Subgroups — C₂×#20272542 #27273471

Order 13568.

982 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×#88322	C₂	1	2	yes	maximal
2	#20272542	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₄	1	4	yes	—
5	C₂×C₄×#88322	—	2	4	no	—
6	C₂×C₄×#88322	—	2	4	no	—
7	C₂×C₄×#88322	C₄	1	4	yes	—
8	C₂×C₄×#88322	C₂²	1	4	yes	—
9	C₂³×C₈×C₅₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	—	2	4	no	—
12	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂²×#88322	—	2	8	no	—
15	C₂²×#88322	—	2	8	no	—
16	C₂²×#88322	C₈	1	16	yes	complemented
17	C₂²×#88322	C₂×C₄	1	8	yes	—
18	C₄×#88322	—	2	8	no	—
19	C₄×#88322	—	2	4	no	—
20	C₄×#88322	—	2	4	no	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	C₈	1	8	yes	complemented
26	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	4	yes	—
31	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₅₃	—	2	4	no	—
34	C₄×C₈×C₅₃	—	2	4	no	—
35	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
37	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
45	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	16	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
52	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×#88322	—	2	32	no	—
60	C₂×#88322	—	2	8	no	—
61	C₂×#88322	—	2	8	no	—
62	C₂×#88322	C₂×C₈	1	16	yes	complemented
63	C₂³×C₅₃	#95	1	2	yes	—
64	C₂³×C₅₃	C₄×C₈	1	4	yes	—
65	C₂³×C₅₃	C₄×C₈	1	2	yes	—
66	C₂³×C₅₃	#97	1	4	yes	—
67	C₂³×C₅₃	#97	1	2	yes	—
68	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#95	1	4	yes	—
70	C₂×C₄×C₅₃	#95	1	2	yes	—
71	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
72	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
73	C₂×C₄×C₅₃	#97	1	4	yes	—
74	C₂×C₄×C₅₃	#97	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²⋊C₈	1	16	yes	—
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
77	C₂×C₄×C₅₃	C₄⋊C₈	1	16	yes	—
78	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
86	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
87	C₈×C₅₃	—	2	8	no	—
88	#88322	—	2	16	no	—
89	C₂×#1273	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	C₈²	1	4	yes	—
91	C₂²×C₅₃	#322	1	4	yes	—
92	C₂²×C₅₃	#322	1	8	yes	—
93	C₂²×C₅₃	#336	1	8	yes	—
94	C₂²×C₅₃	#336	1	4	yes	—
95	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
96	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
97	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₂×#97	1	2	yes	—
99	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
100	C₄×C₅₃	#336	1	8	yes	—
101	C₄×C₅₃	#336	1	4	yes	—
102	C₄×C₅₃	C₂×C₂²⋊C₈	1	4	yes	—
103	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
104	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
105	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
106	C₂²×C₄×C₈	—	53	2	no	—
107	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
108	#1273	—	53	4	no	—
109	C₂×C₅₃	#1273	1	8	yes	complemented
110	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#322	1	2	yes	—
113	C₂×C₅₃	C₂×#336	1	2	yes	—
114	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
115	C₂³×C₈	—	53	4	no	—
116	C₂³×C₈	Dic₅₃	1	2	yes	—
117	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	—	2	4	no	—
119	C₂×C₄×C₈	—	53	4	no	—
120	C₂×C₄×C₈	—	53	4	no	—
121	C₂×C₄×C₈	—	106	4	no	—
122	C₂×C₄×C₈	—	106	4	no	—
123	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
125	C₅₃	C₂×#1273	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	C₄×D₅₃	1	2	yes	—
127	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
128	C₂²×C₈	—	2	4	no	—
129	C₂²×C₈	—	2	4	no	—
130	C₂²×C₈	—	53	8	no	—
131	C₂²×C₈	—	53	16	no	—
132	C₂²×C₈	—	106	8	no	—
133	C₂²×C₈	—	106	8	no	—
134	C₂²×C₈	#88322	1	8	yes	complemented
