Subgroups — #22569037

Order 8576.

506 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	C₂×#14524068	C₂	1	1	yes	maximal, characteristic
2	#16863870	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863506	C₂	1	2	yes	maximal, complemented
4	#16863494	C₂	1	2	yes	maximal, complemented
5	C₄×#11761615	C₂	1	2	yes	maximal, complemented
6	C₄×#11761613	C₂	1	2	yes	maximal, complemented
7	#16863436	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863392	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523963	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863352	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
13	#14524068	C₄	1	8	yes	complemented
14	C₂×#11761615	C₂²	1	2	yes	—
15	C₂×#11761613	C₂²	1	2	yes	—
16	C₂×#11761612	C₂²	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×D₂₆₈	C₂²	1	2	yes	—
19	C₄×D₂₆₈	C₂²	1	2	yes	—
20	#14524005	—	2	1	no	—
21	#14524005	—	2	1	no	—
22	#14524002	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×D₆₇	—	2	2	no	—
24	#14523976	C₂²	1	2	yes	complemented
25	#14523974	C₂²	1	2	yes	complemented
26	#14523967	C₂²	1	1	yes	characteristic, complemented
27	#14523967	C₂²	1	1	yes	characteristic, complemented
28	#14523964	C₂²	1	1	yes	characteristic, complemented
29	#14523964	C₂²	1	1	yes	characteristic, complemented
30	#14523963	C₂²	1	2	yes	—
31	#14523963	C₂²	1	2	yes	—
32	#14523954	C₂²	1	2	yes	complemented
33	C₄×C₆₇⋊C₈	C₂²	1	2	yes	complemented
34	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
36	C₄×D₄×C₆₇	—	2	2	no	—
37	C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
38	#117×C₆₇	—	2	1	no	—
39	#117×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	C₂³×D₆₇	—	2	1	no	—
43	C₂×D₂₆₈	—	2	1	no	—
44	C₂×D₂₆₈	—	2	2	no	—
45	C₂×D₂₆₈	C₂×C₄	1	4	yes	complemented
46	C₂×D₂₆₈	C₂³	1	2	yes	—
47	C₂×C₄×D₆₇	—	2	2	no	—
48	C₂×C₄×D₆₇	—	2	2	no	—
49	#11761615	C₂×C₄	1	8	yes	complemented
50	#11761613	C₂×C₄	1	8	yes	complemented
51	#11761612	C₂×C₄	1	4	yes	complemented
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
56	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₆₇	—	2	1	no	—
59	(C₂×D₆₇)⋊C₄	—	2	2	no	—
60	(C₂×D₆₇)⋊C₄	—	2	2	no	—
61	C₄×Dic₆₇	—	2	1	no	—
62	#41×C₆₇	—	2	2	no	—
63	#41×C₆₇	C₂×C₄	1	4	yes	complemented
64	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₆₇	—	2	1	no	—
66	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₆₇	—	2	1	no	—
81	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₆₇	—	2	2	no	—
83	C₂²×D₆₇	—	2	4	no	—
84	C₂²×D₆₇	—	8	1	no	—
85	C₂²×D₆₇	—	8	1	no	—
86	Q₈×C₆₇	—	2	1	no	—
87	Q₈×C₆₇	C₂²×C₄	1	2	yes	—
88	D₄×C₆₇	—	2	4	no	—
89	D₄×C₆₇	—	2	1	no	—
90	D₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	8	2	no	—
95	C₂×C₄×C₆₇	—	8	1	no	—
96	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#41	1	2	yes	—
101	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	1	no	—
103	C₂×Dic₆₇	—	2	1	no	—
104	D₂₆₈	—	2	4	no	—
105	D₂₆₈	—	2	2	no	—
106	D₂₆₈	C₂²×C₄	1	4	yes	—
107	C₄×D₆₇	—	8	2	no	—
108	C₄×D₆₇	—	8	2	no	—
109	C₆₇⋊C₈	—	2	1	no	—
110	C₆₇⋊C₈	—	2	1	no	—
111	C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	4	1	no	—
116	C₂²×C₆₇	—	4	1	no	—
117	C₂²×C₆₇	C₄×D₄	1	2	yes	—
118	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₆₇	—	2	4	no	—
120	C₂×D₆₇	—	4	2	no	—
121	C₂×D₆₇	—	4	2	no	—
122	C₂×D₆₇	—	8	4	no	—
123	C₄×C₆₇	—	2	2	no	—
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	—	2	2	no	—
126	C₄×C₆₇	—	4	2	no	—
127	C₄×C₆₇	C₄×D₄	1	2	yes	—
128	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
130	Dic₆₇	—	4	1	no	—
131	Dic₆₇	—	4	1	no	—
132	C₂×C₆₇	—	2	2	no	—
133	C₂×C₆₇	—	4	2	no	—
134	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₆₇	#576	1	1	yes	characteristic
136	C₂×C₆₇	#577	1	1	yes	characteristic
137	D₆₇	—	4	4	no	—
138	#2946	—	67	1	no	maximal, Sylow
139	C₆₇	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	—	67	1	no	—
141	#520	D₆₇	1	1	yes	characteristic, complemented
142	#518	—	67	1	no	—
143	#442	—	67	2	no	—
144	#440	—	67	2	no	—
145	C₄×SD₁₆	—	67	2	no	—
146	C₄×D₈	—	67	2	no	—
147	#428	—	67	1	no	—
148	#416	—	67	1	no	—
