Subgroups — C₄×#16810672 #27382199

Order 13568.

1034 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	2	yes	maximal
2	C₂×C₄×#11719917	C₂	1	2	yes	maximal
3	#20274621	C₂	1	4	yes	maximal, complemented
4	C₄×#13133153	C₂	1	4	yes	maximal, complemented
5	#20274442	C₂	1	2	yes	maximal
6	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	4	yes	—
8	#16810672	C₄	1	16	yes	direct product factor
9	C₂×#13133153	C₂²	1	8	yes	—
10	C₄×#11719917	—	2	2	no	—
11	C₄×#11719917	—	2	2	no	—
12	C₄×#11719917	C₂²	1	4	yes	—
13	C₄×#11719917	C₂²	1	4	yes	—
14	C₂×#13133128	—	2	2	no	—
15	C₂×C₄×#88322	—	2	2	no	—
16	#16810388	C₂²	1	8	yes	—
17	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₅₃	C₄	1	8	yes	—
21	C₂×#97×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
23	C₂×#11719917	—	2	4	no	—
24	C₂×#11719917	—	2	8	no	—
25	C₂×#11719917	C₂×C₄	1	16	yes	—
26	C₂×#11719917	C₂³	1	8	yes	—
27	C₂²×#88322	—	2	4	no	—
28	#13133153	C₂×C₄	1	32	yes	complemented
29	#13133128	—	2	4	no	—
30	#13133128	—	2	4	no	—
31	C₄×#88322	—	2	4	no	—
32	C₄×#88322	—	2	4	no	—
33	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	4	no	—
36	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
37	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	#97×C₅₃	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
49	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	4	4	no	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	8	no	—
62	M₁₆×C₅₃	C₄²	1	16	yes	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	#11719917	—	2	16	no	—
75	#11719917	—	2	8	no	—
76	#11719917	C₂²×C₄	1	16	yes	—
77	C₂×#88322	—	2	16	no	—
78	C₂×#88322	—	2	8	no	—
79	C₂³×C₅₃	—	4	2	no	—
80	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
82	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	8	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₅₃	#117	1	4	yes	—
93	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
94	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
99	C₈×C₅₃	—	2	4	no	—
100	C₈×C₅₃	C₂×C₄²	1	8	yes	—
101	#88322	—	2	16	no	—
102	C₄×#430	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	4	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	4	4	no	—
106	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
107	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
109	C₄×C₅₃	—	2	4	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
112	C₄×C₅₃	#430	1	4	yes	complemented
113	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
114	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
115	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
116	C₂×#430	—	53	2	no	—
117	C₂×C₄×M₁₆	—	53	2	no	—
118	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
119	#1778	—	53	4	no	—
120	C₄×#108	—	53	4	no	—
121	#1738	—	53	2	no	—
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	—	2	4	no	—
124	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#430	1	2	yes	—
126	C₂²×M₁₆	—	53	4	no	—
127	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
128	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
129	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	#430	—	53	16	no	—
131	C₂×#108	—	53	8	no	—
132	C₄×M₁₆	—	53	4	no	—
133	C₄×M₁₆	—	53	4	no	—
134	C₄×M₁₆	—	106	2	no	—
135	C₄×M₁₆	—	106	2	no	—
136	C₄×M₁₆	Dic₅₃	1	8	yes	—
137	C₂×#97	—	106	2	no	—
138	C₂×#97	C₂×D₅₃	1	2	yes	—
139	C₂×C₄×C₈	—	106	2	no	—
140	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
141	#341	—	53	8	no	—
142	C₅₃	C₄×#430	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₄	Dic₁₀₆	1	2	yes	—
144	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
145	C₂×M₁₆	—	2	4	no	—
146	C₂×M₁₆	—	53	16	no	—
147	C₂×M₁₆	—	53	8	no	—
148	C₂×M₁₆	—	106	4	no	—
149	C₂×M₁₆	—	106	8	no	—
150	C₂×M₁₆	C₄×D₅₃	1	8	yes	—
151	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
152	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
153	C₂²×C₈	—	106	4	no	—
154	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
155	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
156	C₂×C₄²	Dic₁₀₆	1	2	yes	—
157	C₂×C₄²	D₂₁₂	1	2	yes	—
158	C₂×C₄²	D₂₁₂	1	2	yes	—
159	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
160	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	#108	—	53	32	no	—
163	#97	—	106	4	no	—
164	#97	—	106	4	no	—
165	#97	C₂×Dic₅₃	1	8	yes	—
166	C₄×C₈	—	106	4	no	—
167	C₄×C₈	—	106	4	no	—
168	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
169	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
172	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
173	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
174	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
175	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
177	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
178	C₂²×C₄	#11719945	1	4	yes	—
179	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
180	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
181	M₁₆	—	2	8	no	—
182	M₁₆	—	53	16	no	—
183	M₁₆	—	106	16	no	—
184	M₁₆	—	106	8	no	—
185	M₁₆	C₄×Dic₅₃	1	16	yes	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	106	16	no	—
188	C₂×C₈	—	106	8	no	—
189	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
190	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
191	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
192	C₄²	C₄⋊Dic₅₃	1	4	yes	—
193	C₄²	C₄⋊Dic₅₃	1	4	yes	—
194	C₄²	C₂×D₂₁₂	1	2	yes	—
195	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
196	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
197	C₄²	C₂²×Dic₅₃	1	2	yes	—
198	C₄²	C₂²×Dic₅₃	1	2	yes	—
199	C₂³	—	4	2	no	—
200	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
201	C₂³	C₄×Dic₁₀₆	1	4	yes	—
202	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
211	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
212	C₂×C₄	#13133176	1	4	yes	—
213	C₂×C₄	C₄×D₂₁₂	1	8	yes	—
214	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
215	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
216	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719945	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	106	16	no	—
221	C₈	C₂×C₄×Dic₅₃	1	8	yes	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₄×C₄⋊Dic₅₃	1	4	yes	—
226	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
227	C₂²	C₂×C₄×Dic₁₀₆	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	C₄×C₄⋊Dic₅₃	1	4	yes	—
231	C₄	#16810672	1	4	yes	direct product factor
232	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
233	C₄	C₂×#13133176	1	1	yes	characteristic
234	C₄	C₂×C₄×D₂₁₂	1	2	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	2	4	no	—
237	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
238	C₂	C₂×#16810672	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₄×#430
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20274621	C₂	semidirect
4. C₄×#13133153	C₂	semidirect
8. #16810672	C₄	direct (N × H)
28. #13133153	C₂×C₄	semidirect
112. C₄×C₅₃	#430	semidirect
142. C₅₃	C₄×#430	semidirect
231. C₄	#16810672	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×#16810672	2
C₂×C₄×#11719917	2
#20274621	2
C₄×#13133153	2
#20274442	2
C₂×C₄×M₁₆×C₅₃	2
C₄×#430	53

Subgroups — C4×#16810672 #27382199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810672 #27382199