Subgroups — #27353241

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	2	yes	maximal
2	#20273490	C₂	1	8	yes	maximal, complemented
3	#20273474	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	2	no	—
6	#16810647	—	2	2	no	—
7	#16810647	—	2	2	no	—
8	#16810647	C₂²	1	4	yes	—
9	#16810647	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	—	2	2	no	—
11	C₂×C₄⋊#88322	C₂²	1	4	yes	—
12	#16810376	C₂²	1	16	yes	complemented
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#431	C₄	1	4	yes	—
15	C₅₃×#431	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	4	no	—
21	C₂×#11719917	—	4	8	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	4	no	—
25	C₄⋊#88322	—	2	4	no	—
26	C₄⋊#88322	C₂³	1	8	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
40	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	4	yes	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	D₈	1	4	yes	—
53	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₃	—	4	4	no	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
62	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₃	C₂×D₄	1	2	yes	—
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
70	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
72	#11719917	—	4	8	no	—
73	#11719917	—	4	8	no	—
74	C₂×#88322	—	2	8	no	—
75	C₂×#88322	—	4	8	no	—
76	C₂³×C₅₃	—	4	1	no	—
77	C₂³×C₅₃	—	4	1	no	—
78	C₂³×C₅₃	#107	1	4	yes	—
79	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	#107	1	4	yes	—
90	C₂×C₄×C₅₃	#107	1	8	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
98	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₅₃	—	2	8	no	—
105	#88322	—	4	8	no	—
106	#17814	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	4	4	no	—
112	C₂²×C₅₃	C₂×#107	1	4	yes	—
113	C₂²×C₅₃	C₂×#107	1	2	yes	—
114	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	C₂×#107	1	4	yes	—
120	C₄×C₅₃	C₂×#107	1	2	yes	—
121	C₄×C₅₃	#430	1	2	yes	—
122	C₄×C₅₃	#568	1	2	yes	—
123	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
125	C₂×#431	D₅₃	1	1	yes	characteristic
126	C₂×#423	—	53	2	no	—
127	#1571	—	53	8	no	—
128	#1567	—	53	4	no	—
129	C₂×C₅₃	—	2	4	no	—
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	#2905	1	1	yes	characteristic
132	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
134	C₂²×M₁₆	—	106	2	no	—
135	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
136	#431	Dic₅₃	1	4	yes	—
137	#431	C₂×D₅₃	1	4	yes	—
138	#423	—	53	4	no	—
139	#423	—	53	4	no	—
140	#423	—	106	2	no	—
141	#423	—	106	2	no	—
142	C₂×C₄⋊C₈	—	53	4	no	—
143	C₂×C₄⋊C₈	—	106	2	no	—
144	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
145	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
147	#335	—	53	16	no	—
148	C₅₃	#17814	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
150	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
151	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂×M₁₆	—	106	4	no	—
153	C₂×M₁₆	—	106	4	no	—
154	C₂×M₁₆	—	212	8	no	—
155	C₂²×C₈	—	106	4	no	—
156	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
157	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
158	C₂×C₄²	Dic₁₀₆	1	2	yes	—
159	C₂×C₄²	D₂₁₂	1	2	yes	—
160	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
163	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
165	C₄⋊C₈	—	53	8	no	—
166	C₄⋊C₈	—	106	4	no	—
167	C₄⋊C₈	—	106	4	no	—
168	C₄⋊C₈	—	106	4	no	—
169	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
170	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
171	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
172	C₄×C₈	C₂²×D₅₃	1	4	yes	—
173	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	D₄₂₄	1	4	yes	—
177	C₂²×C₄	Dic₂₁₂	1	4	yes	—
178	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
179	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
180	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
184	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
185	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
186	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
187	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
190	M₁₆	—	212	8	no	—
191	M₁₆	—	212	8	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	—	106	8	no	—
194	C₂×C₈	—	212	8	no	—
195	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
196	C₄²	C₄⋊Dic₅₃	1	2	yes	—
197	C₄²	C₄⋊Dic₅₃	1	2	yes	—
198	C₄²	C₄⋊Dic₅₃	1	2	yes	—
199	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
200	C₄²	C₂×Dic₁₀₆	1	2	yes	—
201	C₄²	C₂×D₂₁₂	1	2	yes	—
202	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
203	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
204	C₄²	C₂²×Dic₅₃	1	2	yes	—
205	C₄²	C₂³×D₅₃	1	1	yes	characteristic
206	C₂³	—	4	1	no	—
207	C₂³	—	4	1	no	—
208	C₂³	#13133151	1	4	yes	—
209	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
210	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	#13133151	1	4	yes	—
220	C₂×C₄	#13133151	1	8	yes	—
221	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
228	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
229	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
230	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
231	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
234	C₈	—	2	8	no	—
235	C₈	—	212	8	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×#13133151	1	4	yes	—
242	C₂²	C₂×#13133151	1	2	yes	—
243	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	C₂×#13133151	1	4	yes	—
249	C₄	C₂×#13133151	1	2	yes	—
250	C₄	#16810672	1	2	yes	—
251	C₄	#16811390	1	2	yes	—
252	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
253	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	4	no	—
256	C₂	#20283279	1	1	yes	characteristic
257	C₂	C₂²×#13133151	1	1	yes	characteristic
258	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#17814
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #20273490	C₂	semidirect
3. #20273474	C₂	semidirect
12. #16810376	C₂²	semidirect
148. C₅₃	#17814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
#20273490	2
#20273474	2
C₂×C₅₃×#431	2
#17814	53