Subgroups — C₂×#20274889 #27384551

Order 13568.

1006 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133152	C₂	1	4	yes	maximal
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133128	C₂	1	1	yes	maximal, characteristic
4	#20274889	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂³×#88322	C₂²	1	2	yes	—
8	C₂×#13133152	—	2	16	no	—
9	C₂×#13133152	C₂²	1	16	yes	—
10	#16810647	—	2	2	no	—
11	#16810647	C₂²	1	4	yes	—
12	C₂×C₄⋊#88322	—	2	2	no	—
13	C₂×#13133128	C₄	1	4	yes	—
14	C₂×#13133128	C₄	1	4	yes	—
15	C₂×#13133128	C₂²	1	4	yes	—
16	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#423	—	2	2	no	—
19	C₅₃×#423	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	8	no	—
23	C₂×#11719917	C₂³	1	4	yes	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₂²×#88322	C₂×C₄	1	4	yes	—
27	C₂²×#88322	D₄	1	8	yes	—
28	C₂²×#88322	Q₈	1	8	yes	—
29	C₂²×#88322	C₂³	1	4	yes	—
30	#13133152	—	2	32	no	—
31	C₄⋊#88322	—	2	8	no	—
32	#13133128	C₂×C₄	1	16	yes	—
33	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	8	no	—
37	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
45	C₄⋊C₈×C₅₃	—	2	8	no	—
46	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	8	no	—
60	M₁₆×C₅₃	—	4	4	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	4	4	no	—
63	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
64	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
65	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
66	C₄²×C₅₃	C₂×D₄	1	4	yes	—
67	#11719917	—	2	8	no	—
68	#11719917	—	4	4	no	—
69	C₂×#88322	—	2	4	no	—
70	C₂×#88322	—	4	4	no	—
71	C₂×#88322	C₄⋊C₄	1	16	yes	—
72	C₂×#88322	C₄⋊C₄	1	16	yes	—
73	C₂×#88322	C₂²×C₄	1	8	yes	—
74	C₂×#88322	C₂×D₄	1	8	yes	—
75	C₂×#88322	C₂×Q₈	1	8	yes	—
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
80	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	—	4	4	no	—
83	C₂×C₄×C₅₃	—	4	4	no	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#127	1	2	yes	—
90	C₂×C₄×C₅₃	#127	1	2	yes	—
91	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
98	C₈×C₅₃	—	4	4	no	—
99	#88322	—	4	4	no	—
100	#88322	C₂×C₄⋊C₄	1	16	yes	—
101	C₂×#1849	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	8	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	#384	1	2	yes	—
106	C₂²×C₅₃	#384	1	2	yes	—
107	C₂²×C₅₃	#430	1	4	yes	—
108	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
110	C₄×C₅₃	—	2	8	no	—
111	C₄×C₅₃	#384	1	2	yes	—
112	C₄×C₅₃	#384	1	2	yes	—
113	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
114	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
115	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×#108	—	53	4	no	—
117	C₂×#423	—	53	1	no	—
118	C₂×#423	D₅₃	1	1	yes	characteristic
119	C₂²×#97	—	53	1	no	—
120	#1849	—	53	8	no	—
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	#1849	1	4	yes	complemented
124	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#430	1	2	yes	—
126	C₂²×M₁₆	—	53	2	no	—
127	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
128	C₂³×C₈	—	53	2	no	—
129	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	C₂×#108	—	53	16	no	—
131	C₂×#108	—	106	16	no	—
132	#423	—	2	2	no	—
133	#423	—	53	4	no	—
134	#423	—	106	2	no	—
135	#423	C₂×D₅₃	1	4	yes	—
136	C₂×C₄⋊C₈	—	2	2	no	—
137	C₂×C₄⋊C₈	—	106	2	no	—
138	C₂×#97	—	53	4	no	—
