Subgroups — C₂×#20291104 #28328342

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#20291104	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×#16811509	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	#16811509	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	1	yes	characteristic
32	C₂×#11719910	D₄	1	1	yes	characteristic
33	C₂×#11719910	D₄	1	2	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	2	yes	—
36	C₂×#11719945	C₂×C₄	1	2	yes	—
37	#13133239	C₈	1	4	yes	—
38	#13133239	C₂×C₄	1	4	yes	—
39	C₂×Dic₂₁₂	C₈	1	1	yes	characteristic
40	C₂×#11719913	C₈	1	2	yes	—
41	C₂×D₄₂₄	C₈	1	1	yes	characteristic
42	#13133225	—	2	2	no	—
43	#13133225	—	2	2	no	—
44	#13133225	C₂×C₄	1	4	yes	—
45	#13133225	C₂³	1	4	yes	—
46	C₂×#11719912	—	2	2	no	—
47	C₂²×#88322	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₅₃	—	2	2	no	—
49	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	#13133143	C₂×C₄	1	8	yes	—
51	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₅₃	C₂³	1	2	yes	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#108×C₅₃	C₂×C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₃	—	2	2	no	—
58	C₂²×Dic₅₃	—	2	1	no	—
59	#11719945	—	2	2	no	—
60	#11719945	—	2	2	no	—
61	#11719945	C₂×C₈	1	4	yes	—
62	#11719945	C₂²×C₄	1	4	yes	—
63	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₅₃	—	2	2	no	—
65	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
66	M₁₆×C₅₃	—	2	2	no	—
67	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄×Dic₅₃	—	2	1	no	—
74	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
75	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
76	C₂×#88322	C₂×D₄	1	1	yes	characteristic
77	C₂×#88322	C₂×D₄	1	1	yes	characteristic
78	C₂×#88322	#41	1	1	yes	characteristic
79	C₂×#88322	#41	1	1	yes	characteristic
80	C₂×#88322	C₂⁴	1	1	yes	characteristic
81	Dic₂₁₂	C₂×C₈	1	4	yes	—
82	D₄₂₄	C₂×C₈	1	4	yes	—
83	#11719913	C₂×C₈	1	8	yes	—
84	#11719912	—	2	4	no	—
85	#11719912	—	2	4	no	—
86	C₈×D₅₃	—	2	4	no	—
87	C₈×D₅₃	—	2	4	no	—
88	#11719910	—	2	4	no	—
89	#11719910	—	2	2	no	—
90	#11719910	C₂×D₄	1	2	yes	—
91	#11719910	C₂×D₄	1	2	yes	—
92	C₂³×C₅₃	#131	1	1	yes	characteristic
93	C₂²×D₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂²⋊D₅₃	—	2	4	no	—
101	C₂²⋊D₅₃	—	2	4	no	—
102	C₂×Dic₅₃	—	2	4	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	2	2	no	—
105	D₂₁₂	—	2	2	no	—
106	D₂₁₂	C₂²×C₈	1	4	yes	—
107	C₄×D₅₃	—	2	4	no	—
108	C₄×D₅₃	—	2	4	no	—
109	Dic₁₀₆	—	2	2	no	—
110	Dic₁₀₆	C₂²×C₈	1	4	yes	—
111	C₈×C₅₃	—	2	4	no	—
112	C₈×C₅₃	C₂²×C₈	1	2	yes	—
113	C₈×C₅₃	C₂²×C₈	1	2	yes	—
114	#88322	C₄×D₄	1	1	yes	characteristic
115	#88322	C₄×D₄	1	1	yes	characteristic
116	#88322	C₂²×D₄	1	1	yes	characteristic
117	#88322	C₂×#41	1	1	yes	characteristic
118	C₂×#2170	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
123	C₂×D₅₃	—	2	4	no	—
124	C₂×D₅₃	—	4	4	no	—
125	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
126	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
127	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
129	Dic₅₃	—	2	4	no	—
130	Dic₅₃	—	2	4	no	—
131	C₂×#504	—	53	2	no	—
132	C₂×#443	—	53	1	no	—
133	#2170	—	53	8	no	—
134	C₂×#364	—	53	1	no	—
135	C₂×#350	—	53	2	no	—
136	C₂×C₄×C₁₆	—	53	1	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	#2170	1	2	yes	complemented
140	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
141	D₅₃	—	4	4	no	—
142	C₂×#135	—	53	1	no	—
143	C₂×#131	—	53	2	no	—
144	#504	—	53	8	no	—
145	#504	—	106	4	no	—
146	C₂×M₃₂	—	53	2	no	—
147	C₂×M₃₂	—	106	2	no	—
148	C₂²×C₁₆	—	53	1	no	—
149	C₂²×C₁₆	—	53	1	no	—
150	C₂²×C₁₆	—	106	2	no	—
151	#443	—	53	4	no	—
152	#443	—	53	4	no	—
153	C₂×#108	#6157	1	1	yes	characteristic
154	C₂×#104	—	53	2	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	#364	—	53	4	no	—
157	#350	—	53	8	no	—
