Subgroups — C₂²×#16811026 #28253688

Order 13568.

2398 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811026	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133150	C₂	1	4	yes	maximal
4	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₆	—	2	1	no	—
7	C₂³×#88322	C₂²	1	2	yes	—
8	C₂×#13133217	—	2	6	no	—
9	C₂×#13133217	C₂²	1	12	yes	—
10	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	#16811026	C₂²	1	64	yes	direct product factor
13	C₂×#13133150	C₂²	1	48	yes	—
14	C₂×C₄×#88322	C₂²	1	12	yes	—
15	C₂³×Q₈×C₅₃	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₅₃	—	2	6	no	—
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	12	yes	—
18	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₀₆	—	2	12	no	—
22	C₂²×Dic₁₀₆	—	4	8	no	—
23	C₂³×Dic₅₃	—	2	2	no	—
24	C₂²×#88322	Q₈	1	16	yes	complemented
25	C₂²×#88322	C₂³	1	12	yes	—
26	#13133217	—	2	24	no	—
27	#13133217	C₂³	1	16	yes	—
28	C₂×C₄⋊Dic₅₃	—	2	12	no	—
29	C₂×C₄⋊Dic₅₃	—	2	6	no	—
30	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
31	#13133150	C₂³	1	64	yes	—
32	C₄×#88322	C₂³	1	16	yes	—
33	C₂²×Q₈×C₅₃	—	2	12	no	—
34	C₂²×Q₈×C₅₃	—	4	8	no	—
35	C₂³×C₄×C₅₃	—	2	2	no	—
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₅₃	—	2	24	no	—
40	C₄⋊Q₈×C₅₃	C₂³	1	16	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
44	C₂×C₄²×C₅₃	D₄	1	6	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
46	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Q₈×C₅₃	—	2	16	no	—
48	C₂×Q₈×C₅₃	—	4	48	no	—
49	C₂²×C₄×C₅₃	—	2	12	no	—
50	C₂²×C₄×C₅₃	—	4	8	no	—
51	C₂²×C₄×C₅₃	D₈	1	8	yes	—
52	C₂²×C₄×C₅₃	Q₁₆	1	8	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
58	C₂²×Dic₅₃	—	2	12	no	—
59	C₂²×Dic₅₃	—	4	8	no	—
60	C₂×Dic₁₀₆	—	2	16	no	—
61	C₂×Dic₁₀₆	—	4	48	no	—
62	C₄⋊C₄×C₅₃	—	2	16	no	—
63	C₄⋊C₄×C₅₃	—	2	12	no	—
64	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
65	C₄²×C₅₃	C₂×D₄	1	12	yes	—
66	C₄²×C₅₃	C₂⁴	1	4	yes	—
67	C₄⋊Dic₅₃	—	2	16	no	—
68	C₄⋊Dic₅₃	—	2	12	no	—
69	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
70	C₂×#88322	C₂×Q₈	1	48	yes	complemented
71	C₂×#88322	C₂⁴	1	8	yes	—
72	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
73	C₂³×C₅₃	C₂×D₈	1	4	yes	—
74	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
75	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
76	Q₈×C₅₃	—	4	32	no	—
77	C₂×C₄×C₅₃	—	2	8	no	—
78	C₂×C₄×C₅₃	—	4	24	no	—
79	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	24	yes	—
81	C₂×C₄×C₅₃	C₂×Q₁₆	1	24	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
85	C₂×C₄×C₅₃	C₂²×Q₈	1	12	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	8	no	—
88	C₂×Dic₅₃	—	4	24	no	—
89	Dic₁₀₆	—	4	32	no	—
90	#88322	C₂²×Q₈	1	16	yes	complemented
91	C₂²×#500	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	#500	1	16	yes	complemented
93	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	6	yes	—
95	C₂²×C₅₃	C₂²×Q₁₆	1	6	yes	—
96	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
97	C₄×C₅₃	—	4	8	no	—
98	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
99	C₄×C₅₃	C₂²×D₈	1	8	yes	—
100	C₄×C₅₃	C₂²×Q₁₆	1	8	yes	—
101	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
102	Dic₅₃	—	4	8	no	—
103	C₂²×C₄⋊Q₈	—	53	1	no	—
104	C₂²×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
105	C₂×#500	—	53	24	no	—
106	C₂²×#107	—	53	4	no	—
107	C₂²×C₄×C₈	—	53	1	no	—
108	C₂×C₅₃	C₂×#500	1	12	yes	complemented
109	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
111	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
112	C₂³×Q₈	—	2	1	no	—
113	C₂³×Q₈	—	106	1	no	—
114	C₂³×C₈	—	53	2	no	—
115	C₂×C₄⋊Q₈	—	2	6	no	—
116	C₂×C₄⋊Q₈	—	53	12	no	—
117	C₂×C₄⋊Q₈	—	106	6	no	—
118	C₂×C₄⋊Q₈	C₂×D₅₃	1	12	yes	—
119	C₂²×C₄⋊C₄	—	2	1	no	—
