Subgroups — C₂²×#16811339 #28291112

Order 13568.

8710 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×#16811339	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133206	C₂	1	4	yes	maximal, complemented
3	C₂²×#13133192	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×D₂₁₂	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133175	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₁₂	C₂²	1	2	yes	—
8	C₂³×D₂₁₂	C₂²	1	4	yes	—
9	C₂³×C₄×D₅₃	C₂²	1	4	yes	—
10	#16811339	C₂²	1	256	yes	direct product factor
11	C₂×#13133206	C₂²	1	96	yes	complemented
12	C₂×#13133192	C₂²	1	96	yes	complemented
13	C₂×C₄×D₂₁₂	C₂²	1	48	yes	complemented
14	C₂×#13133175	C₂²	1	96	yes	complemented
15	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
16	C₂²×#11719921	C₂²	1	8	yes	—
17	C₂²×#11719921	C₂²	1	4	yes	—
18	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
19	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂⁴×D₅₃	C₂³	1	4	yes	—
24	C₂²×D₂₁₂	—	2	16	no	—
25	C₂²×D₂₁₂	—	4	16	no	—
26	C₂²×D₂₁₂	C₂³	1	24	yes	—
27	C₂²×D₂₁₂	C₂³	1	48	yes	—
28	C₂³×Dic₅₃	C₂³	1	4	yes	—
29	C₂²×C₄×D₅₃	—	2	32	no	—
30	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
31	#13133206	C₂³	1	256	yes	complemented
32	#13133192	C₂³	1	256	yes	complemented
33	C₄×D₂₁₂	C₂³	1	128	yes	complemented
34	#13133175	C₂³	1	256	yes	complemented
35	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
36	C₂×#11719921	C₂³	1	96	yes	—
37	C₂×#11719921	C₂³	1	48	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	48	yes	—
39	C₂²×Q₈×C₅₃	—	4	8	no	—
40	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
47	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂³×D₅₃	—	2	16	no	—
49	C₂³×D₅₃	—	2	16	no	—
50	C₂³×D₅₃	—	2	16	no	—
51	C₂³×D₅₃	C₂⁴	1	24	yes	—
52	C₂×Q₈×C₅₃	—	4	48	no	—
53	C₂×Q₈×C₅₃	C₂⁴	1	32	yes	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	2	16	no	—
56	C₂²×C₄×C₅₃	#41	1	16	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
60	C₂²×Dic₅₃	—	2	16	no	—
61	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
62	C₂×D₂₁₂	—	2	96	no	—
63	C₂×D₂₁₂	—	4	96	no	—
64	C₂×D₂₁₂	C₂⁴	1	32	yes	—
65	C₂×D₂₁₂	C₂⁴	1	64	yes	—
66	C₂×C₄×D₅₃	—	2	192	no	—
67	C₂×C₄×D₅₃	C₂⁴	1	64	yes	—
68	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
69	C₄²×C₅₃	C₂⁴	1	16	yes	—
70	#11719921	C₂⁴	1	128	yes	—
71	#11719921	C₂⁴	1	64	yes	—
72	C₄⋊Dic₅₃	C₂⁴	1	32	yes	—
73	C₄×Dic₅₃	C₂⁴	1	64	yes	—
74	C₂³×C₅₃	C₂×#41	1	8	yes	—
75	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
76	C₂³×C₅₃	C₂⁵	1	3	yes	—
77	C₂²×D₅₃	—	2	48	no	—
78	C₂²×D₅₃	—	2	48	no	—
79	C₂²×D₅₃	—	2	48	no	—
80	C₂²×D₅₃	—	4	64	no	—
81	C₂²×D₅₃	C₂⁵	1	16	yes	—
82	Q₈×C₅₃	—	4	32	no	—
83	C₂×C₄×C₅₃	—	2	48	no	—
84	C₂×C₄×C₅₃	—	2	12	no	—
85	C₂×C₄×C₅₃	C₂×#41	1	48	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
