Subgroups — C₂×#20274832 #27384235

Order 13568.

1246 conjugacy classes of subgroups.

The subgroup lattice has 246 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	#20274832	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810378	C₂	1	4	yes	maximal
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₂²	1	4	yes	—
6	C₂×C₄⋊#88322	—	2	8	no	—
7	C₂×C₄⋊#88322	C₂²	1	8	yes	—
8	#16810378	C₂²	1	16	yes	—
9	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄³×C₅₃	C₄	1	4	yes	—
15	C₄³×C₅₃	C₂²	1	4	yes	—
16	C₂²×#88322	—	2	8	no	—
17	C₂²×#88322	—	2	16	no	—
18	C₂²×#88322	C₂³	1	8	yes	—
19	C₄⋊#88322	—	2	16	no	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	—	2	8	no	—
26	C₂×C₄²×C₅₃	—	2	4	no	—
27	C₂×C₄²×C₅₃	—	2	8	no	—
28	C₂×C₄²×C₅₃	C₈	1	8	yes	complemented
29	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₃	D₄	1	4	yes	—
35	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
49	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₅₃	#41	1	8	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₄²×C₅₃	—	2	8	no	—
60	C₄²×C₅₃	—	2	16	no	—
61	C₄²×C₅₃	—	2	8	no	—
62	C₄²×C₅₃	—	2	8	no	—
63	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
64	C₄²×C₅₃	C₂×C₈	1	16	yes	complemented
65	C₄²×C₅₃	C₂×D₄	1	4	yes	—
66	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
67	C₂×#88322	—	2	8	no	—
68	C₂×#88322	—	2	16	no	—
69	C₂×#88322	—	4	16	no	—
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₅₃	#117	1	4	yes	—
72	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
73	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
75	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	—	2	16	no	—
83	C₂×C₄×C₅₃	—	2	16	no	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	16	yes	—
85	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₅₃	#117	1	4	yes	—
89	C₂×C₄×C₅₃	#124	1	8	yes	—
90	C₂×C₄×C₅₃	#125	1	8	yes	—
91	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
92	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
94	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
98	#88322	—	4	8	no	—
99	C₂×#1831	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	#382	1	4	yes	—
101	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
102	C₂²×C₅₃	#423	1	4	yes	—
103	C₂²×C₅₃	#431	1	8	yes	—
104	C₂²×C₅₃	#432	1	8	yes	—
105	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₅₃	C₂×#117	1	2	yes	—
107	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
108	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
109	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	—	2	8	no	—
112	C₄×C₅₃	—	2	8	no	—
113	C₄×C₅₃	#382	1	4	yes	—
114	C₄×C₅₃	C₂×C₄⋊C₈	1	8	yes	—
115	C₄×C₅₃	C₂×#124	1	2	yes	—
116	C₄×C₅₃	C₂×#125	1	2	yes	—
117	C₂²×C₄⋊C₈	—	53	2	no	—
118	C₂×C₄³	D₅₃	1	1	yes	characteristic
119	#1831	—	53	8	no	—
120	C₂×#336	—	53	4	no	—
121	C₂×C₅₃	#1831	1	8	yes	complemented
122	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
123	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
124	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#431	1	2	yes	—
126	C₂×C₅₃	C₂×#432	1	2	yes	—
127	C₂³×C₈	—	53	4	no	—
128	C₂²×C₄²	Dic₅₃	1	2	yes	—
129	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
130	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
131	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
132	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	53	8	no	—
134	C₂×C₄⋊C₈	—	106	8	no	—
135	C₄³	Dic₅₃	1	4	yes	—
136	C₄³	C₂×D₅₃	1	4	yes	—
137	#336	—	53	16	no	—
138	C₅₃	C₂×#1831	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
140	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
141	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
