Subgroups — #27295316

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 264 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	1	yes	maximal, characteristic, complemented
2	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272758	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	2	yes	—
9	#16810376	—	2	1	no	—
10	#16810374	C₂²	1	2	yes	—
11	C₂×#135×C₅₃	—	2	1	no	—
12	C₂×#135×C₅₃	C₄	1	2	yes	—
13	C₅₃×#532	—	2	1	no	—
14	C₅₃×#495	—	2	1	no	—
15	C₅₃×#495	C₂²	1	2	yes	—
16	C₅₃×#494	C₄	1	1	yes	characteristic
17	C₅₃×#493	C₄	1	1	yes	characteristic
18	C₅₃×#492	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	4	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₂²⋊#88322	—	4	1	no	—
24	C₄×#88322	—	4	1	no	—
25	C₂×#41×C₅₃	—	2	2	no	—
26	#135×C₅₃	—	4	2	no	—
27	#135×C₅₃	—	4	2	no	—
28	#135×C₅₃	—	4	4	no	—
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×SD₁₆×C₅₃	—	2	2	no	—
32	C₂×SD₁₆×C₅₃	—	2	2	no	—
33	C₂×D₈×C₅₃	—	2	1	no	—
34	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₅₃	—	2	1	no	—
38	#127×C₅₃	—	2	1	no	—
39	#124×C₅₃	—	2	2	no	—
40	#121×C₅₃	—	4	1	no	—
41	#121×C₅₃	—	4	1	no	—
42	C₄×D₄×C₅₃	—	4	1	no	—
43	C₄×D₄×C₅₃	—	4	1	no	—
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	#41×C₅₃	—	4	4	no	—
50	#41×C₅₃	—	4	4	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	1	no	—
55	C₂×D₄×C₅₃	—	4	1	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	Q₁₆×C₅₃	—	4	2	no	—
61	Q₁₆×C₅₃	—	4	2	no	—
62	SD₁₆×C₅₃	—	4	4	no	—
63	SD₁₆×C₅₃	—	4	2	no	—
64	SD₁₆×C₅₃	—	4	2	no	—
65	D₈×C₅₃	—	4	2	no	—
66	D₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	D₈	1	2	yes	—
71	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₅₃	—	4	1	no	—
75	C₄⋊C₄×C₅₃	—	4	1	no	—
76	C₂²⋊C₄×C₅₃	—	4	2	no	—
77	C₂²⋊C₄×C₅₃	—	4	2	no	—
78	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅₃	D₈	1	1	yes	characteristic
80	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×#88322	—	4	2	no	—
83	#11719908	—	4	2	no	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	#102	1	1	yes	characteristic
86	Q₈×C₅₃	—	4	2	no	—
87	Q₈×C₅₃	—	4	2	no	—
88	D₄×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	4	2	no	—
90	D₄×C₅₃	—	8	2	no	—
91	D₄×C₅₃	—	8	4	no	—
92	D₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	8	4	no	—
97	C₂×C₄×C₅₃	#100	1	2	yes	—
98	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#102	1	2	yes	—
100	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	#102	1	2	yes	—
108	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
109	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	#88322	—	8	2	no	—
111	#11924	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	8	2	no	—
115	C₂²×C₅₃	—	8	2	no	—
116	C₂²×C₅₃	—	8	2	no	—
117	C₂²×C₅₃	#331	1	2	yes	—
118	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	8	2	no	—
123	C₄×C₅₃	#331	1	2	yes	—
124	C₄×C₅₃	#359	1	2	yes	—
125	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
127	#3147	D₅₃	1	1	yes	characteristic
128	C₂×C₄⋊C₁₆	—	53	1	no	—
129	#1564	—	53	1	no	—
130	#1360	—	53	4	no	—
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	—	16	2	no	—
133	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
135	C₂×C₅₃	#2145	1	1	yes	characteristic
136	C₂×#135	—	2	1	no	—
137	C₂×#135	Dic₅₃	1	2	yes	—
138	#532	—	2	1	no	—
139	C₂²×C₁₆	—	106	1	no	—
140	#495	—	2	1	no	—
141	#495	C₂×D₅₃	1	2	yes	—
142	#494	Dic₅₃	1	1	yes	characteristic
143	#493	Dic₅₃	1	1	yes	characteristic
144	#492	—	2	1	no	—
145	#431	—	106	1	no	—
146	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
147	C₄⋊C₁₆	—	53	2	no	—
148	C₄⋊C₁₆	—	106	1	no	—
149	#335	—	106	1	no	—
150	#334	—	53	2	no	—
151	C₅₃	#11924	1	1	yes	characteristic, complemented, Sylow
152	C₂×#41	—	2	2	no	—
153	#135	—	4	4	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
162	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	1	no	—
165	#127	—	2	1	no	—
166	#124	—	2	2	no	—
167	#121	—	4	1	no	—
168	#121	—	4	1	no	—
169	C₄×D₄	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₁₆	—	106	2	no	—
173	C₂×C₁₆	—	212	2	no	—
174	C₄⋊C₈	—	106	2	no	—
175	C₂²⋊C₈	—	212	1	no	—
176	C₄×C₈	—	212	1	no	—
177	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	#41	—	4	4	no	—
182	#41	—	4	4	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	#11719922	1	1	yes	characteristic
190	C₂²×C₄	#11719923	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
192	Q₁₆	—	4	2	no	—
193	Q₁₆	—	4	2	no	—
194	SD₁₆	—	4	4	no	—
195	SD₁₆	—	4	2	no	—
196	SD₁₆	—	4	2	no	—
197	D₈	—	4	2	no	—
198	D₈	—	4	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	#11719922	1	2	yes	—
202	C₂×C₈	#11719923	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	#11719922	1	1	yes	characteristic
212	C₄²	#11719923	1	1	yes	characteristic
213	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₁₆	—	212	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#13133138	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	#13133134	1	2	yes	—
230	C₂×C₄	#13133138	1	1	yes	characteristic
231	C₂×C₄	#13133138	1	2	yes	—
232	C₂×C₄	#13133138	1	1	yes	characteristic
233	C₂×C₄	#13133138	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	424	2	no	—
240	C₈	#13133138	1	2	yes	—
241	C₈	C₂×#11719922	1	1	yes	characteristic
242	C₈	C₂×#11719923	1	1	yes	characteristic
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	#16810367	1	2	yes	—
249	C₂²	C₂×#13133138	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	#16810367	1	2	yes	—
255	C₄	#16810430	1	2	yes	—
256	C₄	C₂×#13133134	1	1	yes	characteristic
257	C₄	C₂×#13133138	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₂×#16810367	1	1	yes	characteristic
261	C₂	C₂×#16810430	1	1	yes	characteristic
262	C₂	#20276451	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3147	C₂
socle	C₂²×C₅₃	C₂×#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#3147	D₅₃
hypocenter	C₅₃	#11924
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#11719908	C₂	semidirect
2. #20273463	C₂	semidirect
3. #20272758	C₂	semidirect
151. C₅₃	#11924	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
#20273463	2
#20272758	2
C₅₃×#3147	2
#11924	53