Subgroups — C₂×#16811505 #20290424

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133296	C₂	1	1	yes	maximal, characteristic
3	#16811505	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719936	—	2	1	no	—
6	D₅₃⋊M₁₆	—	2	1	no	—
7	D₅₃⋊M₁₆	—	2	1	no	—
8	D₅₃⋊M₁₆	C₄	1	2	yes	complemented
9	D₅₃⋊M₁₆	C₂²	1	2	yes	—
10	C₂×#11719940	C₄	1	2	yes	complemented
11	C₂×D₅₃⋊C₈	—	2	1	no	—
12	#13133296	—	2	1	no	—
13	#13133296	—	2	1	no	—
14	#13133296	C₄	1	2	yes	—
15	#13133296	C₂²	1	2	yes	—
16	C₂×C₄⋊#6157	—	2	1	no	—
17	C₂×C₂²⋊#6157	C₄	1	2	yes	—
18	C₂×C₄×#6157	—	2	1	no	—
19	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719912	—	2	1	no	—
21	C₂×#11719912	—	2	1	no	—
22	C₂×#11719912	—	2	1	no	—
23	C₂×#11719912	—	2	1	no	—
24	C₂×#11719912	C₄	1	2	yes	complemented
25	C₂×#11719912	C₂²	1	2	yes	—
26	C₂²×#88322	C₄	1	1	yes	characteristic, complemented
27	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
28	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#6157	—	2	2	no	—
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
42	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
43	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
44	C₂²⋊#6157	—	2	2	no	—
45	C₂²⋊#6157	C₂×C₄	1	4	yes	—
46	#11719940	—	2	2	no	—
47	#11719940	C₂×C₄	1	4	yes	complemented
48	C₂×#88324	—	2	2	no	—
49	C₄⋊#6157	—	2	2	no	—
50	C₄⋊#6157	—	2	2	no	—
51	C₄×#6157	—	2	2	no	—
52	C₄×#6157	—	2	2	no	—
53	#11719936	—	2	2	no	—
54	#11719936	—	2	2	no	—
55	D₅₃⋊C₈	—	2	2	no	—
56	D₅₃⋊C₈	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	C₂×C₄	1	2	yes	complemented
63	#11719912	—	2	2	no	—
64	#11719912	—	2	2	no	—
65	#11719912	—	2	2	no	—
66	#11719912	—	2	2	no	—
67	C₂³×C₅₃	C₄²	1	1	yes	characteristic
68	C₂²×D₅₃	—	4	2	no	—
69	C₂²×D₅₃	C₄²	1	1	yes	characteristic
70	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×#6157	—	2	4	no	—
76	C₂×#6157	—	4	4	no	—
77	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
105	#88324	—	2	4	no	—
106	C₈×C₅₃	—	2	2	no	—
107	#88322	—	2	2	no	—
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	#95	1	1	yes	characteristic
110	C₂²×C₅₃	#95	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×D₅₃	—	2	2	no	—
113	C₂×D₅₃	—	2	2	no	—
114	C₂×D₅₃	—	4	1	no	—
115	C₂×D₅₃	—	4	1	no	—
116	C₂×D₅₃	—	4	1	no	—
117	C₂×D₅₃	—	4	1	no	—
118	C₂×D₅₃	#95	1	1	yes	characteristic
119	C₂×D₅₃	#95	1	1	yes	characteristic
120	C₂×D₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	#6157	—	4	4	no	—
123	C₄×C₅₃	#95	1	1	yes	characteristic
124	C₄×C₅₃	#95	1	1	yes	characteristic
125	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	Dic₅₃	#95	1	1	yes	characteristic
128	Dic₅₃	#95	1	1	yes	characteristic
129	Dic₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×#344	—	53	1	no	maximal, Sylow
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	#344	1	2	yes	complemented
135	C₂×C₅₃	C₂×#95	1	1	yes	characteristic
136	D₅₃	—	2	1	no	—
137	D₅₃	—	2	1	no	—
138	D₅₃	—	2	1	no	—
139	D₅₃	—	2	1	no	—
140	C₂²×M₁₆	—	53	1	no	—
141	C₂²×M₁₆	—	53	1	no	—
142	C₂×#117	—	53	1	no	—
143	#344	—	53	4	no	—
144	C₅₃	C₂×#344	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	—	53	1	no	—
146	C₂×M₁₆	—	53	2	no	—
147	C₂×M₁₆	—	53	2	no	—
148	C₂×M₁₆	—	53	2	no	—
149	C₂×M₁₆	—	53	2	no	—
150	C₂×M₁₆	—	53	2	no	—
151	C₂×M₁₆	—	106	1	no	—
152	C₂×M₁₆	—	106	1	no	—
153	C₂×M₁₆	—	106	1	no	—
154	C₂×M₁₆	—	106	1	no	—
155	C₂×M₁₆	—	106	1	no	—
156	C₂×M₁₆	—	106	1	no	—
157	C₂×M₁₆	—	106	1	no	—
158	C₂²×C₈	—	53	1	no	—
159	C₂²×C₈	—	106	1	no	—
160	C₂²×C₈	#6157	1	1	yes	characteristic, complemented
161	#117	—	53	2	no	—
162	#117	—	53	2	no	—
163	#117	—	106	1	no	—
164	#117	—	106	1	no	—
165	C₂×C₄⋊C₄	—	106	1	no	—
166	C₂×C₂²⋊C₄	—	53	2	no	—
