Subgroups — #22569107

Order 8576.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 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	#131×D₆₇	C₂	1	2	yes	maximal, complemented
3	#16863518	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863517	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863515	C₂	1	2	yes	maximal, complemented
6	#16863514	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	#16863411	C₂	1	2	yes	maximal, complemented
10	#16863350	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×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	C₂²	1	1	yes	characteristic
25	M₁₆×D₆₇	—	2	2	no	—
26	M₁₆×D₆₇	C₂²	1	2	yes	—
27	C₂×C₈×D₆₇	—	2	2	no	—
28	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
29	#14524000	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	#14523953	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	C₄×C₈×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	C₂³	1	1	yes	characteristic
53	Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
54	#11761618	C₂×C₄	1	2	yes	—
55	#11761617	C₂×C₄	1	2	yes	—
56	#11761616	C₂×C₄	1	2	yes	—
57	#11761615	C₂×C₄	1	2	yes	—
58	D₅₃₆	C₂×C₄	1	1	yes	characteristic
59	#11761613	C₂×C₄	1	2	yes	—
60	#11761612	—	2	2	no	—
61	#11761612	C₂³	1	2	yes	—
62	#11761611	—	2	2	no	—
63	#11761611	—	2	2	no	—
64	#11761611	—	2	2	no	—
65	#11761611	C₂×C₄	1	2	yes	—
66	#11761611	C₂×C₄	1	2	yes	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×D₆₇	—	2	2	no	—
71	C₈×D₆₇	—	2	2	no	—
72	C₈×D₆₇	—	2	2	no	—
73	C₈×D₆₇	D₄	1	2	yes	—
74	C₈×D₆₇	D₄	1	2	yes	—
75	Dic₆₇⋊C₄	—	2	1	no	—
76	(C₂×D₆₇)⋊C₄	—	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
80	SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
81	D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
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	2	no	—
100	C₂×Dic₆₇	—	2	1	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	C₂²×C₄	1	2	yes	—
108	C₄×D₆₇	C₂×D₄	1	2	yes	—
109	Dic₁₃₄	—	2	2	no	—
110	Dic₁₃₄	C₂²×C₄	1	2	yes	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₆₇⋊C₈	—	2	2	no	—
117	C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
118	C₆₇⋊C₈	C₂×D₄	1	2	yes	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂×D₆₇	—	2	2	no	—
122	C₂×D₆₇	—	4	1	no	—
123	C₂×D₆₇	—	4	4	no	—
124	C₂×D₆₇	C₄×D₄	1	2	yes	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	4	1	no	—
127	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	Dic₆₇	—	2	2	no	—
130	Dic₆₇	—	4	1	no	—
131	Dic₆₇	C₄×D₄	1	2	yes	—
132	C₂×C₆₇	—	2	1	no	—
133	C₂×C₆₇	—	4	2	no	—
134	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
135	D₆₇	—	2	2	no	—
136	D₆₇	—	4	2	no	—
137	#2955	—	67	1	no	maximal, Sylow
138	C₆₇	#2955	1	1	yes	characteristic, complemented, Sylow
139	#575	—	67	1	no	—
140	C₂×#131	—	67	2	no	—
141	#444	—	67	1	no	—
142	#444	—	67	1	no	—
143	#444	—	67	2	no	—
144	#443	—	67	2	no	—
145	#443	—	67	1	no	—
146	#443	D₆₇	1	1	yes	characteristic, complemented
147	#430	—	67	1	no	—
148	#421	—	67	2	no	—
149	#405	—	67	1	no	—
150	C₂×#41	—	67	2	no	—
151	#137	—	67	2	no	—
152	#137	—	67	2	no	—
153	AGL(1,ℤ/8)	—	67	2	no	—
154	AGL(1,ℤ/8)	—	67	2	no	—
155	#135	—	67	2	no	—
156	#135	—	67	1	no	—
157	#135	C₂×D₆₇	1	1	yes	characteristic
158	#131	—	67	2	no	—
159	#131	—	67	2	no	—
160	#131	—	67	2	no	—
161	#131	—	134	2	no	—
162	#131	—	134	2	no	—
163	#131	C₂×D₆₇	1	2	yes	complemented
164	C₂×M₁₆	—	67	1	no	—
165	C₂×M₁₆	—	67	2	no	—
166	C₂×M₁₆	—	134	2	no	—
