Subgroups — C₂×#20272550 #27273548

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#20272550	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	—	2	2	no	—
5	C₂×C₄⋊#88322	—	2	4	no	—
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₄	1	8	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	4	no	—
16	C₂²×#88322	—	4	4	no	—
17	C₂²×#88322	—	4	4	no	—
18	C₄⋊#88322	—	2	8	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₄⋊#88322	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	—	2	4	no	—
26	C₂²×C₈×C₅₃	C₈	1	8	yes	complemented
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	4	yes	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₅₃	—	2	4	no	—
35	C₄×C₈×C₅₃	—	2	4	no	—
36	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
47	C₂²×C₄×C₅₃	D₈	1	2	yes	—
48	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
49	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₃	—	2	16	no	—
54	C₂×C₈×C₅₃	—	2	4	no	—
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
57	C₄²×C₅₃	C₄²	1	2	yes	—
58	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×#88322	—	2	8	no	—
64	C₂×#88322	—	4	16	no	—
65	C₂×#88322	—	4	4	no	—
66	C₂×#88322	—	4	4	no	—
67	C₂³×C₅₃	#95	1	2	yes	—
68	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
69	C₂³×C₅₃	#106	1	2	yes	—
70	C₂³×C₅₃	#107	1	2	yes	—
71	C₂³×C₅₃	#108	1	4	yes	—
72	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	#95	1	2	yes	—
77	C₂×C₄×C₅₃	#95	1	2	yes	—
78	C₂×C₄×C₅₃	#95	1	2	yes	—
79	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
80	C₂×C₄×C₅₃	#97	1	4	yes	—
81	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
82	C₂×C₄×C₅₃	#102	1	8	yes	—
83	C₂×C₄×C₅₃	#103	1	8	yes	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
85	C₂×C₄×C₅₃	#106	1	2	yes	—
86	C₂×C₄×C₅₃	#107	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
96	C₈×C₅₃	—	2	8	no	—
97	#88322	—	4	8	no	—
98	C₂×#1277	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	#334	1	8	yes	—
100	C₂²×C₅₃	#335	1	8	yes	—
101	C₂²×C₅₃	#336	1	4	yes	—
102	C₂²×C₅₃	#340	1	4	yes	—
103	C₂²×C₅₃	#341	1	4	yes	—
104	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
106	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#108	1	2	yes	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	#336	1	4	yes	—
112	C₄×C₅₃	#340	1	4	yes	—
113	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
114	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
115	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
116	C₄×C₅₃	C₂×#102	1	2	yes	—
117	C₄×C₅₃	C₂×#103	1	2	yes	—
118	C₂²×C₄⋊C₈	—	53	2	no	—
119	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
120	#1277	—	53	4	no	—
121	C₂×C₅₃	#1277	1	8	yes	complemented
122	C₂×C₅₃	C₂×#334	1	2	yes	—
123	C₂×C₅₃	C₂×#335	1	2	yes	—
124	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
127	C₂³×C₈	—	106	2	no	—
128	C₂³×C₈	Dic₅₃	1	2	yes	—
129	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	53	8	no	—
131	C₂×C₄⋊C₈	—	53	4	no	—
132	C₂×C₄⋊C₈	—	106	2	no	—
133	C₂×C₄⋊C₈	—	106	4	no	—
134	C₂×C₄×C₈	—	2	4	no	—
135	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
136	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
137	C₅₃	C₂×#1277	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
139	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
140	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	—	106	4	no	—
144	C₂²×C₈	—	106	8	no	—
145	C₂²×C₈	—	212	4	no	—
146	C₂²×C₈	—	212	4	no	—
147	C₂²×C₈	#88322	1	8	yes	complemented
148	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
149	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
