Subgroups — #27282865

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	—	2	2	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	Q₈⋊#88322	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#391	C₄	1	2	yes	—
13	C₅₃×#389	—	2	2	no	—
14	C₅₃×#384	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄⋊#88322	—	4	4	no	—
21	C₄×#88322	—	2	8	no	—
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	D₄	1	4	yes	—
24	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
27	C₄×Q₈×C₅₃	—	2	4	no	—
28	C₄×Q₈×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#95×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	—	2	2	no	—
41	C₂×Q₈×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	—	2	4	no	—
43	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	—	2	8	no	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	4	4	no	—
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄²×C₅₃	—	2	4	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	4	2	no	—
59	C₄²×C₅₃	C₂×C₈	1	2	yes	—
60	C₄²×C₅₃	M₁₆	1	2	yes	—
61	C₄²×C₅₃	SD₁₆	1	4	yes	—
62	C₄²×C₅₃	Q₁₆	1	4	yes	—
63	C₄²×C₅₃	C₂×D₄	1	2	yes	—
64	C₄²×C₅₃	#41	1	2	yes	—
65	C₂×#88322	—	2	8	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	4	no	—
69	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	Q₈×C₅₃	—	2	8	no	—
71	Q₈×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
81	C₂×C₄×C₅₃	#103	1	4	yes	—
82	C₂×C₄×C₅₃	#103	1	4	yes	—
83	C₂×C₄×C₅₃	#104	1	4	yes	—
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	#121	1	2	yes	—
86	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#131	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
93	#88322	—	2	8	no	—
94	#88322	—	8	2	no	—
95	#9637	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#390	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
98	C₂²×C₅₃	#408	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×#103	1	2	yes	—
100	C₂²×C₅₃	C₂×#104	1	2	yes	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	4	no	—
103	C₄×C₅₃	—	4	4	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	Q₈⋊C₈	1	8	yes	complemented
106	C₄×C₅₃	C₈×D₄	1	2	yes	—
107	C₄×C₅₃	#436	1	2	yes	—
108	C₄×C₅₃	#487	1	2	yes	—
109	C₄×C₅₃	#492	1	1	yes	characteristic
110	C₄×C₅₃	#494	1	1	yes	characteristic
111	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
112	#1842	—	53	1	no	—
113	C₄²×C₈	—	53	1	no	—
114	C₂×Q₈⋊C₈	—	53	4	no	—
115	C₂×C₅₃	C₂×Q₈⋊C₈	1	2	yes	—
116	C₂×C₅₃	#1493	1	2	yes	—
117	C₂×C₅₃	#1954	1	1	yes	characteristic
118	C₂×C₅₃	#1956	1	1	yes	characteristic
119	C₂×C₅₃	#1962	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
122	C₂×C₄⋊C₈	—	53	2	no	—
123	C₂×C₄⋊C₈	—	106	2	no	—
124	C₂×C₄×C₈	—	53	2	no	—
125	C₂×C₄×C₈	—	106	2	no	—
126	#391	Dic₅₃	1	2	yes	—
127	#389	—	2	2	no	—
128	#384	Dic₅₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	2	2	no	—
130	C₄³	C₂×D₅₃	1	1	yes	characteristic
131	Q₈⋊C₈	—	106	8	no	—
132	C₅₃	#9637	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
134	C₂²×C₈	—	106	2	no	—
135	C₂²×C₈	—	106	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	#88322	1	4	yes	complemented
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	2	4	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
148	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
149	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
150	C₄⋊C₈	—	106	4	no	—
151	C₄⋊C₈	—	212	4	no	—
152	C₄×C₈	—	53	4	no	—
153	C₄×C₈	—	106	8	no	—
154	C₄×C₈	—	106	2	no	—
155	#95	—	2	4	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	C₂×#88322	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
163	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂×C₈	—	106	8	no	—
168	C₂×C₈	—	106	4	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	—	212	4	no	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₂×#88322	1	2	yes	—
180	C₄²	#11719917	1	2	yes	—
181	C₄²	#11719924	1	4	yes	—
182	C₄²	#11719925	1	4	yes	—
183	C₄²	D₄×D₅₃	1	2	yes	—
184	C₄²	#11719947	1	2	yes	—
185	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	Q₈	—	4	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂²⋊#88322	1	4	yes	—
197	C₂×C₄	#13133141	1	4	yes	—
198	C₂×C₄	#13133141	1	4	yes	—
199	C₂×C₄	#13133144	1	4	yes	—
200	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
201	C₂×C₄	#13133188	1	2	yes	—
202	C₂×C₄	#13133204	1	1	yes	characteristic
203	C₂×C₄	#13133213	1	1	yes	characteristic
204	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
206	C₂×C₄	#13133224	1	2	yes	—
207	C₂×C₄	C₂×#11719924	1	2	yes	—
208	C₂×C₄	C₂×#11719925	1	2	yes	—
209	C₈	—	106	8	no	—
210	C₈	—	424	2	no	—
211	C₂²	#16810526	1	1	yes	characteristic
212	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
213	C₂²	#16810590	1	1	yes	characteristic
214	C₂²	C₂×#13133141	1	2	yes	—
215	C₂²	C₂×#13133144	1	2	yes	—
216	C₄	—	2	4	no	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	2	no	—
220	C₄	Q₈⋊#88322	1	8	yes	complemented
221	C₄	D₄×#88322	1	2	yes	—
222	C₄	#16810701	1	2	yes	—
223	C₄	#16810973	1	2	yes	—
224	C₄	#16810997	1	1	yes	characteristic
225	C₄	#16811002	1	1	yes	characteristic
226	C₂	C₂×Q₈⋊#88322	1	2	yes	—
227	C₂	#20273170	1	2	yes	—
228	C₂	#20275510	1	1	yes	characteristic
229	C₂	#20275518	1	1	yes	characteristic
230	C₂	#20275544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#9637
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. C₄⋊Q₈×C₅₃	C₈	semidirect
105. C₄×C₅₃	Q₈⋊C₈	semidirect
132. C₅₃	#9637	semidirect
137. C₄⋊Q₈	#88322	semidirect
220. C₄	Q₈⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄²×#88322	2
C₂×Q₈⋊#88322	2
C₄×C₄⋊Q₈×C₅₃	2
#9637	53