Subgroups — C₄×#16810378 #27289239

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 214 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	2	yes	maximal
2	C₂×#16810378	C₂	1	4	yes	maximal
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₂²	1	4	yes	—
5	C₂×C₄×#88322	—	2	4	no	—
6	C₂×C₄×#88322	—	2	4	no	—
7	C₂×C₄×#88322	C₄	1	8	yes	complemented
8	#16810378	C₄	1	16	yes	direct product factor
9	C₂²×C₄²×C₅₃	C₄	1	4	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄³×C₅₃	—	2	4	no	—
13	C₂²×#88322	—	2	4	no	—
14	C₂²×#88322	—	2	4	no	—
15	C₂²×#88322	—	2	8	no	—
16	C₂²×#88322	C₂×C₄	1	8	yes	—
17	C₂²×#88322	C₂×C₄	1	16	yes	—
18	C₄×#88322	—	2	16	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	—	2	8	no	—
23	C₂×C₄²×C₅₃	—	2	8	no	—
24	C₂×C₄²×C₅₃	—	2	4	no	—
25	C₂×C₄²×C₅₃	—	2	4	no	—
26	C₂×C₄²×C₅₃	C₈	1	16	yes	complemented
27	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	D₄	1	4	yes	—
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
32	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	8	no	—
36	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
37	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	4	yes	—
49	C₄²×C₅₃	—	2	32	no	—
50	C₄²×C₅₃	—	2	16	no	—
51	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
52	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
53	C₂×#88322	—	2	8	no	—
54	C₂×#88322	—	2	8	no	—
55	C₂×#88322	—	2	16	no	—
56	C₂×#88322	—	2	16	no	—
57	C₂×#88322	C₄²	1	16	yes	complemented
58	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
60	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂³×C₅₃	#117	1	2	yes	—
63	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
64	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	—	2	8	no	—
67	C₂×C₄×C₅₃	—	2	16	no	—
68	C₂×C₄×C₅₃	—	2	16	no	—
69	C₂×C₄×C₅₃	#95	1	8	yes	—
70	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
71	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
72	C₂×C₄×C₅₃	#97	1	8	yes	—
73	C₂×C₄×C₅₃	C₂²⋊C₈	1	16	yes	—
74	C₂×C₄×C₅₃	C₄⋊C₈	1	16	yes	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₃	#117	1	4	yes	—
79	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
80	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
82	#88322	—	2	16	no	—
83	C₄×#336	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	C₄³	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
86	C₂²×C₅₃	#376	1	1	yes	characteristic
87	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
88	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
89	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
90	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
91	C₂²×C₅₃	C₂×#97	1	2	yes	—
92	C₂²×C₅₃	C₄×M₁₆	1	4	yes	—
93	C₂²×C₅₃	C₂×C₂²⋊C₈	1	4	yes	—
94	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
95	C₂²×C₅₃	#431	1	8	yes	—
96	C₄×C₅₃	—	2	16	no	—
97	C₄×C₅₃	#336	1	16	yes	complemented
98	C₄×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
99	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
100	C₂²×C₄×C₈	—	53	2	no	—
101	C₂×C₄³	D₅₃	1	1	yes	characteristic
102	C₂×#336	—	53	4	no	—
103	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
104	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
105	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#336	1	2	yes	—
107	C₂×C₅₃	#1744	1	2	yes	—
108	C₂×C₅₃	C₄×C₂²⋊C₈	1	4	yes	—
109	C₂×C₅₃	C₄×C₄⋊C₈	1	4	yes	—
110	C₂³×C₈	—	53	4	no	—
111	C₂²×C₄²	Dic₅₃	1	4	yes	—
112	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
113	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
114	C₂×C₄×C₈	—	53	8	no	—
115	C₂×C₄×C₈	—	106	4	no	—
116	C₂×C₄×C₈	—	106	4	no	—
117	C₄³	—	2	4	no	—
118	#336	—	53	16	no	—
119	C₅₃	C₄×#336	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