139	C₂×#97	—	53	4	no	—
140	C₂×#97	—	53	4	no	—
141	C₅₃	C₂×#1849	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	Dic₁₀₆	1	2	yes	—
143	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	—	53	4	no	—
147	C₂×M₁₆	—	106	2	no	—
148	C₂×M₁₆	—	106	8	no	—
149	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	53	4	no	—
152	C₂²×C₈	—	53	4	no	—
153	C₂²×C₈	—	53	8	no	—
154	C₂²×C₈	—	53	8	no	—
155	C₂²×C₈	—	53	4	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂×C₄²	Dic₁₀₆	1	2	yes	—
158	C₂×C₄²	C₄×D₅₃	1	2	yes	—
159	C₂×C₄²	C₄×D₅₃	1	2	yes	—
160	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
161	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
162	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
163	#108	—	106	32	no	—
164	C₄⋊C₈	—	2	8	no	—
165	C₄⋊C₈	—	106	8	no	—
166	#97	—	53	16	no	—
167	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	#11719919	1	4	yes	—
170	C₂²×C₄	#11719919	1	4	yes	—
171	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
172	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
175	C₂²×C₄	D₄×D₅₃	1	4	yes	—
176	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
180	M₁₆	—	2	8	no	—
181	M₁₆	—	4	4	no	—
182	M₁₆	—	106	8	no	—
183	M₁₆	—	212	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	4	no	—
186	C₂×C₈	—	53	16	no	—
187	C₂×C₈	—	53	16	no	—
188	C₂×C₈	—	53	8	no	—
189	C₂×C₈	—	53	8	no	—
190	C₂×C₈	—	53	8	no	—
191	C₂×C₈	—	106	4	no	—
192	C₂×C₈	—	212	4	no	—
193	C₄²	#11719919	1	4	yes	—
194	C₄²	#11719919	1	4	yes	—
195	C₄²	C₂×C₄×D₅₃	1	4	yes	—
196	C₄²	C₂×C₂²⋊D₅₃	1	4	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	C₂×#11719919	1	1	yes	characteristic
199	C₂³	C₂×#11719919	1	1	yes	characteristic
200	C₂³	#13133216	1	4	yes	—
201	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
206	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
207	C₂×C₄	C₂×#11719919	1	8	yes	—
208	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
210	C₂×C₄	#13133213	1	2	yes	—
211	C₂×C₄	#13133213	1	2	yes	—
212	C₂×C₄	#13133215	1	2	yes	—
213	C₂×C₄	#13133215	1	2	yes	—
214	C₂×C₄	#13133216	1	4	yes	—
215	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
217	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
219	C₈	—	4	4	no	—
220	C₈	—	53	16	no	—
221	C₈	—	212	4	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	8	no	—
225	C₂²	#16810492	1	2	yes	—
226	C₂²	#16810492	1	2	yes	—
227	C₂²	#16810671	1	4	yes	—
228	C₂²	C₂²×#11719919	1	1	yes	characteristic
229	C₂²	C₂×#13133216	1	2	yes	—
230	C₄	—	2	8	no	—
231	C₄	#16810492	1	2	yes	—
232	C₄	#16810492	1	2	yes	—
233	C₄	C₂×C₄⋊C₄×D₅₃	1	2	yes	—
234	C₄	C₂×#13133213	1	1	yes	characteristic
235	C₄	C₂×#13133215	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	2	2	no	—
238	C₂	#20274889	1	4	yes	direct product factor
239	C₂	C₂×#16810492	1	1	yes	characteristic
240	C₂	C₂×#16810671	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	C₂×#1849
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20274889	C₂	direct (N × H)
123. C₂×C₅₃	#1849	semidirect
141. C₅₃	C₂×#1849	semidirect
238. C₂	#20274889	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×#13133152	2
C₂×#16810647	2
C₂²×#13133128	2
#20274889	2
C₂×C₅₃×#423	2
C₂×#1849	53

Subgroups — C2×#20274889 #27384551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274889 #27384551