Subgroups — #30748887

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	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22560211	C₂	1	2	yes	maximal
3	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
4	#22558092	C₂	1	2	yes	maximal
5	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₃₄	—	2	2	no	—
8	C₈×Dic₁₃₄	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×Dic₆₇⋊C₈	—	2	1	no	—
12	C₂×#14523966	—	2	2	no	—
13	C₂×C₈×Dic₆₇	—	2	1	no	—
14	#16863158	—	2	2	no	—
15	#16863149	—	2	2	no	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	—	2	1	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#384×C₆₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	—	2	1	no	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	—	2	1	no	—
25	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₃₄	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
28	C₂×Dic₆₇⋊C₄	—	2	1	no	—
29	C₂×C₄×Dic₆₇	—	2	1	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₆₇⋊C₈	—	2	4	no	—
35	#14523966	—	4	4	no	—
36	C₈×Dic₆₇	—	2	4	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	—	4	2	no	—
44	#107×C₆₇	—	4	2	no	—
45	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	C₂×Dic₁₃₄	—	2	1	no	—
48	C₂×Dic₁₃₄	—	2	1	no	—
49	C₂×Dic₁₃₄	—	2	1	no	—
50	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
51	C₂×Dic₁₃₄	D₈	1	2	yes	—
52	C₂×Dic₁₃₄	Q₁₆	1	2	yes	—
53	C₂²×Dic₆₇	—	2	1	no	—
54	C₂²×Dic₆₇	—	2	1	no	—
55	C₂×C₆₇⋊C₈	—	2	2	no	—
56	C₂×C₆₇⋊C₈	—	4	1	no	—
57	C₂×C₆₇⋊C₈	—	4	1	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	2	1	no	—
62	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
63	Dic₆₇⋊C₄	—	2	2	no	—
64	Dic₆₇⋊C₄	—	2	2	no	—
65	C₄×Dic₆₇	—	2	2	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₆₇	—	2	1	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
76	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
77	C₄⋊C₄×C₆₇	—	4	2	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄⋊C₄×C₆₇	—	4	2	no	—
81	C₄⋊C₄×C₆₇	—	8	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	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	8	2	no	—
90	C₂×C₄×C₆₇	—	8	2	no	—
91	C₂×C₄×C₆₇	—	8	2	no	—
92	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
104	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
106	C₂×Dic₆₇	—	2	2	no	—
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	2	2	no	—
110	C₂×Dic₆₇	—	4	1	no	—
111	C₂×Dic₆₇	—	4	1	no	—
112	Dic₁₃₄	—	2	2	no	—
113	Dic₁₃₄	—	2	2	no	—
114	Dic₁₃₄	—	2	2	no	—
115	Dic₁₃₄	#107	1	4	yes	complemented
116	C₈×C₆₇	—	2	2	no	—
117	C₈×C₆₇	#103	1	4	yes	complemented
118	C₆₇⋊C₈	—	4	2	no	—
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	1	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	16	2	no	—
130	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
131	C₄×C₆₇	#417	1	1	yes	characteristic
132	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
133	C₄×C₆₇	#427	1	1	yes	characteristic
134	C₄×C₆₇	#466	1	1	yes	characteristic
135	C₄×C₆₇	#467	1	1	yes	characteristic
136	C₄×C₆₇	#475	1	1	yes	characteristic
137	C₄×C₆₇	#477	1	1	yes	characteristic
138	Dic₆₇	—	2	4	no	—
139	Dic₆₇	—	4	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	—	67	1	no	—
149	#1845	D₆₇	1	1	yes	characteristic
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	—	67	1	no	—
155	C₈×Q₈	—	67	4	no	—
156	C₈×Q₈	—	134	2	no	—
157	C₂×#107	—	2	1	no	—
158	C₂×#107	—	2	1	no	—
159	C₂×C₄⋊C₈	—	67	1	no	—
160	C₂×C₄⋊C₈	—	67	2	no	—
161	C₂×C₄⋊C₈	—	134	1	no	—
162	C₂×#103	—	134	2	no	—
163	C₂×C₄×C₈	—	134	1	no	—
164	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
165	#384	C₂×D₆₇	1	2	yes	—
166	#340	—	134	2	no	—
167	#335	—	134	2	no	—
168	C₂²×Q₈	—	67	1	no	—
169	C₂²×C₈	—	134	2	no	—
170	C₂²×C₈	—	134	1	no	—
171	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
173	C₄×Q₈	—	67	2	no	—
174	C₄×Q₈	—	67	2	no	—
175	C₄×Q₈	—	134	1	no	—
