Subgroups — C₄×#16810430 #27299316

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810430	C₂	1	2	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272810	C₂	1	2	yes	maximal, complemented
5	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	C₄×#11719917	—	2	1	no	—
9	#16810430	C₄	1	8	yes	direct product factor
10	C₄×#11719908	—	2	2	no	—
11	#16810388	—	2	1	no	—
12	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×#41×C₅₃	—	2	1	no	—
14	C₄×Q₁₆×C₅₃	C₄	1	2	yes	—
15	C₄×SD₁₆×C₅₃	—	2	2	no	—
16	C₄×D₈×C₅₃	C₄	1	2	yes	—
17	C₅₃×#427	C₂²	1	1	yes	characteristic
18	C₅₃×#416	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	C₂×C₄	1	4	yes	complemented
24	#13133153	—	2	2	no	—
25	#13133153	C₂×C₄	1	4	yes	complemented
26	#13133128	—	4	1	no	—
27	C₄×#88322	—	4	1	no	—
28	C₂×#41×C₅₃	—	2	1	no	—
29	#135×C₅₃	—	2	2	no	—
30	#135×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×SD₁₆×C₅₃	—	2	1	no	—
33	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₄×D₄×C₅₃	—	4	2	no	—
39	#117×C₅₃	—	2	1	no	—
40	#117×C₅₃	—	4	1	no	—
41	C₂×C₄²×C₅₃	—	4	1	no	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	#107×C₅₃	C₂×C₄	1	2	yes	—
44	#106×C₅₃	—	2	1	no	—
45	#103×C₅₃	—	2	2	no	—
46	#102×C₅₃	—	2	2	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	#41×C₅₃	—	2	4	no	—
50	#41×C₅₃	—	4	2	no	—
51	C₂×Q₈×C₅₃	—	2	1	no	—
52	C₂×D₄×C₅₃	—	2	1	no	—
53	C₂×D₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	Q₁₆×C₅₃	—	2	1	no	—
59	Q₁₆×C₅₃	C₄²	1	2	yes	—
60	SD₁₆×C₅₃	—	2	4	no	—
61	D₈×C₅₃	—	2	1	no	—
62	D₈×C₅₃	C₄²	1	2	yes	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	2	yes	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₂²⋊C₄×C₅₃	—	4	1	no	—
72	C₂²⋊C₄×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₅₃	D₈	1	1	yes	characteristic
77	C₄²×C₅₃	D₈	1	1	yes	characteristic
78	#11719917	—	2	4	no	—
79	#11719917	—	4	2	no	—
80	C₂×#88322	—	4	2	no	—
81	#11719908	—	2	4	no	—
82	C₂³×C₅₃	—	4	1	no	—
83	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	Q₈×C₅₃	—	2	2	no	—
85	Q₈×C₅₃	—	4	1	no	—
86	D₄×C₅₃	—	2	2	no	—
87	D₄×C₅₃	—	4	4	no	—
88	D₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	#102	1	2	yes	—
95	C₂×C₄×C₅₃	#102	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#135	1	2	yes	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₈×C₅₃	#117	1	1	yes	characteristic
103	C₈×C₅₃	C₄×D₄	1	2	yes	—
104	#88322	—	4	4	no	—
105	C₄×#359	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	4	2	no	—
108	C₂²×C₅₃	—	8	2	no	—
109	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
110	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	#359	1	4	yes	complemented
115	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
116	C₄×C₅₃	#416	1	1	yes	characteristic
117	C₄×C₅₃	C₄×D₈	1	2	yes	—
118	C₄×#135	D₅₃	1	1	yes	characteristic
119	C₂×#359	—	53	2	no	—
120	C₂×C₄×C₁₆	—	53	1	no	—
121	C₄×#108	—	53	1	no	—
122	#1382	—	53	2	no	—
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	—	8	2	no	—
125	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#359	1	2	yes	—
127	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
128	C₄×#41	—	2	1	no	—
129	C₂²×C₁₆	—	53	2	no	—
130	C₄×Q₁₆	Dic₅₃	1	2	yes	—
131	C₄×SD₁₆	—	2	2	no	—
132	C₄×D₈	Dic₅₃	1	2	yes	—
133	C₂×#108	—	53	2	no	—
134	#427	C₂×D₅₃	1	1	yes	characteristic
135	#416	—	2	1	no	—
136	C₄×M₁₆	—	106	1	no	—
137	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
138	#359	—	53	8	no	—
139	C₄×C₁₆	—	106	2	no	—
140	#341	—	106	1	no	—
141	C₅₃	C₄×#359	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	—	2	1	no	—
143	#135	—	2	2	no	—
144	#135	C₄×D₅₃	1	4	yes	—
145	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂×SD₁₆	—	2	1	no	—
147	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂×M₁₆	—	106	2	no	—
149	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
151	C₄×Q₈	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	4	2	no	—
154	#117	—	2	1	no	—
155	#117	—	4	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂×C₁₆	—	53	4	no	—
159	C₂×C₁₆	—	106	2	no	—
160	C₂×C₁₆	—	106	2	no	—
161	#108	—	53	4	no	—
162	#108	—	106	2	no	—
163	#107	C₂×Dic₅₃	1	2	yes	—
164	#106	—	2	1	no	—
165	#103	—	2	2	no	—
166	#102	—	2	2	no	—
167	#97	—	212	1	no	—
168	C₄×C₈	—	212	1	no	—
169	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
170	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
171	#41	—	2	4	no	—
172	#41	—	4	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	#11719923	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	Q₁₆	—	2	1	no	—
181	Q₁₆	C₄×Dic₅₃	1	2	yes	—
182	SD₁₆	—	2	4	no	—
183	D₈	—	2	1	no	—
184	D₈	C₄×Dic₅₃	1	2	yes	—
185	M₁₆	—	106	4	no	—
186	M₁₆	—	212	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
191	C₂×C₈	#11719945	1	2	yes	—
192	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
193	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	4	1	no	—
200	C₄²	#11719922	1	1	yes	characteristic
201	C₄²	#11719922	1	1	yes	characteristic
202	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
203	C₁₆	—	106	4	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₂×#11719923	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	#13133138	1	2	yes	—
217	C₂×C₄	#13133138	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
220	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
221	C₂×C₄	#13133237	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	212	4	no	—
224	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
225	C₈	#13133176	1	1	yes	characteristic
226	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₂×#13133138	1	1	yes	characteristic
231	C₂²	C₄×#11719923	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	2	no	—
235	C₄	#16810430	1	4	yes	direct product factor
236	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₄	#16810619	1	1	yes	characteristic
238	C₄	C₄×#11719922	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	—	8	2	no	—
241	C₂	C₄×#13133138	1	1	yes	characteristic
242	C₂	C₂×#16810430	1	2	yes	—

	subgroup	G/H
center	C₄²	#11719922
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	C₄×#359
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11719908	C₂	semidirect
3. C₄×#13133153	C₂	semidirect
4. #20272810	C₂	semidirect
9. #16810430	C₄	direct (N × H)
23. C₂×#11719908	C₂×C₄	semidirect
25. #13133153	C₂×C₄	semidirect
114. C₄×C₅₃	#359	semidirect
141. C₅₃	C₄×#359	semidirect
235. C₄	#16810430	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810430	2
C₂×C₄×#11719908	2
C₄×#13133153	2
#20272810	2
C₄×#135×C₅₃	2
C₄×#359	53

Subgroups — C4×#16810430 #27299316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810430 #27299316