₂²×C₈×C₅₃	—	2	1	no	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	C₄×C₈×C₅₃	C₂³	1	2	yes	—
55	C₄×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂×Dic₁₀₆	—	2	2	no	—
60	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	#41	1	1	yes	characteristic
73	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂²⋊Dic₅₃	—	2	4	no	—
75	C₄⋊Dic₅₃	—	2	2	no	—
76	C₄⋊Dic₅₃	—	2	2	no	—
77	C₄⋊Dic₅₃	—	2	2	no	—
78	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
79	#11719919	—	2	4	no	—
80	#11719919	—	2	4	no	—
81	C₄×Dic₅₃	—	2	4	no	—
82	#11719917	—	4	4	no	—
83	C₂×#88322	—	2	4	no	—
84	Dic₂₁₂	—	4	4	no	—
85	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₅₃	#135	1	2	yes	—
93	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₅₃	—	2	4	no	—
97	C₂×Dic₅₃	—	2	4	no	—
98	Dic₁₀₆	—	2	8	no	—
99	Dic₁₀₆	—	4	4	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	4	2	no	—
104	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
105	#88322	—	4	4	no	—
106	#17977	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
110	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
114	C₄×C₅₃	#444	1	2	yes	—
115	C₄×C₅₃	#568	1	2	yes	—
116	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
118	Dic₅₃	—	4	4	no	—
119	Dic₅₃	—	4	4	no	—
120	#3170	—	53	1	no	—
121	#2960	—	53	2	no	—
122	#1584	—	53	8	no	—
123	#1567	—	53	1	no	—
124	#1493	—	53	2	no	—
125	C₂×#322	D₅₃	1	1	yes	characteristic
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	#2932	1	1	yes	characteristic
128	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
130	#533	—	53	2	no	—
131	#500	—	53	1	no	—
132	#494	—	53	1	no	—
133	#487	—	53	2	no	—
134	#467	—	53	4	no	—
135	#446	—	53	4	no	—
136	#446	—	106	2	no	—
137	C₈×Q₈	—	53	4	no	—
138	C₈×Q₈	—	106	2	no	—
139	C₄×Q₁₆	—	53	4	no	—
140	#424	—	106	2	no	—
141	#423	—	53	2	no	—
142	#405	—	106	2	no	—
143	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#335	—	53	4	no	—
147	Q₈⋊C₈	—	53	8	no	—
148	#322	C₂×D₅₃	1	4	yes	—
149	C₅₃	#17977	1	1	yes	characteristic, complemented, Sylow
150	C₂×Q₁₆	—	53	4	no	—
151	C₂×M₁₆	—	106	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₄×D₅₃	1	2	yes	—
154	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
155	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
156	C₄⋊Q₈	—	53	2	no	—
157	C₄⋊Q₈	—	53	2	no	—
158	#125	—	106	2	no	—
159	C₂²⋊Q₈	—	53	4	no	—
160	C₂²⋊Q₈	—	106	2	no	—
161	C₄×Q₈	—	53	4	no	—
162	C₄×Q₈	—	106	2	no	—
163	#117	—	106	2	no	—
164	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	#107	—	53	4	no	—
166	C₄⋊C₈	—	53	4	no	—
167	C₄⋊C₈	—	106	4	no	—
168	C₄⋊C₈	—	106	4	no	—
169	C₄⋊C₈	—	106	2	no	—
170	#103	—	53	8	no	—
171	#97	—	106	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	106	4	no	—
174	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
175	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
176	C₄×C₈	D₂₁₂	1	2	yes	—
177	C₄×C₈	C₂²×D₅₃	1	2	yes	—
178	C₄×C₈	C₂²×D₅₃	1	2	yes	—
179	C₄×C₈	C₂²×D₅₃	1	2	yes	—
180	C₂×Q₈	—	53	4	no	—
181	C₂×Q₈	—	106	2	no	—
182	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
184	C₂²×C₄	#11719945	1	1	yes	characteristic
185	Q₁₆	—	212	4	no	—
186	M₁₆	—	212	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	106	4	no	—
192	C₂×C₈	Dic₂₁₂	1	4	yes	complemented
193	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
195	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
196	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
197	C₄⋊C₄	—	53	4	no	—
198	C₄⋊C₄	—	106	2	no	—
199	C₄⋊C₄	—	106	2	no	—
200	C₄⋊C₄	—	106	4	no	—
201	C₄⋊C₄	—	106	2	no	—
202	C₄⋊C₄	—	106	4	no	—
203	C₂²⋊C₄	—	106	4	no	—
204	C₄²	—	106	4	no	—
205	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
207	C₄²	#11719945	1	1	yes	characteristic
208	C₄²	C₂³×D₅₃	1	1	yes	characteristic
209	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
210	Q₈	—	106	8	no	—
211	Q₈	—	212	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	106	4	no	—
215	C₂×C₄	—	106	4	no	—
216	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
217	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
218	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
220	C₂×C₄	#13133239	1	2	yes	—
221	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	212	4	no	—
229	C₈	C₂×Dic₂₁₂	1	4	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	C₄×Dic₂₁₂	1	2	yes	—
233	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	212	4	no	—
237	C₄	—	212	4	no	—
238	C₄	C₄×Dic₂₁₂	1	2	yes	—
239	C₄	#16810746	1	2	yes	—
240	C₄	#16811389	1	2	yes	—
241	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
242	C₄	C₂×#13133239	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#20283434	1	1	yes	characteristic
245	C₂	C₂×C₄×Dic₂₁₂	1	1	yes	characteristic
246	C₂	C₂×#16810746	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#17977
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #20273567	C₂	semidirect
65. C₂×C₈×C₅₃	Q₁₆	semidirect
149. C₅₃	#17977	semidirect
192. C₂×C₈	Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20283649	2
#20273567	2
#20273474	2
#20273171	2
C₂×C₅₃×#322	2
#17977	53