Subgroups — C₂×#20291107 #28328346

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811682	C₂	1	2	yes	maximal, complemented
2	#20291107	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16811545	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16811522	C₂	1	2	yes	maximal, complemented
5	C₂×#16811512	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810739	C₂	1	1	yes	maximal, characteristic
7	#16811682	—	2	4	no	—
8	#16811682	C₂²	1	8	yes	complemented
9	C₂×#13133292	—	2	2	no	—
10	C₂×#13133291	C₂²	1	2	yes	—
11	C₂×D₅₃⋊C₁₆	—	2	2	no	—
12	C₂²×#11719910	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	1	yes	characteristic
14	#16811545	C₂²	1	4	yes	complemented
15	#16811522	C₂²	1	8	yes	complemented
16	#16811512	C₂²	1	4	yes	complemented
17	C₂×#13133239	C₄	1	1	yes	characteristic
18	C₂×#13133225	C₂²	1	2	yes	—
19	#16810739	C₄	1	4	yes	—
20	#16810739	C₂²	1	4	yes	—
21	C₂×#13133143	C₄	1	2	yes	—
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
24	#13133292	—	2	8	no	—
25	#13133291	—	2	2	no	—
26	#13133291	C₂³	1	4	yes	—
27	D₅₃⋊C₁₆	—	2	8	no	—
28	C₂×#11719910	—	2	2	no	—
29	C₂×#11719910	—	2	1	no	—
30	C₂×#11719910	—	2	1	no	—
31	C₂×#11719910	D₄	1	2	yes	—
32	C₂×#11719910	D₄	1	2	yes	—
33	C₂×#11719910	C₂³	1	2	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719945	C₂×C₄	1	2	yes	—
36	#13133239	C₈	1	4	yes	—
37	#13133239	C₂×C₄	1	4	yes	—
38	C₂×Dic₂₁₂	C₈	1	1	yes	characteristic
39	C₂×#11719913	C₈	1	2	yes	—
40	C₂×D₄₂₄	C₈	1	1	yes	characteristic
41	#13133225	—	2	2	no	—
42	#13133225	—	2	2	no	—
43	#13133225	C₂×C₄	1	4	yes	—
44	#13133225	C₂³	1	4	yes	—
45	C₂×#11719912	—	2	2	no	—
46	C₂²×#88322	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₅₃	—	2	2	no	—
48	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	#13133143	C₂×C₄	1	8	yes	—
50	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₅₃	C₂³	1	2	yes	—
52	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	#108×C₅₃	C₂×C₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₅₃	—	2	2	no	—
57	C₂²×Dic₅₃	—	2	1	no	—
58	#11719945	—	2	2	no	—
59	#11719945	—	2	2	no	—
60	#11719945	C₂×C₈	1	4	yes	—
61	#11719945	C₂²×C₄	1	4	yes	—
62	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₅₃	—	2	2	no	—
64	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
65	M₁₆×C₅₃	—	2	2	no	—
66	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄×Dic₅₃	—	2	1	no	—
73	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
74	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
75	C₂×#88322	C₂×D₄	1	1	yes	characteristic
76	C₂×#88322	C₂×D₄	1	1	yes	characteristic
77	C₂×#88322	#41	1	1	yes	characteristic
78	C₂×#88322	#41	1	1	yes	characteristic
79	C₂×#88322	C₂⁴	1	1	yes	characteristic
80	Dic₂₁₂	C₂×C₈	1	4	yes	—
81	D₄₂₄	C₂×C₈	1	4	yes	—
82	#11719913	C₂×C₈	1	8	yes	—
83	#11719912	—	2	4	no	—
84	#11719912	—	2	4	no	—
85	C₈×D₅₃	—	2	4	no	—
86	C₈×D₅₃	—	2	4	no	—
87	#11719910	—	2	4	no	—
88	#11719910	—	2	2	no	—
89	#11719910	C₂×D₄	1	4	yes	—
90	C₂³×C₅₃	#131	1	1	yes	characteristic
91	C₂²×D₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂²⋊D₅₃	—	2	4	no	—
99	C₂²⋊D₅₃	—	2	4	no	—
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	4	2	no	—
103	D₂₁₂	—	2	2	no	—
104	D₂₁₂	C₂²×C₈	1	4	yes	—
105	C₄×D₅₃	—	2	4	no	—
106	C₄×D₅₃	—	2	4	no	—
107	Dic₁₀₆	—	2	2	no	—
108	Dic₁₀₆	C₂²×C₈	1	4	yes	—
109	C₈×C₅₃	—	2	4	no	—
110	C₈×C₅₃	C₂²×C₈	1	2	yes	—
111	C₈×C₅₃	C₂²×C₈	1	2	yes	—
112	#88322	C₄×D₄	1	1	yes	characteristic
113	#88322	C₄×D₄	1	1	yes	characteristic
114	#88322	C₂²×D₄	1	1	yes	characteristic
115	#88322	C₂×#41	1	1	yes	characteristic
116	C₂×#2171	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
121	C₂×D₅₃	—	2	4	no	—
122	C₂×D₅₃	—	4	4	no	—
123	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
124	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
127	Dic₅₃	—	2	4	no	—
128	Dic₅₃	—	4	2	no	—
129	C₂×#504	—	53	2	no	—
130	C₂×#443	—	53	1	no	—
131	#2171	—	53	8	no	—
132	C₂×#364	—	53	1	no	—
133	C₂×#350	—	53	2	no	—
134	C₂×#346	—	53	1	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	#2171	1	2	yes	complemented
138	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
139	D₅₃	—	4	4	no	—
140	C₂×#135	—	53	1	no	—
141	C₂×#131	—	53	2	no	—
