Subgroups — C₂²×#16811254 #28290862

Order 13568.

7630 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811254	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133216	C₂	1	4	yes	maximal, complemented
3	C₂²×#13133208	C₂	1	4	yes	maximal, complemented
4	C₂²×#13133196	C₂	1	4	yes	maximal
5	C₂²×#13133176	C₂	1	2	yes	maximal
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	C₂²	1	4	yes	—
8	#16811254	C₂²	1	256	yes	direct product factor
9	C₂×#13133216	C₂²	1	96	yes	complemented
10	C₂×#13133208	C₂²	1	96	yes	complemented
11	C₂×#13133196	C₂²	1	96	yes	—
12	C₂×#13133176	C₂²	1	48	yes	—
13	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
14	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
15	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
16	C₂²×#11719919	C₂²	1	8	yes	—
17	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
18	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
19	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
20	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	24	yes	—
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂²×Dic₁₀₆	—	4	16	no	—
24	C₂²×Dic₁₀₆	C₂³	1	48	yes	—
25	C₂³×Dic₅₃	C₂³	1	4	yes	—
26	C₂³×Dic₅₃	C₂³	1	4	yes	—
27	#13133216	C₂³	1	256	yes	complemented
28	#13133208	C₂³	1	256	yes	complemented
29	#13133196	C₂³	1	256	yes	—
30	#13133176	C₂³	1	128	yes	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	48	yes	—
34	C₂×#11719919	C₂³	1	96	yes	—
35	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	—
36	C₂×C₄×Dic₅₃	C₂³	1	48	yes	—
37	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	—	4	8	no	—
39	C₂³×C₄×C₅₃	Q₈	1	16	yes	complemented
40	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	64	yes	—
46	C₂⁴×C₅₃	—	2	4	no	—
47	C₂⁴×C₅₃	—	2	4	no	—
48	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
49	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
50	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	16	no	—
52	C₂²×C₄×C₅₃	—	4	48	no	—
53	C₂²×C₄×C₅₃	C₂×Q₈	1	16	yes	—
54	C₂²×C₄×C₅₃	C₂×Q₈	1	96	yes	complemented
55	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
59	C₂²×Dic₅₃	—	2	16	no	—
60	C₂²×Dic₅₃	—	2	16	no	—
61	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
62	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
63	C₂×Dic₁₀₆	—	4	96	no	—
64	C₂×Dic₁₀₆	C₂⁴	1	64	yes	—
65	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
66	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
67	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
68	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
69	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
70	#11719919	C₂⁴	1	128	yes	—
71	C₄×Dic₅₃	C₂⁴	1	64	yes	—
72	C₂³×C₅₃	—	2	32	no	—
73	C₂³×C₅₃	—	2	12	no	—
74	C₂³×C₅₃	—	2	12	no	—
75	C₂³×C₅₃	C₂²×Q₈	1	24	yes	—
76	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
77	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
78	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
79	C₂³×C₅₃	C₂⁵	1	4	yes	—
80	C₂³×C₅₃	C₂⁵	1	3	yes	—
81	C₂×C₄×C₅₃	—	2	48	no	—
82	C₂×C₄×C₅₃	—	4	32	no	—
83	C₂×C₄×C₅₃	C₂²×Q₈	1	48	yes	—
84	C₂×C₄×C₅₃	C₂²×Q₈	1	64	yes	complemented
85	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
87	C₂×Dic₅₃	—	2	48	no	—
88	C₂×Dic₅₃	—	2	48	no	—
89	C₂×Dic₅₃	C₂⁵	1	16	yes	—
90	C₂×Dic₅₃	C₂⁵	1	16	yes	—
91	Dic₁₀₆	—	4	64	no	—
92	C₂²×#544	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	4	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	—	2	48	no	—
96	C₂²×C₅₃	#544	1	16	yes	complemented
97	C₂²×C₅₃	C₂³×Q₈	1	8	yes	—
98	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
99	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
100	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	6	yes	—
101	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
102	C₄×C₅₃	—	2	16	no	—
103	C₄×C₅₃	C₂³×Q₈	1	16	yes	—
104	Dic₅₃	—	2	16	no	—
105	Dic₅₃	—	2	16	no	—
106	C₂×#544	—	53	48	no	—
107	C₂²×C₄⋊Q₈	—	53	4	no	—
108	C₂²×#125	—	53	4	no	—
109	C₂²×C₂²⋊Q₈	—	53	4	no	—
110	C₂²×#117	—	53	2	no	—
111	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
112	C₂×C₅₃	—	2	8	no	—
113	C₂×C₅₃	C₂×#544	1	12	yes	complemented
114	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
115	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
117	C₂³×Q₈	—	53	4	no	—
118	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
119	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
