Subgroups — #27292696

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	3	yes	maximal
2	C₂×#16810378	C₂	1	3	yes	maximal
3	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₂²	1	3	yes	—
5	C₂³×#88322	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊#88322	—	2	12	no	—
7	C₂×C₄⋊#88322	—	2	6	no	—
8	#16810378	—	2	6	no	—
9	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
10	C₂×#95×C₅₃	C₄	1	3	yes	—
11	C₂×#95×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×#88322	—	2	3	no	—
13	C₂²×#88322	—	2	3	no	—
14	C₂²×#88322	—	2	6	no	—
15	C₂²×#88322	—	2	6	no	—
16	C₂²×#88322	—	2	3	no	—
17	C₂²×#88322	—	2	3	no	—
18	C₂²×#88322	D₄	1	6	yes	—
19	C₂²×#88322	Q₈	1	2	yes	—
20	C₄⋊#88322	—	2	24	no	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	—	2	6	no	—
25	C₂×C₄²×C₅₃	—	2	6	no	—
26	C₂×C₄²×C₅₃	D₄	1	6	yes	—
27	C₂×C₄²×C₅₃	Q₈	1	6	yes	—
28	#95×C₅₃	—	2	6	no	—
29	#95×C₅₃	C₈	1	4	yes	complemented
30	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	—	2	6	no	—
32	C₂²×C₄×C₅₃	—	2	6	no	—
33	C₂²×C₄×C₅₃	—	2	6	no	—
34	C₂²×C₄×C₅₃	—	2	6	no	—
35	C₂²×C₄×C₅₃	—	2	3	no	—
36	C₂²×C₄×C₅₃	—	2	3	no	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
39	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
40	C₂²×C₄×C₅₃	M₁₆	1	6	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	3	yes	—
47	C₄²×C₅₃	—	2	6	no	—
48	C₄²×C₅₃	—	2	12	no	—
49	C₄²×C₅₃	—	2	6	no	—
50	C₂×#88322	—	2	12	no	—
51	C₂×#88322	—	2	6	no	—
52	C₂×#88322	—	2	6	no	—
53	C₂×#88322	—	4	3	no	—
54	C₂×#88322	—	4	6	no	—
55	C₂×#88322	—	4	3	no	—
56	C₂×#88322	—	4	3	no	—
57	C₂×#88322	—	4	1	no	—
58	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
59	C₂³×C₅₃	C₄×D₄	1	3	yes	—
60	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
63	C₂³×C₅₃	#131	1	3	yes	—
64	C₂³×C₅₃	#131	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	6	no	—
66	C₂×C₄×C₅₃	—	2	6	no	—
67	C₂×C₄×C₅₃	—	2	12	no	—
68	C₂×C₄×C₅₃	—	2	12	no	—
69	C₂×C₄×C₅₃	—	2	6	no	—
70	C₂×C₄×C₅₃	—	2	6	no	—
71	C₂×C₄×C₅₃	—	2	6	no	—
72	C₂×C₄×C₅₃	—	4	6	no	—
73	C₂×C₄×C₅₃	—	4	6	no	—
74	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	complemented
75	C₂×C₄×C₅₃	#120	1	3	yes	—
76	C₂×C₄×C₅₃	#121	1	3	yes	—
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	6	yes	—
78	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
79	C₂×C₄×C₅₃	#123	1	3	yes	—
80	C₂×C₄×C₅₃	#125	1	3	yes	—
81	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
83	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
84	#88322	—	4	6	no	—
85	#88322	—	4	2	no	—
86	#11322	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	#384	1	1	yes	characteristic
88	C₂²×C₅₃	#385	1	3	yes	—
89	C₂²×C₅₃	#389	1	3	yes	—
90	C₂²×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
91	C₂²×C₅₃	#423	1	3	yes	—
92	C₂²×C₅₃	#424	1	3	yes	—
93	C₂²×C₅₃	#424	1	3	yes	—
94	C₂²×C₅₃	C₈×D₄	1	3	yes	—
95	C₂²×C₅₃	#435	1	6	yes	—
96	C₂²×C₅₃	#436	1	3	yes	—
97	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	3	yes	—
99	C₄×C₅₃	—	2	12	no	—
100	C₄×C₅₃	—	4	6	no	—
101	C₄×C₅₃	#392	1	1	yes	characteristic
102	C₄×C₅₃	#395	1	3	yes	—
103	C₄×C₅₃	#398	1	3	yes	—
104	C₄×C₅₃	#399	1	1	yes	characteristic
105	C₂²×C₄⋊C₈	—	53	3	no	—
106	C₂×#336	—	53	3	no	—
107	C₄×#95	D₅₃	1	1	yes	characteristic
108	C₂×C₅₃	#1444	1	1	yes	characteristic
109	C₂×C₅₃	#1842	1	1	yes	characteristic
110	C₂×C₅₃	#1843	1	1	yes	characteristic
111	C₂×C₅₃	#1850	1	3	yes	—
112	C₂×C₅₃	#1851	1	3	yes	—
113	C₂×C₅₃	#1939	1	3	yes	—
114	C₂×C₅₃	#1940	1	3	yes	—
115	C₂³×C₈	—	53	3	no	—
116	C₂³×C₈	—	53	1	no	—
