Subgroups — #30760874

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561015	C₂	1	1	yes	maximal, characteristic
2	#22561015	C₂	1	1	yes	maximal, characteristic
3	#22561015	C₂	1	1	yes	maximal, characteristic
4	#22560369	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560366	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560337	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524040	C₂²	1	1	yes	characteristic
9	C₂×#14523977	—	2	2	no	—
10	C₂×#14523975	—	2	2	no	—
11	C₂×#14523965	—	2	2	no	—
12	C₂×#14523965	—	2	2	no	—
13	C₂×#14523965	—	2	2	no	—
14	#16863266	C₂²	1	1	yes	characteristic
15	#16863266	C₂²	1	1	yes	characteristic
16	#16863266	C₂²	1	1	yes	characteristic
17	#16863159	—	2	2	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×D₂₆₈	—	2	1	no	—
22	C₂²×D₂₆₈	—	2	1	no	—
23	C₂²×D₂₆₈	—	2	1	no	—
24	#14524040	—	2	1	no	—
25	#14524040	—	2	1	no	—
26	#14524040	—	2	1	no	—
27	#14524040	Q₈	1	2	yes	—
28	C₂×C₄⋊Dic₆₇	—	2	1	no	—
29	C₂×C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₄⋊Dic₆₇	—	2	1	no	—
33	C₂×C₄⋊Dic₆₇	—	2	1	no	—
34	C₂×C₄⋊Dic₆₇	—	2	1	no	—
35	C₂×C₄⋊Dic₆₇	—	2	1	no	—
36	C₂×C₄⋊Dic₆₇	—	2	1	no	—
37	#14523977	—	4	4	no	—
38	#14523975	—	4	4	no	—
39	#14523965	—	4	4	no	—
40	#14523965	—	4	4	no	—
41	#14523965	—	4	4	no	—
42	C₂²×C₈×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₆₇	—	2	1	no	—
47	C₄×C₈×C₆₇	—	2	1	no	—
48	C₄×C₈×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	—	2	1	no	—
50	C₄×C₈×C₆₇	—	2	1	no	—
51	C₄×C₈×C₆₇	—	2	1	no	—
52	C₂³×D₆₇	—	4	1	no	—
53	C₂×D₂₆₈	—	2	2	no	—
54	C₂×D₂₆₈	—	2	2	no	—
55	C₂×D₂₆₈	—	2	2	no	—
56	C₂×D₂₆₈	—	4	2	no	—
57	C₂×D₂₆₈	—	4	2	no	—
58	C₂×D₂₆₈	—	4	2	no	—
59	C₂×D₂₆₈	—	4	2	no	—
60	C₂×D₂₆₈	—	4	2	no	—
61	C₂×D₂₆₈	—	4	2	no	—
62	C₂×D₂₆₈	—	4	1	no	—
63	C₂×D₂₆₈	—	4	1	no	—
64	C₂×D₂₆₈	—	4	1	no	—
65	C₂²×Dic₆₇	—	4	1	no	—
66	C₂²×Dic₆₇	—	4	1	no	—
67	C₂²×Dic₆₇	—	4	1	no	—
68	C₄⋊Dic₆₇	—	4	2	no	—
69	C₄⋊Dic₆₇	—	4	2	no	—
70	C₄⋊Dic₆₇	—	4	2	no	—
71	C₄⋊Dic₆₇	—	4	2	no	—
72	C₄⋊Dic₆₇	—	4	2	no	—
73	C₄⋊Dic₆₇	—	4	2	no	—
74	C₄⋊Dic₆₇	—	4	2	no	—
75	C₄⋊Dic₆₇	—	4	2	no	—
76	C₄⋊Dic₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	D₈	1	4	yes	—
88	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
89	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₆₇	#41	1	1	yes	characteristic
91	C₄²×C₆₇	#41	1	1	yes	characteristic
92	C₄²×C₆₇	#41	1	1	yes	characteristic
93	C₂³×C₆₇	#127	1	1	yes	characteristic
94	C₂²×D₆₇	—	4	2	no	—
95	C₂²×D₆₇	—	8	1	no	—
96	C₂²×D₆₇	—	8	1	no	—
97	C₂²×D₆₇	—	8	1	no	—
98	C₂²×D₆₇	—	8	1	no	—
99	C₂²×D₆₇	—	8	1	no	—
100	C₂²×D₆₇	—	8	1	no	—
101	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₆₇	#135	1	2	yes	—
110	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₆₇	#137	1	2	yes	—
113	C₂×Dic₆₇	—	8	1	no	—
114	C₂×Dic₆₇	—	8	1	no	—
115	C₂×Dic₆₇	—	8	1	no	—
116	C₂×Dic₆₇	—	8	1	no	—
117	C₂×Dic₆₇	—	8	1	no	—
118	C₂×Dic₆₇	—	8	1	no	—
119	C₂×Dic₆₇	—	8	1	no	—
120	C₂×Dic₆₇	—	8	1	no	—
121	C₂×Dic₆₇	—	8	1	no	—
122	D₂₆₈	—	4	4	no	—
123	D₂₆₈	—	4	4	no	—