158	C₄×C₁₆	—	53	4	no	—
159	C₅₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
160	C₂×#41	—	53	2	no	—
161	#135	—	53	4	no	—
162	#135	—	53	4	no	—
163	C₂×Q₁₆	—	53	1	no	—
164	C₂×SD₁₆	—	53	2	no	—
165	C₂×D₈	—	53	1	no	—
166	#131	—	53	4	no	—
167	#131	—	53	4	no	—
168	#131	—	106	2	no	—
169	#131	—	106	2	no	—
170	C₂×M₁₆	—	106	2	no	—
171	C₂×M₁₆	C₂×#6157	1	2	yes	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	106	2	no	—
174	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
175	C₂×C₄²	—	53	1	no	—
176	M₃₂	—	53	4	no	—
177	M₃₂	—	106	8	no	—
178	M₃₂	—	106	2	no	—
179	C₂×C₁₆	—	53	2	no	—
180	C₂×C₁₆	—	53	1	no	—
181	C₂×C₁₆	—	53	1	no	—
182	C₂×C₁₆	—	53	2	no	—
183	C₂×C₁₆	—	53	2	no	—
184	C₂×C₁₆	—	106	8	no	—
185	C₂×C₁₆	—	106	1	no	—
186	C₂×C₁₆	—	106	1	no	—
187	C₂×C₁₆	—	106	2	no	—
188	#108	C₂×#6157	1	4	yes	—
189	#104	—	53	8	no	—
190	C₄×C₈	—	53	2	no	—
191	C₄×C₈	—	53	2	no	—
192	#41	—	53	4	no	—
193	#41	—	53	4	no	—
194	#41	—	106	2	no	—
195	#41	—	106	2	no	—
196	C₂×Q₈	—	53	2	no	—
197	C₂×D₄	—	53	2	no	—
198	C₂×D₄	—	106	2	no	—
199	C₂²×C₄	—	106	1	no	—
200	C₂²×C₄	—	106	2	no	—
201	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
202	Q₁₆	—	53	4	no	—
203	SD₁₆	—	53	8	no	—
204	D₈	—	53	4	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	106	4	no	—
207	M₁₆	—	106	4	no	—
208	M₁₆	C₂²×#6157	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	53	1	no	—
211	C₂×C₈	—	53	1	no	—
212	C₂×C₈	—	53	1	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	53	1	no	—
216	C₂×C₈	—	106	4	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
219	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
220	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
221	C₂×C₈	C₂²×#6157	1	2	yes	—
222	C₄²	—	53	2	no	—
223	C₄²	—	106	1	no	—
224	C₁₆	—	53	2	no	—
225	C₁₆	—	53	2	no	—
226	C₁₆	—	106	2	no	—
227	C₁₆	—	106	4	no	—
228	C₂³	—	106	2	no	—
229	C₂³	#13133305	1	1	yes	characteristic
230	Q₈	—	53	4	no	—
231	Q₈	—	106	2	no	—
232	D₄	—	53	4	no	—
233	D₄	—	106	4	no	—
234	D₄	—	106	4	no	—
235	D₄	—	106	2	no	—
236	C₂×C₄	—	106	4	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	106	2	no	—
239	C₂×C₄	—	106	4	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	D₄×#6157	1	1	yes	characteristic
242	C₂×C₄	D₄×#6157	1	1	yes	characteristic
243	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
244	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
245	C₂×C₄	#13133305	1	1	yes	characteristic
246	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	—	53	1	no	—
249	C₈	—	53	1	no	—
250	C₈	—	53	1	no	—
251	C₈	—	53	1	no	—
252	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
253	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	106	4	no	—
256	C₂²	—	212	4	no	—
257	C₂²	#16811651	1	1	yes	characteristic
258	C₂²	#16811651	1	1	yes	characteristic
259	C₂²	C₂×#13133305	1	1	yes	characteristic
260	C₄	—	106	4	no	—
261	C₄	—	106	4	no	—
262	C₄	#16811651	1	1	yes	characteristic
263	C₄	#16811651	1	1	yes	characteristic
264	C₄	C₂×D₄×#6157	1	1	yes	characteristic
265	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	212	4	no	—
269	C₂	#20291104	1	2	yes	direct product factor
270	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×#2170
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. #20291104	C₂	direct (N × H)
3. C₂×#16811545	C₂	semidirect
4. C₂×#16811522	C₂	semidirect
5. C₂×#16811509	C₂	semidirect
8. #16811682	C₂²	semidirect
14. #16811545	C₂²	semidirect
15. #16811522	C₂²	semidirect
16. #16811509	C₂²	semidirect
139. C₂×C₅₃	#2170	semidirect
159. C₅₃	C₂×#2170	semidirect
269. C₂	#20291104	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
#20291104	2
C₂×#16811545	2
C₂×#16811522	2
C₂×#16811509	2
C₂×#16810739	2
C₂×#2170	53

Subgroups — C2×#20291104 #28328342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20291104 #28328342