176	C₄×Q₈	—	134	1	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	67	1	no	—
181	C₂×C₄⋊C₄	—	134	1	no	—
182	C₂×C₄²	—	134	1	no	—
183	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
184	#107	—	4	2	no	—
185	#107	—	4	2	no	—
186	C₄⋊C₈	—	67	2	no	—
187	C₄⋊C₈	—	134	4	no	—
188	C₄⋊C₈	—	134	2	no	—
189	C₄⋊C₈	—	134	1	no	—
190	C₄⋊C₈	—	134	2	no	—
191	#103	—	268	4	no	—
192	C₄×C₈	—	134	4	no	—
193	C₄×C₈	C₄×D₆₇	1	2	yes	—
194	C₄×C₈	D₂₆₈	1	2	yes	—
195	C₂×Q₈	—	67	2	no	—
196	C₂×Q₈	—	67	2	no	—
197	C₂×Q₈	—	67	2	no	—
198	C₂×Q₈	—	134	1	no	—
199	C₂×Q₈	—	134	1	no	—
200	C₂×Q₈	—	134	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	134	1	no	—
203	C₂²×C₄	—	134	1	no	—
204	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₄	#11761627	1	1	yes	characteristic
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	268	1	no	—
210	C₂×C₈	—	268	1	no	—
211	C₂×C₈	—	268	2	no	—
212	C₂×C₈	—	268	2	no	—
213	C₂×C₈	—	268	2	no	—
214	C₂×C₈	—	134	2	no	—
215	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
216	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
217	C₂×C₈	#11761616	1	2	yes	—
218	C₂×C₈	#11761618	1	2	yes	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	8	2	no	—
224	C₄⋊C₄	—	67	2	no	—
225	C₄⋊C₄	—	134	2	no	—
226	C₄⋊C₄	—	134	1	no	—
227	C₄⋊C₄	—	134	2	no	—
228	C₄²	—	134	2	no	—
229	C₄²	—	134	2	no	—
230	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
231	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
232	C₄²	Q₈×D₆₇	1	1	yes	characteristic
233	C₄²	#11761629	1	1	yes	characteristic
234	C₂³	#14524014	1	1	yes	characteristic
235	Q₈	—	67	4	no	—
236	Q₈	—	134	2	no	—
237	Q₈	—	134	2	no	—
238	Q₈	—	134	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	268	1	no	—
245	C₂×C₄	—	268	1	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	—	134	2	no	—
249	C₂×C₄	—	134	2	no	—
250	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
252	C₂×C₄	#14524001	1	1	yes	characteristic
253	C₂×C₄	#14524018	1	1	yes	characteristic
254	C₂×C₄	#14524019	1	1	yes	characteristic
255	C₂×C₄	#14524023	1	1	yes	characteristic
256	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
258	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
260	C₂×C₄	#14524062	1	1	yes	characteristic
261	C₂×C₄	#14524066	1	1	yes	characteristic
262	C₂×C₄	#14524069	1	1	yes	characteristic
263	C₂×C₄	#14524074	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	268	2	no	—
266	C₈	—	536	2	no	—
267	C₈	#14523966	1	4	yes	complemented
268	C₂²	#16863253	1	1	yes	characteristic
269	C₂²	#16863593	1	1	yes	characteristic
270	C₂²	#16863603	1	1	yes	characteristic
271	C₂²	#16863629	1	1	yes	characteristic
272	C₂²	#16863634	1	1	yes	characteristic
273	C₂²	#16863663	1	1	yes	characteristic
274	C₂²	#16863668	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	268	2	no	—
278	C₄	—	16	2	no	—
279	C₄	—	134	4	no	—
280	C₄	C₂×#14523966	1	1	yes	characteristic
281	C₄	#16863395	1	1	yes	characteristic
282	C₄	#107×D₆₇	1	1	yes	characteristic
283	C₄	#16863433	1	1	yes	characteristic
284	C₄	#16863633	1	1	yes	characteristic
285	C₄	#16863638	1	1	yes	characteristic
286	C₄	#16863678	1	1	yes	characteristic
287	C₄	#16863688	1	1	yes	characteristic
288	C₂	#22559566	1	1	yes	characteristic
289	C₂	#22559576	1	1	yes	characteristic
290	C₂	#22559600	1	1	yes	characteristic
291	C₂	#22559610	1	1	yes	characteristic
292	C₂	#22560211	1	1	yes	characteristic
293	C₂	#22560235	1	1	yes	characteristic
294	C₂	#22560274	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524014
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#8645
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₃₄	C₄	semidirect
115. Dic₁₃₄	#107	semidirect
117. C₈×C₆₇	#103	semidirect
153. C₆₇	#8645	semidirect
267. C₈	#14523966	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22560211	2
C₂×#16863149	2
#22558092	2
C₆₇×#1845	2
#8645	67