142	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
144	C₂²×C₈	—	53	8	no	—
145	C₂²×C₈	—	106	8	no	—
146	C₂²×C₈	—	106	16	no	—
147	C₂×C₄²	—	2	8	no	—
148	C₂×C₄²	—	2	8	no	—
149	C₂×C₄²	—	2	4	no	—
150	C₂×C₄²	#88322	1	8	yes	complemented
151	C₂×C₄²	Dic₁₀₆	1	4	yes	—
152	C₂×C₄²	D₂₁₂	1	4	yes	—
153	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
154	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
155	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
156	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
157	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
158	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
159	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
160	C₄⋊C₈	—	106	16	no	—
161	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂×#88322	1	8	yes	—
168	C₂²×C₄	C₂×#88322	1	4	yes	—
169	C₂²×C₄	#11719917	1	8	yes	—
170	C₂²×C₄	#11719917	1	4	yes	—
171	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
172	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
173	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
174	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
175	C₂²×C₄	#11719945	1	8	yes	—
176	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
179	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
181	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
182	C₂×C₈	—	106	8	no	—
183	C₂×C₈	—	106	16	no	—
184	C₂×C₈	—	212	16	no	—
185	C₄²	—	2	8	no	—
186	C₄²	—	2	16	no	—
187	C₄²	—	2	8	no	—
188	C₄²	—	2	8	no	—
189	C₄²	C₂×#88322	1	16	yes	complemented
190	C₄²	C₄⋊Dic₅₃	1	8	yes	—
191	C₄²	C₂×Dic₁₀₆	1	4	yes	—
192	C₄²	C₂×D₂₁₂	1	4	yes	—
193	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
194	C₂³	#13133176	1	4	yes	—
195	C₂³	C₂²×#88322	1	2	yes	—
196	C₂³	C₂×#11719917	1	2	yes	—
197	C₂³	C₂×#11719917	1	4	yes	—
198	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	C₄⋊#88322	1	16	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
211	C₂×C₄	#13133176	1	4	yes	—
212	C₂×C₄	#13133205	1	8	yes	—
213	C₂×C₄	#13133209	1	8	yes	—
214	C₂×C₄	C₂²×#88322	1	8	yes	—
215	C₂×C₄	C₂²×#88322	1	4	yes	—
216	C₂×C₄	C₂×#11719917	1	8	yes	—
217	C₂×C₄	C₂×#11719917	1	4	yes	—
218	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
219	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719945	1	4	yes	—
221	C₈	—	212	8	no	—
222	C₂²	#16810489	1	4	yes	—
223	C₂²	C₂×C₄⋊#88322	1	4	yes	—
224	C₂²	#16810647	1	4	yes	—
225	C₂²	#16810676	1	8	yes	—
226	C₂²	#16810680	1	8	yes	—
227	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
228	C₂²	C₂×#13133176	1	2	yes	—
229	C₂²	C₂³×#88322	1	1	yes	characteristic
230	C₂²	C₂²×#11719917	1	1	yes	characteristic
231	C₂²	C₂²×#11719917	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	8	no	—
234	C₄	—	2	8	no	—
235	C₄	#16810489	1	4	yes	—
236	C₄	C₂×C₄⋊#88322	1	8	yes	—
237	C₄	C₂×#13133205	1	2	yes	—
238	C₄	C₂×#13133209	1	2	yes	—
239	C₂	#20274832	1	8	yes	direct product factor
240	C₂	C₂×#16810489	1	1	yes	characteristic
241	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
242	C₂	C₂×#16810647	1	1	yes	characteristic
243	C₂	C₂×#16810676	1	2	yes	—
244	C₂	C₂×#16810680	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₂×#1831
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20274832	C₂	direct (N × H)
28. C₂×C₄²×C₅₃	C₈	semidirect
64. C₄²×C₅₃	C₂×C₈	semidirect
121. C₂×C₅₃	#1831	semidirect
138. C₅₃	C₂×#1831	semidirect
150. C₂×C₄²	#88322	semidirect
189. C₄²	C₂×#88322	semidirect
239. C₂	#20274832	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
#20274832	2
C₂×#16810378	2
C₂×C₄³×C₅₃	2
C₂×#1831	53

Subgroups — C2×#20274832 #27384235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274832 #27384235