Subgroups — C₂²×#16810753 #28248449

Order 13568.

2222 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810753	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13133146	C₂	1	2	yes	maximal
5	C₂²×#13133127	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	C₄	1	1	yes	characteristic
8	C₂³×#88322	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₀₆	C₄	1	12	yes	—
10	C₂×C₄×Dic₁₀₆	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
12	C₂²×#11719919	C₄	1	2	yes	—
13	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
14	#16810753	C₂²	1	64	yes	direct product factor
15	C₂×C₄⋊#88322	C₂²	1	12	yes	—
16	C₂×#13133146	C₂²	1	24	yes	—
17	C₂×#13133127	C₂²	1	24	yes	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	—
22	C₂²×Dic₁₀₆	—	2	8	no	—
23	C₂²×Dic₁₀₆	C₂×C₄	1	12	yes	—
24	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂²×#88322	—	2	8	no	—
27	C₂²×#88322	C₂³	1	12	yes	—
28	C₄×Dic₁₀₆	C₂×C₄	1	48	yes	—
29	C₄×Dic₁₀₆	C₂³	1	16	yes	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	12	yes	—
31	C₂×#11719919	C₂×C₄	1	24	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
34	C₄⋊#88322	C₂³	1	16	yes	—
35	#13133146	C₂³	1	32	yes	—
36	#13133127	C₂³	1	32	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	—	2	4	no	—
41	C₂²×C₈×C₅₃	Q₈	1	8	yes	complemented
42	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
46	C₄×C₈×C₅₃	C₂³	1	16	yes	—
47	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₅₃	#41	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
56	C₂²×Dic₅₃	—	2	8	no	—
57	C₂²×Dic₅₃	—	2	8	no	—
58	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
59	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
60	C₂×Dic₁₀₆	—	2	48	no	—
61	C₂×Dic₁₀₆	C₂²×C₄	1	16	yes	—
62	C₂×C₈×C₅₃	—	2	12	no	—
63	C₂×C₈×C₅₃	C₂×Q₈	1	24	yes	complemented
64	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
65	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
66	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
67	C₄²×C₅₃	C₂⁴	1	4	yes	—
68	C₄⋊Dic₅₃	C₂²×C₄	1	16	yes	—
69	#11719919	C₂²×C₄	1	32	yes	—
70	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
71	C₄×Dic₅₃	C₂⁴	1	8	yes	—
72	C₂×#88322	—	2	24	no	—
73	C₂×#88322	C₂⁴	1	8	yes	—
74	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
75	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
76	C₂³×C₅₃	#131	1	4	yes	—
77	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
78	C₂×C₄×C₅₃	—	2	12	no	—
79	C₂×C₄×C₅₃	C₄×Q₈	1	12	yes	—
80	C₂×C₄×C₅₃	C₂×M₁₆	1	24	yes	—
81	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
82	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
85	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	24	no	—
88	C₂×Dic₅₃	—	2	24	no	—
89	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
90	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
91	Dic₁₀₆	—	2	32	no	—
92	C₈×C₅₃	—	2	4	no	—
93	C₈×C₅₃	C₂²×Q₈	1	8	yes	complemented
94	#88322	—	2	8	no	—
95	C₂²×#446	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#446	1	16	yes	complemented
97	C₂²×C₅₃	C₂×C₄×Q₈	1	6	yes	—
98	C₂²×C₅₃	C₂²×M₁₆	1	6	yes	—
99	C₂²×C₅₃	C₂×#131	1	6	yes	—
100	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₅₃	—	2	4	no	—
102	C₄×C₅₃	C₂×C₄×Q₈	1	6	yes	—
103	C₄×C₅₃	C₂²×M₁₆	1	8	yes	—
104	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
105	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
106	Dic₅₃	—	2	8	no	—
107	Dic₅₃	—	2	8	no	—
108	C₂²×C₄×Q₈	—	53	1	no	—
109	C₂×#446	—	53	24	no	—
110	C₂²×C₄⋊C₈	—	53	1	no	—
111	C₂²×C₄⋊C₈	—	53	2	no	—
112	C₂²×#97	—	53	2	no	—
113	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#446	1	12	yes	complemented
115	C₂×C₅₃	C₂²×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
117	C₂×C₅₃	C₂²×#131	1	1	yes	characteristic
118	C₂³×Q₈	—	53	1	no	—
119	C₂³×C₈	—	53	2	no	—
120	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	53	12	no	—
123	C₂×C₄×Q₈	—	53	12	no	—
124	C₂²×C₄⋊C₄	—	53	2	no	—
125	C₂²×C₄⋊C₄	—	53	1	no	—
126	C₂²×C₄²	—	53	2	no	—
127	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
