Subgroups — #22571996

Order 8576.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864213	C₂	1	1	yes	maximal, characteristic, complemented
2	#16864203	C₂	1	2	yes	maximal, complemented
3	#16864201	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863780	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863671	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863666	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863660	C₂	1	2	yes	maximal, complemented
8	#16863575	C₂	1	2	yes	maximal, complemented
9	#16863570	C₂	1	2	yes	maximal, complemented
10	#16863514	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#14524090	C₂²	1	2	yes	complemented
13	#14524089	C₂²	1	2	yes	—
14	#14524076	—	2	2	no	—
15	#14524076	C₂²	1	2	yes	complemented
16	#14524075	—	2	2	no	—
17	#14524073	C₂²	1	2	yes	—
18	#14524071	—	2	2	no	—
19	#14524071	C₂²	1	2	yes	—
20	#14524068	C₂²	1	2	yes	complemented
21	#14524067	—	2	2	no	—
22	#14524065	—	2	1	no	—
23	#14524065	—	2	1	no	—
24	#14524065	C₂²	1	1	yes	characteristic
25	#14524064	—	2	2	no	—
26	C₂×Dic₂₆₈	—	2	1	no	—
27	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×#11761617	C₂²	1	2	yes	complemented
29	SD₁₆×D₆₇	—	2	2	no	—
30	C₂×D₅₃₆	—	2	1	no	—
31	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
32	#14524049	C₂²	1	2	yes	complemented
33	#14524031	C₂²	1	2	yes	complemented
34	#14523979	C₂²	1	1	yes	characteristic, complemented
35	#14523971	C₂²	1	2	yes	complemented
36	#14523961	C₂²	1	2	yes	complemented
37	#14523959	C₂²	1	2	yes	complemented
38	#135×C₆₇	C₂²	1	1	yes	characteristic
39	#131×C₆₇	C₂²	1	1	yes	characteristic
40	#131×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	#104×C₆₇	C₂²	1	1	yes	characteristic
43	#104×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#11761629	—	2	2	no	—
46	#11761628	—	2	2	no	—
47	#11761628	—	2	2	no	—
48	#11761628	C₂³	1	2	yes	—
49	#11761627	—	2	2	no	—
50	#11761627	—	4	2	no	—
51	C₂×Dic₁₃₄	—	2	2	no	—
52	C₂×Dic₁₃₄	C₂³	1	2	yes	—
53	Q₈×D₆₇	—	2	2	no	—
54	C₂×D₂₆₈	—	2	2	no	—
55	C₂×D₂₆₈	C₂³	1	2	yes	—
56	D₄×D₆₇	—	2	2	no	—
57	D₄×D₆₇	—	4	2	no	—
58	Dic₂₆₈	—	2	2	no	—
59	Dic₂₆₈	—	2	1	no	—
60	Dic₂₆₈	—	2	1	no	—
61	Dic₂₆₈	D₄	1	1	yes	characteristic
62	#11761618	—	2	2	no	—
63	#11761617	—	2	2	no	—
64	#11761617	—	2	2	no	—
65	#11761617	—	2	2	no	—
66	#11761617	—	2	2	no	—
67	#11761617	D₄	1	2	yes	—
68	#11761616	—	2	2	no	—
69	#11761615	—	2	2	no	—
70	D₅₃₆	—	2	2	no	—
71	D₅₃₆	—	2	1	no	—
72	D₅₃₆	—	2	1	no	—
73	D₅₃₆	D₄	1	1	yes	characteristic
74	#11761613	—	2	2	no	—
75	#11761612	C₂³	1	2	yes	—
76	#11761611	—	2	2	no	—
77	C₈×D₆₇	—	2	2	no	—
78	(C₂×D₆₇)⋊C₄	—	4	2	no	—
79	#41×C₆₇	C₂³	1	1	yes	characteristic
80	#41×C₆₇	C₂³	1	1	yes	characteristic
81	Q₁₆×C₆₇	D₄	1	1	yes	characteristic
82	SD₁₆×C₆₇	D₄	1	1	yes	characteristic
83	SD₁₆×C₆₇	D₄	1	1	yes	characteristic
84	D₈×C₆₇	D₄	1	1	yes	characteristic
85	M₁₆×C₆₇	—	2	1	no	—
86	M₁₆×C₆₇	—	2	1	no	—
87	M₁₆×C₆₇	C₂³	1	1	yes	characteristic
88	M₁₆×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×D₆₇	—	2	2	no	—
