Subgroups — #22569117

Order 8576.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864193	C₂	1	1	yes	maximal, characteristic
2	#16864161	C₂	1	2	yes	maximal, complemented
3	#16863519	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863515	C₂	1	2	yes	maximal, complemented
5	#16863513	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863512	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863511	C₂	1	2	yes	maximal, complemented
8	#16863443	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₆₇	C₂	1	2	yes	maximal, complemented
10	#16863348	C₂	1	1	yes	maximal, characteristic, complemented
11	#444×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₆₇	C₂²	1	2	yes	—
13	#14524075	C₄	1	2	yes	complemented
14	#14524072	C₄	1	2	yes	complemented
15	#14524070	C₄	1	2	yes	complemented
16	#14524067	C₄	1	2	yes	complemented
17	#14524065	C₂²	1	1	yes	characteristic
18	#14524063	C₂²	1	2	yes	—
19	#14524051	—	2	2	no	—
20	#14524051	C₂²	1	2	yes	complemented
21	#14524050	C₂²	1	2	yes	complemented
22	#14524049	—	2	2	no	—
23	#14524049	C₂²	1	2	yes	complemented
24	C₂×#11761611	—	2	2	no	—
25	C₂×#11761611	C₂²	1	1	yes	characteristic
26	C₂×#11761611	C₂²	1	1	yes	characteristic
27	M₁₆×D₆₇	—	2	2	no	—
28	M₁₆×D₆₇	C₂²	1	2	yes	—
29	C₄²×D₆₇	C₂²	1	1	yes	characteristic
30	#14523979	C₂²	1	1	yes	characteristic, complemented
31	#14523978	C₂²	1	2	yes	complemented
32	#14523971	C₂²	1	2	yes	complemented
33	#14523970	C₂²	1	2	yes	complemented
34	#14523969	C₂²	1	2	yes	complemented
35	#14523954	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
38	#135×C₆₇	C₂²	1	1	yes	characteristic
39	#131×C₆₇	C₂²	1	2	yes	complemented
40	#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	#104×C₆₇	C₂²	1	2	yes	complemented
42	#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	#11761629	C₂×C₄	1	2	yes	—
44	#11761628	—	2	2	no	—
45	#11761628	C₂³	1	2	yes	—
46	#11761627	—	2	2	no	—
47	#11761627	C₂×C₄	1	2	yes	—
48	Q₈×D₆₇	C₂×C₄	1	2	yes	—
49	D₄×D₆₇	—	2	2	no	—
50	D₄×D₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₆₇	—	2	2	no	—
52	C₂×C₄×D₆₇	—	2	1	no	—
53	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
54	Dic₂₆₈	C₂×C₄	1	1	yes	characteristic, complemented
55	#11761618	C₂×C₄	1	2	yes	complemented
56	#11761617	C₂×C₄	1	2	yes	complemented
57	#11761616	C₂×C₄	1	2	yes	complemented
58	#11761615	C₂×C₄	1	2	yes	complemented
59	D₅₃₆	C₂×C₄	1	1	yes	characteristic, complemented
60	#11761613	C₂×C₄	1	2	yes	complemented
61	#11761612	—	2	2	no	—
62	#11761612	C₂³	1	2	yes	—
63	#11761611	—	2	2	no	—
64	#11761611	—	2	2	no	—
65	#11761611	—	2	2	no	—
66	#11761611	C₂×C₄	1	2	yes	—
67	#11761611	C₂×C₄	1	2	yes	—
68	#11761611	D₄	1	2	yes	—
69	#11761611	D₄	1	2	yes	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₆₇	—	2	2	no	—
74	C₈×D₆₇	—	2	2	no	—
75	C₈×D₆₇	—	2	2	no	—
76	C₄×Dic₆₇	—	2	1	no	—
77	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
78	#41×C₆₇	C₂³	1	2	yes	—
79	Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
80	SD₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
81	D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
82	M₁₆×C₆₇	—	2	2	no	—
83	M₁₆×C₆₇	C₂³	1	2	yes	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×D₆₇	—	2	2	no	—