167	C₂²×C₈	—	67	1	no	—
168	C₂²×C₈	—	134	2	no	—
169	#117	—	67	1	no	—
170	#108	—	67	2	no	—
171	#108	—	67	1	no	—
172	#108	C₂×D₆₇	1	1	yes	characteristic, complemented
173	#104	—	67	2	no	—
174	#104	—	67	2	no	—
175	#104	—	67	2	no	—
176	#104	C₂×D₆₇	1	2	yes	complemented
177	#97	—	67	1	no	—
178	#97	—	67	1	no	—
179	C₄×C₈	—	67	1	no	—
180	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
181	#41	—	67	2	no	—
182	#41	—	67	2	no	—
183	#41	—	67	2	no	—
184	#41	—	134	2	no	—
185	#41	—	134	2	no	—
186	#41	C₂²×D₆₇	1	2	yes	—
187	C₂×Q₈	—	67	2	no	—
188	C₂×D₄	—	67	2	no	—
189	C₂×D₄	—	134	2	no	—
190	C₂²×C₄	—	67	1	no	—
191	C₂²×C₄	—	134	2	no	—
192	Q₁₆	—	67	2	no	—
193	Q₁₆	—	67	1	no	—
194	Q₁₆	C₄×D₆₇	1	1	yes	characteristic
195	SD₁₆	—	67	2	no	—
196	SD₁₆	—	67	2	no	—
197	SD₁₆	—	67	2	no	—
198	SD₁₆	C₄×D₆₇	1	2	yes	—
199	D₈	—	67	2	no	—
200	D₈	—	67	1	no	—
201	D₈	C₄×D₆₇	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	67	2	no	—
204	M₁₆	—	67	2	no	—
205	M₁₆	—	67	2	no	—
206	M₁₆	—	134	2	no	—
207	M₁₆	—	134	2	no	—
208	M₁₆	—	134	2	no	—
209	M₁₆	—	134	2	no	—
210	M₁₆	C₂²×D₆₇	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	67	1	no	—
213	C₂×C₈	—	67	1	no	—
214	C₂×C₈	—	67	2	no	—
215	C₂×C₈	—	67	2	no	—
216	C₂×C₈	—	134	2	no	—
217	C₂×C₈	—	134	2	no	—
218	C₂×C₈	—	134	2	no	—
219	C₂×C₈	—	134	2	no	—
220	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
222	C₄⋊C₄	—	134	1	no	—
223	C₂²⋊C₄	—	134	1	no	—
224	C₄²	—	67	1	no	—
225	C₄²	C₂²×D₆₇	1	1	yes	characteristic
226	C₂³	—	67	1	no	—
227	C₂³	—	134	2	no	—
228	Q₈	—	67	2	no	—
229	Q₈	—	134	2	no	—
230	Q₈	C₂×C₄×D₆₇	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	67	2	no	—
233	D₄	—	134	2	no	—
234	D₄	—	134	2	no	—
235	D₄	—	134	2	no	—
236	D₄	—	134	2	no	—
237	D₄	C₂×C₄×D₆₇	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	67	1	no	—
241	C₂×C₄	—	67	2	no	—
242	C₂×C₄	—	67	2	no	—
243	C₂×C₄	—	134	2	no	—
244	C₂×C₄	—	134	2	no	—
245	C₂×C₄	—	134	1	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	67	2	no	—
251	C₈	—	67	2	no	—
252	C₈	—	134	2	no	—
253	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
254	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
255	C₈	D₄×D₆₇	1	1	yes	characteristic
256	C₈	D₄×D₆₇	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	268	1	no	—
259	C₂²	—	268	4	no	—
260	C₂²	—	67	2	no	—
261	C₂²	—	134	2	no	—
262	C₂²	#41×D₆₇	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	268	1	no	—
266	C₄	—	67	2	no	—
267	C₄	—	134	2	no	—
268	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
269	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	268	2	no	—
273	C₂	—	134	2	no	—
274	C₂	C₄×D₄×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#443×C₆₇	C₂
socle	C₂×C₆₇	C₂×C₄×D₄
hypercenter	#443	D₆₇
hypocenter	C₆₇	#2955
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₆₇	C₂	semidirect
3. #16863518	C₂	semidirect
4. #16863517	C₂	semidirect
5. #16863515	C₂	semidirect
6. #16863514	C₂	semidirect
7. #16863511	C₂	semidirect
8. #16863443	C₂	semidirect
9. #16863411	C₂	semidirect
10. #16863350	C₂	semidirect
11. #443×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. #14523953	C₂²	semidirect
37. C₈×Dic₆₇	C₂²	semidirect
39. #131×C₆₇	C₂²	semidirect
40. #108×C₆₇	C₂²	semidirect
41. #104×C₆₇	C₂²	semidirect
42. C₄×C₈×C₆₇	C₂²	semidirect
138. C₆₇	#2955	semidirect
146. #443	D₆₇	semidirect
163. #131	C₂×D₆₇	semidirect
172. #108	C₂×D₆₇	semidirect
176. #104	C₂×D₆₇	semidirect
180. C₄×C₈	C₂×D₆₇	semidirect