117	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
118	C₂×C₄⋊C₈	—	106	12	no	—
119	C₂×C₄⋊C₈	—	106	6	no	—
120	C₂×#95	Dic₅₃	1	3	yes	—
121	C₂×#95	Dic₅₃	1	1	yes	characteristic
122	#336	—	106	6	no	—
123	C₅₃	#11322	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
125	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
126	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	C₂²×C₈	—	53	6	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	106	3	no	—
130	C₂²×C₈	—	106	3	no	—
131	C₂²×C₈	—	106	6	no	—
132	C₂²×C₈	—	106	6	no	—
133	C₂²×C₈	—	106	3	no	—
134	C₂²×C₈	—	106	3	no	—
135	C₂×C₄²	—	2	6	no	—
136	C₂×C₄²	—	2	6	no	—
137	C₂×C₄²	Dic₁₀₆	1	6	yes	—
138	C₂×C₄²	D₂₁₂	1	6	yes	—
139	C₄⋊C₈	—	106	24	no	—
140	#95	—	2	6	no	—
141	#95	#88322	1	4	yes	complemented
142	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	6	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	3	no	—
148	C₂²×C₄	—	2	3	no	—
149	C₂²×C₄	C₂×#88322	1	6	yes	—
150	C₂²×C₄	#11719917	1	6	yes	—
151	C₂²×C₄	C₄⋊Dic₅₃	1	6	yes	—
152	C₂²×C₄	C₄⋊Dic₅₃	1	6	yes	—
153	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
154	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
155	C₂²×C₄	D₄×D₅₃	1	3	yes	—
156	C₂²×C₄	#11719947	1	3	yes	—
157	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	#11719949	1	1	yes	characteristic
159	C₂×C₈	—	106	12	no	—
160	C₂×C₈	—	106	6	no	—
161	C₂×C₈	—	106	6	no	—
162	C₂×C₈	—	212	3	no	—
163	C₂×C₈	—	212	6	no	—
164	C₂×C₈	—	212	3	no	—
165	C₂×C₈	—	212	3	no	—
166	C₂×C₈	—	212	1	no	—
167	C₄²	—	2	6	no	—
168	C₄²	—	2	12	no	—
169	C₄²	—	2	6	no	—
170	C₂³	C₂×C₄⋊Dic₅₃	1	3	yes	—
171	C₂³	D₄×Dic₅₃	1	3	yes	—
172	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
173	C₂³	C₂²×#88322	1	1	yes	characteristic
174	C₂³	C₂×#11719917	1	3	yes	—
175	C₂³	#13133224	1	3	yes	—
176	C₂³	#13133224	1	1	yes	characteristic
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	4	6	no	—
185	C₂×C₄	—	4	6	no	—
186	C₂×C₄	C₄⋊#88322	1	12	yes	complemented
187	C₂×C₄	#13133187	1	3	yes	—
188	C₂×C₄	#13133192	1	3	yes	—
189	C₂×C₄	#13133194	1	6	yes	—
190	C₂×C₄	#13133197	1	3	yes	—
191	C₂×C₄	#13133201	1	3	yes	—
192	C₂×C₄	#13133208	1	3	yes	—
193	C₂×C₄	#13133214	1	1	yes	characteristic
194	C₂×C₄	#13133216	1	3	yes	—
195	C₂×C₄	#13133217	1	3	yes	—
196	C₈	—	212	6	no	—
197	C₈	—	212	2	no	—
198	C₂²	#16810493	1	1	yes	characteristic
199	C₂²	#16810498	1	3	yes	—
200	C₂²	#16810522	1	3	yes	—
201	C₂²	C₂×C₄⋊#88322	1	3	yes	—
202	C₂²	#16810647	1	3	yes	—
203	C₂²	#16810650	1	3	yes	—
204	C₂²	#16810650	1	3	yes	—
205	C₂²	D₄×#88322	1	3	yes	—
206	C₂²	#16810695	1	6	yes	—
207	C₂²	#16810701	1	3	yes	—
208	C₂²	Q₈×#88322	1	1	yes	characteristic
209	C₂²	#16810754	1	3	yes	—
210	C₄	—	2	12	no	—
211	C₄	—	4	6	no	—
212	C₄	#16810533	1	1	yes	characteristic
213	C₄	#16810544	1	3	yes	—
214	C₄	#16810555	1	3	yes	—
215	C₄	#16810560	1	1	yes	characteristic
216	C₂	#20272980	1	1	yes	characteristic
217	C₂	#20274867	1	1	yes	characteristic
218	C₂	#20274869	1	1	yes	characteristic
219	C₂	#20274895	1	3	yes	—
220	C₂	#20274900	1	3	yes	—
221	C₂	#20275434	1	3	yes	—
222	C₂	#20275439	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Fitting	C₄×#95×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₄×#95	D₅₃
hypocenter	C₅₃	#11322
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
29. #95×C₅₃	C₈	semidirect
74. C₂×C₄×C₅₃	C₄⋊C₈	semidirect
123. C₅₃	#11322	semidirect
141. #95	#88322	semidirect
186. 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
C₂²×C₄⋊#88322	2
C₂×#16810378	2
C₄×#95×C₅₃	2
#11322	53