89	C₂²×D₆₇	#41	1	1	yes	characteristic
90	Q₈×C₆₇	C₂²×C₄	1	2	yes	—
91	D₄×C₆₇	—	2	2	no	—
92	D₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₆₇	—	2	2	no	—
97	C₂²⋊D₆₇	—	2	2	no	—
98	C₂²⋊D₆₇	—	2	2	no	—
99	C₂×Dic₆₇	—	2	1	no	—
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	#41	1	1	yes	characteristic
102	D₂₆₈	—	2	2	no	—
103	D₂₆₈	C₂²×C₄	1	2	yes	—
104	C₄×D₆₇	—	2	2	no	—
105	C₄×D₆₇	—	2	2	no	—
106	C₄×D₆₇	—	2	2	no	—
107	C₄×D₆₇	—	4	2	no	—
108	C₄×D₆₇	C₂²×C₄	1	2	yes	—
109	C₄×D₆₇	C₂×D₄	1	2	yes	—
110	Dic₁₃₄	—	2	2	no	—
111	Dic₁₃₄	C₂²×C₄	1	2	yes	—
112	C₈×C₆₇	—	2	2	no	—
113	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₆₇⋊C₈	—	2	2	no	—
117	C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
118	C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
119	C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₆₇	—	2	2	no	—
121	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
122	C₂×D₆₇	—	2	2	no	—
123	C₂×D₆₇	—	4	1	no	—
124	C₂×D₆₇	—	8	2	no	—
125	C₂×D₆₇	C₄×D₄	1	2	yes	—
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	4	1	no	—
128	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	Dic₆₇	—	2	2	no	—
131	Dic₆₇	—	2	2	no	—
132	Dic₆₇	C₄×D₄	1	2	yes	—
133	C₂×C₆₇	—	2	1	no	—
134	C₂×C₆₇	—	4	2	no	—
135	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
136	D₆₇	—	2	2	no	—
137	D₆₇	—	4	2	no	—
138	#2956	—	67	1	no	maximal, Sylow
139	C₆₇	#2956	1	1	yes	characteristic, complemented, Sylow
140	#575	—	67	1	no	—
141	#568	—	67	2	no	—
142	#444	—	67	2	no	—
143	#444	—	67	1	no	—
144	#444	D₆₇	1	1	yes	characteristic, complemented
145	#443	—	67	1	no	—
146	#443	—	67	2	no	—
147	#443	—	67	1	no	—
148	#430	—	67	1	no	—
149	C₂×#104	—	67	2	no	—
150	C₄×M₁₆	—	67	1	no	—
151	C₂×#41	—	67	2	no	—
152	#137	—	67	2	no	—
153	#137	—	67	2	no	—
154	AGL(1,ℤ/8)	—	67	2	no	—
155	AGL(1,ℤ/8)	—	67	2	no	—
156	#135	—	67	2	no	—
157	#135	—	67	1	no	—
158	#135	C₂×D₆₇	1	1	yes	characteristic
159	#131	—	67	2	no	—
160	#131	—	67	2	no	—
161	#131	—	67	2	no	—
162	#131	—	134	2	no	—
163	#131	—	134	2	no	—
164	#131	C₂×D₆₇	1	2	yes	complemented
165	C₂×M₁₆	—	67	1	no	—
166	C₂×M₁₆	—	67	1	no	—
167	C₂×M₁₆	—	67	2	no	—
168	C₂×M₁₆	—	134	2	no	—
169	C₂×M₁₆	—	134	2	no	—
170	C₂×C₄²	—	67	1	no	—
171	#108	—	67	2	no	—
172	#108	—	67	1	no	—
173	#108	C₂×D₆₇	1	1	yes	characteristic, complemented
174	#104	—	67	2	no	—
175	#104	—	67	2	no	—
176	#104	—	67	2	no	—
177	#104	C₂×D₆₇	1	2	yes	complemented
178	#97	—	67	1	no	—
179	#97	C₂×D₆₇	1	1	yes	characteristic, complemented
180	C₄×C₈	—	67	1	no	—
181	C₄×C₈	—	67	1	no	—
182	#41	—	67	2	no	—
183	#41	—	67	2	no	—
184	#41	—	67	2	no	—
185	#41	—	134	2	no	—
186	#41	—	134	2	no	—
187	#41	C₂²×D₆₇	1	2	yes	—
188	C₂×Q₈	—	67	2	no	—
189	C₂×D₄	—	67	2	no	—
190	C₂×D₄	—	134	2	no	—
191	C₂²×C₄	—	67	1	no	—
192	C₂²×C₄	—	134	1	no	—
193	C₂²×C₄	—	134	2	no	—
194	Q₁₆	—	67	2	no	—
195	Q₁₆	—	67	1	no	—
196	Q₁₆	C₄×D₆₇	1	1	yes	characteristic, complemented
197	SD₁₆	—	67	2	no	—
198	SD₁₆	—	67	2	no	—
199	SD₁₆	—	67	2	no	—
200	SD₁₆	C₄×D₆₇	1	2	yes	complemented
201	D₈	—	67	2	no	—
202	D₈	—	67	1	no	—
203	D₈	C₄×D₆₇	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	67	2	no	—
206	M₁₆	—	67	2	no	—
207	M₁₆	—	67	2	no	—
208	M₁₆	—	67	2	no	—
