Subgroups — #22569042

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	#16863879	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863871	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863508	C₂	1	2	yes	maximal, complemented
4	#16863497	C₂	1	2	yes	maximal, complemented
5	SD₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
6	D₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
7	#16863438	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863392	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523964	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863354	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
14	D₄×Dic₆₇	—	2	1	no	—
15	D₄×Dic₆₇	—	2	1	no	—
16	D₄×Dic₆₇	—	2	2	no	—
17	D₄×Dic₆₇	C₂²	1	2	yes	—
18	D₄×Dic₆₇	C₂²	1	2	yes	—
19	D₄×Dic₆₇	C₂²	1	2	yes	complemented
20	#14524002	—	2	1	no	—
21	#14524002	C₂²	1	1	yes	characteristic
22	#14524002	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₆₇	—	2	1	no	—
24	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
26	#14523977	C₂²	1	2	yes	complemented
27	#14523975	C₂²	1	2	yes	complemented
28	#14523967	C₂²	1	1	yes	characteristic, complemented
29	#14523967	C₂²	1	1	yes	characteristic, complemented
30	#14523964	C₂²	1	2	yes	—
31	#14523964	C₂²	1	2	yes	—
32	#14523964	C₂²	1	1	yes	characteristic, complemented
33	#14523964	C₂²	1	1	yes	characteristic, complemented
34	#14523953	C₂²	1	2	yes	complemented
35	C₈×Dic₆₇	C₂²	1	2	yes	complemented
36	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	AGL(1,ℤ/8)×C₆₇	C₄	1	8	yes	complemented
39	C₂×SD₁₆×C₆₇	C₂²	1	2	yes	—
40	C₂×D₈×C₆₇	C₂²	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	C₂²×Dic₆₇	—	2	1	no	—
43	C₂²×Dic₆₇	—	2	2	no	—
44	C₂²×Dic₆₇	—	2	2	no	—
45	C₂²×Dic₆₇	—	2	2	no	—
46	C₂×C₆₇⋊C₈	—	2	2	no	—
47	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₆₇	—	2	1	no	—
50	C₄⋊Dic₆₇	—	2	2	no	—
51	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
52	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
53	C₄⋊Dic₆₇	C₂³	1	2	yes	—
54	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₆₇	—	2	1	no	—
57	C₂²⋊Dic₆₇	—	2	2	no	—
58	C₂²⋊Dic₆₇	—	2	2	no	—
59	C₂²⋊Dic₆₇	—	2	2	no	—
60	C₂²⋊Dic₆₇	D₄	1	2	yes	—
61	C₄×Dic₆₇	—	2	1	no	—
62	C₄×Dic₆₇	—	2	2	no	—
63	C₄×Dic₆₇	C₂³	1	2	yes	—
64	C₂⁴×C₆₇	—	2	1	no	—
65	#41×C₆₇	—	2	2	no	—
66	#41×C₆₇	C₂×C₄	1	4	yes	complemented
67	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×D₄×C₆₇	—	2	1	no	—
69	C₂×D₄×C₆₇	—	2	1	no	—
70	C₂×D₄×C₆₇	—	2	2	no	—
71	C₂×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
72	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
77	D₈×C₆₇	C₂×C₄	1	8	yes	complemented
78	M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₆₇	—	2	1	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	—	2	4	no	—
83	C₂³×C₆₇	—	8	1	no	—
84	C₂³×C₆₇	—	8	1	no	—