87	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
88	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
89	C₂×Dic₅₃	—	2	48	no	—
90	C₂×Dic₅₃	C₂⁵	1	16	yes	—
91	D₂₁₂	—	2	64	no	—
92	D₂₁₂	—	4	64	no	—
93	C₄×D₅₃	—	2	128	no	—
94	C₂²×#555	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	#555	1	16	yes	complemented
96	C₂²×C₅₃	C₂²×#41	1	12	yes	—
97	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
98	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
99	C₂×D₅₃	—	2	16	no	—
100	C₂×D₅₃	—	2	16	no	—
101	C₂×D₅₃	—	2	16	no	—
102	C₂×D₅₃	—	4	96	no	—
103	C₄×C₅₃	—	2	16	no	—
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	C₂²×#41	1	16	yes	—
106	Dic₅₃	—	2	16	no	—
107	C₂×#555	—	53	48	no	—
108	C₂²×C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
109	C₂²×#124	—	53	4	no	—
110	C₂²×#121	—	53	4	no	—
111	C₂²×C₄×D₄	—	53	2	no	—
112	C₂²×#117	—	53	4	no	—
113	C₂×C₅₃	C₂×#555	1	12	yes	complemented
114	C₂×C₅₃	C₂³×#41	1	2	yes	—
115	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
116	D₅₃	—	4	16	no	—
117	C₂³×Q₈	C₂×D₅₃	1	2	yes	—
118	C₂³×D₄	—	53	2	no	—
119	C₂³×D₄	—	53	4	no	—
120	C₂⁴×C₄	—	53	4	no	—
121	#555	—	53	256	no	—
122	C₂×C₄⋊Q₈	C₂×D₅₃	1	24	yes	complemented
123	C₂×#124	—	53	96	no	—
124	C₂×#121	—	53	96	no	—
125	C₂×C₄×D₄	—	53	48	no	—
126	C₂×#117	—	53	96	no	—
127	C₂²×C₄⋊C₄	—	53	2	no	—
128	C₂²×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
129	C₂²×C₂²⋊C₄	—	53	8	no	—
130	C₂²×C₂²⋊C₄	—	53	4	no	—
131	C₂²×C₄²	—	53	4	no	—
132	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
133	C₅₃	C₂²×#555	1	1	yes	characteristic, complemented, Sylow
134	C₂⁵	—	53	4	no	—
135	C₂²×Q₈	—	4	8	no	—
136	C₂²×Q₈	C₂²×D₅₃	1	24	yes	—
137	C₂²×D₄	—	53	24	no	—
138	C₂²×D₄	—	53	48	no	—
139	C₂²×D₄	—	106	16	no	—
140	C₂²×D₄	—	212	16	no	—
141	C₂³×C₄	—	53	4	no	—
142	C₂³×C₄	—	53	48	no	—
143	C₂³×C₄	—	106	32	no	—
144	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
145	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
146	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
147	C₄⋊Q₈	C₂²×D₅₃	1	64	yes	complemented
148	#124	—	53	256	no	—
149	#121	—	53	256	no	—
150	C₄×D₄	—	53	128	no	—
151	#117	—	53	256	no	—
152	C₂×C₄⋊C₄	—	53	24	no	—
153	C₂×C₄⋊C₄	C₂²×D₅₃	1	48	yes	—
154	C₂×C₂²⋊C₄	—	53	96	no	—
155	C₂×C₂²⋊C₄	—	53	48	no	—
156	C₂×C₄²	—	53	48	no	—
157	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
158	C₂⁴	—	53	24	no	—
159	C₂⁴	—	106	16	no	—
160	C₂⁴	—	106	16	no	—
161	C₂⁴	—	106	16	no	—
162	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
163	C₂×Q₈	—	4	48	no	—
164	C₂×Q₈	C₂³×D₅₃	1	32	yes	—
165	C₂×D₄	—	53	32	no	—
166	C₂×D₄	—	53	64	no	—
167	C₂×D₄	—	106	96	no	—
168	C₂×D₄	—	212	96	no	—
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	53	24	no	—
172	C₂²×C₄	—	53	64	no	—
173	C₂²×C₄	—	106	16	no	—
174	C₂²×C₄	—	106	192	no	—