167	C₂×C₄²	—	106	1	no	—
168	C₂⁴	—	53	1	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	1	no	—
171	C₂²×C₄	—	53	1	no	—
172	C₂²×C₄	—	53	1	no	—
173	C₂²×C₄	—	53	1	no	—
174	C₂²×C₄	—	53	1	no	—
175	C₂²×C₄	—	53	1	no	—
176	C₂²×C₄	—	53	1	no	—
177	C₂²×C₄	—	53	1	no	—
178	C₂²×C₄	—	53	1	no	—
179	C₂²×C₄	—	53	1	no	—
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	1	no	—
182	C₂²×C₄	—	106	2	no	—
183	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
184	M₁₆	—	53	4	no	—
185	M₁₆	—	106	2	no	—
186	M₁₆	—	106	2	no	—
187	M₁₆	—	106	2	no	—
188	M₁₆	—	106	2	no	—
189	M₁₆	—	106	2	no	—
190	M₁₆	—	106	2	no	—
191	M₁₆	—	106	2	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	106	2	no	—
196	C₂×C₈	—	106	2	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	—	106	1	no	—
200	C₂×C₈	C₂×#6157	1	2	yes	complemented
201	C₄⋊C₄	—	106	2	no	—
202	C₄⋊C₄	—	106	2	no	—
203	C₂²⋊C₄	—	53	4	no	—
204	C₂²⋊C₄	—	106	2	no	—
205	C₄²	—	106	2	no	—
206	C₄²	—	106	2	no	—
207	C₂³	—	53	1	no	—
208	C₂³	—	53	1	no	—
209	C₂³	—	53	1	no	—
210	C₂³	—	53	1	no	—
211	C₂³	—	53	1	no	—
212	C₂³	—	53	1	no	—
213	C₂³	—	212	2	no	—
214	C₂³	C₄×#6157	1	1	yes	characteristic
215	C₂×C₄	—	53	1	no	—
216	C₂×C₄	—	53	1	no	—
217	C₂×C₄	—	53	1	no	—
218	C₂×C₄	—	53	1	no	—
219	C₂×C₄	—	53	1	no	—
220	C₂×C₄	—	53	1	no	—
221	C₂×C₄	—	53	1	no	—
222	C₂×C₄	—	53	1	no	—
223	C₂×C₄	—	53	1	no	—
224	C₂×C₄	—	53	1	no	—
225	C₂×C₄	—	53	1	no	—
226	C₂×C₄	—	53	1	no	—
227	C₂×C₄	—	53	1	no	—
228	C₂×C₄	—	53	1	no	—
229	C₂×C₄	—	53	1	no	—
230	C₂×C₄	—	53	1	no	—
231	C₂×C₄	—	53	1	no	—
232	C₂×C₄	—	53	1	no	—
233	C₂×C₄	—	53	1	no	—
234	C₂×C₄	—	53	1	no	—
235	C₂×C₄	—	53	1	no	—
236	C₂×C₄	—	53	1	no	—
237	C₂×C₄	—	106	4	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	C₄×#6157	1	1	yes	characteristic
240	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
241	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
242	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
244	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	106	4	no	—
247	C₈	—	106	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	53	1	no	—
250	C₂²	—	53	1	no	—
251	C₂²	—	53	1	no	—
252	C₂²	—	53	1	no	—
253	C₂²	—	106	2	no	—
254	C₂²	—	106	2	no	—
255	C₂²	—	212	1	no	—
256	C₂²	—	212	1	no	—
257	C₂²	—	212	1	no	—
258	C₂²	—	212	1	no	—
259	C₂²	(C₂×#6157)⋊C₄	1	1	yes	characteristic
260	C₂²	(C₂×#6157)⋊C₄	1	1	yes	characteristic
261	C₂²	C₂×C₄×#6157	1	1	yes	characteristic
262	C₄	—	53	1	no	—
263	C₄	—	53	1	no	—
264	C₄	—	53	1	no	—
265	C₄	—	53	1	no	—
266	C₄	—	212	4	no	—
267	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
268	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
269	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
270	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	106	1	no	—
274	C₂	—	106	1	no	—
275	C₂	—	106	1	no	—
276	C₂	—	106	1	no	—
277	C₂	#16811505	1	2	yes	direct product factor
278	C₂	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#6157
commutator	C₂²×C₅₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂²×C₈×C₅₃	C₄
socle	C₂²×C₅₃	#95
hypercenter	C₂²×C₈	#6157
hypocenter	C₅₃	C₂×#344
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #16811505	C₂	direct (N × H)
8. D₅₃⋊M₁₆	C₄	semidirect
10. C₂×#11719940	C₄	semidirect
24. C₂×#11719912	C₄	semidirect
26. C₂²×#88322	C₄	semidirect
27. C₂²×C₈×C₅₃	C₄	semidirect
47. #11719940	C₂×C₄	semidirect
59. C₂×C₈×C₅₃	C₂×C₄	semidirect
62. C₂×#88322	C₂×C₄	semidirect
134. C₂×C₅₃	#344	semidirect
144. C₅₃	C₂×#344	semidirect
160. C₂²×C₈	#6157	semidirect
200. C₂×C₈	C₂×#6157	semidirect
277. C₂	#16811505	direct (N × H)

Subgroups — C2×#16811505 #20290424

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811505 #20290424