Subgroups — #28298027

Order 13568.

338 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	C₄×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
2	#20290429	C₂	1	1	yes	maximal, characteristic
3	#20274570	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88324	—	2	1	no	—
5	C₂×C₄×#88324	—	2	1	no	—
6	C₂×C₄×#88324	—	2	1	no	—
7	#16811489	—	2	1	no	—
8	#16811489	C₄	1	2	yes	—
9	#16811489	C₄	1	2	yes	—
10	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
11	C₂×#13133127	—	2	1	no	—
12	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810377	—	2	1	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×#88324	—	2	2	no	—
16	C₂²×#88324	—	2	2	no	—
17	C₄⋊#88324	—	2	4	no	—
18	C₄×#88324	—	2	2	no	—
19	C₄×#88324	—	2	2	no	—
20	C₄×#88324	—	4	1	no	—
21	C₄×#88324	—	4	1	no	—
22	C₂²×#88322	—	2	1	no	—
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
29	C₄⋊#88322	—	2	1	no	—
30	C₄⋊#88322	C₈	1	2	yes	complemented
31	#13133127	—	4	1	no	—
32	#13133127	—	4	1	no	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	1	no	—
36	C₄⋊C₈×C₅₃	C₈	1	2	yes	complemented
37	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#88324	—	2	4	no	—
45	C₂×#88324	—	2	4	no	—
46	C₂×#88324	—	4	2	no	—
47	C₂×#88324	—	4	2	no	—
48	C₂×#88324	—	4	2	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	—	4	1	no	—
53	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
56	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
57	C₄×Dic₅₃	—	2	1	no	—
58	C₄×Dic₅₃	—	2	1	no	—
59	C₄×Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	—	4	1	no	—
62	C₄×Dic₅₃	—	4	1	no	—
63	C₄×Dic₅₃	D₈	1	1	yes	characteristic
64	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
65	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
66	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
67	C₂×#88322	—	2	2	no	—
68	C₂×#88322	—	4	1	no	—
69	C₂×#88322	—	4	1	no	—
70	C₂³×C₅₃	#95	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
85	C₂×Dic₅₃	—	2	1	no	—
86	C₂×Dic₅₃	—	2	1	no	—
87	C₂×Dic₅₃	—	2	1	no	—
88	C₂×Dic₅₃	—	2	1	no	—
89	C₂×Dic₅₃	—	2	1	no	—
90	C₂×Dic₅₃	—	2	1	no	—
91	C₂×Dic₅₃	—	2	1	no	—
92	C₂×Dic₅₃	—	2	1	no	—
93	C₂×Dic₅₃	#102	1	1	yes	characteristic
94	C₂×Dic₅₃	#103	1	1	yes	characteristic
95	C₂×Dic₅₃	#104	1	2	yes	—
96	C₂×Dic₅₃	#106	1	1	yes	characteristic
97	C₂×Dic₅₃	#107	1	1	yes	characteristic
98	C₂×Dic₅₃	#108	1	2	yes	—
99	#88324	—	4	4	no	—
100	#88324	—	4	4	no	—
101	C₈×C₅₃	—	4	2	no	—
102	#88322	—	4	2	no	—
103	#7027	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	#336	1	1	yes	characteristic
105	C₂²×C₅₃	#339	1	1	yes	characteristic
106	C₂²×C₅₃	#339	1	1	yes	characteristic
107	C₂²×C₅₃	#340	1	1	yes	characteristic
108	C₂²×C₅₃	#341	1	1	yes	characteristic
109	C₂²×C₅₃	#344	1	1	yes	characteristic
110	C₂²×C₅₃	#344	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
116	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
117	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
118	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
119	Dic₅₃	—	2	2	no	—
120	Dic₅₃	—	2	1	no	—
121	Dic₅₃	—	2	1	no	—
122	Dic₅₃	—	2	2	no	—
123	Dic₅₃	—	2	1	no	—
124	Dic₅₃	—	2	1	no	—
125	#1767	—	53	1	no	—
126	C₄×C₄⋊C₈	—	53	1	no	—
127	#1744	—	53	1	no	—
128	C₂×C₅₃	#1278	1	1	yes	characteristic
129	C₂×C₅₃	#1279	1	1	yes	characteristic
130	C₂×C₅₃	#1281	1	1	yes	characteristic
131	C₂×C₅₃	#1300	1	1	yes	characteristic
132	C₂×C₅₃	#1301	1	1	yes	characteristic
133	C₂×C₅₃	#1302	1	1	yes	characteristic
134	C₂×C₅₃	#1303	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	53	1	no	—
136	C₂×C₄⋊C₈	—	106	1	no	—
137	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
138	C₂×#97	—	106	1	no	—
