Subgroups — C₂×#16811634 #20291367

Order 6784.

536 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133296	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
3	#16811634	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133286	C₂	1	2	yes	maximal, complemented
5	C₂×#13133285	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#6157	—	2	1	no	—
8	#13133296	—	2	2	no	—
9	#13133296	C₂²	1	4	yes	—
10	C₂×C₄⋊#6157	—	2	1	no	—
11	C₂×C₄⋊#6157	—	2	2	no	—
12	C₂×C₄⋊#6157	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#6157	C₂²	1	2	yes	—
14	C₂×C₄⋊#6157	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#6157	C₂²	1	4	yes	—
16	C₂×C₂²⋊#6157	—	2	1	no	—
17	C₂×C₄×#6157	—	2	1	no	—
18	#13133286	C₂²	1	8	yes	complemented
19	#13133285	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
21	C₂×#11719917	C₄	1	1	yes	characteristic, complemented
22	C₂×#11719912	C₄	1	2	yes	complemented
23	M₁₆×D₅₃	C₄	1	4	yes	complemented
24	M₁₆×D₅₃	C₂²	1	4	yes	—
25	C₂×C₈×D₅₃	C₂²	1	2	yes	—
26	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
27	C₂³×D₅₃	D₄	1	1	yes	characteristic
28	C₂²×#6157	—	2	1	no	—
29	C₂²×#6157	—	2	1	no	—
30	C₂²×#6157	—	2	4	no	—
31	C₂²×#6157	—	2	2	no	—
32	C₂²×#6157	—	4	4	no	—
33	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
35	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄×D₅₃	D₄	1	2	yes	—
39	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₄×D₅₃	Q₈	1	2	yes	—
41	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₅₃	C₂³	1	2	yes	—
43	C₂²⋊#6157	—	2	4	no	—
44	C₄⋊#6157	—	2	4	no	—
45	C₄⋊#6157	—	2	2	no	—
46	C₄⋊#6157	—	2	1	no	—
47	C₄⋊#6157	—	2	1	no	—
48	C₄⋊#6157	C₂³	1	2	yes	—
49	C₄⋊#6157	C₂³	1	4	yes	—
50	C₄⋊#6157	C₂³	1	2	yes	—
51	C₄×#6157	—	2	4	no	—
52	M₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
54	#11719917	C₂×C₄	1	4	yes	complemented
55	C₂×#88322	C₂×C₄	1	2	yes	complemented
56	#11719912	C₂×C₄	1	8	yes	complemented
57	C₈×D₅₃	C₂×C₄	1	4	yes	complemented
58	C₈×D₅₃	C₂³	1	4	yes	—
59	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×D₅₃	—	2	2	no	—
61	C₂²×D₅₃	—	2	2	no	—
62	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×D₅₃	C₂×D₄	1	2	yes	—
65	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×#6157	—	2	2	no	—
67	C₂×#6157	—	2	2	no	—
68	C₂×#6157	—	2	4	no	—
69	C₂×#6157	—	4	8	no	—
70	C₂×#6157	—	4	2	no	—
71	C₂×#6157	—	4	4	no	—
72	C₂×#6157	—	4	2	no	—
73	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
78	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×Dic₅₃	C₂×Q₈	1	2	yes	—
80	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
81	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
82	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
83	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
84	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₅₃	C₂²×C₄	1	2	yes	—
86	C₄×D₅₃	C₂×D₄	1	2	yes	—
87	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
88	C₄×D₅₃	C₂×Q₈	1	2	yes	—
89	C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₅₃	C₂²×C₄	1	4	yes	complemented
92	#88322	C₂²×C₄	1	4	yes	complemented
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×D₅₃	—	2	2	no	—
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	2	2	no	—
99	C₂×D₅₃	—	2	4	no	—
100	C₂×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×D₅₃	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×D₅₃	#137	1	2	yes	—
104	C₂×D₅₃	C₂²×D₄	1	1	yes	characteristic
105	#6157	—	4	2	no	—
106	#6157	—	4	4	no	—
107	#6157	—	4	2	no	—
108	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×#428	—	53	1	no	maximal, Sylow
115	C₂×C₅₃	—	2	2	no	—
116	C₂×C₅₃	#428	1	2	yes	complemented
117	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	D₅₃	—	2	2	no	—
119	D₅₃	#428	1	2	yes	complemented
120	D₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	D₅₃	C₂×#137	1	1	yes	characteristic
122	C₂²×M₁₆	—	53	1	no	—
123	C₂×#117	—	53	1	no	—
124	C₂²×C₄⋊C₄	—	53	1	no	—
125	#428	—	53	8	no	—