85	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₆₇	—	2	1	no	—
87	Q₈×C₆₇	C₂²×C₄	1	2	yes	—
88	D₄×C₆₇	—	2	4	no	—
89	D₄×C₆₇	—	2	4	no	—
90	D₄×C₆₇	—	2	2	no	—
91	D₄×C₆₇	—	2	1	no	—
92	D₄×C₆₇	C₂²×C₄	1	2	yes	—
93	D₄×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#41	1	2	yes	—
99	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	1	no	—
101	C₂×Dic₆₇	—	2	1	no	—
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	8	2	no	—
104	C₂×Dic₆₇	—	8	2	no	—
105	C₂×Dic₆₇	—	8	2	no	—
106	C₂×Dic₆₇	—	8	1	no	—
107	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
108	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
109	C₈×C₆₇	C₂²×C₄	1	4	yes	—
110	C₆₇⋊C₈	—	2	1	no	—
111	C₆₇⋊C₈	—	2	1	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	2	4	no	—
116	C₂²×C₆₇	—	4	1	no	—
117	C₂²×C₆₇	—	4	2	no	—
118	C₂²×C₆₇	—	4	2	no	—
119	C₂²×C₆₇	—	4	1	no	—
120	C₂²×C₆₇	—	8	4	no	—
121	C₂²×C₆₇	C₄×D₄	1	2	yes	—
122	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₆₇	—	2	2	no	—
124	C₄×C₆₇	C₄×D₄	1	2	yes	—
125	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
127	Dic₆₇	—	2	2	no	—
128	Dic₆₇	—	2	2	no	—
129	Dic₆₇	—	4	1	no	—
130	Dic₆₇	—	4	2	no	—
131	Dic₆₇	—	4	1	no	—
132	C₂×C₆₇	—	2	2	no	—
133	C₂×C₆₇	—	4	2	no	—
134	C₂×C₆₇	—	4	4	no	—
135	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₆₇	#576	1	1	yes	characteristic
137	C₂×C₆₇	#577	1	1	yes	characteristic
138	#2946	—	67	1	no	maximal, Sylow
139	C₆₇	#2946	1	1	yes	characteristic, complemented, Sylow
140	C₂×AGL(1,ℤ/8)	D₆₇	1	1	yes	characteristic
141	#520	—	67	1	no	—
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	C₂×D₆₇	1	1	yes	characteristic
153	AGL(1,ℤ/8)	Dic₆₇	1	8	yes	complemented
154	C₂×SD₁₆	C₂×D₆₇	1	2	yes	—
155	C₂×D₈	C₂×D₆₇	1	2	yes	—
156	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
157	C₂²×C₈	—	67	1	no	—
158	C₄×Q₈	—	67	2	no	—
159	C₄×D₄	—	67	2	no	—
160	C₄×D₄	—	67	2	no	—
161	C₄×D₄	—	67	2	no	—
162	C₄×D₄	—	134	2	no	—
163	C₄×D₄	—	134	1	no	—
164	C₄×D₄	—	134	1	no	—
165	#117	—	67	1	no	—
166	#117	—	67	1	no	—
167	#117	—	134	1	no	—
168	C₂×C₄⋊C₄	—	67	1	no	—
169	C₂×C₄⋊C₄	—	134	1	no	—
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	2	no	—
176	#102	—	67	2	no	—
177	#102	—	67	1	no	—
178	#102	—	67	1	no	—
179	#97	—	67	2	no	—
180	C₄×C₈	—	67	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	2	2	no	—
183	#41	C₂×Dic₆₇	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂×Dic₆₇	1	4	yes	complemented
189	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
190	C₂×D₄	C₂²×D₆₇	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	134	2	no	—
193	C₂²×C₄	—	134	2	no	—
194	C₂²×C₄	—	134	2	no	—
195	C₂²×C₄	—	134	1	no	—
196	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
197	SD₁₆	C₂×Dic₆₇	1	8	yes	complemented
198	D₈	C₂×Dic₆₇	1	8	yes	complemented
199	M₁₆	C₂×Dic₆₇	1	4	yes	complemented
200	C₂×C₈	—	67	1	no	—
