Subgroups — #117×#88322 #27382297

Order 13568.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×#88322	C₂	1	4	yes	maximal, complemented
4	C₂²⋊C₄×#88322	C₂	1	4	yes	maximal
5	#20274461	C₂	1	2	yes	maximal, complemented
6	C₄²×#88322	C₂	1	2	yes	maximal, complemented
7	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×#88322	C₂²	1	2	yes	—
9	#16810676	C₄	1	8	yes	complemented
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₄	1	8	yes	complemented
13	C₂×C₄×#88322	C₂²	1	8	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	2	yes	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	#16810378	C₂²	1	8	yes	—
19	C₂×#117×C₅₃	C₄	1	2	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#376	C₄	1	2	yes	—
26	C₄³×C₅₃	C₂²	1	2	yes	—
27	C₂²×#88322	—	2	8	no	—
28	C₂²×#88322	C₂×C₄	1	8	yes	—
29	C₂²×#88322	C₂³	1	4	yes	—
30	C₂²×#88322	C₂³	1	2	yes	—
31	C₂²×#88322	C₂³	1	2	yes	—
32	C₄⋊#88322	C₂×C₄	1	8	yes	—
33	C₂²⋊#88322	C₂×C₄	1	8	yes	—
34	C₄×#88322	—	2	16	no	—
35	C₄×#88322	C₂×C₄	1	4	yes	—
36	C₄×#88322	C₂×C₄	1	8	yes	—
37	C₄×#88322	C₂×C₄	1	4	yes	—
38	C₄×#88322	C₂×C₄	1	8	yes	—
39	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	#117×C₅₃	C₈	1	16	yes	complemented
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#95×C₅₃	C₂×C₄	1	4	yes	—
55	#95×C₅₃	C₂×C₄	1	4	yes	—
56	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
60	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₅₃	C₂×C₈	1	16	yes	complemented
72	C₂²⋊C₄×C₅₃	C₂×C₈	1	16	yes	—
73	C₄²×C₅₃	—	2	8	no	—
74	C₄²×C₅₃	—	2	8	no	—
75	C₄²×C₅₃	C₄²	1	4	yes	—
76	C₄²×C₅₃	C₄²	1	4	yes	—
77	C₄²×C₅₃	C₂×C₈	1	16	yes	complemented
78	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×#88322	—	2	8	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	16	no	—
85	C₂×#88322	C₂²×C₄	1	4	yes	—
86	C₂×#88322	C₂²×C₄	1	8	yes	—
87	C₂×#88322	C₂²×C₄	1	4	yes	—
88	C₂×#88322	#41	1	8	yes	—
89	C₂³×C₅₃	—	2	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
92	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
93	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	8	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	2	8	no	—
101	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
102	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	16	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₅₃	#131	1	4	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
114	#88322	—	2	4	no	—
115	#88322	—	2	8	no	—
116	#88322	#117	1	8	yes	direct product factor
117	C₈×#117	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	8	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
122	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
123	C₂²×C₅₃	C₄×#41	1	2	yes	—
124	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
125	C₂²×C₅₃	C₂×#131	1	2	yes	—
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	16	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
131	C₄×C₅₃	#405	1	4	yes	—
132	C₄×C₅₃	C₂×#117	1	2	yes	—
133	C₄×C₅₃	C₄×#41	1	2	yes	—
134	C₂×#431	—	53	1	no	—
135	C₂²×C₄×C₈	—	53	1	no	—
136	C₄×#117	D₅₃	1	1	yes	characteristic
137	C₈×C₄⋊C₄	—	53	4	no	—
138	C₈×C₂²⋊C₄	—	53	4	no	—
139	#1744	—	53	2	no	—
140	C₄²×C₈	—	53	2	no	—
141	C₂×C₅₃	—	2	4	no	—
142	C₂×C₅₃	C₄×#117	1	1	yes	characteristic
143	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
144	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
145	C₂×C₅₃	C₈×#41	1	4	yes	—
146	C₂³×C₈	—	53	2	no	—
147	C₂×#117	Dic₅₃	1	2	yes	—
148	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
149	#431	—	53	8	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₂²⋊C₈	—	53	2	no	—
152	C₂×C₄×C₈	—	53	8	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	53	2	no	—
157	C₂×C₄×C₈	—	53	8	no	—
158	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