124	D₂₆₈	—	4	4	no	—
125	D₂₆₈	—	8	2	no	—
126	D₂₆₈	—	8	2	no	—
127	D₂₆₈	—	8	2	no	—
128	C₈×C₆₇	—	2	4	no	—
129	C₈×C₆₇	—	2	4	no	—
130	C₈×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	#399	1	1	yes	characteristic
132	C₂²×C₆₇	#492	1	1	yes	characteristic
133	C₂²×C₆₇	#493	1	1	yes	characteristic
134	C₂²×C₆₇	#495	1	1	yes	characteristic
135	C₂²×C₆₇	#496	1	1	yes	characteristic
136	C₂²×C₆₇	#496	1	1	yes	characteristic
137	C₂²×C₆₇	#497	1	1	yes	characteristic
138	C₂×D₆₇	—	8	2	no	—
139	C₂×D₆₇	—	8	2	no	—
140	C₂×D₆₇	—	8	2	no	—
141	C₂×D₆₇	—	16	1	no	—
142	C₂×D₆₇	—	16	1	no	—
143	C₂×D₆₇	—	16	1	no	—
144	C₂×D₆₇	—	16	1	no	—
145	C₄×C₆₇	#474	1	2	yes	—
146	C₄×C₆₇	#476	1	2	yes	—
147	C₄×C₆₇	#478	1	2	yes	—
148	C₄×C₆₇	#480	1	2	yes	—
149	C₄×C₆₇	#481	1	2	yes	—
150	C₄×C₆₇	#482	1	2	yes	—
151	Dic₆₇	—	16	1	no	—
152	Dic₆₇	—	16	1	no	—
153	Dic₆₇	—	16	1	no	—
154	#10672	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	#1706	1	1	yes	characteristic
156	C₂×C₆₇	#1714	1	1	yes	characteristic
157	C₂×C₆₇	#1720	1	1	yes	characteristic
158	C₂×C₆₇	#1721	1	1	yes	characteristic
159	C₂×C₆₇	#2092	1	1	yes	characteristic
160	C₂×C₆₇	#2093	1	1	yes	characteristic
161	C₂×C₆₇	#2094	1	1	yes	characteristic
162	D₆₇	—	16	2	no	—
163	#1957	—	67	1	no	—
164	#1957	—	67	1	no	—
165	#1957	—	67	1	no	—
166	#1845	—	67	1	no	—
167	#1844	—	67	1	no	—
168	#1834	—	67	1	no	—
169	#1273	D₆₇	1	1	yes	characteristic, complemented
170	C₆₇	#10672	1	1	yes	characteristic, complemented, Sylow
171	C₂×#127	—	67	1	no	—
172	C₂×#107	—	134	2	no	—
173	C₂×#106	—	134	2	no	—
174	C₂×#102	—	134	2	no	—
175	C₂×#102	—	134	2	no	—
176	C₂×#102	—	134	2	no	—
177	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
180	#384	—	67	1	no	—
181	#384	—	67	1	no	—
182	#384	—	67	1	no	—
183	#340	—	134	2	no	—
184	C₂²×D₄	—	134	1	no	—
185	C₂²×D₄	—	134	1	no	—
186	C₂²×D₄	—	134	1	no	—
187	C₂²×C₈	D₂₆₈	1	2	yes	—
188	C₂²×C₈	D₂₆₈	1	2	yes	—
189	C₂²×C₈	D₂₆₈	1	2	yes	—
190	#127	—	67	2	no	—
191	#127	—	134	1	no	—
192	#127	—	134	1	no	—
193	#127	—	134	1	no	—
194	C₂×C₄⋊C₄	—	134	1	no	—
195	C₂×C₄⋊C₄	—	134	1	no	—
196	C₂×C₄⋊C₄	—	134	1	no	—
197	C₂×C₄⋊C₄	—	134	1	no	—
198	C₂×C₄⋊C₄	—	134	1	no	—
199	C₂×C₄⋊C₄	—	134	1	no	—
200	C₂×C₄⋊C₄	—	134	1	no	—
201	C₂×C₄⋊C₄	—	134	1	no	—
202	C₂×C₄⋊C₄	—	134	1	no	—
203	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
204	#107	—	268	4	no	—
205	#106	—	268	4	no	—
206	#102	—	268	4	no	—
207	#102	—	268	4	no	—
208	#102	—	268	4	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₂⁴	—	268	1	no	—
216	C₂×D₄	—	268	1	no	—
217	C₂×D₄	—	268	1	no	—
218	C₂×D₄	—	268	1	no	—
219	C₂×D₄	—	268	2	no	—
220	C₂×D₄	—	268	2	no	—
221	C₂×D₄	—	268	2	no	—
222	C₂×D₄	—	268	2	no	—
223	C₂×D₄	—	268	2	no	—
224	C₂×D₄	—	268	2	no	—
225	C₂×D₄	—	134	2	no	—
226	C₂×D₄	—	134	2	no	—
227	C₂×D₄	—	134	2	no	—
228	C₂²×C₄	—	268	1	no	—
229	C₂²×C₄	—	268	1	no	—
230	C₂²×C₄	—	268	1	no	—
231	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	D₅₃₆	1	4	yes	—
242	C₂×C₈	#11761617	1	4	yes	—
243	C₄⋊C₄	—	268	2	no	—
