Subgroups — #27282984

Order 13568.

482 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	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20275651	C₂	1	2	yes	maximal
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810355	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
8	C₄×Dic₂₁₂	—	2	2	no	—
9	C₄×Dic₂₁₂	—	2	2	no	—
10	C₂×#13133151	C₄	1	1	yes	characteristic
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×#13133142	C₄	1	2	yes	—
13	#16810377	—	2	4	no	—
14	#16810376	—	2	2	no	—
15	#16810355	—	2	4	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#322	—	2	2	no	—
20	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₄×Dic₁₀₆	—	2	2	no	—
25	C₄×Dic₁₀₆	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×#11719919	—	2	2	no	—
30	C₂×C₄×Dic₅₃	—	2	2	no	—
31	#13133151	—	2	2	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	—	2	2	no	—
34	#13133142	—	2	4	no	—
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₃	—	2	1	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	2	no	—
50	C₂²×Dic₅₃	—	2	2	no	—
51	C₂×Dic₁₀₆	—	2	2	no	—
52	C₂×Dic₁₀₆	—	2	2	no	—
53	C₂×Dic₁₀₆	—	2	2	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	2	2	no	—
56	C₂×Dic₁₀₆	M₁₆	1	4	yes	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	#41	1	1	yes	characteristic
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄⋊Dic₅₃	—	2	2	no	—
72	#11719919	—	4	2	no	—
73	#11719919	—	4	2	no	—
74	C₄×Dic₅₃	—	4	2	no	—
75	C₄×Dic₅₃	—	4	2	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	4	no	—
78	Dic₂₁₂	—	2	8	no	—
79	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₅₃	#131	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₅₃	#135	1	2	yes	—
91	C₂×Dic₅₃	—	2	4	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	C₂×Dic₅₃	—	4	4	no	—
96	Dic₁₀₆	—	2	4	no	—
97	Dic₁₀₆	—	2	4	no	—
98	Dic₁₀₆	—	4	2	no	—
99	Dic₁₀₆	—	4	2	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	4	no	—
103	C₈×C₅₃	—	4	2	no	—
104	#88322	—	8	2	no	—
105	#9662	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#390	1	1	yes	characteristic
107	C₂²×C₅₃	#435	1	2	yes	—
108	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₅₃	#444	1	2	yes	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	#407	1	1	yes	characteristic
113	C₄×C₅₃	#408	1	1	yes	characteristic
114	C₄×C₅₃	#435	1	2	yes	—
115	C₄×C₅₃	#467	1	2	yes	—
116	C₄×C₅₃	#494	1	1	yes	characteristic
117	C₄×C₅₃	#495	1	1	yes	characteristic
118	Dic₅₃	—	4	4	no	—
119	Dic₅₃	—	8	2	no	—
120	C₂×C₄×Q₁₆	—	53	1	no	—
121	#1984	—	53	2	no	—
122	C₂×#335	—	53	1	no	—
123	C₂×Q₈⋊C₈	—	53	2	no	—
124	C₂×#322	D₅₃	1	1	yes	characteristic
125	C₂×C₅₃	#1584	1	4	yes	—
126	C₂×C₅₃	#1955	1	1	yes	characteristic
127	C₂×C₅₃	#1969	1	1	yes	characteristic
128	C₂×C₅₃	#1975	1	1	yes	characteristic
129	C₂²×Q₁₆	—	53	1	no	—
130	C₂×C₄×Q₈	—	53	2	no	—
131	C₄×Q₁₆	—	106	2	no	—
132	C₄×Q₁₆	—	106	2	no	—
133	C₂×#107	—	53	1	no	—
134	C₂×C₄⋊C₈	—	53	2	no	—
135	C₂×#103	—	53	2	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	#336	—	106	4	no	—
140	#335	—	106	2	no	—
141	Q₈⋊C₈	—	106	4	no	—
142	#322	—	2	2	no	—
