Subgroups — #30827657

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570788	C₂	1	1	yes	maximal, characteristic
2	#22568955	C₂	1	2	yes	maximal, complemented
3	#22559124	C₂	1	2	yes	maximal, complemented
4	#22559075	C₂	1	4	yes	maximal, complemented
5	#22559034	C₂	1	2	yes	maximal, complemented
6	C₂×#16863147	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558609	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524051	—	2	2	no	—
10	#16863946	C₂²	1	2	yes	—
11	#16863801	C₄	1	1	yes	characteristic
12	#16863773	C₄	1	1	yes	characteristic
13	#16863743	C₄	1	2	yes	—
14	#16863632	C₄	1	2	yes	—
15	#16863632	C₄	1	2	yes	—
16	C₄×D₅₃₆	C₂²	1	2	yes	—
17	C₄×D₅₃₆	C₂²	1	2	yes	—
18	#16863475	C₂²	1	4	yes	—
19	#16863469	—	2	4	no	—
20	C₈×D₂₆₈	C₂²	1	4	yes	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
22	#16863364	—	2	2	no	—
23	#16863147	C₂²	1	1	yes	characteristic
24	#16863147	C₂²	1	2	yes	—
25	#16863147	C₂²	1	1	yes	characteristic
26	#16863125	C₂²	1	4	yes	—
27	#16863125	C₂²	1	4	yes	—
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#322×C₆₇	C₂²	1	2	yes	—
32	#322×C₆₇	C₂²	1	2	yes	—
33	C₂×#11761628	—	2	2	no	—
34	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
35	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
36	#14524051	—	4	8	no	—
37	C₂×#11761611	—	2	4	no	—
38	C₂²×C₆₇⋊C₈	—	2	2	no	—
39	C₂×C₈×D₆₇	—	2	4	no	—
40	#14524043	C₂×C₄	1	2	yes	—
41	#14524040	C₂×C₄	1	2	yes	—
42	#14524031	—	2	2	no	—
43	#14524017	—	2	2	no	—
44	#14524017	C₂×C₄	1	4	yes	—
45	C₄×D₂₆₈	—	2	2	no	—
46	C₄×D₂₆₈	C₂³	1	4	yes	—
47	#14523977	C₂×C₄	1	2	yes	—
48	#14523977	C₂×C₄	1	2	yes	—
49	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
51	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
52	#14523965	C₂×C₄	1	4	yes	—
53	#14523965	C₂×C₄	1	4	yes	—
54	(C₂×D₆₇)⋊C₈	—	2	4	no	—
55	(C₂×D₆₇)⋊C₈	—	2	4	no	—
56	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₂²⋊C₈×C₆₇	—	2	1	no	—
62	#97×C₆₇	D₄	1	2	yes	—
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₆₇	C₂³	1	2	yes	—
66	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₆₇	C₂³	1	2	yes	—
69	#11761628	—	4	4	no	—
70	#11761628	—	4	4	no	—
71	C₂×Dic₁₃₄	—	2	2	no	—
72	C₂×D₂₆₈	—	2	2	no	—
73	C₂×D₂₆₈	—	2	2	no	—
74	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
75	C₂×C₂²⋊D₆₇	—	2	4	no	—
76	C₂×C₄×D₆₇	—	2	4	no	—
77	D₅₃₆	—	2	4	no	—
78	D₅₃₆	—	4	2	no	—
79	#11761611	—	4	8	no	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	—	4	4	no	—
83	C₈×D₆₇	—	4	8	no	—
84	C₄⋊Dic₆₇	—	2	2	no	—
85	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
86	(C₂×D₆₇)⋊C₄	—	2	4	no	—
87	(C₂×D₆₇)⋊C₄	—	2	4	no	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	M₁₆×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	4	1	no	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	#41	1	1	yes	characteristic
103	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	4	1	no	—
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
113	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₂²⋊D₆₇	—	4	8	no	—
117	C₂×Dic₆₇	—	2	4	no	—
118	D₂₆₈	—	2	4	no	—
119	D₂₆₈	—	2	4	no	—
120	D₂₆₈	—	4	4	no	—
121	D₂₆₈	—	8	2	no	—
122	C₄×D₆₇	—	4	4	no	—
123	C₄×D₆₇	—	4	4	no	—
124	Dic₁₃₄	—	4	4	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	—	2	1	no	—
128	C₈×C₆₇	—	2	4	no	—
129	C₈×C₆₇	—	4	2	no	—
130	C₈×C₆₇	#131	1	2	yes	complemented
131	C₆₇⋊C₈	—	4	2	no	—
132	C₆₇⋊C₈	—	4	2	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	#442	1	2	yes	—
136	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×D₆₇	—	4	4	no	—
138	C₂×D₆₇	—	4	4	no	—
139	C₂×D₆₇	—	4	4	no	—
140	C₄×C₆₇	—	2	1	no	—
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	#442	1	2	yes	—
144	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
145	C₄×C₆₇	#568	1	1	yes	characteristic
146	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
147	Dic₆₇	—	4	4	no	—
148	#18073	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	#2935	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
152	C₂×C₆₇	#2957	1	1	yes	characteristic
153	D₆₇	—	8	4	no	—
154	#3169	—	67	1	no	—
155	#2935	—	67	2	no	—
156	#1594	—	67	2	no	—
157	#1586	—	67	4	no	—
158	#1580	—	67	2	no	—
159	C₂×#334	—	67	1	no	—
160	#1480	—	67	2	no	—
161	#1451	D₆₇	1	1	yes	characteristic, complemented
162	C₆₇	#18073	1	1	yes	characteristic, complemented, Sylow
163	C₂×#131	—	134	2	no	—