201	C₂×C₈	—	67	1	no	—
202	C₂×C₈	—	134	2	no	—
203	C₂×C₈	C₂²×D₆₇	1	2	yes	—
204	C₄⋊C₄	—	67	1	no	—
205	C₄⋊C₄	—	67	1	no	—
206	C₄⋊C₄	—	67	2	no	—
207	C₄⋊C₄	—	67	1	no	—
208	C₄⋊C₄	—	67	1	no	—
209	C₄⋊C₄	—	134	2	no	—
210	C₄⋊C₄	—	134	1	no	—
211	C₂²⋊C₄	—	67	2	no	—
212	C₂²⋊C₄	—	134	2	no	—
213	C₂²⋊C₄	—	134	2	no	—
214	C₂²⋊C₄	—	134	2	no	—
215	C₂²⋊C₄	—	134	1	no	—
216	C₄²	—	67	2	no	—
217	C₄²	—	134	2	no	—
218	C₄²	—	134	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	D₄×D₆₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂²×Dic₆₇	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	1	no	—
231	D₄	C₂²×Dic₆₇	1	2	yes	—
232	D₄	C₂²×Dic₆₇	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	536	1	no	—
236	C₂×C₄	—	536	2	no	—
237	C₂×C₄	—	536	2	no	—
238	C₂×C₄	—	536	2	no	—
239	C₂×C₄	—	67	2	no	—
240	C₂×C₄	—	67	2	no	—
241	C₂×C₄	—	134	1	no	—
242	C₂×C₄	—	134	1	no	—
243	C₂×C₄	—	134	2	no	—
244	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
245	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
246	C₂×C₄	#11761627	1	2	yes	—
247	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
248	C₈	—	134	1	no	—
249	C₈	—	134	1	no	—
250	C₈	C₂²×Dic₆₇	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	4	no	—
260	C₂²	D₄×Dic₆₇	1	2	yes	—
261	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	268	2	no	—
264	C₄	—	268	1	no	—
265	C₄	—	268	1	no	—
266	C₄	—	134	2	no	—
267	C₄	—	134	2	no	—
268	C₄	D₄×Dic₆₇	1	2	yes	—
269	C₄	C₂³×Dic₆₇	1	1	yes	characteristic
270	C₄	C₂×#11761627	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×D₄×Dic₆₇	1	1	yes	characteristic
275	C₂	#16864200	1	1	yes	characteristic
276	C₂	#16864206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×AGL(1,ℤ/8)×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×AGL(1,ℤ/8)	D₆₇
hypocenter	C₆₇	#2946
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #16863879	C₂	semidirect
2. #16863871	C₂	semidirect
3. #16863508	C₂	semidirect
4. #16863497	C₂	semidirect
5. SD₁₆×Dic₆₇	C₂	semidirect
6. D₈×Dic₆₇	C₂	semidirect
7. #16863438	C₂	semidirect
8. #16863392	C₂	semidirect
9. C₂×#14523964	C₂	semidirect
10. #16863354	C₂	semidirect
13. Q₈×Dic₆₇	C₂²	semidirect
19. D₄×Dic₆₇	C₂²	semidirect
26. #14523977	C₂²	semidirect
27. #14523975	C₂²	semidirect
28. #14523967	C₂²	semidirect
29. #14523967	C₂²	semidirect
32. #14523964	C₂²	semidirect
33. #14523964	C₂²	semidirect
34. #14523953	C₂²	semidirect
35. C₈×Dic₆₇	C₂²	semidirect
38. AGL(1,ℤ/8)×C₆₇	C₄	semidirect
66. #41×C₆₇	C₂×C₄	semidirect
71. C₂×D₄×C₆₇	C₂×C₄	semidirect
76. SD₁₆×C₆₇	C₂×C₄	semidirect
77. D₈×C₆₇	C₂×C₄	semidirect
78. M₁₆×C₆₇	C₂×C₄	semidirect
139. C₆₇	#2946	semidirect
153. AGL(1,ℤ/8)	Dic₆₇	semidirect
183. #41	C₂×Dic₆₇	semidirect
188. C₂×D₄	C₂×Dic₆₇	semidirect
197. SD₁₆	C₂×Dic₆₇	semidirect
198. D₈	C₂×Dic₆₇	semidirect
199. M₁₆	C₂×Dic₆₇	semidirect