95	C₂²×D₆₇	—	4	2	no	—
96	Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	D₄×C₆₇	—	2	1	no	—
99	D₄×C₆₇	—	2	1	no	—
100	D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₆₇	—	2	2	no	—
107	C₂²⋊D₆₇	—	4	2	no	—
108	C₂²⋊D₆₇	—	4	2	no	—
109	C₂×Dic₆₇	—	2	2	no	—
110	C₂×Dic₆₇	—	4	2	no	—
111	D₂₆₈	—	2	2	no	—
112	D₂₆₈	—	2	2	no	—
113	D₂₆₈	—	4	2	no	—
114	D₂₆₈	C₂×D₄	1	2	yes	—
115	C₄×D₆₇	—	2	2	no	—
116	C₄×D₆₇	—	4	2	no	—
117	C₄×D₆₇	—	4	2	no	—
118	Dic₁₃₄	—	2	2	no	—
119	Dic₁₃₄	—	2	2	no	—
120	Dic₁₃₄	—	4	2	no	—
121	Dic₁₃₄	C₂×D₄	1	2	yes	—
122	C₈×C₆₇	—	2	1	no	—
123	C₈×C₆₇	—	2	1	no	—
124	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₆₇⋊C₈	—	2	2	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₆₇	—	2	2	no	—
133	C₂×D₆₇	—	4	2	no	—
134	C₂×D₆₇	—	8	2	no	—
135	C₂×D₆₇	—	16	2	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	—	4	1	no	—
139	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	Dic₆₇	—	2	2	no	—
142	Dic₆₇	—	4	2	no	—
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	—	4	1	no	—
145	C₂×C₆₇	—	4	1	no	—
146	C₂×C₆₇	D₄²	1	1	yes	characteristic
147	D₆₇	—	4	2	no	—
148	D₆₇	—	8	2	no	—
149	#3290	—	67	1	no	maximal, Sylow
150	C₆₇	#3290	1	1	yes	characteristic, complemented, Sylow
151	#578	—	67	1	no	—
152	#577	—	67	2	no	—
153	#576	—	67	1	no	—
154	#495	—	67	1	no	—
155	#473	—	67	1	no	—
156	#472	—	67	1	no	—
157	#471	—	67	2	no	—
158	#455	—	67	2	no	—
159	#454	—	67	2	no	—
160	#443	—	67	1	no	—
161	#443	D₆₇	1	1	yes	characteristic, complemented
162	2₋¹⁺⁴	—	67	2	no	—
163	H(5,𝔽₂)	—	67	2	no	—
164	#137	—	67	2	no	—
165	#137	—	67	2	no	—
166	#137	—	134	2	no	—
167	#137	—	134	2	no	—
168	AGL(1,ℤ/8)	—	67	2	no	—
169	AGL(1,ℤ/8)	—	67	2	no	—
170	AGL(1,ℤ/8)	—	134	2	no	—
171	AGL(1,ℤ/8)	—	134	2	no	—
172	#135	—	67	1	no	—
173	#135	—	134	1	no	—
174	#135	—	134	1	no	—
175	#135	—	134	2	no	—
176	#135	C₂×D₆₇	1	1	yes	characteristic
177	C₂×Q₁₆	—	67	1	no	—
178	C₂×Q₁₆	—	134	1	no	—
179	C₂×SD₁₆	—	67	2	no	—
180	C₂×SD₁₆	—	134	2	no	—
181	C₂×D₈	—	67	1	no	—
182	C₂×D₈	—	134	1	no	—
183	#131	—	67	2	no	—
184	#131	C₂×D₆₇	1	1	yes	characteristic
185	#131	C₂×D₆₇	1	1	yes	characteristic, complemented
186	#124	—	67	2	no	—
187	#108	—	67	1	no	—
188	#108	C₂×D₆₇	1	1	yes	characteristic, complemented
189	#104	—	67	2	no	—
190	#104	C₂×D₆₇	1	1	yes	characteristic
191	#104	C₂×D₆₇	1	1	yes	characteristic, complemented
192	#101	—	67	2	no	—
193	#100	—	67	2	no	—
194	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
195	#41	—	268	2	no	—
196	#41	—	67	2	no	—
197	#41	—	134	2	no	—
198	#41	—	134	2	no	—
199	#41	—	134	2	no	—
200	#41	—	134	2	no	—
201	#41	C₂²×D₆₇	1	1	yes	characteristic
202	#41	C₂²×D₆₇	1	1	yes	characteristic
203	C₂×Q₈	—	67	2	no	—
204	C₂×Q₈	—	134	2	no	—
205	C₂×Q₈	—	134	2	no	—
206	C₂×D₄	—	268	2	no	—
207	C₂×D₄	—	67	2	no	—
208	C₂×D₄	—	134	2	no	—
209	C₂×D₄	—	134	2	no	—
210	Q₁₆	—	67	1	no	—
211	Q₁₆	—	134	2	no	—
212	Q₁₆	—	134	2	no	—
213	Q₁₆	—	134	1	no	—
214	Q₁₆	—	134	1	no	—
215	Q₁₆	D₂₆₈	1	1	yes	characteristic