126	C₂×#107	—	53	2	no	—
127	C₂×#106	—	53	2	no	—
128	C₅₃	C₂×#428	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₄	—	53	1	no	—
130	C₂³×C₄	—	106	1	no	—
131	C₂×M₁₆	—	53	1	no	—
132	C₂×M₁₆	—	53	4	no	—
133	C₂×M₁₆	—	53	4	no	—
134	C₂×M₁₆	—	53	2	no	—
135	C₂×M₁₆	#6157	1	1	yes	characteristic, complemented
136	C₂²×C₈	—	53	2	no	—
137	#117	—	53	4	no	—
138	#117	—	106	2	no	—
139	C₂×C₄⋊C₄	—	53	2	no	—
140	C₂×C₄⋊C₄	—	53	4	no	—
141	C₂×C₄⋊C₄	—	53	1	no	—
142	C₂×C₄⋊C₄	—	53	1	no	—
143	C₂×C₄⋊C₄	—	106	1	no	—
144	C₂×C₄⋊C₄	—	106	2	no	—
145	C₂×C₂²⋊C₄	—	106	1	no	—
146	C₂×C₄²	—	106	1	no	—
147	#107	—	53	8	no	—
148	#106	—	53	8	no	—
149	C₂⁴	—	53	1	no	—
150	C₂²×C₄	—	53	1	no	—
151	C₂²×C₄	—	53	1	no	—
152	C₂²×C₄	—	53	2	no	—
153	C₂²×C₄	—	53	2	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	2	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	2	no	—
159	C₂²×C₄	—	106	1	no	—
160	C₂²×C₄	—	106	1	no	—
161	C₂²×C₄	—	106	2	no	—
162	C₂²×C₄	—	106	4	no	—
163	C₂²×C₄	—	212	4	no	—
164	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
165	M₁₆	—	53	4	no	—
166	M₁₆	—	53	8	no	—
167	M₁₆	C₂×#6157	1	4	yes	complemented
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	C₂×#6157	1	2	yes	complemented
172	C₄⋊C₄	—	53	2	no	—
173	C₄⋊C₄	—	53	4	no	—
174	C₄⋊C₄	—	53	2	no	—
175	C₄⋊C₄	—	106	4	no	—
176	C₄⋊C₄	—	106	2	no	—
177	C₄⋊C₄	—	106	1	no	—
178	C₄⋊C₄	—	106	1	no	—
179	C₂²⋊C₄	—	106	4	no	—
180	C₄²	—	106	4	no	—
181	C₂³	—	53	1	no	—
182	C₂³	—	53	2	no	—
183	C₂³	—	53	2	no	—
184	C₂³	—	53	1	no	—
185	C₂³	—	106	2	no	—
186	C₂³	—	106	2	no	—
187	C₂³	C₄⋊#6157	1	1	yes	characteristic
188	C₂×C₄	—	53	2	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	2	no	—
193	C₂×C₄	—	53	2	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	53	2	no	—
199	C₂×C₄	—	53	1	no	—
200	C₂×C₄	—	53	2	no	—
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	106	2	no	—
203	C₂×C₄	—	106	2	no	—
204	C₂×C₄	—	106	4	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	—	212	4	no	—
208	C₂×C₄	—	212	8	no	—
209	C₂×C₄	C₄⋊#6157	1	2	yes	—
210	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
211	C₂×C₄	C₂²×#6157	1	2	yes	—
212	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
213	C₈	—	53	4	no	—
214	C₈	C₂²×#6157	1	4	yes	complemented
215	C₂²	—	2	2	no	—
216	C₂²	—	53	2	no	—
217	C₂²	—	53	2	no	—
218	C₂²	—	53	1	no	—
219	C₂²	—	53	1	no	—
220	C₂²	—	53	2	no	—
221	C₂²	—	106	2	no	—
222	C₂²	—	106	2	no	—
223	C₂²	—	106	4	no	—
224	C₂²	—	106	2	no	—
225	C₂²	C₂×C₄⋊#6157	1	2	yes	—
226	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
227	C₄	—	53	1	no	—
228	C₄	—	53	2	no	—
229	C₄	—	53	1	no	—
230	C₄	—	212	4	no	—
231	C₄	—	212	2	no	—
232	C₄	—	212	2	no	—
233	C₄	C₂×C₄⋊#6157	1	2	yes	—
234	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
235	C₄	C₂³×#6157	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	—	53	1	no	—
238	C₂	—	53	1	no	—
239	C₂	—	53	2	no	—
240	C₂	—	106	2	no	—
241	C₂	#16811634	1	2	yes	direct product factor
242	C₂	C₂²×C₄⋊#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	C₂×#428
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #16811634	C₂	direct (N × H)
4. C₂×#13133286	C₂	semidirect
5. C₂×#13133285	C₂	semidirect
18. #13133286	C₂²	semidirect
19. #13133285	C₂²	semidirect
21. C₂×#11719917	C₄	semidirect
22. C₂×#11719912	C₄	semidirect
23. M₁₆×D₅₃	C₄	semidirect
26. C₂×M₁₆×C₅₃	C₄	semidirect
52. M₁₆×C₅₃	C₂×C₄	semidirect
53. C₂×C₈×C₅₃	C₂×C₄	semidirect
54. #11719917	C₂×C₄	semidirect
55. C₂×#88322	C₂×C₄	semidirect
56. #11719912	C₂×C₄	semidirect
57. C₈×D₅₃	C₂×C₄	semidirect
91. C₈×C₅₃	C₂²×C₄	semidirect
92. #88322	C₂²×C₄	semidirect
116. C₂×C₅₃	#428	semidirect
119. D₅₃	#428	semidirect
128. C₅₃	C₂×#428	semidirect
135. C₂×M₁₆	#6157	semidirect
167. M₁₆	C₂×#6157	semidirect
171. C₂×C₈	C₂×#6157	semidirect
214. C₈	C₂²×#6157	semidirect
241. C₂	#16811634	direct (N × H)

Subgroups — C2×#16811634 #20291367

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811634 #20291367