209	M₁₆	—	67	2	no	—
210	M₁₆	—	134	2	no	—
211	M₁₆	—	134	2	no	—
212	M₁₆	—	134	2	no	—
213	M₁₆	—	134	2	no	—
214	M₁₆	C₂²×D₆₇	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	67	1	no	—
217	C₂×C₈	—	67	1	no	—
218	C₂×C₈	—	134	2	no	—
219	C₂×C₈	—	134	2	no	—
220	C₂×C₈	—	134	2	no	—
221	C₂×C₈	—	134	2	no	—
222	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
224	C₄²	—	67	1	no	—
225	C₄²	—	134	1	no	—
226	C₄²	C₂²×D₆₇	1	1	yes	characteristic
227	C₂³	—	67	1	no	—
228	C₂³	—	134	2	no	—
229	Q₈	—	67	2	no	—
230	Q₈	—	134	2	no	—
231	Q₈	C₂×C₄×D₆₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	67	2	no	—
234	D₄	—	134	2	no	—
235	D₄	—	134	2	no	—
236	D₄	—	134	2	no	—
237	D₄	—	134	2	no	—
238	D₄	C₂×C₄×D₆₇	1	2	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	268	2	no	—
242	C₂×C₄	—	67	1	no	—
243	C₂×C₄	—	67	2	no	—
244	C₂×C₄	—	67	2	no	—
245	C₂×C₄	—	134	2	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	—	134	1	no	—
249	C₂×C₄	—	134	2	no	—
250	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	67	2	no	—
253	C₈	—	67	1	no	—
254	C₈	—	67	1	no	—
255	C₈	—	134	2	no	—
256	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
257	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₈	D₄×D₆₇	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	268	1	no	—
261	C₂²	—	536	2	no	—
262	C₂²	—	67	2	no	—
263	C₂²	—	134	2	no	—
264	C₂²	#41×D₆₇	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	67	2	no	—
268	C₄	—	134	2	no	—
269	C₄	—	134	2	no	—
270	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
271	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	268	2	no	—
275	C₂	—	134	2	no	—
276	C₂	C₄×D₄×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#444×C₆₇	C₂
socle	C₂×C₆₇	C₂×C₄×D₄
hypercenter	#444	D₆₇
hypocenter	C₆₇	#2956
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #16864161	C₂	semidirect
3. #16863519	C₂	semidirect
4. #16863515	C₂	semidirect
5. #16863513	C₂	semidirect
6. #16863512	C₂	semidirect
7. #16863511	C₂	semidirect
8. #16863443	C₂	semidirect
9. #104×D₆₇	C₂	semidirect
10. #16863348	C₂	semidirect
11. #444×C₆₇	C₂	semidirect
13. #14524075	C₄	semidirect
14. #14524072	C₄	semidirect
15. #14524070	C₄	semidirect
16. #14524067	C₄	semidirect
20. #14524051	C₂²	semidirect
21. #14524050	C₂²	semidirect
23. #14524049	C₂²	semidirect
30. #14523979	C₂²	semidirect
31. #14523978	C₂²	semidirect
32. #14523971	C₂²	semidirect
33. #14523970	C₂²	semidirect
34. #14523969	C₂²	semidirect
35. #14523954	C₂²	semidirect
36. C₈×Dic₆₇	C₂²	semidirect
37. C₈×Dic₆₇	C₂²	semidirect
39. #131×C₆₇	C₂²	semidirect
40. #108×C₆₇	C₂²	semidirect
41. #104×C₆₇	C₂²	semidirect
42. #97×C₆₇	C₂²	semidirect
54. Dic₂₆₈	C₂×C₄	semidirect
55. #11761618	C₂×C₄	semidirect
56. #11761617	C₂×C₄	semidirect
57. #11761616	C₂×C₄	semidirect
58. #11761615	C₂×C₄	semidirect
59. D₅₃₆	C₂×C₄	semidirect
60. #11761613	C₂×C₄	semidirect
79. Q₁₆×C₆₇	C₂×C₄	semidirect
80. SD₁₆×C₆₇	C₂×C₄	semidirect
81. D₈×C₆₇	C₂×C₄	semidirect
139. C₆₇	#2956	semidirect
144. #444	D₆₇	semidirect
164. #131	C₂×D₆₇	semidirect
173. #108	C₂×D₆₇	semidirect
177. #104	C₂×D₆₇	semidirect
179. #97	C₂×D₆₇	semidirect
196. Q₁₆	C₄×D₆₇	semidirect
200. SD₁₆	C₄×D₆₇	semidirect
203. D₈	C₄×D₆₇	semidirect