128	#446	—	53	64	no	—
129	C₂×C₄⋊C₈	—	53	24	no	—
130	C₂×C₄⋊C₈	—	53	12	no	—
131	C₂×#97	—	53	24	no	—
132	C₂×C₄×C₈	C₂×D₅₃	1	12	yes	—
133	C₅₃	C₂²×#446	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₈	—	53	12	no	—
135	C₂²×Q₈	—	106	8	no	—
136	C₂³×C₄	—	53	2	no	—
137	C₂³×C₄	—	53	2	no	—
138	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
139	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	53	12	no	—
143	C₂²×C₈	—	106	8	no	—
144	C₂²×C₈	Dic₁₀₆	1	8	yes	complemented
145	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
146	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
147	C₄×Q₈	—	53	48	no	—
148	C₄×Q₈	—	53	16	no	—
149	C₂×C₄⋊C₄	—	53	24	no	—
150	C₂×C₄⋊C₄	—	53	12	no	—
151	C₂×C₄²	—	53	12	no	—
152	C₂×C₄²	—	53	12	no	—
153	C₂×C₄²	C₄×D₅₃	1	6	yes	—
154	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
155	C₄⋊C₈	—	53	32	no	—
156	C₄⋊C₈	—	53	16	no	—
157	#97	—	53	32	no	—
158	C₄×C₈	C₂²×D₅₃	1	16	yes	—
159	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
160	C₂×Q₈	—	53	16	no	—
161	C₂×Q₈	—	106	48	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	53	12	no	—
164	C₂²×C₄	—	53	12	no	—
165	C₂²×C₄	—	106	8	no	—
166	C₂²×C₄	—	106	8	no	—
167	C₂²×C₄	#11719912	1	8	yes	—
168	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
169	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
170	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
171	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
172	C₂²×C₄	#11719945	1	4	yes	—
173	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
174	C₂×C₈	—	2	12	no	—
175	C₂×C₈	—	53	8	no	—
176	C₂×C₈	—	106	24	no	—
177	C₂×C₈	C₂×Dic₁₀₆	1	24	yes	complemented
178	C₂×C₈	C₂³×D₅₃	1	4	yes	—
179	C₂×C₈	C₂³×D₅₃	1	4	yes	—
180	C₄⋊C₄	—	53	32	no	—
181	C₄⋊C₄	—	53	16	no	—
182	C₄²	—	53	24	no	—
183	C₄²	—	53	8	no	—
184	C₄²	C₂×C₄×D₅₃	1	12	yes	—
185	C₄²	C₂³×D₅₃	1	4	yes	—
186	C₂³	C₄×Dic₁₀₆	1	4	yes	—
187	C₂³	C₂×#11719912	1	4	yes	—
188	C₂³	#13133223	1	4	yes	—
189	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
190	Q₈	—	106	32	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	53	8	no	—
193	C₂×C₄	—	53	8	no	—
194	C₂×C₄	—	106	24	no	—
195	C₂×C₄	—	106	24	no	—
196	C₂×C₄	C₄×Dic₁₀₆	1	12	yes	—
197	C₂×C₄	C₂×#11719912	1	24	yes	—
198	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
199	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
200	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
201	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
202	C₂×C₄	C₂×#11719945	1	6	yes	—
203	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
204	C₈	—	2	4	no	—
205	C₈	—	106	8	no	—
206	C₈	C₂²×Dic₁₀₆	1	8	yes	complemented
207	C₂²	#16810753	1	16	yes	direct product factor
208	C₂²	C₂×C₄×Dic₁₀₆	1	6	yes	—
209	C₂²	C₂²×#11719912	1	6	yes	—
210	C₂²	C₂×#13133223	1	6	yes	—
211	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	106	8	no	—
214	C₄	—	106	8	no	—
215	C₄	C₂×C₄×Dic₁₀₆	1	6	yes	—
216	C₄	C₂²×#11719912	1	8	yes	—
217	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
218	C₄	C₂²×#11719945	1	1	yes	characteristic
219	C₂	C₂×#16810753	1	12	yes	direct product factor
220	C₂	C₂²×C₄×Dic₁₀₆	1	1	yes	characteristic
221	C₂	C₂³×#11719912	1	1	yes	characteristic
222	C₂	C₂²×#13133223	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂²×#446
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810753	C₂	direct (N × H)
14. #16810753	C₂²	direct (N × H)
41. C₂²×C₈×C₅₃	Q₈	semidirect
63. C₂×C₈×C₅₃	C₂×Q₈	semidirect
93. C₈×C₅₃	C₂²×Q₈	semidirect
96. C₂²×C₅₃	#446	semidirect
114. C₂×C₅₃	C₂×#446	semidirect
133. C₅₃	C₂²×#446	semidirect
144. C₂²×C₈	Dic₁₀₆	semidirect
177. C₂×C₈	C₂×Dic₁₀₆	semidirect
206. C₈	C₂²×Dic₁₀₆	semidirect
207. C₂²	#16810753	direct (N × H)
219. C₂	C₂×#16810753	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×Dic₁₀₆	2
C₂×#16810753	2
C₂²×C₄⋊#88322	2
C₂²×#13133146	2
C₂²×#13133127	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#446	53

Subgroups — C22×#16810753 #28248449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810753 #28248449