Subgroups — #27300554

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
4	#20272793	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	—	2	4	no	—
7	C₂²×#11719908	—	2	2	no	—
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₄⋊#11719908	—	2	4	no	—
12	C₂²⋊#11719908	—	2	2	no	—
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
17	C₅₃×#336	C₄	1	2	yes	—
18	C₂×#11719908	—	2	8	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	—	2	1	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	D₄	1	4	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²⋊C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	#95×C₅₃	C₈	1	2	yes	—
41	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₅₃	#41	1	2	yes	—
62	C₂²⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	C₁₆	1	4	yes	complemented
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	#11719908	—	2	4	no	—
67	#11719908	—	4	1	no	—
68	#11719908	—	4	1	no	—
69	C₂³×C₅₃	—	2	2	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
72	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
73	C₂³×C₅₃	M₃₂	1	2	yes	—
74	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	complemented
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#131	1	2	yes	—
90	C₈×C₅₃	—	2	2	no	—
91	C₈×C₅₃	—	4	2	no	—
92	C₈×C₅₃	#120	1	1	yes	characteristic
93	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₅₃	#123	1	1	yes	characteristic
95	#12912	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	C₄⋊C₁₆	1	4	yes	complemented
101	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₅₃	C₈×D₄	1	2	yes	—
103	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
105	C₂²×C₅₃	#504	1	2	yes	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	4	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	#385	1	1	yes	characteristic
110	C₄×C₅₃	#424	1	1	yes	characteristic
111	C₄×C₅₃	#435	1	2	yes	—
112	C₂³×C₁₆	—	53	1	no	—
113	C₂×C₄⋊C₁₆	—	53	2	no	—
114	C₂×C₂²⋊C₁₆	—	53	1	no	—
115	C₈×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
116	#1374	—	53	2	no	—
117	C₂×C₅₃	—	2	4	no	—
118	C₂×C₅₃	#1850	1	1	yes	characteristic
119	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
120	C₂×C₅₃	#2151	1	1	yes	characteristic
121	C₂×C₅₃	D₄×C₁₆	1	2	yes	—
122	C₂×C₅₃	#2168	1	2	yes	—
123	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
124	C₂²×C₁₆	—	53	2	no	—
125	C₂²×C₁₆	—	53	1	no	—
126	C₂²×C₁₆	—	53	1	no	—
127	C₂²×C₁₆	—	106	4	no	—
128	C₂²×C₁₆	—	106	2	no	—
129	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
131	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	106	4	no	—
133	C₂²⋊C₁₆	—	106	2	no	—
134	#336	Dic₅₃	1	2	yes	—
135	C₅₃	#12912	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	Dic₁₀₆	1	2	yes	—
140	C₂²×C₈	D₂₁₂	1	2	yes	—
141	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
142	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
147	C₂×C₁₆	—	53	4	no	—
148	C₂×C₁₆	—	106	8	no	—
149	C₂×C₁₆	—	106	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	C₂×C₁₆	—	106	1	no	—
152	C₂×C₁₆	—	106	1	no	—
153	C₂×C₁₆	—	106	2	no	—
154	C₂×C₁₆	—	106	2	no	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	#95	#88322	1	2	yes	—
159	C₂⁴	C₂×#88322	1	1	yes	characteristic
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂×#88322	1	2	yes	—
163	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
164	C₂²×C₄	C₂×#88322	1	2	yes	—
165	C₂²×C₄	#11719917	1	2	yes	—
166	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
167	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
176	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
177	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
178	C₂×C₈	D₄×D₅₃	1	2	yes	—
179	C₂×C₈	#11719947	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	#11719908	1	4	yes	complemented
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₁₆	—	106	4	no	—
185	C₁₆	—	212	1	no	—
186	C₁₆	—	212	1	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	C₄⋊#88322	1	2	yes	—
190	C₂³	C₂×#11719908	1	2	yes	—
191	C₂³	#13133157	1	2	yes	—
192	C₂³	C₂²×#88322	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	C₄⋊#88322	1	2	yes	—
203	C₂×C₄	C₂×#11719908	1	4	yes	complemented
204	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
206	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
207	C₂×C₄	#13133224	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	#13133187	1	1	yes	characteristic
211	C₈	#13133194	1	2	yes	—
212	C₈	#13133201	1	1	yes	characteristic
213	C₂²	—	2	4	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	C₄⋊#11719908	1	4	yes	complemented
218	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
219	C₂²	D₄×#88322	1	2	yes	—
220	C₂²	C₂²×#11719908	1	1	yes	characteristic
221	C₂²	C₂×#13133157	1	1	yes	characteristic
222	C₂²	#16811040	1	2	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	#16810498	1	1	yes	characteristic
227	C₄	#16810650	1	1	yes	characteristic
228	C₄	#16810695	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	#20274895	1	1	yes	characteristic
231	C₂	C₂×C₄⋊#11719908	1	1	yes	characteristic
232	C₂	#20276474	1	1	yes	characteristic
233	C₂	D₄×#11719908	1	2	yes	—
234	C₂	#20276541	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₈×C₂²⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₈×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#12912
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#11719908	C₂	semidirect
4. #20272793	C₂	semidirect
63. C₂²⋊C₄×C₅₃	C₁₆	semidirect
85. C₂×C₄×C₅₃	C₂×C₁₆	semidirect
100. C₂²×C₅₃	C₄⋊C₁₆	semidirect
135. C₅₃	#12912	semidirect
181. C₂²⋊C₄	#11719908	semidirect
203. C₂×C₄	C₂×#11719908	semidirect
217. C₂²	C₄⋊#11719908	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂×C₄⋊#11719908	2
C₂×C₂²⋊#11719908	2
#20272793	2
C₈×C₂²⋊C₄×C₅₃	2
#12912	53