244	C₄⋊C₄	—	268	2	no	—
245	C₄⋊C₄	—	268	2	no	—
246	C₄⋊C₄	—	268	2	no	—
247	C₄⋊C₄	—	268	2	no	—
248	C₄⋊C₄	—	268	2	no	—
249	C₄⋊C₄	—	268	2	no	—
250	C₄⋊C₄	—	268	2	no	—
251	C₄⋊C₄	—	268	2	no	—
252	C₄²	Q₈×D₆₇	1	1	yes	characteristic
253	C₄²	#11761627	1	1	yes	characteristic
254	C₄²	#11761627	1	1	yes	characteristic
255	C₄²	#11761627	1	1	yes	characteristic
256	C₂³	—	268	2	no	—
257	C₂³	—	536	1	no	—
258	C₂³	—	536	1	no	—
259	C₂³	—	536	1	no	—
260	C₂³	—	536	1	no	—
261	C₂³	—	536	1	no	—
262	C₂³	—	536	1	no	—
263	C₂³	#14524040	1	1	yes	characteristic
264	D₄	—	268	4	no	—
265	D₄	—	268	4	no	—
266	D₄	—	268	4	no	—
267	D₄	—	536	2	no	—
268	D₄	—	536	2	no	—
269	D₄	—	536	2	no	—
270	C₂×C₄	—	536	1	no	—
271	C₂×C₄	—	536	1	no	—
272	C₂×C₄	—	536	1	no	—
273	C₂×C₄	—	536	1	no	—
274	C₂×C₄	—	536	1	no	—
275	C₂×C₄	—	536	1	no	—
276	C₂×C₄	—	536	1	no	—
277	C₂×C₄	—	536	1	no	—
278	C₂×C₄	—	536	1	no	—
279	C₂×C₄	#14524023	1	1	yes	characteristic
280	C₂×C₄	#14524023	1	1	yes	characteristic
281	C₂×C₄	#14524023	1	1	yes	characteristic
282	C₂×C₄	#14524027	1	1	yes	characteristic
283	C₂×C₄	#14524027	1	1	yes	characteristic
284	C₂×C₄	#14524027	1	1	yes	characteristic
285	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
286	C₂×C₄	C₂×#11761617	1	2	yes	—
287	C₂×C₄	#14524065	1	2	yes	—
288	C₂×C₄	#14524071	1	2	yes	—
289	C₂×C₄	#14524071	1	2	yes	—
290	C₂×C₄	#14524076	1	2	yes	—
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	536	2	no	—
295	C₂²	—	536	2	no	—
296	C₂²	—	536	2	no	—
297	C₂²	—	1072	1	no	—
298	C₂²	—	1072	1	no	—
299	C₂²	—	1072	1	no	—
300	C₂²	—	1072	1	no	—
301	C₂²	#16863333	1	1	yes	characteristic
302	C₂²	#16863769	1	1	yes	characteristic
303	C₂²	#16863773	1	1	yes	characteristic
304	C₂²	#16863780	1	1	yes	characteristic
305	C₂²	#16863785	1	1	yes	characteristic
306	C₂²	#16863785	1	1	yes	characteristic
307	C₂²	#16863790	1	1	yes	characteristic
308	C₄	—	1072	1	no	—
309	C₄	—	1072	1	no	—
310	C₄	—	1072	1	no	—
311	C₄	#16863677	1	2	yes	—
312	C₄	#16863687	1	2	yes	—
313	C₄	#16863697	1	2	yes	—
314	C₄	#16863710	1	2	yes	—
315	C₄	#16863715	1	2	yes	—
316	C₄	#16863721	1	2	yes	—
317	C₂	—	1072	2	no	—
318	C₂	#22559801	1	1	yes	characteristic
319	C₂	#22559831	1	1	yes	characteristic
320	C₂	#22559869	1	1	yes	characteristic
321	C₂	#22559875	1	1	yes	characteristic
322	C₂	#22561729	1	1	yes	characteristic
323	C₂	#22561736	1	1	yes	characteristic
324	C₂	#22561741	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524040
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1273	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#14524040	C₂²
hypercenter	#1273	D₆₇
hypocenter	C₆₇	#10672
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
4. #22560369	C₂	semidirect
5. #22560366	C₂	semidirect
6. #22560337	C₂	semidirect
7. C₆₇×#1273	C₂	semidirect
169. #1273	D₆₇	semidirect
170. C₆₇	#10672	semidirect

Maximal subgroups

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

subgroup	index
#22561015	2
#22560369	2
#22560366	2
#22560337	2
C₆₇×#1273	2
#10672	67