Subgroups — C₂²×#16810378 #27381443

Order 13568.

3086 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#88322	C₂	1	2	yes	maximal
2	C₂×#16810378	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	—	2	8	no	—
5	C₂³×#88322	—	2	8	no	—
6	C₂³×#88322	C₄	1	16	yes	complemented
7	C₂³×#88322	C₂²	1	12	yes	—
8	#16810378	C₂²	1	16	yes	direct product factor
9	C₂⁴×C₄×C₅₃	C₄	1	2	yes	—
10	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅₃	—	2	8	no	—
12	C₂²×C₄²×C₅₃	C₄	1	6	yes	—
13	C₂²×C₄²×C₅₃	C₂²	1	6	yes	—
14	C₂²×#88322	—	2	64	no	—
15	C₂²×#88322	—	2	24	no	—
16	C₂²×#88322	—	2	24	no	—
17	C₂²×#88322	C₂×C₄	1	48	yes	complemented
18	C₂²×#88322	C₂³	1	8	yes	—
19	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	—	2	8	no	—
21	C₂³×C₄×C₅₃	—	2	8	no	—
22	C₂³×C₄×C₅₃	C₈	1	16	yes	complemented
23	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₅₃	D₄	1	4	yes	—
27	C₂³×C₄×C₅₃	D₄	1	8	yes	—
28	C₂³×C₄×C₅₃	Q₈	1	4	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
30	C₂×C₄²×C₅₃	—	2	24	no	—
31	C₂×C₄²×C₅₃	—	2	24	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
34	C₂⁴×C₅₃	C₄²	1	4	yes	—
35	C₂⁴×C₅₃	C₂²⋊C₄	1	4	yes	—
36	C₂⁴×C₅₃	C₄⋊C₄	1	4	yes	—
37	C₂⁴×C₅₃	C₂×C₈	1	8	yes	—
38	C₂⁴×C₅₃	M₁₆	1	8	yes	—
39	C₂⁴×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₅₃	—	2	64	no	—
41	C₂²×C₄×C₅₃	—	2	24	no	—
42	C₂²×C₄×C₅₃	—	2	24	no	—
43	C₂²×C₄×C₅₃	C₄²	1	12	yes	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	32	yes	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	32	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	48	yes	complemented
49	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
54	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄²×C₅₃	—	2	24	no	—
57	C₄²×C₅₃	—	2	8	no	—
58	C₂×#88322	—	2	96	no	—
59	C₂×#88322	—	2	8	no	—
60	C₂×#88322	—	2	8	no	—
61	C₂×#88322	C₂²×C₄	1	16	yes	complemented
62	C₂³×C₅₃	#95	1	16	yes	—
63	C₂³×C₅₃	C₄×C₈	1	16	yes	—
64	C₂³×C₅₃	#97	1	16	yes	—
65	C₂³×C₅₃	C₂²⋊C₈	1	32	yes	—
66	C₂³×C₅₃	C₄⋊C₈	1	32	yes	—
67	C₂³×C₅₃	C₂×C₄²	1	6	yes	—
68	C₂³×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
69	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
70	C₂³×C₅₃	C₂²×C₈	1	12	yes	—
71	C₂³×C₅₃	C₂×M₁₆	1	12	yes	—
72	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	8	no	—
75	C₂×C₄×C₅₃	—	2	96	no	—
76	C₂×C₄×C₅₃	#95	1	48	yes	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₅₃	C₂²×C₈	1	16	yes	complemented
83	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
87	#88322	—	2	16	no	—
88	C₂²×#336	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#336	1	64	yes	complemented
90	C₂²×C₅₃	C₂×#95	1	12	yes	—
91	C₂²×C₅₃	C₂×C₄×C₈	1	12	yes	—
92	C₂²×C₅₃	C₂×#97	1	12	yes	—
93	C₂²×C₅₃	C₂×C₂²⋊C₈	1	24	yes	—
94	C₂²×C₅₃	C₂×C₄⋊C₈	1	24	yes	—
95	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
99	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
100	C₄×C₅₃	—	2	16	no	—
101	C₄×C₅₃	C₂×#95	1	12	yes	—
102	C₄×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
103	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
104	C₂⁴×C₈	—	53	2	no	—
105	C₂³×C₄²	D₅₃	1	1	yes	characteristic
106	C₂×#336	—	53	12	no	—
107	C₂×C₅₃	C₂×#336	1	24	yes	complemented
108	C₂×C₅₃	C₂²×#95	1	1	yes	characteristic
109	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
111	C₂×C₅₃	C₂²×C₂²⋊C₈	1	2	yes	—
112	C₂×C₅₃	C₂²×C₄⋊C₈	1	2	yes	—
113	C₂⁴×C₄	Dic₅₃	1	2	yes	—
114	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
115	C₂³×C₈	—	53	16	no	—
116	C₂³×C₈	—	53	12	no	—
117	C₂³×C₈	—	106	8	no	—
118	C₂³×C₈	—	106	8	no	—
119	C₂²×C₄²	—	2	8	no	—
120	C₂²×C₄²	Dic₅₃	1	6	yes	—
121	C₂²×C₄²	C₂×D₅₃	1	6	yes	—
122	#336	—	53	16	no	—
123	C₅₃	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	—	2	8	no	—
127	C₂³×C₄	#88322	1	16	yes	complemented
128	C₂³×C₄	Dic₁₀₆	1	4	yes	—
129	C₂³×C₄	C₄×D₅₃	1	8	yes	—
130	C₂³×C₄	D₂₁₂	1	4	yes	—
131	C₂³×C₄	C₂×Dic₅₃	1	12	yes	—
132	C₂³×C₄	C₂×Dic₅₃	1	3	yes	—