120	C₂²×C₄⋊C₄	—	53	2	no	—
121	C₂²×C₄⋊C₄	—	106	1	no	—
122	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
123	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
124	#500	—	53	64	no	—
125	C₂×#107	—	53	48	no	—
126	C₂×C₄×C₈	—	53	12	no	—
127	C₅₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	2	12	no	—
129	C₂²×Q₈	—	4	8	no	—
130	C₂²×Q₈	—	106	12	no	—
131	C₂²×Q₈	—	212	8	no	—
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	—	106	2	no	—
134	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
137	C₂²×C₈	—	53	16	no	—
138	C₂²×C₈	—	53	12	no	—
139	C₄⋊Q₈	—	2	24	no	—
140	C₄⋊Q₈	—	53	16	no	—
141	C₄⋊Q₈	—	106	24	no	—
142	C₄⋊Q₈	C₂²×D₅₃	1	16	yes	—
143	C₂×C₄⋊C₄	—	2	12	no	—
144	C₂×C₄⋊C₄	—	2	6	no	—
145	C₂×C₄⋊C₄	—	53	12	no	—
146	C₂×C₄⋊C₄	—	106	6	no	—
147	C₂×C₄⋊C₄	—	106	12	no	—
148	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
149	C₂×C₄²	C₂²⋊D₅₃	1	6	yes	—
150	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
151	#107	—	53	64	no	—
152	C₄×C₈	—	53	16	no	—
153	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂×Q₈	—	2	16	no	—
155	C₂×Q₈	—	4	48	no	—
156	C₂×Q₈	—	106	16	no	—
157	C₂×Q₈	—	212	48	no	—
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	—	106	12	no	—
161	C₂²×C₄	—	212	8	no	—
162	C₂²×C₄	#11719922	1	8	yes	—
163	C₂²×C₄	#11719925	1	8	yes	—
164	C₂²×C₄	Q₈×D₅₃	1	8	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
166	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
167	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
168	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
169	C₂×C₈	—	53	48	no	—
170	C₂×C₈	—	53	8	no	—
171	C₄⋊C₄	—	2	16	no	—
172	C₄⋊C₄	—	2	12	no	—
173	C₄⋊C₄	—	53	8	no	—
174	C₄⋊C₄	—	106	12	no	—
175	C₄⋊C₄	—	106	16	no	—
176	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
177	C₄²	C₂×C₂²⋊D₅₃	1	12	yes	—
178	C₄²	C₂³×D₅₃	1	4	yes	—
179	C₂³	#13133215	1	4	yes	—
180	C₂³	C₂×#11719922	1	4	yes	—
181	C₂³	C₂×#11719925	1	4	yes	—
182	C₂³	C₂²×C₂²⋊D₅₃	1	3	yes	—
183	Q₈	—	4	32	no	—
184	Q₈	—	212	32	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	4	24	no	—
187	C₂×C₄	—	106	8	no	—
188	C₂×C₄	—	212	24	no	—
189	C₂×C₄	#13133215	1	12	yes	—
190	C₂×C₄	C₂×#11719922	1	24	yes	—
191	C₂×C₄	C₂×#11719925	1	24	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₅₃	1	4	yes	—
193	C₂×C₄	C₂²×C₂²⋊D₅₃	1	4	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₅₃	1	3	yes	—
195	C₂×C₄	C₂×Q₈×D₅₃	1	12	yes	—
196	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
197	C₈	—	53	16	no	—
198	C₂²	#16811026	1	16	yes	direct product factor
199	C₂²	C₂×#13133215	1	6	yes	—
200	C₂²	C₂²×#11719922	1	6	yes	—
201	C₂²	C₂²×#11719925	1	6	yes	—
202	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄	—	4	8	no	—
204	C₄	—	212	8	no	—
205	C₄	C₂×#13133215	1	6	yes	—
206	C₄	C₂²×#11719922	1	8	yes	—
207	C₄	C₂²×#11719925	1	8	yes	—
208	C₄	C₂²×Q₈×D₅₃	1	2	yes	—
209	C₂	C₂×#16811026	1	12	yes	direct product factor
210	C₂	C₂²×#13133215	1	1	yes	characteristic
211	C₂	C₂³×#11719922	1	1	yes	characteristic
212	C₂	C₂³×#11719925	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂²×#500
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16811026	C₂	direct (N × H)
12. #16811026	C₂²	direct (N × H)
24. C₂²×#88322	Q₈	semidirect
70. C₂×#88322	C₂×Q₈	semidirect
90. #88322	C₂²×Q₈	semidirect
92. C₂²×C₅₃	#500	semidirect
108. C₂×C₅₃	C₂×#500	semidirect
127. C₅₃	C₂²×#500	semidirect
198. C₂²	#16811026	direct (N × H)
209. C₂	C₂×#16811026	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×#13133217	2
C₂×#16811026	2
C₂²×#13133150	2
C₂²×C₄×#88322	2
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×#500	53

Subgroups — C22×#16811026 #28253688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811026 #28253688