164	#532	—	67	2	no	—
165	#500	—	67	1	no	—
166	#493	—	67	1	no	—
167	#486	—	67	2	no	—
168	#466	—	67	2	no	—
169	#466	—	67	2	no	—
170	C₄×D₈	—	67	2	no	—
171	C₄×D₈	—	67	2	no	—
172	#436	—	67	4	no	—
173	#435	—	134	4	no	—
174	C₈×D₄	—	67	4	no	—
175	#431	C₂×D₆₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	—	67	2	no	—
177	#408	—	134	2	no	—
178	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
180	#334	—	67	1	no	—
181	#334	—	67	2	no	—
182	#334	—	67	1	no	—
183	D₄⋊C₈	—	67	4	no	—
184	D₄⋊C₈	—	67	4	no	—
185	#322	C₂×D₆₇	1	2	yes	—
186	#322	C₂×D₆₇	1	2	yes	—
187	C₂×#41	—	134	2	no	—
188	C₂×D₈	—	67	2	no	—
189	C₂×D₈	—	67	2	no	—
190	#131	—	268	8	no	—
191	C₂×M₁₆	—	134	4	no	—
192	C₂×M₁₆	D₂₆₈	1	2	yes	—
193	C₂²×C₈	—	134	2	no	—
194	C₂²×C₈	—	134	4	no	—
195	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
197	C₄⋊Q₈	—	67	2	no	—
198	#127	—	67	2	no	—
199	#124	—	134	2	no	—
200	#121	—	67	4	no	—
201	#121	—	134	2	no	—
202	C₄×D₄	—	67	4	no	—
203	C₄×D₄	—	134	2	no	—
204	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
205	#107	—	67	2	no	—
206	#107	—	67	2	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	67	2	no	—
209	C₄⋊C₈	—	67	2	no	—
210	C₄⋊C₈	—	134	2	no	—
211	#102	—	67	4	no	—
212	#102	—	67	4	no	—
213	C₂²⋊C₈	—	2	1	no	—
214	C₂²⋊C₈	—	134	4	no	—
215	C₂²⋊C₈	—	134	4	no	—
216	#97	D₂₆₈	1	2	yes	—
217	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
218	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₆₇	1	2	yes	—
220	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₆₇	1	2	yes	—
223	#41	—	268	4	no	—
224	#41	—	268	4	no	—
225	C₂×Q₈	—	134	2	no	—
226	C₂×D₄	—	67	4	no	—
227	C₂×D₄	—	134	4	no	—
228	C₂×D₄	—	134	2	no	—
229	C₂×D₄	—	134	2	no	—
230	C₂²×C₄	—	134	4	no	—
231	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
233	C₂²×C₄	#11761628	1	1	yes	characteristic
234	D₈	—	268	2	no	—
235	D₈	—	134	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	268	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	268	8	no	—
242	C₂×C₈	—	268	4	no	—
243	C₂×C₈	—	134	2	no	—
244	C₂×C₈	—	134	2	no	—
245	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
250	C₄⋊C₄	—	67	4	no	—
251	C₄⋊C₄	—	134	2	no	—
252	C₂²⋊C₄	—	134	4	no	—
253	C₂²⋊C₄	—	134	4	no	—
254	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
255	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
256	C₄²	#11761628	1	1	yes	characteristic
257	C₄²	C₂³×D₆₇	1	1	yes	characteristic
258	C₂³	—	134	4	no	—
259	C₂³	C₄×D₂₆₈	1	1	yes	characteristic
260	Q₈	—	268	4	no	—
261	D₄	—	268	8	no	—
262	D₄	—	268	4	no	—
263	D₄	—	536	2	no	—
264	D₄	—	134	4	no	—
265	D₄	—	134	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	268	4	no	—
270	C₂×C₄	—	268	4	no	—
271	C₂×C₄	—	134	4	no	—
272	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
273	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
274	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
275	C₂×C₄	#14524071	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
278	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	268	2	no	—
285	C₈	—	268	2	no	—
286	C₈	#14524051	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	268	4	no	—
290	C₂²	—	268	4	no	—
291	C₂²	—	268	4	no	—
292	C₂²	#16863509	1	2	yes	—
293	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	268	4	no	—
298	C₄	#16863509	1	2	yes	—
299	C₄	C₂×#14524051	1	1	yes	characteristic
300	C₄	#16864162	1	1	yes	characteristic
301	C₄	C₂×#14524071	1	2	yes	—
302	C₂	—	2	2	no	—
303	C₂	—	536	4	no	—
304	C₂	#22568955	1	1	yes	characteristic
305	C₂	C₂×#16863509	1	1	yes	characteristic
306	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1451	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570788	C₂
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1451	D₆₇
hypocenter	C₆₇	#18073
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #22568955	C₂	semidirect
3. #22559124	C₂	semidirect
4. #22559075	C₂	semidirect
5. #22559034	C₂	semidirect
6. C₂×#16863147	C₂	semidirect
7. #22558609	C₂	semidirect
8. C₆₇×#1451	C₂	semidirect
130. C₈×C₆₇	#131	semidirect
161. #1451	D₆₇	semidirect
162. C₆₇	#18073	semidirect
286. C₈	#14524051	semidirect

Maximal subgroups

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

subgroup	index
#22570788	2
#22568955	2
#22559124	2
#22559075	2
#22559034	2
C₂×#16863147	2
#22558609	2
C₆₇×#1451	2
#18073	67