143	C₅₃	#9662	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	53	2	no	—
145	C₂×Q₁₆	—	106	4	no	—
146	C₂×Q₁₆	—	106	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	106	2	no	—
149	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
151	C₂²×C₈	D₂₁₂	1	2	yes	—
152	C₄×Q₈	—	106	2	no	—
153	C₄×Q₈	—	106	2	no	—
154	C₄×Q₈	—	106	2	no	—
155	C₄×Q₈	—	106	2	no	—
156	C₂×C₄⋊C₄	—	53	2	no	—
157	C₂×C₄⋊C₄	—	106	2	no	—
158	C₂×C₄²	—	106	2	no	—
159	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
160	#107	—	106	2	no	—
161	C₄⋊C₈	—	106	2	no	—
162	C₄⋊C₈	—	106	2	no	—
163	#103	—	106	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	D₂₁₂	1	2	yes	—
168	C₄×C₈	D₂₁₂	1	2	yes	—
169	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
170	C₂×Q₈	—	53	4	no	—
171	C₂×Q₈	—	106	2	no	—
172	C₂×Q₈	—	106	2	no	—
173	C₂×Q₈	—	106	2	no	—
174	C₂×Q₈	—	106	2	no	—
175	C₂×Q₈	—	106	2	no	—
176	C₂²×C₄	—	106	2	no	—
177	C₂²×C₄	—	106	2	no	—
178	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
180	C₂²×C₄	#11719945	1	1	yes	characteristic
181	Q₁₆	—	106	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	—	212	4	no	—
190	C₂×C₈	Dic₂₁₂	1	4	yes	complemented
191	C₂×C₈	#11719921	1	2	yes	—
192	C₂×C₈	#11719921	1	2	yes	—
193	C₄⋊C₄	—	106	2	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄²	—	212	2	no	—
198	C₄²	—	212	2	no	—
199	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
200	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
201	C₄²	#11719945	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
204	Q₈	—	106	4	no	—
205	Q₈	—	106	4	no	—
206	Q₈	—	212	2	no	—
207	Q₈	—	212	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	106	4	no	—
210	C₂×C₄	—	212	2	no	—
211	C₂×C₄	—	212	4	no	—
212	C₂×C₄	—	212	2	no	—
213	C₂×C₄	—	212	2	no	—
214	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
216	C₂×C₄	#13133191	1	1	yes	characteristic
217	C₂×C₄	#13133191	1	1	yes	characteristic
218	C₂×C₄	#13133205	1	1	yes	characteristic
219	C₂×C₄	#13133214	1	1	yes	characteristic
220	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
221	C₂×C₄	#13133225	1	2	yes	—
222	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
223	C₂×C₄	#13133239	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₈	—	424	2	no	—
229	C₂²	#16810525	1	1	yes	characteristic
230	C₂²	#16810696	1	2	yes	—
231	C₂²	C₄×Dic₂₁₂	1	2	yes	—
232	C₂²	#16810746	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	424	2	no	—
236	C₄	—	212	4	no	—
237	C₄	#16810586	1	1	yes	characteristic
238	C₄	#16810591	1	1	yes	characteristic
239	C₄	#16810696	1	2	yes	—
240	C₄	#16810864	1	2	yes	—
241	C₄	#16811003	1	1	yes	characteristic
242	C₄	#16811007	1	1	yes	characteristic
243	C₂	#20273567	1	4	yes	—
244	C₂	#20275513	1	1	yes	characteristic
245	C₂	#20275576	1	1	yes	characteristic
246	C₂	#20275608	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#9662
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
65. C₂×C₈×C₅₃	Q₁₆	semidirect
143. C₅₃	#9662	semidirect
190. 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
C₂×C₄×Dic₂₁₂	2
#20275651	2
C₂×#16810376	2
C₂×#16810355	2
C₂×C₅₃×#322	2
#9662	53