150	C₂×C₄²	C₄×D₅₃	1	4	yes	—
151	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
154	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	53	8	no	—
156	C₄⋊C₈	—	106	4	no	—
157	C₄⋊C₈	—	106	8	no	—
158	C₄×C₈	—	2	4	no	—
159	C₄×C₈	—	2	4	no	—
160	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	#11719913	1	4	yes	—
163	C₂²×C₄	D₄₂₄	1	2	yes	—
164	C₂²×C₄	Dic₂₁₂	1	2	yes	—
165	C₂²×C₄	C₂×#88322	1	4	yes	—
166	C₂²×C₄	#11719917	1	4	yes	—
167	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
168	C₂²×C₄	#11719919	1	4	yes	—
169	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₂²×C₄	#11719921	1	4	yes	—
173	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
174	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
177	C₂×C₈	—	2	16	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	106	8	no	—
181	C₂×C₈	—	212	16	no	—
182	C₂×C₈	—	212	4	no	—
183	C₂×C₈	—	212	4	no	—
184	C₂×C₈	C₂×#88322	1	8	yes	complemented
185	C₄²	C₄×Dic₅₃	1	2	yes	—
186	C₄²	#11719919	1	4	yes	—
187	C₄²	#11719921	1	4	yes	—
188	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
189	C₄²	C₂×C₄×D₅₃	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
191	C₂³	#13133124	1	2	yes	—
192	C₂³	C₄⋊#88322	1	4	yes	—
193	C₂³	#13133149	1	2	yes	—
194	C₂³	#13133151	1	2	yes	—
195	C₂³	#13133153	1	4	yes	—
196	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	#13133124	1	2	yes	—
201	C₂×C₄	#13133124	1	2	yes	—
202	C₂×C₄	#13133124	1	2	yes	—
203	C₂×C₄	C₄×#88322	1	4	yes	—
204	C₂×C₄	#13133128	1	4	yes	—
205	C₂×C₄	C₂²⋊#88322	1	8	yes	—
206	C₂×C₄	#13133139	1	8	yes	—
207	C₂×C₄	#13133142	1	8	yes	—
208	C₂×C₄	C₄⋊#88322	1	4	yes	—
209	C₂×C₄	#13133149	1	2	yes	—
210	C₂×C₄	#13133151	1	2	yes	—
211	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂×#11719921	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂×#11719919	1	2	yes	—
215	C₂×C₄	C₂²×#88322	1	2	yes	—
216	C₂×C₄	C₂×#11719917	1	2	yes	—
217	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719913	1	2	yes	—
219	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
220	C₈	—	2	8	no	—
221	C₈	—	212	8	no	—
222	C₂²	#16810374	1	8	yes	—
223	C₂²	#16810376	1	8	yes	—
224	C₂²	#16810378	1	4	yes	—
225	C₂²	#16810386	1	4	yes	—
226	C₂²	#16810388	1	4	yes	—
227	C₂²	C₂×#13133124	1	1	yes	characteristic
228	C₂²	C₂×C₄⋊#88322	1	2	yes	—
229	C₂²	C₂×#13133149	1	1	yes	characteristic
230	C₂²	C₂×#13133151	1	1	yes	characteristic
231	C₂²	C₂×#13133153	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	#16810378	1	4	yes	—
235	C₄	#16810386	1	4	yes	—
236	C₄	C₂×C₄×#88322	1	1	yes	characteristic
237	C₄	C₂×#13133128	1	1	yes	characteristic
238	C₄	C₂×C₂²⋊#88322	1	2	yes	—
239	C₄	C₂×#13133139	1	2	yes	—
240	C₄	C₂×#13133142	1	2	yes	—
241	C₂	#20272550	1	8	yes	direct product factor
242	C₂	C₂×#16810374	1	2	yes	—
243	C₂	C₂×#16810376	1	2	yes	—
244	C₂	C₂×#16810378	1	1	yes	characteristic
245	C₂	C₂×#16810386	1	1	yes	characteristic
246	C₂	C₂×#16810388	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₁₂
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1277
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20272550	C₂	direct (N × H)
26. C₂²×C₈×C₅₃	C₈	semidirect
56. C₂×C₈×C₅₃	C₂×C₈	semidirect
121. C₂×C₅₃	#1277	semidirect
137. C₅₃	C₂×#1277	semidirect
147. C₂²×C₈	#88322	semidirect
184. C₂×C₈	C₂×#88322	semidirect
241. C₂	#20272550	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
#20272550	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1277	53

Subgroups — C2×#20272550 #27273548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272550 #27273548