Subgroups — #28328278

Order 13568.

520 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291091	C₂	1	4	yes	maximal, complemented
2	#20291087	C₂	1	4	yes	maximal, complemented
3	#20291058	C₂	1	2	yes	maximal, complemented
4	#20290953	C₂	1	2	yes	maximal
5	C₂×#16811516	C₂	1	2	yes	maximal, complemented
6	#20283142	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133291	C₂²	1	4	yes	—
8	#16811546	C₂²	1	8	yes	—
9	#16811520	C₂²	1	8	yes	—
10	#16811516	C₂²	1	4	yes	—
11	#16811516	C₂²	1	4	yes	—
12	C₂×#13133225	C₂²	1	1	yes	characteristic
13	C₂×#13133223	C₄	1	1	yes	characteristic
14	#16811091	C₄	1	1	yes	characteristic
15	#16810580	C₂²	1	2	yes	—
16	#16810579	C₂²	1	1	yes	characteristic
17	#16810577	C₄	1	2	yes	—
18	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
19	C₄×#11719912	C₂²	1	2	yes	—
20	C₂×#13133127	C₂²	1	1	yes	characteristic
21	#97×D₅₃	C₄	1	2	yes	—
22	C₅₃×#405	C₄	1	1	yes	characteristic
23	#13133291	—	2	4	no	—
24	#13133291	C₂³	1	8	yes	—
25	C₂×#11719910	—	2	4	no	—
26	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
27	#13133225	—	2	4	no	—
28	#13133223	—	2	4	no	—
29	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719912	C₂×C₄	1	2	yes	—
31	C₂×#11719912	C₂³	1	2	yes	—
32	C₂²×#88322	C₂³	1	1	yes	characteristic
33	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	#13133182	—	2	1	no	—
36	#13133179	—	2	1	no	—
37	#13133177	—	2	2	no	—
38	#13133174	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×D₅₃	C₂×C₄	1	2	yes	—
41	#13133128	C₂×C₄	1	2	yes	—
42	#13133127	C₂×C₄	1	1	yes	characteristic
43	#13133127	C₂×C₄	1	2	yes	—
44	#13133127	C₂×C₄	1	1	yes	characteristic
45	#13133127	C₂³	1	1	yes	characteristic
46	#13133127	C₂³	1	1	yes	characteristic
47	C₄×#88322	C₂³	1	2	yes	—
48	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
52	#97×C₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₅₃	M₁₆	1	2	yes	—
56	C₂²×Dic₅₃	—	2	1	no	—
57	#11719945	—	2	2	no	—
58	#11719945	—	2	2	no	—
59	C₂×D₂₁₂	M₁₆	1	1	yes	characteristic
60	C₂×C₄×D₅₃	—	2	2	no	—
61	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
62	C₂×Dic₁₀₆	M₁₆	1	1	yes	characteristic
63	M₁₆×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	#11719921	—	2	2	no	—
72	#11719919	—	2	2	no	—
73	C₄×Dic₅₃	—	2	1	no	—
74	C₄×Dic₅₃	—	2	2	no	—
75	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
77	#11719917	M₁₆	1	4	yes	—
78	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
79	C₂×#88322	C₂²×C₄	1	2	yes	—
80	C₂×#88322	#41	1	4	yes	—
81	C₂×#88322	C₂⁴	1	1	yes	characteristic
82	#11719912	—	2	4	no	—
83	#11719912	—	2	4	no	—
84	C₈×D₅₃	—	2	4	no	—
85	C₈×D₅₃	—	2	4	no	—
86	#11719910	—	2	8	no	—
87	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂²×D₅₃	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	#117	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₂²⋊D₅₃	—	2	4	no	—
95	C₂×Dic₅₃	—	2	1	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	8	1	no	—
98	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
99	D₂₁₂	—	2	2	no	—
100	C₄×D₅₃	—	2	2	no	—
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	8	2	no	—
103	Dic₁₀₆	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	#88322	#117	1	2	yes	—
108	#88322	C₂×M₁₆	1	4	yes	—
109	#88322	C₂×#41	1	2	yes	—
110	#33426	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	#432	1	2	yes	—
113	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
114	C₂×D₅₃	—	2	2	no	—
115	C₂×D₅₃	—	4	2	no	—
116	C₄×C₅₃	—	4	2	no	—
117	C₄×C₅₃	#432	1	2	yes	—
118	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
119	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
120	Dic₅₃	—	2	2	no	—
121	Dic₅₃	—	4	1	no	—
122	Dic₅₃	—	4	1	no	—
123	#2875	—	53	1	no	—
124	#2166	—	53	4	no	—
125	#2165	—	53	4	no	—
126	#2153	—	53	2	no	—
127	#2115	—	53	2	no	—
