Subgroups — #27302631

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
2	#20276471	C₂	1	1	yes	maximal, characteristic
3	#20272798	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	#16810753	—	2	1	no	—
8	#16810753	—	2	1	no	—
9	#16810753	—	2	1	no	—
10	#16810753	—	2	1	no	—
11	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
12	C₂×#13133146	C₄	1	2	yes	—
13	C₂×#13133127	C₂²	1	2	yes	—
14	C₄⋊#11719908	—	2	1	no	—
15	C₄⋊#11719908	—	2	1	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#346	—	2	1	no	—
19	C₄×C₁₆×C₅₃	—	2	1	no	—
20	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₀₆	—	2	1	no	—
23	C₄×Dic₁₀₆	—	2	1	no	—
24	C₄×Dic₁₀₆	—	2	1	no	—
25	C₄×Dic₁₀₆	—	2	1	no	—
26	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
27	C₂×#11719919	C₈	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	#13133157	—	2	4	no	—
30	C₂×#11719908	—	2	2	no	—
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	#13133146	—	2	2	no	—
34	#13133146	—	2	2	no	—
35	#13133127	—	2	2	no	—
36	#13133127	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₅₃	—	2	2	no	—
41	M₃₂×C₅₃	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
51	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
52	C₂×Dic₁₀₆	—	2	1	no	—
53	C₂×Dic₁₀₆	—	2	1	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₅₃	#41	1	2	yes	—
62	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
65	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
66	C₄⋊Dic₅₃	—	2	1	no	—
67	C₄⋊Dic₅₃	—	2	1	no	—
68	#11719919	—	2	2	no	—
69	#11719919	—	2	2	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	2	4	no	—
74	C₁₆×C₅₃	—	2	4	no	—
75	#11719908	—	4	2	no	—
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#131	1	2	yes	—
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	2	2	no	—
89	C₂×Dic₅₃	—	4	2	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	Dic₁₀₆	—	4	2	no	—
92	Dic₁₀₆	—	4	2	no	—
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
96	C₈×C₅₃	#124	1	1	yes	characteristic
97	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
98	#88322	—	4	2	no	—
99	#13268	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	#349	1	2	yes	—
108	C₄×C₅₃	#391	1	1	yes	characteristic
109	C₄×C₅₃	#408	1	1	yes	characteristic
110	C₄×C₅₃	#446	1	2	yes	—
111	Dic₅₃	—	4	2	no	—
112	Dic₅₃	—	4	2	no	—
113	C₂×#446	—	53	1	no	—
114	#2150	—	53	1	no	—
115	C₄×M₃₂	D₅₃	1	1	yes	characteristic
116	#1377	—	53	4	no	—
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	#1984	1	1	yes	characteristic
119	C₂×C₅₃	C₂×#349	1	1	yes	characteristic
120	C₂×C₅₃	#2115	1	1	yes	characteristic
121	C₂×C₄×Q₈	—	53	1	no	—
122	C₂×M₃₂	—	53	2	no	—
123	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
124	#446	—	106	1	no	—
125	#446	—	106	1	no	—
126	#446	—	106	1	no	—
127	#446	—	106	1	no	—
128	C₂×C₄⋊C₈	—	53	2	no	—
129	C₂×C₄⋊C₈	—	53	1	no	—
130	C₂×#97	—	53	2	no	—
131	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	106	1	no	—
133	C₄⋊C₁₆	—	106	1	no	—
134	#346	—	2	1	no	—
135	C₄×C₁₆	—	2	1	no	—
136	C₅₃	#13268	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	53	1	no	—
138	C₂²×C₈	—	53	2	no	—
139	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₄×Q₈	—	106	1	no	—
142	C₄×Q₈	—	106	1	no	—
143	C₄×Q₈	—	106	1	no	—
144	C₄×Q₈	—	106	1	no	—
145	C₂×C₄⋊C₄	—	53	2	no	—
146	C₂×C₄⋊C₄	—	53	1	no	—
147	C₂×C₄²	—	53	2	no	—
148	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
149	M₃₂	—	2	2	no	—
150	M₃₂	—	106	4	no	—
151	M₃₂	Dic₁₀₆	1	4	yes	complemented
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	106	2	no	—
154	C₄⋊C₈	—	106	2	no	—
155	C₄⋊C₈	—	106	2	no	—
156	C₄⋊C₈	—	106	1	no	—
157	C₄⋊C₈	—	106	1	no	—
158	#97	—	106	2	no	—
159	#97	—	106	2	no	—
160	C₄×C₈	D₂₁₂	1	1	yes	characteristic
161	C₄×C₈	D₂₁₂	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂×Q₈	—	106	1	no	—
165	C₂×Q₈	—	106	1	no	—
166	C₂×Q₈	—	212	2	no	—
167	C₂²×C₄	—	53	2	no	—
168	C₂²×C₄	—	53	2	no	—
169	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	106	2	no	—
174	C₂×C₈	—	106	4	no	—
175	C₂×C₈	#11719921	1	1	yes	characteristic
176	C₂×C₈	#11719921	1	1	yes	characteristic
177	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
178	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
179	C₂×C₈	#11719945	1	2	yes	—
180	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄⋊C₄	—	106	2	no	—
182	C₄⋊C₄	—	106	2	no	—
183	C₄⋊C₄	—	106	1	no	—
184	C₄⋊C₄	—	106	1	no	—
185	C₄²	—	106	2	no	—
186	C₄²	—	106	2	no	—
187	C₄²	#11719912	1	1	yes	characteristic
188	C₄²	#11719912	1	1	yes	characteristic
189	C₄²	#11719921	1	1	yes	characteristic
190	C₄²	#11719921	1	1	yes	characteristic
191	C₁₆	—	2	4	no	—
192	C₁₆	—	212	2	no	—
193	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
194	Q₈	—	212	2	no	—
195	Q₈	—	212	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	106	2	no	—
199	C₂×C₄	—	106	2	no	—
200	C₂×C₄	—	212	2	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	#13133129	1	1	yes	characteristic
203	C₂×C₄	#13133129	1	1	yes	characteristic
204	C₂×C₄	#13133129	1	1	yes	characteristic
205	C₂×C₄	#13133129	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
207	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
208	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
209	C₂×C₄	#13133223	1	2	yes	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	212	2	no	—
213	C₈	#13133196	1	2	yes	—
214	C₈	#13133205	1	1	yes	characteristic
215	C₈	#13133217	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×#13133129	1	1	yes	characteristic
219	C₂²	C₈×Dic₁₀₆	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	212	2	no	—
224	C₄	—	212	2	no	—
225	C₄	#16810404	1	2	yes	—
226	C₄	#16810529	1	1	yes	characteristic
227	C₄	#16810589	1	1	yes	characteristic
228	C₄	#16810753	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	#20275651	1	1	yes	characteristic
231	C₂	C₂×#16810404	1	1	yes	characteristic
232	C₂	#20276335	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator, prime radical	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#13268
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
41. M₃₂×C₅₃	Q₈	semidirect
136. C₅₃	#13268	semidirect
151. M₃₂	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810753	2
#20276471	2
#20272798	2
C₄×M₃₂×C₅₃	2
#13268	53