159	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
160	C₄×C₂²⋊C₄	Dic₅₃	1	2	yes	—
161	C₄×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
162	#376	Dic₅₃	1	2	yes	—
163	C₄³	C₂×D₅₃	1	2	yes	—
164	#336	—	53	8	no	—
165	C₅₃	C₈×#117	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	8	no	—
171	C₂²×C₈	—	53	2	no	—
172	C₂²×C₈	—	53	4	no	—
173	C₂²×C₈	—	106	8	no	—
174	#117	#88322	1	16	yes	direct product factor
175	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
176	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
177	C₂×C₄²	C₄×D₅₃	1	4	yes	—
178	C₂×C₄²	C₄×D₅₃	1	4	yes	—
179	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
180	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
181	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
184	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	C₄⋊C₈	—	53	8	no	—
187	C₂²⋊C₈	—	53	8	no	—
188	C₄×C₈	—	53	8	no	—
189	C₄×C₈	—	53	4	no	—
190	C₄×C₈	—	53	4	no	—
191	C₄×C₈	—	53	8	no	—
192	C₄×C₈	—	106	16	no	—
193	#95	C₂×Dic₅₃	1	4	yes	—
194	#95	C₂×Dic₅₃	1	4	yes	—
195	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂×#88322	1	8	yes	—
199	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
200	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
201	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
206	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
209	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	—	53	8	no	—
212	C₂×C₈	—	53	4	no	—
213	C₂×C₈	—	53	8	no	—
214	C₂×C₈	—	106	16	no	—
215	C₂×C₈	—	106	8	no	—
216	C₂×C₈	—	106	4	no	—
217	C₄⋊C₄	C₂×#88322	1	16	yes	complemented
218	C₂²⋊C₄	C₂×#88322	1	16	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	C₂×#88322	1	16	yes	complemented
222	C₄²	C₄×Dic₅₃	1	4	yes	—
223	C₄²	C₄×Dic₅₃	1	4	yes	—
224	C₄²	C₂×C₄×D₅₃	1	2	yes	—
225	C₄²	C₂×C₄×D₅₃	1	2	yes	—
226	C₄²	C₂×C₄×D₅₃	1	2	yes	—
227	C₄²	C₂×C₄×D₅₃	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
231	C₂³	C₂²×#88322	1	4	yes	—
232	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₄×#88322	1	16	yes	complemented
241	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
242	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
246	C₂×C₄	C₂²×#88322	1	16	yes	—
247	C₂×C₄	C₂²×#88322	1	4	yes	—
248	C₂×C₄	C₂²×#88322	1	4	yes	—
249	C₂×C₄	#13133224	1	4	yes	—
250	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₂×C₄	#41×D₅₃	1	4	yes	—
253	C₈	—	53	8	no	—
254	C₈	—	106	4	no	—
255	C₈	—	106	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×C₄×#88322	1	4	yes	—
260	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
261	C₂²	#41×Dic₅₃	1	2	yes	—
262	C₂²	C₂³×#88322	1	2	yes	—
263	C₂²	C₂×#13133224	1	2	yes	—
264	C₄	—	2	16	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	C₂×C₄×#88322	1	4	yes	—
268	C₄	C₂×C₄×#88322	1	4	yes	—
269	C₄	#16810579	1	4	yes	—
270	C₄	#117×D₅₃	1	2	yes	—
271	C₄	#16811091	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	#117×Dic₅₃	1	1	yes	characteristic
274	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
275	C₂	C₂×#16810579	1	1	yes	characteristic
276	C₂	#41×#88322	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	C₈×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×#88322	C₂	semidirect
5. #20274461	C₂	semidirect
6. C₄²×#88322	C₂	semidirect
9. #16810676	C₄	semidirect
12. C₂×C₄×#88322	C₄	semidirect
42. #117×C₅₃	C₈	semidirect
71. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
77. C₄²×C₅₃	C₂×C₈	semidirect
101. C₂×C₄×C₅₃	C₄×C₈	semidirect
116. #88322	#117	direct (N × H)
165. C₅₃	C₈×#117	semidirect
174. #117	#88322	direct (N × H)
217. C₄⋊C₄	C₂×#88322	semidirect
221. C₄²	C₂×#88322	semidirect
240. C₂×C₄	C₄×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄×#88322	2
C₄⋊C₄×#88322	2
C₂²⋊C₄×#88322	2
#20274461	2
C₄²×#88322	2
C₄×#117×C₅₃	2
C₈×#117	53