Subgroups — #30748890

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560369	C₂	1	1	yes	maximal, characteristic
2	#22560214	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	#22558093	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523977	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14523967	—	2	2	no	—
10	#16863266	C₂²	1	2	yes	—
11	#16863158	—	2	2	no	—
12	#16863149	—	2	2	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₆₇	—	2	2	no	—
15	C₈×Q₈×C₆₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	C₂×C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×C₄⋊Dic₆₇	—	2	2	no	—
24	#14523977	—	4	2	no	—
25	#14523977	—	4	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	#14523967	—	4	4	no	—
29	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	—	2	1	no	—
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₆₇	—	2	1	no	—
34	C₄×Q₈×C₆₇	—	2	1	no	—
35	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₆₇	Q₈	1	2	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	—	2	1	no	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	4	no	—
42	C₄⋊C₈×C₆₇	—	2	1	no	—
43	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₆₇	—	2	4	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	C₂²×Dic₆₇	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₄⋊Dic₆₇	—	4	2	no	—
52	C₄⋊Dic₆₇	—	4	2	no	—
53	C₄⋊Dic₆₇	—	4	2	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	C₄⋊Dic₆₇	—	8	2	no	—
56	C₂×Q₈×C₆₇	—	2	1	no	—
57	C₂×Q₈×C₆₇	—	2	1	no	—
58	C₂×Q₈×C₆₇	—	2	1	no	—
59	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
60	C₂×Q₈×C₆₇	D₈	1	2	yes	—
61	C₂×Q₈×C₆₇	Q₁₆	1	2	yes	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	1	no	—
68	C₂×C₈×C₆₇	—	2	1	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	4	1	no	—
71	C₂×C₈×C₆₇	—	4	1	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
75	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	1	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₆₇	#41	1	1	yes	characteristic
86	C₂³×C₆₇	#121	1	1	yes	characteristic
87	Q₈×C₆₇	—	2	2	no	—
88	Q₈×C₆₇	—	2	2	no	—
89	Q₈×C₆₇	—	2	2	no	—
90	Q₈×C₆₇	#107	1	4	yes	complemented
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	4	1	no	—
98	C₂×C₄×C₆₇	—	4	1	no	—
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
111	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
113	C₂×Dic₆₇	—	8	2	no	—
114	C₂×Dic₆₇	—	8	2	no	—
115	C₂×Dic₆₇	—	8	2	no	—
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	—	4	2	no	—
118	C₈×C₆₇	#103	1	4	yes	complemented
119	C₆₇⋊C₈	—	8	2	no	—
120	C₂²×C₆₇	#380	1	1	yes	characteristic
121	C₂²×C₆₇	#460	1	1	yes	characteristic
122	C₂²×C₆₇	#462	1	1	yes	characteristic
123	C₂²×C₆₇	#466	1	1	yes	characteristic
124	C₂²×C₆₇	#467	1	1	yes	characteristic
125	C₂²×C₆₇	#472	1	1	yes	characteristic
126	C₂²×C₆₇	#473	1	1	yes	characteristic
127	C₄×C₆₇	—	2	4	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	4	2	no	—
131	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
132	C₄×C₆₇	#417	1	1	yes	characteristic
133	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
134	C₄×C₆₇	#427	1	1	yes	characteristic
135	C₄×C₆₇	#466	1	1	yes	characteristic
136	C₄×C₆₇	#467	1	1	yes	characteristic
137	C₄×C₆₇	#475	1	1	yes	characteristic
138	C₄×C₆₇	#477	1	1	yes	characteristic
139	Dic₆₇	—	16	2	no	—
140	#8645	—	67	1	no	maximal, Sylow
141	C₂×C₆₇	#1668	1	1	yes	characteristic
142	C₂×C₆₇	#1670	1	1	yes	characteristic
143	C₂×C₆₇	#1674	1	1	yes	characteristic
144	C₂×C₆₇	#1676	1	1	yes	characteristic
145	C₂×C₆₇	#1803	1	1	yes	characteristic
146	C₂×C₆₇	#1808	1	1	yes	characteristic
147	C₂×C₆₇	#1816	1	1	yes	characteristic
148	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
149	#1845	—	67	1	no	—
150	#1803	—	67	2	no	—
151	C₂×#335	—	67	1	no	—
152	#1296	—	67	2	no	—
153	C₆₇	#8645	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
155	C₈×Q₈	—	2	2	no	—
156	C₈×Q₈	Dic₆₇	1	4	yes	complemented
157	C₂×#107	—	134	1	no	—
158	C₂×#107	—	134	1	no	—
159	C₂×C₄⋊C₈	—	2	1	no	—
160	C₂×C₄⋊C₈	—	67	2	no	—