135	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
136	C₂×C₄²	Dic₁₀₆	1	2	yes	—
137	C₂×C₄²	C₄×D₅₃	1	2	yes	—
138	C₂×C₄²	D₂₁₂	1	2	yes	—
139	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
141	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
142	C₄×C₈	—	2	4	no	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	—	106	4	no	—
145	C₄×C₈	—	106	4	no	—
146	C₄×C₈	—	106	8	no	—
147	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₈×D₅₃	1	8	yes	—
149	C₂²×C₄	#11719912	1	8	yes	—
150	C₂²×C₄	C₂×#88322	1	4	yes	—
151	C₂²×C₄	#11719917	1	4	yes	—
152	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
153	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₄	#11719919	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
156	C₂²×C₄	#11719921	1	4	yes	—
157	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
158	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
159	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	53	16	no	—
164	C₂×C₈	—	106	32	no	—
165	C₂×C₈	—	106	8	no	—
166	C₂×C₈	—	106	8	no	—
167	C₂×C₈	C₂×#88322	1	8	yes	complemented
168	C₄²	#11719919	1	4	yes	—
169	C₄²	C₄⋊Dic₅₃	1	2	yes	—
170	C₄²	#11719921	1	4	yes	—
171	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
172	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
173	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
175	C₂³	#13133124	1	2	yes	—
176	C₂³	C₈×Dic₅₃	1	4	yes	—
177	C₂³	C₄×#88322	1	2	yes	—
178	C₂³	#13133127	1	4	yes	—
179	C₂³	#13133128	1	2	yes	—
180	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	#13133124	1	4	yes	—
182	C₂×C₄	#13133124	1	2	yes	—
183	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
184	C₂×C₄	C₄×#88322	1	2	yes	—
185	C₂×C₄	#13133127	1	4	yes	—
186	C₂×C₄	#13133128	1	2	yes	—
187	C₂×C₄	#13133129	1	16	yes	—
188	C₂×C₄	C₂²⋊#88322	1	8	yes	—
189	C₂×C₄	#13133146	1	16	yes	—
190	C₂×C₄	C₄⋊#88322	1	8	yes	—
191	C₂×C₄	C₂×#11719921	1	2	yes	—
192	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719919	1	2	yes	—
194	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
196	C₂×C₄	C₂²×#88322	1	2	yes	—
197	C₂×C₄	C₂×#11719912	1	4	yes	—
198	C₂×C₄	C₂×#11719917	1	2	yes	—
199	C₈	—	2	8	no	—
200	C₈	—	106	16	no	—
201	C₂²	C₈×#88322	1	4	yes	—
202	C₂²	#16810344	1	4	yes	—
203	C₂²	#16810345	1	8	yes	—
204	C₂²	#16810377	1	8	yes	—
205	C₂²	#16810378	1	4	yes	—
206	C₂²	C₂×#13133124	1	1	yes	characteristic
207	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
208	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
209	C₂²	C₂×#13133127	1	2	yes	—
210	C₂²	C₂×#13133128	1	1	yes	characteristic
211	C₄	#16810377	1	8	yes	—
212	C₄	#16810378	1	4	yes	—
213	C₄	C₂×#13133129	1	4	yes	—
214	C₄	C₂×C₂²⋊#88322	1	2	yes	—
215	C₄	C₂×#13133146	1	4	yes	—
216	C₄	C₂×C₄⋊#88322	1	2	yes	—
217	C₂	#20272542	1	8	yes	direct product factor
218	C₂	C₂×C₈×#88322	1	1	yes	characteristic
219	C₂	C₂×#16810344	1	1	yes	characteristic
220	C₂	C₂×#16810345	1	2	yes	—
221	C₂	C₂×#16810377	1	2	yes	—
222	C₂	C₂×#16810378	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1273
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20272542	C₂	direct (N × H)
16. C₂²×#88322	C₈	semidirect
25. C₂²×C₈×C₅₃	C₈	semidirect
51. C₂×C₈×C₅₃	C₂×C₈	semidirect
62. C₂×#88322	C₂×C₈	semidirect
109. C₂×C₅₃	#1273	semidirect
125. C₅₃	C₂×#1273	semidirect
134. C₂²×C₈	#88322	semidirect
167. C₂×C₈	C₂×#88322	semidirect
217. C₂	#20272542	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×#88322	2
#20272542	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1273	53

Subgroups — C2×#20272542 #27273471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272542 #27273471