139	C₂×C₄×C₈	—	106	1	no	—
140	C₂×C₄×C₈	—	106	1	no	—
141	C₄³	—	53	1	no	—
142	#336	—	53	2	no	—
143	#336	—	53	2	no	—
144	#336	—	106	1	no	—
145	#336	—	106	1	no	—
146	C₅₃	#7027	1	1	yes	characteristic, complemented, Sylow
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	106	2	no	—
149	C₂²×C₈	—	106	2	no	—
150	C₂²×C₈	—	106	1	no	—
151	C₂×C₄²	—	53	1	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	—	53	1	no	—
154	C₂×C₄²	—	53	1	no	—
155	C₂×C₄²	—	53	1	no	—
156	C₂×C₄²	—	53	1	no	—
157	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	53	2	no	—
160	C₄⋊C₈	—	106	4	no	—
161	C₄⋊C₈	—	106	1	no	—
162	C₄⋊C₈	#88324	1	2	yes	complemented
163	#97	—	212	1	no	—
164	#97	—	212	1	no	—
165	C₄×C₈	—	106	2	no	—
166	C₄×C₈	—	106	2	no	—
167	C₄×C₈	—	212	1	no	—
168	C₄×C₈	—	212	1	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	1	no	—
171	C₂²×C₄	—	53	1	no	—
172	C₂²×C₄	—	53	1	no	—
173	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	106	4	no	—
180	C₂×C₈	—	106	4	no	—
181	C₂×C₈	—	106	2	no	—
182	C₂×C₈	—	212	2	no	—
183	C₂×C₈	—	212	2	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	1	no	—
187	C₂×C₈	—	212	1	no	—
188	C₄²	—	53	1	no	—
189	C₄²	—	53	1	no	—
190	C₄²	—	53	1	no	—
191	C₄²	—	53	1	no	—
192	C₄²	—	106	2	no	—
193	C₄²	—	106	2	no	—
194	C₄²	—	106	1	no	—
195	C₄²	—	106	1	no	—
196	C₄²	—	212	1	no	—
197	C₄²	—	212	1	no	—
198	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
199	C₄²	#11719936	1	1	yes	characteristic
200	C₄²	C₂×#88324	1	1	yes	characteristic
201	C₄²	#11719940	1	1	yes	characteristic
202	C₂³	(C₂×#6157)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	1	no	—
211	C₂×C₄	—	53	1	no	—
212	C₂×C₄	—	53	2	no	—
213	C₂×C₄	—	106	1	no	—
214	C₂×C₄	—	106	1	no	—
215	C₂×C₄	—	106	1	no	—
216	C₂×C₄	—	106	1	no	—
217	C₂×C₄	—	106	1	no	—
218	C₂×C₄	—	106	1	no	—
219	C₂×C₄	—	106	1	no	—
220	C₂×C₄	—	106	1	no	—
221	C₂×C₄	C₄×#88324	1	1	yes	characteristic
222	C₂×C₄	#13133272	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
224	C₂×C₄	#13133274	1	1	yes	characteristic
225	C₂×C₄	#13133279	1	1	yes	characteristic
226	C₂×C₄	#13133280	1	1	yes	characteristic
227	C₂×C₄	#13133281	1	1	yes	characteristic
228	C₂×C₄	#13133282	1	1	yes	characteristic
229	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
230	C₂×C₄	#13133284	1	1	yes	characteristic
231	C₈	—	4	2	no	—
232	C₈	—	212	4	no	—
233	C₈	—	212	4	no	—
234	C₈	—	212	2	no	—
235	C₂²	#16811489	1	1	yes	characteristic
236	C₂²	#16811493	1	1	yes	characteristic
237	C₂²	#16811495	1	1	yes	characteristic
238	C₂²	#16811496	1	1	yes	characteristic
239	C₂²	#16811498	1	1	yes	characteristic
240	C₂²	#16811505	1	1	yes	characteristic
241	C₂²	#16811505	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	106	2	no	—
247	C₄	—	106	1	no	—
248	C₄	—	106	1	no	—
249	C₄	—	106	2	no	—
250	C₄	—	106	1	no	—
251	C₄	—	106	1	no	—
252	C₄	D₄⋊#88324	1	1	yes	characteristic
253	C₄	#16811471	1	1	yes	characteristic
254	C₄	Q₈⋊#88324	1	1	yes	characteristic
255	C₄	#16811473	1	1	yes	characteristic
256	C₂	#20289548	1	1	yes	characteristic
257	C₂	#20289551	1	1	yes	characteristic
258	C₂	#20289555	1	1	yes	characteristic
259	C₂	#20289602	1	1	yes	characteristic
260	C₂	#20289605	1	1	yes	characteristic
261	C₂	#20289608	1	1	yes	characteristic
262	C₂	#20289611	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄²×Dic₅₃	C₂²
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	#7027
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
30. C₄⋊#88322	C₈	semidirect
36. C₄⋊C₈×C₅₃	C₈	semidirect
146. C₅₃	#7027	semidirect
162. C₄⋊C₈	#88324	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88324	2
#20290429	2
#20274570	2
#7027	53