121	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
122	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
123	C₂²×C₈	—	53	16	no	—
124	C₂²×C₈	—	53	8	no	—
125	C₂²×C₈	—	106	4	no	—
126	C₂²×C₈	—	106	8	no	—
127	C₂²×C₈	—	106	4	no	—
128	C₂×C₄²	—	2	4	no	—
129	C₂×C₄²	—	2	4	no	—
130	C₂×C₄²	—	2	8	no	—
131	C₂×C₄²	—	2	8	no	—
132	C₂×C₄²	#88322	1	16	yes	complemented
133	C₂×C₄²	Dic₁₀₆	1	2	yes	—
134	C₂×C₄²	C₄×D₅₃	1	8	yes	—
135	C₂×C₄²	C₄×D₅₃	1	4	yes	—
136	C₂×C₄²	D₂₁₂	1	2	yes	—
137	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
138	C₄×C₈	—	106	16	no	—
139	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	C₂×#88322	1	16	yes	—
144	C₂²×C₄	C₂×#88322	1	8	yes	—
145	C₂²×C₄	#11719917	1	8	yes	—
146	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
147	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
148	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
149	C₂²×C₄	C₂²⋊Dic₅₃	1	4	yes	—
150	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
153	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
154	C₂²×C₄	#11719945	1	4	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
156	C₂×C₈	—	53	16	no	—
157	C₂×C₈	—	106	8	no	—
158	C₂×C₈	—	106	8	no	—
159	C₂×C₈	—	106	16	no	—
160	C₂×C₈	—	106	16	no	—
161	C₄²	—	2	32	no	—
162	C₄²	—	2	16	no	—
163	C₄²	#11719919	1	8	yes	—
164	C₄²	#11719921	1	8	yes	—
165	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
166	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
167	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
168	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
169	C₂³	#13133176	1	2	yes	—
170	C₂³	C₂²×#88322	1	4	yes	—
171	C₂³	C₂×#11719917	1	4	yes	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	16	no	—
174	C₂×C₄	—	2	16	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	#13133124	1	8	yes	—
177	C₂×C₄	C₄×#88322	1	8	yes	—
178	C₂×C₄	C₄×#88322	1	16	yes	complemented
179	C₂×C₄	#13133128	1	8	yes	—
180	C₂×C₄	C₂²⋊#88322	1	16	yes	—
181	C₂×C₄	C₄⋊#88322	1	16	yes	—
182	C₂×C₄	C₄²×D₅₃	1	4	yes	—
183	C₂×C₄	C₂×#11719921	1	2	yes	—
184	C₂×C₄	C₂×#11719919	1	2	yes	—
185	C₂×C₄	#13133174	1	4	yes	—
186	C₂×C₄	C₄×C₂²⋊D₅₃	1	8	yes	—
187	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
188	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
189	C₈	—	106	16	no	—
190	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
191	C₂²	C₂×#13133124	1	1	yes	characteristic
192	C₂²	#16810469	1	1	yes	characteristic
193	C₂²	C₄×C₂²⋊Dic₅₃	1	2	yes	—
194	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
195	C₂²	C₂×C₄×#88322	1	2	yes	—
196	C₂²	C₂×C₄×#88322	1	4	yes	—
197	C₂²	C₂×#13133128	1	2	yes	—
198	C₂²	C₄×#11719917	1	4	yes	—
199	C₂²	C₂×C₂²⋊#88322	1	4	yes	—
200	C₂²	C₂×C₄⋊#88322	1	4	yes	—
201	C₂²	#16810676	1	8	yes	—
202	C₄	—	2	16	no	—
203	C₄	#16810378	1	16	yes	direct product factor
204	C₄	C₄×#11719921	1	4	yes	—
205	C₄	C₄×#11719919	1	4	yes	—
206	C₂	C₄×#13133124	1	1	yes	characteristic
207	C₂	C₄²×#88322	1	1	yes	characteristic
208	C₂	C₄×#13133128	1	1	yes	characteristic
209	C₂	C₂×#16810378	1	2	yes	—
210	C₂	#20274461	1	2	yes	—
211	C₂	C₄×C₂²⋊#88322	1	4	yes	—
212	C₂	C₄×C₄⋊#88322	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₄×#336
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
7. C₂×C₄×#88322	C₄	semidirect
8. #16810378	C₄	direct (N × H)
26. C₂×C₄²×C₅₃	C₈	semidirect
57. C₂×#88322	C₄²	semidirect
71. C₂×C₄×C₅₃	C₄×C₈	semidirect
97. C₄×C₅₃	#336	semidirect
119. C₅₃	C₄×#336	semidirect
132. C₂×C₄²	#88322	semidirect
178. C₂×C₄	C₄×#88322	semidirect
203. C₄	#16810378	direct (N × H)

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
C₂×C₄³×C₅₃	2
C₄×#336	53

Subgroups — C4×#16810378 #27289239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810378 #27289239