120	#544	—	53	256	no	—
121	C₂×C₄⋊Q₈	—	53	96	no	—
122	C₂×#125	—	53	96	no	—
123	C₂×C₂²⋊Q₈	—	53	96	no	—
124	C₂×#117	—	53	48	no	—
125	C₂²×C₄⋊C₄	—	53	8	no	—
126	C₂²×C₄⋊C₄	—	53	2	no	—
127	C₂²×C₄⋊C₄	—	53	2	no	—
128	C₂²×C₄⋊C₄	—	53	4	no	—
129	C₂²×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
130	C₂²×C₄⋊C₄	C₂×D₅₃	1	24	yes	—
131	C₂²×C₂²⋊C₄	—	53	4	no	—
132	C₂²×C₄²	—	53	4	no	—
133	C₅₃	C₂²×#544	1	1	yes	characteristic, complemented, Sylow
134	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
135	C₂²×Q₈	—	53	48	no	—
136	C₂²×Q₈	—	212	16	no	—
137	C₂³×C₄	—	4	8	no	—
138	C₂³×C₄	—	53	4	no	—
139	C₂³×C₄	—	53	4	no	—
140	C₂³×C₄	Dic₁₀₆	1	16	yes	complemented
141	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
142	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
143	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
144	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
145	C₄⋊Q₈	—	53	256	no	—
146	#125	—	53	256	no	—
147	C₂²⋊Q₈	—	53	256	no	—
148	#117	—	53	128	no	—
149	C₂×C₄⋊C₄	—	53	24	no	—
150	C₂×C₄⋊C₄	—	53	24	no	—
151	C₂×C₄⋊C₄	—	53	48	no	—
152	C₂×C₄⋊C₄	—	53	96	no	—
153	C₂×C₄⋊C₄	C₂²×D₅₃	1	48	yes	—
154	C₂×C₄⋊C₄	C₂²×D₅₃	1	64	yes	—
155	C₂×C₂²⋊C₄	—	53	48	no	—
156	C₂×C₄²	—	53	48	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	—	2	4	no	—
159	C₂⁴	C₂×Dic₁₀₆	1	8	yes	—
160	C₂⁴	C₂³×D₅₃	1	6	yes	—
161	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
162	C₂×Q₈	—	53	64	no	—
163	C₂×Q₈	—	212	96	no	—
164	C₂²×C₄	—	2	16	no	—
165	C₂²×C₄	—	4	48	no	—
166	C₂²×C₄	—	53	24	no	—
167	C₂²×C₄	—	53	24	no	—
168	C₂²×C₄	—	106	16	no	—
169	C₂²×C₄	—	106	16	no	—
170	C₂²×C₄	C₂×Dic₁₀₆	1	16	yes	—
171	C₂²×C₄	C₂×Dic₁₀₆	1	96	yes	complemented
172	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
173	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
174	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
175	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
176	C₄⋊C₄	—	53	32	no	—
177	C₄⋊C₄	—	53	32	no	—
178	C₄⋊C₄	—	53	64	no	—
179	C₄⋊C₄	—	53	128	no	—
180	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
181	C₂²⋊C₄	—	53	64	no	—
182	C₄²	—	53	64	no	—
183	C₂³	—	2	12	no	—
184	C₂³	—	2	12	no	—
185	C₂³	—	2	32	no	—
186	C₂³	C₂²×Dic₁₀₆	1	24	yes	—
187	C₂³	C₂²×Dic₁₀₆	1	4	yes	—
188	C₂³	#13133262	1	4	yes	—
189	C₂³	#13133265	1	4	yes	—
190	C₂³	C₂⁴×D₅₃	1	4	yes	—
191	C₂³	C₂⁴×D₅₃	1	3	yes	—
192	Q₈	—	212	64	no	—
193	C₂×C₄	—	2	48	no	—
194	C₂×C₄	—	4	32	no	—
195	C₂×C₄	—	53	16	no	—
196	C₂×C₄	—	53	16	no	—
197	C₂×C₄	—	106	48	no	—
198	C₂×C₄	—	106	48	no	—
199	C₂×C₄	C₂²×Dic₁₀₆	1	48	yes	—
200	C₂×C₄	C₂²×Dic₁₀₆	1	64	yes	complemented
201	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
202	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
203	C₂²	—	2	48	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	#16811254	1	16	yes	direct product factor
207	C₂²	C₂³×Dic₁₀₆	1	8	yes	—
208	C₂²	C₂³×Dic₁₀₆	1	6	yes	—
209	C₂²	C₂×#13133262	1	6	yes	—
210	C₂²	C₂×#13133265	1	6	yes	—
211	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
212	C₄	—	2	16	no	—
213	C₄	—	106	16	no	—
214	C₄	—	106	16	no	—
215	C₄	C₂³×Dic₁₀₆	1	16	yes	—
216	C₂	—	2	8	no	—
217	C₂	C₂×#16811254	1	12	yes	direct product factor
218	C₂	C₂⁴×Dic₁₀₆	1	1	yes	characteristic
219	C₂	C₂²×#13133262	1	1	yes	characteristic
220	C₂	C₂²×#13133265	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×#544
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16811254	C₂	direct (N × H)
2. C₂²×#13133216	C₂	semidirect
3. C₂²×#13133208	C₂	semidirect
8. #16811254	C₂²	direct (N × H)
9. C₂×#13133216	C₂²	semidirect
10. C₂×#13133208	C₂²	semidirect
27. #13133216	C₂³	semidirect
28. #13133208	C₂³	semidirect
39. C₂³×C₄×C₅₃	Q₈	semidirect
54. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
84. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
96. C₂²×C₅₃	#544	semidirect
113. C₂×C₅₃	C₂×#544	semidirect
133. C₅₃	C₂²×#544	semidirect
140. C₂³×C₄	Dic₁₀₆	semidirect
171. C₂²×C₄	C₂×Dic₁₀₆	semidirect
200. C₂×C₄	C₂²×Dic₁₀₆	semidirect
206. C₂²	#16811254	direct (N × H)
217. C₂	C₂×#16811254	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×#16811254	2
C₂²×#13133216	2
C₂²×#13133208	2
C₂²×#13133196	2
C₂²×#13133176	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#544	53

Subgroups — C22×#16811254 #28290862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811254 #28290862