142	#504	—	53	8	no	—
143	#504	—	106	4	no	—
144	C₂×M₃₂	—	53	2	no	—
145	C₂×M₃₂	—	106	2	no	—
146	C₂²×C₁₆	—	53	1	no	—
147	C₂²×C₁₆	—	53	1	no	—
148	C₂²×C₁₆	—	106	2	no	—
149	#443	—	53	4	no	—
150	#443	—	53	4	no	—
151	C₂×#108	#6157	1	1	yes	characteristic
152	C₂×#104	—	53	2	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	#364	—	53	4	no	—
155	#350	—	53	8	no	—
156	#346	—	53	4	no	—
157	C₅₃	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	53	2	no	—
159	#135	—	53	4	no	—
160	#135	—	53	4	no	—
161	C₂×Q₁₆	—	53	1	no	—
162	C₂×SD₁₆	—	53	2	no	—
163	C₂×D₈	—	53	1	no	—
164	#131	—	53	4	no	—
165	#131	—	53	4	no	—
166	#131	—	106	2	no	—
167	#131	—	106	2	no	—
168	C₂×M₁₆	—	106	2	no	—
169	C₂×M₁₆	C₂×#6157	1	2	yes	—
170	C₂²×C₈	—	53	1	no	—
171	C₂²×C₈	—	106	2	no	—
172	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
173	C₂×C₄²	—	53	1	no	—
174	M₃₂	—	53	4	no	—
175	M₃₂	—	106	8	no	—
176	M₃₂	—	106	2	no	—
177	C₂×C₁₆	—	53	2	no	—
178	C₂×C₁₆	—	53	2	no	—
179	C₂×C₁₆	—	53	2	no	—
180	C₂×C₁₆	—	53	2	no	—
181	C₂×C₁₆	—	106	8	no	—
182	C₂×C₁₆	—	106	1	no	—
183	C₂×C₁₆	—	106	1	no	—
184	C₂×C₁₆	—	106	2	no	—
185	#108	C₂×#6157	1	4	yes	—
186	#104	—	53	8	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	53	2	no	—
189	#41	—	53	4	no	—
190	#41	—	53	4	no	—
191	#41	—	106	2	no	—
192	#41	—	106	2	no	—
193	C₂×Q₈	—	53	2	no	—
194	C₂×D₄	—	53	2	no	—
195	C₂×D₄	—	106	2	no	—
196	C₂²×C₄	—	106	2	no	—
197	C₂²×C₄	—	106	1	no	—
198	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
199	Q₁₆	—	53	4	no	—
200	SD₁₆	—	53	8	no	—
201	D₈	—	53	4	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	106	4	no	—
204	M₁₆	—	106	4	no	—
205	M₁₆	C₂²×#6157	1	4	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	53	1	no	—
208	C₂×C₈	—	53	1	no	—
209	C₂×C₈	—	53	1	no	—
210	C₂×C₈	—	53	1	no	—
211	C₂×C₈	—	53	1	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	—	106	4	no	—
215	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
216	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
217	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
218	C₂×C₈	C₂²×#6157	1	2	yes	—
219	C₄²	—	53	2	no	—
220	C₄²	—	106	1	no	—
221	C₁₆	—	53	4	no	—
222	C₁₆	—	106	2	no	—
223	C₁₆	—	106	4	no	—
224	C₂³	—	106	2	no	—
225	C₂³	#13133305	1	1	yes	characteristic
226	Q₈	—	53	4	no	—
227	Q₈	—	106	2	no	—
228	D₄	—	53	4	no	—
229	D₄	—	106	4	no	—
230	D₄	—	106	4	no	—
231	D₄	—	106	2	no	—
232	C₂×C₄	—	106	2	no	—
233	C₂×C₄	—	106	4	no	—
234	C₂×C₄	—	106	4	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	—	212	2	no	—
237	C₂×C₄	D₄×#6157	1	1	yes	characteristic
238	C₂×C₄	D₄×#6157	1	1	yes	characteristic
239	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
241	C₂×C₄	#13133305	1	1	yes	characteristic
242	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	—	53	1	no	—
245	C₈	—	53	1	no	—
246	C₈	—	53	1	no	—
247	C₈	—	53	1	no	—
248	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
249	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	106	4	no	—
252	C₂²	—	212	4	no	—
253	C₂²	#16811651	1	1	yes	characteristic
254	C₂²	#16811651	1	1	yes	characteristic
255	C₂²	C₂×#13133305	1	1	yes	characteristic
256	C₄	—	106	4	no	—
257	C₄	—	212	2	no	—
258	C₄	#16811651	1	1	yes	characteristic
259	C₄	#16811651	1	1	yes	characteristic
260	C₄	C₂×D₄×#6157	1	1	yes	characteristic
261	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	212	4	no	—
265	C₂	#20291107	1	2	yes	direct product factor
266	C₂	C₂×#16811651	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×#13133239	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	C₂×#2171
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
1. C₂×#16811682	C₂	semidirect
2. #20291107	C₂	direct (N × H)
3. C₂×#16811545	C₂	semidirect
4. C₂×#16811522	C₂	semidirect
5. C₂×#16811512	C₂	semidirect
8. #16811682	C₂²	semidirect
14. #16811545	C₂²	semidirect
15. #16811522	C₂²	semidirect
16. #16811512	C₂²	semidirect
137. C₂×C₅₃	#2171	semidirect
157. C₅₃	C₂×#2171	semidirect
265. C₂	#20291107	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16811682	2
#20291107	2
C₂×#16811545	2
C₂×#16811522	2
C₂×#16811512	2
C₂×#16810739	2
C₂×#2171	53

Subgroups — C2×#20291107 #28328346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291107 #28328346