Subgroups — #27300017

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal
3	#20272793	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	4	yes	—
6	C₄⋊#11719908	—	2	4	no	—
7	C₂²⋊#11719908	—	2	4	no	—
8	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
11	C₅₃×#405	—	2	4	no	—
12	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
13	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719908	—	2	4	no	—
17	C₂×#11719908	—	2	4	no	—
18	C₂×#11719908	—	2	4	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	#117×C₅₃	—	2	2	no	—
32	#117×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
34	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#97×C₅₃	—	2	2	no	—
38	#97×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	D₄	1	4	yes	—
41	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₅₃	—	2	8	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₂²⋊C₄×C₅₃	—	2	4	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₅₃	M₁₆	1	2	yes	—
74	#11719908	—	4	4	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	—	2	4	no	—
78	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
79	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	#121	1	2	yes	—
101	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
102	#12784	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	4	no	—
105	C₂²×C₅₃	—	4	2	no	—
106	C₂²×C₅₃	—	4	2	no	—
107	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₂²×C₅₃	#431	1	1	yes	characteristic
110	C₂²×C₅₃	#504	1	4	yes	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	4	no	—
113	C₄×C₅₃	—	4	2	no	—
114	C₄×C₅₃	#380	1	1	yes	characteristic
115	C₄×C₅₃	#407	1	1	yes	characteristic
116	C₄×C₅₃	#423	1	1	yes	characteristic
117	C₄×C₅₃	#432	1	1	yes	characteristic
118	C₂×#405	D₅₃	1	1	yes	characteristic
119	C₂×C₄⋊C₁₆	—	53	2	no	—
120	C₂×C₂²⋊C₁₆	—	53	2	no	—
121	#1374	—	53	2	no	—
122	C₂×C₅₃	—	4	2	no	—
123	C₂×C₅₃	#1797	1	1	yes	characteristic
124	C₂×C₅₃	#2113	1	2	yes	—
125	C₂×C₅₃	#2151	1	2	yes	—
126	C₂×C₅₃	#2164	1	2	yes	—
127	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
128	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₂×#117	Dic₅₃	1	1	yes	characteristic
130	C₂²×C₁₆	—	53	4	no	—
131	#405	—	2	4	no	—
132	C₂×#97	Dic₅₃	1	1	yes	characteristic
133	C₂×#97	Dic₅₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
136	C₄⋊C₁₆	—	106	4	no	—
137	C₂²⋊C₁₆	—	106	4	no	—
138	C₅₃	#12784	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	Dic₁₀₆	1	2	yes	—
146	C₂²×C₈	D₂₁₂	1	2	yes	—
147	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
151	#117	—	2	2	no	—
152	#117	—	2	2	no	—
153	C₂×C₄⋊C₄	#88322	1	2	yes	—
154	C₂×C₂²⋊C₄	#88322	1	2	yes	—
155	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂×C₁₆	—	106	4	no	—
158	C₂×C₁₆	—	106	4	no	—
159	C₂×C₁₆	—	106	4	no	—
160	#97	—	2	2	no	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	C₂²⋊D₅₃	1	4	yes	—
164	C₂⁴	C₂×#88322	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	C₂×#88322	1	2	yes	—
170	C₂²×C₄	C₂×#88322	1	2	yes	—
171	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
172	C₂²×C₄	#11719917	1	2	yes	—
173	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
175	M₁₆	—	2	8	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	#11719917	1	2	yes	—
185	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
186	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
187	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
188	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
189	C₂×C₈	#11719945	1	2	yes	—
190	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
191	C₄⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	#11719917	1	2	yes	—
196	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
197	C₁₆	—	212	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₄⋊#88322	1	2	yes	—
202	C₂³	C₂²×#88322	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	C₂²⋊#88322	1	4	yes	—
213	C₂×C₄	C₄⋊#88322	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
216	C₂×C₄	#13133176	1	1	yes	characteristic
217	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
219	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	#13133191	1	2	yes	—
224	C₈	#13133196	1	2	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
230	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
231	C₂²	#16810676	1	1	yes	characteristic
232	C₂²	#16811040	1	4	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	#16810483	1	1	yes	characteristic
237	C₄	#16810587	1	1	yes	characteristic
238	C₄	#16810647	1	1	yes	characteristic
239	C₄	#16810680	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	#20274695	1	1	yes	characteristic
242	C₂	#20276329	1	2	yes	—
243	C₂	#20276474	1	2	yes	—
244	C₂	#20276522	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×#405	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#12784
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#11719908	C₂	semidirect
3. #20272793	C₂	semidirect
138. C₅₃	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×C₂²⋊#11719908	2
#20272793	2
C₂×C₅₃×#405	2
#12784	53