Subgroups — C₄×#16810372 #27275036

Order 13568.

540 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88322	C₂	1	2	yes	maximal
2	C₂×#16810372	C₂	1	2	yes	maximal
3	#20272598	C₂	1	2	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	C₂×C₄×#88322	—	2	2	no	—
7	#16810378	—	2	2	no	—
8	#16810372	C₄	1	8	yes	direct product factor
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
11	C₅₃×#391	C₄	1	2	yes	—
12	C₅₃×#389	—	2	2	no	—
13	C₅₃×#384	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	4	no	—
17	C₄⋊#88322	—	2	4	no	—
18	C₄⋊#88322	C₂×C₄	1	8	yes	—
19	C₄×#88322	—	4	4	no	—
20	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄⋊Q₈×C₅₃	—	2	2	no	—
22	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₄×Q₈×C₅₃	—	4	4	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#95×C₅₃	—	2	4	no	—
35	C₂×Q₈×C₅₃	—	2	2	no	—
36	C₂×Q₈×C₅₃	—	4	2	no	—
37	C₂×Q₈×C₅₃	—	4	2	no	—
38	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₄⋊C₄×C₅₃	—	2	8	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	—	4	1	no	—
53	C₄²×C₅₃	—	4	4	no	—
54	C₄²×C₅₃	SD₁₆	1	4	yes	—
55	C₄²×C₅₃	Q₁₆	1	4	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₂×#88322	—	2	8	no	—
59	C₂×#88322	—	4	4	no	—
60	C₂×#88322	—	4	4	no	—
61	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₅₃	—	4	4	no	—
63	Q₈×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	#100	1	2	yes	—
76	C₂×C₄×C₅₃	#103	1	4	yes	—
77	C₂×C₄×C₅₃	#103	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₅₃	#135	1	4	yes	—
84	#88322	—	4	8	no	—
85	C₄×#333	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
88	C₂²×C₅₃	#413	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×#103	1	2	yes	—
90	C₂²×C₅₃	#416	1	2	yes	—
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	4	1	no	—
95	C₄×C₅₃	—	4	4	no	—
96	C₄×C₅₃	—	8	2	no	—
97	C₄×C₅₃	#333	1	4	yes	complemented
98	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
99	C₄×C₅₃	C₄×Q₁₆	1	4	yes	—
100	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
101	C₄×C₄⋊C₈	—	53	2	no	—
102	C₂×#333	—	53	2	no	—
103	#1296	—	53	2	no	—
104	C₂×C₅₃	C₂×#333	1	1	yes	characteristic
105	C₂×C₅₃	#1542	1	2	yes	—
106	C₂×C₅₃	#1543	1	1	yes	characteristic
107	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
108	C₂×C₅₃	C₄×#103	1	2	yes	—
109	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
110	C₂×C₄×Q₈	Dic₅₃	1	2	yes	—
111	C₂×C₄⋊C₈	—	53	4	no	—
112	C₂×C₄×C₈	—	106	2	no	—
113	#391	Dic₅₃	1	2	yes	—
114	#389	—	2	2	no	—
115	#384	C₂×D₅₃	1	1	yes	characteristic
116	C₄×C₄⋊C₄	—	2	2	no	—
117	C₄³	C₂×D₅₃	1	1	yes	characteristic
118	#336	—	106	2	no	—
119	#333	—	53	8	no	—
120	C₅₃	C₄×#333	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
122	C₂²×C₈	—	106	4	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	C₄×D₅₃	1	4	yes	—
125	C₄×Q₈	—	4	4	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
132	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
134	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
135	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊C₈	—	53	8	no	—
137	C₄⋊C₈	—	106	4	no	—
138	C₄×C₈	—	212	4	no	—
139	#95	—	2	4	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
148	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
151	C₂×C₈	—	106	8	no	—
152	C₂×C₈	—	212	4	no	—
153	C₂×C₈	—	212	4	no	—
154	C₄⋊C₄	—	2	8	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	4	no	—
162	C₄²	#11719924	1	4	yes	—
163	C₄²	#11719925	1	4	yes	—
164	C₄²	C₂×C₄×D₅₃	1	2	yes	—
165	C₄²	#11719945	1	2	yes	—
166	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
167	Q₈	—	4	4	no	—
168	Q₈	—	4	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	#13133134	1	2	yes	—
181	C₂×C₄	#13133141	1	4	yes	—
182	C₂×C₄	#13133141	1	4	yes	—
183	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	#13133176	1	1	yes	characteristic
185	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
186	C₂×C₄	C₂×#11719924	1	2	yes	—
187	C₂×C₄	C₂×#11719925	1	2	yes	—
188	C₂×C₄	#13133237	1	4	yes	—
189	C₈	—	212	8	no	—
190	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂²	C₂×#13133134	1	1	yes	characteristic
192	C₂²	#16810607	1	1	yes	characteristic
193	C₂²	C₂×#13133141	1	2	yes	—
194	C₂²	#16810619	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	4	no	—
198	C₄	—	4	4	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	#16810372	1	4	yes	direct product factor
202	C₄	C₄×#11719924	1	4	yes	—
203	C₄	C₄×#11719925	1	4	yes	—
204	C₂	C₂×#16810372	1	1	yes	characteristic
205	C₂	#20273372	1	2	yes	—
206	C₂	#20273377	1	1	yes	characteristic
207	C₂	C₄×#13133134	1	1	yes	characteristic
208	C₂	C₄×#13133141	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
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	C₄×#333
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×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
8. #16810372	C₄	direct (N × H)
97. C₄×C₅₃	#333	semidirect
120. C₅₃	C₄×#333	semidirect
201. C₄	#16810372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88322	2
C₂×#16810372	2
#20272598	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#333	53

Subgroups — C4×#16810372 #27275036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810372 #27275036