133	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
134	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
135	C₂²×C₈	—	53	48	no	—
136	C₂²×C₈	—	53	8	no	—
137	C₂²×C₈	—	106	24	no	—
138	C₂²×C₈	—	106	24	no	—
139	C₂²×C₈	—	106	64	no	—
140	C₂×C₄²	—	2	24	no	—
141	C₂×C₄²	—	2	24	no	—
142	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
143	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
144	C₂⁴	C₂×#88322	1	8	yes	—
145	C₂⁴	#11719917	1	8	yes	—
146	C₂⁴	C₄×Dic₅₃	1	4	yes	—
147	C₂⁴	C₄⋊Dic₅₃	1	4	yes	—
148	C₂⁴	C₂²⋊Dic₅₃	1	4	yes	—
149	C₂⁴	C₂²×Dic₅₃	1	3	yes	—
150	C₂²×C₄	—	2	64	no	—
151	C₂²×C₄	—	2	24	no	—
152	C₂²×C₄	—	2	24	no	—
153	C₂²×C₄	C₂×#88322	1	48	yes	complemented
154	C₂²×C₄	C₄×Dic₅₃	1	12	yes	—
155	C₂²×C₄	#11719919	1	32	yes	—
156	C₂²×C₄	C₄⋊Dic₅₃	1	12	yes	—
157	C₂²×C₄	#11719921	1	32	yes	—
158	C₂²×C₄	C₂²⋊Dic₅₃	1	12	yes	—
159	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
160	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
161	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
162	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
163	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
165	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
166	C₂×C₈	—	53	16	no	—
167	C₂×C₈	—	106	96	no	—
168	C₂×C₈	—	106	8	no	—
169	C₂×C₈	—	106	8	no	—
170	C₄²	—	2	24	no	—
171	C₄²	—	2	8	no	—
172	C₂³	#13133124	1	16	yes	—
173	C₂³	C₄×#88322	1	16	yes	—
174	C₂³	#13133128	1	16	yes	—
175	C₂³	C₂²⋊#88322	1	32	yes	—
176	C₂³	C₄⋊#88322	1	32	yes	—
177	C₂³	C₂×C₄×Dic₅₃	1	6	yes	—
178	C₂³	C₂×C₂²⋊Dic₅₃	1	6	yes	—
179	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
180	C₂³	C₂²×#88322	1	12	yes	—
181	C₂³	C₂×#11719917	1	12	yes	—
182	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	96	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	#13133124	1	48	yes	—
187	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
188	C₂×C₄	C₂×#11719921	1	24	yes	—
189	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	6	yes	—
190	C₂×C₄	C₂×#11719919	1	24	yes	—
191	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
192	C₂×C₄	C₂²×#88322	1	16	yes	complemented
193	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₅₃	1	2	yes	—
195	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
197	C₈	—	106	16	no	—
198	C₂²	#16810378	1	64	yes	direct product factor
199	C₂²	C₂×#13133124	1	12	yes	—
200	C₂²	C₂×C₄×#88322	1	12	yes	—
201	C₂²	C₂×#13133128	1	12	yes	—
202	C₂²	C₂×C₂²⋊#88322	1	24	yes	—
203	C₂²	C₂×C₄⋊#88322	1	24	yes	—
204	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
205	C₂²	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
207	C₂²	C₂³×#88322	1	2	yes	—
208	C₂²	C₂²×#11719917	1	2	yes	—
209	C₄	—	2	16	no	—
210	C₄	C₂×#13133124	1	12	yes	—
211	C₄	C₂²×#11719921	1	2	yes	—
212	C₄	C₂²×#11719919	1	2	yes	—
213	C₂	C₂×#16810378	1	24	yes	direct product factor
214	C₂	C₂²×#13133124	1	1	yes	characteristic
215	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
216	C₂	C₂²×#13133128	1	1	yes	characteristic
217	C₂	C₂²×C₂²⋊#88322	1	2	yes	—
218	C₂	C₂²×C₄⋊#88322	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#336
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#16810378	C₂	direct (N × H)
6. C₂³×#88322	C₄	semidirect
8. #16810378	C₂²	direct (N × H)
17. C₂²×#88322	C₂×C₄	semidirect
22. C₂³×C₄×C₅₃	C₈	semidirect
48. C₂²×C₄×C₅₃	C₂×C₈	semidirect
61. C₂×#88322	C₂²×C₄	semidirect
82. C₂×C₄×C₅₃	C₂²×C₈	semidirect
89. C₂²×C₅₃	#336	semidirect
107. C₂×C₅₃	C₂×#336	semidirect
123. C₅₃	C₂²×#336	semidirect
127. C₂³×C₄	#88322	semidirect
153. C₂²×C₄	C₂×#88322	semidirect
192. C₂×C₄	C₂²×#88322	semidirect
198. C₂²	#16810378	direct (N × H)
213. C₂	C₂×#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
C₂⁴×#88322	2
C₂×#16810378	2
C₂³×C₄²×C₅₃	2
C₂²×#336	53

Subgroups — C22×#16810378 #27381443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810378 #27381443