Subgroups — #28303535

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290648	C₂	1	1	yes	maximal, characteristic
2	#20290632	C₂	1	1	yes	maximal, characteristic
3	#20289545	C₂	1	2	yes	maximal
4	#20289545	C₂	1	2	yes	maximal
5	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133284	—	2	2	no	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	C₂×C₄⋊#88324	C₂²	1	2	yes	—
9	#16811489	—	2	2	no	—
10	C₂×#13133128	C₄	1	1	yes	characteristic
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×#13133127	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×#88324	—	2	2	no	—
18	C₂²×#88324	—	2	2	no	—
19	#13133284	—	4	4	no	—
20	C₄⋊#88324	—	2	2	no	—
21	C₄⋊#88324	—	2	2	no	—
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	—	2	2	no	—
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
28	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
29	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
30	#13133128	—	2	2	no	—
31	#13133127	—	2	2	no	—
32	#13133127	C₈	1	4	yes	complemented
33	C₈×Dic₅₃	—	2	1	no	—
34	C₈×Dic₅₃	—	2	1	no	—
35	C₈×Dic₅₃	—	2	1	no	—
36	C₈×Dic₅₃	—	2	1	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	#97×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
44	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×#88324	—	4	2	no	—
47	C₂×#88324	—	4	2	no	—
48	C₂×#88324	—	4	4	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
57	C₄²×C₅₃	—	2	1	no	—
58	C₄²×C₅₃	—	2	1	no	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	—	4	2	no	—
62	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
63	C₄×Dic₅₃	M₁₆	1	2	yes	—
64	C₄×Dic₅₃	#41	1	1	yes	characteristic
65	C₄×Dic₅₃	#41	1	1	yes	characteristic
66	C₄×Dic₅₃	#41	1	1	yes	characteristic
67	C₄×Dic₅₃	#41	1	1	yes	characteristic
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	—	2	1	no	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	C₂×C₈	1	2	yes	—
73	C₂×#88322	M₁₆	1	2	yes	—
74	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	2	no	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
88	C₂×Dic₅₃	#124	1	1	yes	characteristic
89	C₂×Dic₅₃	#124	1	1	yes	characteristic
90	C₂×Dic₅₃	#125	1	1	yes	characteristic
91	C₂×Dic₅₃	#125	1	1	yes	characteristic
92	C₂×Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×Dic₅₃	#137	1	2	yes	—
94	#88324	—	8	2	no	—
95	#88324	—	8	2	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	2	2	no	—
98	#88322	—	2	2	no	—
99	#88322	—	2	2	no	—
100	#8786	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	#382	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂²×C₅₃	#423	1	1	yes	characteristic
104	C₂²×C₅₃	#428	1	2	yes	—
105	C₂²×C₅₃	#430	1	2	yes	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	#431	1	1	yes	characteristic
108	C₄×C₅₃	#431	1	1	yes	characteristic
109	C₄×C₅₃	#432	1	1	yes	characteristic
110	C₄×C₅₃	#432	1	1	yes	characteristic
111	Dic₅₃	—	2	4	no	—
112	Dic₅₃	—	2	1	no	—
113	Dic₅₃	—	2	1	no	—
114	Dic₅₃	#489	1	2	yes	—
115	Dic₅₃	#490	1	1	yes	characteristic
116	Dic₅₃	#491	1	1	yes	characteristic
117	#1842	—	53	1	no	—
118	#1831	—	53	1	no	—
119	C₄×#97	—	53	1	no	—
120	#1277	—	53	2	no	—
121	#1277	—	53	2	no	—
122	C₂×C₅₃	#1565	1	2	yes	—
123	C₂×C₅₃	#1569	1	2	yes	—
124	C₂×C₅₃	#1831	1	1	yes	characteristic
125	C₂×C₅₃	#1836	1	1	yes	characteristic
126	C₂×C₅₃	#1838	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	53	2	no	—
128	C₂×C₄⋊C₈	—	53	2	no	—