161	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
162	C₂×#103	—	134	2	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
165	#384	—	67	2	no	—
166	#340	—	134	2	no	—
167	#335	—	134	2	no	—
168	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	134	2	no	—
171	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₄×Q₈	—	2	1	no	—
174	C₄×Q₈	—	2	1	no	—
175	C₄×Q₈	Dic₁₃₄	1	2	yes	—
176	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	134	2	no	—
179	C₂×C₄⋊C₄	—	134	2	no	—
180	C₂×C₄⋊C₄	—	134	2	no	—
181	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
184	#107	—	268	2	no	—
185	#107	—	268	2	no	—
186	C₄⋊C₈	—	2	4	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	134	2	no	—
189	C₄⋊C₈	—	134	2	no	—
190	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
191	#103	—	268	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
194	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
199	C₂×Q₈	D₅₃₆	1	2	yes	—
200	C₂×Q₈	Dic₂₆₈	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	268	2	no	—
204	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
206	C₂²×C₄	#11761628	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	268	2	no	—
213	C₂×C₈	—	268	2	no	—
214	C₂×C₈	—	268	2	no	—
215	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
216	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
217	C₂×C₈	#11761615	1	2	yes	—
218	C₂×C₈	#11761618	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	268	2	no	—
223	C₄⋊C₄	—	268	2	no	—
224	C₄⋊C₄	—	268	2	no	—
225	C₄⋊C₄	—	268	2	no	—
226	C₄⋊C₄	—	536	2	no	—
227	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
232	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
233	C₄²	#11761628	1	1	yes	characteristic
234	C₂³	#14524017	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	2	2	no	—
238	Q₈	#14523977	1	4	yes	complemented
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	536	2	no	—
248	C₂×C₄	—	536	2	no	—
249	C₂×C₄	—	536	2	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
252	C₂×C₄	#14524002	1	1	yes	characteristic
253	C₂×C₄	#14524017	1	1	yes	characteristic
254	C₂×C₄	#14524022	1	1	yes	characteristic
255	C₂×C₄	#14524022	1	1	yes	characteristic
256	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
257	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
258	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
260	C₂×C₄	#14524065	1	1	yes	characteristic
261	C₂×C₄	#14524065	1	1	yes	characteristic
262	C₂×C₄	#14524068	1	1	yes	characteristic
263	C₂×C₄	#14524073	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	536	2	no	—
267	C₈	#14523967	1	4	yes	complemented
268	C₂²	#16863256	1	1	yes	characteristic
269	C₂²	#16863596	1	1	yes	characteristic
270	C₂²	#16863606	1	1	yes	characteristic
271	C₂²	#16863632	1	1	yes	characteristic
272	C₂²	#16863637	1	1	yes	characteristic
273	C₂²	#16863666	1	1	yes	characteristic
274	C₂²	#16863671	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	4	2	no	—
279	C₄	—	1072	2	no	—
280	C₄	C₂×#14523967	1	1	yes	characteristic
281	C₄	#16863397	1	1	yes	characteristic
282	C₄	C₂×#14523977	1	1	yes	characteristic
283	C₄	#16863434	1	1	yes	characteristic
284	C₄	#16863632	1	1	yes	characteristic
285	C₄	#16863637	1	1	yes	characteristic
286	C₄	#16863681	1	1	yes	characteristic
287	C₄	#16863691	1	1	yes	characteristic
288	C₂	#22559569	1	1	yes	characteristic
289	C₂	#22559579	1	1	yes	characteristic
290	C₂	#22559603	1	1	yes	characteristic
291	C₂	#22559613	1	1	yes	characteristic
292	C₂	#22560214	1	1	yes	characteristic
293	C₂	#22560238	1	1	yes	characteristic
294	C₂	#22560273	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	#8645
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₆₇	C₄	semidirect
90. Q₈×C₆₇	#107	semidirect
118. C₈×C₆₇	#103	semidirect
153. C₆₇	#8645	semidirect
156. C₈×Q₈	Dic₆₇	semidirect
238. Q₈	#14523977	semidirect
267. C₈	#14523967	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22560369	2
#22560214	2
C₂×#16863149	2
#22558093	2
C₂×C₈×Q₈×C₆₇	2
#8645	67