128	C₂×#347	—	53	2	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	C₂×#432	1	1	yes	characteristic
132	D₅₃	—	4	2	no	—
133	C₂×#131	—	53	1	no	—
134	C₂×#131	—	53	1	no	—
135	C₄×#41	—	53	1	no	—
136	C₂×M₃₂	—	53	4	no	—
137	#405	—	53	1	no	—
138	#405	—	53	2	no	—
139	#405	—	53	2	no	—
140	#405	#6157	1	1	yes	characteristic
141	C₄×M₁₆	—	53	1	no	—
142	C₄×M₁₆	—	53	2	no	—
143	C₂×#97	—	53	1	no	—
144	C₂×#97	—	53	2	no	—
145	#364	—	53	8	no	—
146	#349	—	53	8	no	—
147	#347	—	53	4	no	—
148	#347	—	53	4	no	—
149	C₅₃	#33426	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	53	1	no	—
151	#131	—	106	4	no	—
152	#131	—	106	4	no	—
153	C₂×M₁₆	—	53	1	no	—
154	C₂×M₁₆	—	53	2	no	—
155	C₂×M₁₆	—	53	2	no	—
156	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
157	C₂²×C₈	—	53	1	no	—
158	C₂²×C₈	—	53	2	no	—
159	C₂²×C₈	—	53	2	no	—
160	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
161	C₄×Q₈	—	106	1	no	—
162	C₄×D₄	—	106	2	no	—
163	C₄×D₄	—	106	1	no	—
164	#117	—	53	2	no	—
165	#117	C₂×#6157	1	1	yes	characteristic
166	C₂×C₄²	—	53	1	no	—
167	C₂×C₄²	—	53	2	no	—
168	M₃₂	—	53	8	no	—
169	M₃₂	—	106	4	no	—
170	C₂×C₁₆	—	106	4	no	—
171	#97	—	53	1	no	—
172	#97	—	53	2	no	—
173	#97	—	53	1	no	—
174	#97	—	53	1	no	—
175	#97	—	53	1	no	—
176	#97	—	53	2	no	—
177	#97	C₂×#6157	1	2	yes	—
178	C₄×C₈	—	53	2	no	—
179	C₄×C₈	—	53	2	no	—
180	C₄×C₈	C₂×#6157	1	2	yes	—
181	#41	—	106	2	no	—
182	#41	—	106	2	no	—
183	C₂×Q₈	—	53	1	no	—
184	C₂×D₄	—	53	2	no	—
185	C₂×D₄	—	53	1	no	—
186	C₂²×C₄	—	53	2	no	—
187	C₂²×C₄	—	106	2	no	—
188	C₂²×C₄	—	106	1	no	—
189	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
190	M₁₆	—	2	2	no	—
191	M₁₆	—	53	4	no	—
192	M₁₆	—	106	4	no	—
193	M₁₆	—	106	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	53	4	no	—
196	C₂×C₈	—	53	1	no	—
197	C₂×C₈	—	53	2	no	—
198	C₂×C₈	—	53	1	no	—
199	C₂×C₈	—	106	4	no	—
200	C₂×C₈	—	106	4	no	—
201	C₂×C₈	C₂²×#6157	1	2	yes	—
202	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
203	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	106	2	no	—
208	C₄²	—	53	1	no	—
209	C₄²	—	53	1	no	—
210	C₄²	—	106	2	no	—
211	C₄²	—	106	1	no	—
212	C₄²	C₂²×#6157	1	2	yes	—
213	C₁₆	—	106	8	no	—
214	C₂³	—	53	2	no	—
215	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
216	Q₈	—	106	2	no	—
217	D₄	—	106	4	no	—
218	D₄	—	106	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	53	2	no	—
221	C₂×C₄	—	106	2	no	—
222	C₂×C₄	—	106	2	no	—
223	C₂×C₄	—	106	1	no	—
224	C₂×C₄	—	106	1	no	—
225	C₂×C₄	—	424	2	no	—
226	C₂×C₄	—	424	1	no	—
227	C₂×C₄	#13133296	1	2	yes	—
228	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
229	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
230	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	53	2	no	—
235	C₈	—	53	2	no	—
236	C₈	—	53	4	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	106	2	no	—
239	C₂²	—	212	2	no	—
240	C₂²	#16811646	1	2	yes	—
241	C₂²	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
242	C₄	—	4	2	no	—
243	C₄	—	106	2	no	—
244	C₄	—	212	1	no	—
245	C₄	—	212	1	no	—
246	C₄	#16811646	1	2	yes	—
247	C₄	C₂×#13133296	1	1	yes	characteristic
248	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	212	2	no	—
252	C₂	C₂×#16811646	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133296
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂×C₅₃	C₂×#432
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#33426
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #20291091	C₂	semidirect
2. #20291087	C₂	semidirect
3. #20291058	C₂	semidirect
5. C₂×#16811516	C₂	semidirect
149. C₅₃	#33426	semidirect

Maximal subgroups

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

subgroup	index
#20291091	2
#20291087	2
#20291058	2
#20290953	2
C₂×#16811516	2
#20283142	2
#33426	53