216	SD₁₆	—	67	2	no	—
217	SD₁₆	—	134	2	no	—
218	SD₁₆	—	134	2	no	—
219	SD₁₆	—	134	2	no	—
220	SD₁₆	—	134	2	no	—
221	SD₁₆	—	134	2	no	—
222	SD₁₆	—	134	2	no	—
223	SD₁₆	D₂₆₈	1	1	yes	characteristic
224	SD₁₆	D₂₆₈	1	1	yes	characteristic
225	D₈	—	67	1	no	—
226	D₈	—	134	2	no	—
227	D₈	—	134	2	no	—
228	D₈	—	134	1	no	—
229	D₈	—	134	1	no	—
230	D₈	D₂₆₈	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	67	2	no	—
234	M₁₆	—	134	2	no	—
235	M₁₆	C₂²×D₆₇	1	1	yes	characteristic
236	M₁₆	C₂²×D₆₇	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	134	2	no	—
240	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
242	C₂²⋊C₄	—	268	2	no	—
243	C₄²	C₂²×D₆₇	1	1	yes	characteristic
244	C₂³	—	268	2	no	—
245	C₂³	—	134	2	no	—
246	Q₈	—	268	2	no	—
247	Q₈	—	67	2	no	—
248	Q₈	—	134	2	no	—
249	Q₈	—	134	2	no	—
250	Q₈	C₂×D₂₆₈	1	1	yes	characteristic
251	Q₈	C₂×D₂₆₈	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	268	2	no	—
255	D₄	—	268	2	no	—
256	D₄	—	268	2	no	—
257	D₄	—	67	2	no	—
258	D₄	—	134	2	no	—
259	D₄	—	134	2	no	—
260	D₄	—	134	2	no	—
261	D₄	C₂×D₂₆₈	1	1	yes	characteristic
262	D₄	C₂×D₂₆₈	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	268	2	no	—
267	C₂×C₄	—	268	2	no	—
268	C₂×C₄	—	268	2	no	—
269	C₂×C₄	—	134	2	no	—
270	C₂×C₄	—	134	2	no	—
271	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	134	2	no	—
275	C₈	D₄×D₆₇	1	1	yes	characteristic
276	C₈	D₄×D₆₇	1	1	yes	characteristic
277	C₈	C₂×D₂₆₈	1	1	yes	characteristic
278	C₈	C₂×D₂₆₈	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	268	2	no	—
282	C₂²	—	536	2	no	—
283	C₂²	—	1072	2	no	—
284	C₂²	—	134	2	no	—
285	C₂²	#14524089	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	268	2	no	—
290	C₄	—	134	2	no	—
291	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
292	C₄	C₂²×D₂₆₈	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	268	2	no	—
297	C₂	—	536	2	no	—
298	C₂	D₄×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#443×C₆₇	C₂
socle	C₂×C₆₇	D₄²
hypercenter	#443	D₆₇
hypocenter	C₆₇	#3290
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #16864213	C₂	semidirect
2. #16864203	C₂	semidirect
3. #16864201	C₂	semidirect
4. #16863780	C₂	semidirect
5. #16863671	C₂	semidirect
6. #16863666	C₂	semidirect
7. #16863660	C₂	semidirect
8. #16863575	C₂	semidirect
9. #16863570	C₂	semidirect
10. #16863514	C₂	semidirect
11. #443×C₆₇	C₂	semidirect
12. #14524090	C₂²	semidirect
15. #14524076	C₂²	semidirect
20. #14524068	C₂²	semidirect
27. C₂×Dic₂₆₈	C₂²	semidirect
28. C₂×#11761617	C₂²	semidirect
32. #14524049	C₂²	semidirect
33. #14524031	C₂²	semidirect
34. #14523979	C₂²	semidirect
35. #14523971	C₂²	semidirect
36. #14523961	C₂²	semidirect
37. #14523959	C₂²	semidirect
40. #131×C₆₇	C₂²	semidirect
41. #108×C₆₇	C₂²	semidirect
43. #104×C₆₇	C₂²	semidirect
44. C₄×C₈×C₆₇	C₂²	semidirect
150. C₆₇	#3290	semidirect
161. #443	D₆₇	semidirect
185. #131	C₂×D₆₇	semidirect
188. #108	C₂×D₆₇	semidirect
191. #104	C₂×D₆₇	semidirect
194. C₄×C₈	C₂×D₆₇	semidirect