175	C₂²×C₄	#11719949	1	16	yes	—
176	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
177	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
178	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
179	C₄⋊C₄	—	53	32	no	—
180	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
181	C₂²⋊C₄	—	53	128	no	—
182	C₂²⋊C₄	—	53	64	no	—
183	C₄²	—	53	64	no	—
184	C₄²	C₂³×D₅₃	1	16	yes	—
185	C₂³	—	53	16	no	—
186	C₂³	—	106	48	no	—
187	C₂³	—	106	48	no	—
188	C₂³	—	106	48	no	—
189	C₂³	—	212	64	no	—
190	C₂³	C₂×#11719949	1	8	yes	—
191	C₂³	#13133262	1	4	yes	—
192	C₂³	C₂⁴×D₅₃	1	3	yes	—
193	Q₈	—	4	32	no	—
194	D₄	—	106	64	no	—
195	D₄	—	212	64	no	—
196	C₂×C₄	—	2	48	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	53	16	no	—
199	C₂×C₄	—	106	48	no	—
200	C₂×C₄	—	106	128	no	—
201	C₂×C₄	C₂×#11719949	1	48	yes	—
202	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
203	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
204	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
205	C₂²	—	106	16	no	—
206	C₂²	—	106	16	no	—
207	C₂²	—	106	16	no	—
208	C₂²	—	212	96	no	—
209	C₂²	#16811339	1	16	yes	direct product factor
210	C₂²	C₂²×#11719949	1	12	yes	—
211	C₂²	C₂×#13133262	1	6	yes	—
212	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
213	C₄	—	2	16	no	—
214	C₄	—	2	4	no	—
215	C₄	—	106	16	no	—
216	C₄	C₂²×#11719949	1	16	yes	—
217	C₂	—	212	16	no	—
218	C₂	C₂×#16811339	1	12	yes	direct product factor
219	C₂	C₂³×#11719949	1	2	yes	—
220	C₂	C₂²×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄⋊Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂²×#555
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16811339	C₂	direct (N × H)
2. C₂²×#13133206	C₂	semidirect
3. C₂²×#13133192	C₂	semidirect
4. C₂²×C₄×D₂₁₂	C₂	semidirect
5. C₂²×#13133175	C₂	semidirect
6. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
10. #16811339	C₂²	direct (N × H)
11. C₂×#13133206	C₂²	semidirect
12. C₂×#13133192	C₂²	semidirect
13. C₂×C₄×D₂₁₂	C₂²	semidirect
14. C₂×#13133175	C₂²	semidirect
20. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
31. #13133206	C₂³	semidirect
32. #13133192	C₂³	semidirect
33. C₄×D₂₁₂	C₂³	semidirect
34. #13133175	C₂³	semidirect
44. C₄⋊Q₈×C₅₃	C₂³	semidirect
95. C₂²×C₅₃	#555	semidirect
108. C₂²×C₄⋊Q₈	D₅₃	semidirect
113. C₂×C₅₃	C₂×#555	semidirect
122. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
133. C₅₃	C₂²×#555	semidirect
147. C₄⋊Q₈	C₂²×D₅₃	semidirect
209. C₂²	#16811339	direct (N × H)
218. C₂	C₂×#16811339	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×#16811339	2
C₂²×#13133206	2
C₂²×#13133192	2
C₂²×C₄×D₂₁₂	2
C₂²×#13133175	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×#555	53

Subgroups — C22×#16811339 #28291112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811339 #28291112