149	C₂×#102	—	67	1	no	—
150	#405	—	67	1	no	—
151	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
152	C₂²×D₄	—	67	1	no	—
153	AGL(1,ℤ/8)	—	67	8	no	—
154	C₂×SD₁₆	—	67	2	no	—
155	C₂×D₈	—	67	2	no	—
156	C₂×M₁₆	—	67	1	no	—
157	C₂²×C₈	—	67	1	no	—
158	C₄×Q₈	C₂×D₆₇	1	2	yes	complemented
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	67	2	no	—
161	C₄×D₄	—	67	2	no	—
162	C₄×D₄	—	134	1	no	—
163	C₄×D₄	—	134	1	no	—
164	C₄×D₄	C₂×D₆₇	1	2	yes	complemented
165	#117	—	2	1	no	—
166	#117	—	67	1	no	—
167	#117	C₂×D₆₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	134	2	no	—
171	#107	—	67	2	no	—
172	#106	—	67	2	no	—
173	#103	—	67	1	no	—
174	#103	—	67	1	no	—
175	#102	—	67	1	no	—
176	#102	—	67	1	no	—
177	#102	—	67	2	no	—
178	#102	—	67	2	no	—
179	#97	—	67	2	no	—
180	C₄×C₈	—	67	2	no	—
181	C₂⁴	—	134	1	no	—
182	#41	—	2	2	no	—
183	#41	C₄×D₆₇	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	67	4	no	—
187	C₂×D₄	—	67	2	no	—
188	C₂×D₄	—	134	1	no	—
189	C₂×D₄	—	134	2	no	—
190	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	134	2	no	—
195	C₂²×C₄	—	134	2	no	—
196	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
197	SD₁₆	—	67	8	no	—
198	D₈	—	67	8	no	—
199	M₁₆	—	67	4	no	—
200	C₂×C₈	—	67	2	no	—
201	C₂×C₈	—	67	1	no	—
202	C₂×C₈	—	67	1	no	—
203	C₂×C₈	—	134	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	67	1	no	—
207	C₄⋊C₄	—	67	1	no	—
208	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
209	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	134	1	no	—
213	C₂²⋊C₄	—	134	2	no	—
214	C₂²⋊C₄	—	134	2	no	—
215	C₂²⋊C₄	C₂²⋊D₆₇	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	134	1	no	—
218	C₄²	C₂²×D₆₇	1	2	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	—	536	1	no	—
221	C₂³	—	536	1	no	—
222	C₂³	—	134	2	no	—
223	C₂³	—	134	4	no	—
224	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂×C₄×D₆₇	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	1	no	—
229	D₄	—	67	4	no	—
230	D₄	—	134	4	no	—
231	D₄	—	134	2	no	—
232	D₄	C₂×C₄×D₆₇	1	2	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	536	2	no	—
239	C₂×C₄	—	536	2	no	—
240	C₂×C₄	—	134	1	no	—
241	C₂×C₄	—	134	1	no	—
242	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
246	C₂×C₄	#11761628	1	2	yes	—
247	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
248	C₈	—	67	4	no	—
249	C₈	—	134	1	no	—
250	C₈	—	134	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	268	2	no	—
257	C₂²	—	268	2	no	—
258	C₂²	—	536	4	no	—
259	C₂²	—	134	4	no	—
260	C₂²	C₄×C₂²⋊D₆₇	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	2	no	—
266	C₄	—	268	1	no	—
267	C₄	—	268	1	no	—
268	C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
269	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
270	C₄	C₂×#11761628	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	2	no	—
273	C₂	—	268	4	no	—
274	C₂	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
275	C₂	#16864199	1	1	yes	characteristic
276	C₂	#16864204	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#520×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#520	D₆₇
hypocenter	C₆₇	#2946
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #16863870	C₂	semidirect
3. #16863506	C₂	semidirect
4. #16863494	C₂	semidirect
5. C₄×#11761615	C₂	semidirect
6. C₄×#11761613	C₂	semidirect
7. #16863436	C₂	semidirect
8. #16863392	C₂	semidirect
9. C₂×#14523963	C₂	semidirect
10. #16863352	C₂	semidirect
11. #520×C₆₇	C₂	semidirect
13. #14524068	C₄	semidirect
24. #14523976	C₂²	semidirect
25. #14523974	C₂²	semidirect
26. #14523967	C₂²	semidirect
27. #14523967	C₂²	semidirect
28. #14523964	C₂²	semidirect
29. #14523964	C₂²	semidirect
32. #14523954	C₂²	semidirect
33. C₄×C₆₇⋊C₈	C₂²	semidirect
35. C₄×Q₈×C₆₇	C₂²	semidirect
37. C₄×D₄×C₆₇	C₂²	semidirect
45. C₂×D₂₆₈	C₂×C₄	semidirect
49. #11761615	C₂×C₄	semidirect
50. #11761613	C₂×C₄	semidirect
51. #11761612	C₂×C₄	semidirect
63. #41×C₆₇	C₂×C₄	semidirect
139. C₆₇	#2946	semidirect
141. #520	D₆₇	semidirect
158. C₄×Q₈	C₂×D₆₇	semidirect
164. C₄×D₄	C₂×D₆₇	semidirect
183. #41	C₄×D₆₇	semidirect