129	C₂×C₄⋊C₈	—	106	2	no	—
130	C₂×#97	—	53	1	no	—
131	C₂×#97	—	53	1	no	—
132	C₂×#97	—	53	1	no	—
133	C₂×#97	#6157	1	1	yes	characteristic
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	—	53	1	no	—
136	C₄³	—	53	1	no	—
137	#336	—	106	2	no	—
138	C₅₃	#8786	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₈	—	53	1	no	—
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	—	106	2	no	—
142	C₂²×C₈	—	106	2	no	—
143	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
144	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	53	1	no	—
147	C₂×C₄²	—	53	2	no	—
148	C₂×C₄²	—	53	2	no	—
149	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
150	C₄⋊C₈	—	106	2	no	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	106	2	no	—
153	C₄⋊C₈	—	106	2	no	—
154	C₄⋊C₈	—	212	4	no	—
155	#97	—	2	2	no	—
156	#97	—	53	4	no	—
157	#97	—	106	2	no	—
158	#97	—	106	2	no	—
159	C₄×C₈	—	106	1	no	—
160	C₄×C₈	—	106	1	no	—
161	C₄×C₈	—	106	1	no	—
162	C₄×C₈	—	106	1	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	2	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	C₄⋊#6157	1	2	yes	—
167	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	53	2	no	—
173	C₂×C₈	—	53	2	no	—
174	C₂×C₈	—	106	1	no	—
175	C₂×C₈	—	106	1	no	—
176	C₂×C₈	—	106	1	no	—
177	C₂×C₈	—	106	1	no	—
178	C₂×C₈	—	212	2	no	—
179	C₂×C₈	—	212	2	no	—
180	C₂×C₈	—	212	4	no	—
181	C₂×C₈	—	212	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
184	C₂×C₈	#11719940	1	2	yes	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	53	1	no	—
188	C₄²	—	53	1	no	—
189	C₄²	—	53	1	no	—
190	C₄²	—	53	1	no	—
191	C₄²	—	53	2	no	—
192	C₄²	—	53	2	no	—
193	C₄²	—	106	2	no	—
194	C₄²	—	106	2	no	—
195	C₄²	—	212	2	no	—
196	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	53	1	no	—
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	53	2	no	—
203	C₂×C₄	—	53	2	no	—
204	C₂×C₄	—	53	1	no	—
205	C₂×C₄	—	53	1	no	—
206	C₂×C₄	—	53	4	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	106	2	no	—
209	C₂×C₄	—	106	2	no	—
210	C₂×C₄	#13133296	1	1	yes	characteristic
211	C₂×C₄	#13133296	1	1	yes	characteristic
212	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
214	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
215	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	106	2	no	—
219	C₈	—	106	2	no	—
220	C₈	—	424	2	no	—
221	C₈	—	424	2	no	—
222	C₂²	#16811567	1	1	yes	characteristic
223	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
224	C₂²	#16811626	1	1	yes	characteristic
225	C₂²	#16811634	1	2	yes	—
226	C₂²	#16811638	1	2	yes	—
227	C₄	—	4	2	no	—
228	C₄	—	53	1	no	—
229	C₄	—	53	2	no	—
230	C₄	—	53	1	no	—
231	C₄	—	106	1	no	—
232	C₄	—	106	1	no	—
233	C₄	—	106	4	no	—
234	C₄	#16811640	1	1	yes	characteristic
235	C₄	#16811641	1	1	yes	characteristic
236	C₄	#16811643	1	1	yes	characteristic
237	C₄	#16811646	1	1	yes	characteristic
238	C₂	#20290267	1	2	yes	—
239	C₂	#20290279	1	2	yes	—
240	C₂	#20290631	1	1	yes	characteristic
241	C₂	#20290638	1	1	yes	characteristic
242	C₂	#20290641	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#97×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#8786
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
32. #13133127	C₈	semidirect
138. C₅₃	#8786	semidirect

Maximal subgroups

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

subgroup	index
#20290648	2
#20290632	2
#20289545	2
#97×Dic₅₃	2
#8786	53