Subgroups — #30864683

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570721	C₂	1	1	yes	maximal, characteristic
2	#22569147	C₂	1	2	yes	maximal
3	#22561188	C₂	1	2	yes	maximal
4	#22560961	C₂	1	2	yes	maximal
5	#22560958	C₂	1	1	yes	maximal, characteristic
6	#22560955	C₂	1	1	yes	maximal, characteristic
7	#22560379	C₂	1	1	yes	maximal, characteristic
8	#22560094	C₂	1	2	yes	maximal
9	#22560088	C₂	1	1	yes	maximal, characteristic
10	#22559895	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic
12	#16863949	C₂²	1	2	yes	—
13	#16863800	C₄	1	2	yes	complemented
14	#16863797	C₄	1	2	yes	complemented
15	#16863796	C₄	1	2	yes	complemented
16	#16863793	C₄	1	2	yes	complemented
17	#16863526	—	2	4	no	—
18	#16863524	—	2	4	no	—
19	C₂×#14523979	C₂²	1	1	yes	characteristic
20	C₂×#14523978	C₂²	1	2	yes	—
21	#16863434	C₂²	1	1	yes	characteristic
22	#16863432	C₂²	1	2	yes	—
23	#16863424	C₂²	1	2	yes	—
24	#16863423	C₂²	1	2	yes	—
25	#16863419	C₂²	1	2	yes	—
26	#16863351	C₂²	1	2	yes	—
27	C₂×#14523954	C₂²	1	1	yes	characteristic
28	C₂×#14523953	C₂²	1	1	yes	characteristic
29	C₂×#14523953	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
32	#16863242	C₂²	1	1	yes	characteristic
33	#16863157	C₂²	1	2	yes	—
34	#16863156	C₂²	1	2	yes	—
35	#16863155	C₂²	1	2	yes	—
36	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
37	#427×C₆₇	C₂²	1	1	yes	characteristic
38	#405×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	#339×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761612	C₂³	1	2	yes	—
42	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
44	#14524042	C₂×C₄	1	4	yes	—
45	#14524034	C₂×C₄	1	4	yes	—
46	#14524020	—	2	4	no	—
47	C₄×Dic₁₃₄	—	2	4	no	—
48	#14524002	C₂³	1	2	yes	—
49	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
51	#14523979	—	2	2	no	—
52	#14523978	—	2	4	no	—
53	#14523977	C₂×C₄	1	2	yes	—
54	#14523976	C₂×C₄	1	4	yes	—
55	#14523975	C₂×C₄	1	2	yes	—
56	#14523974	C₂×C₄	1	4	yes	—
57	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
58	Dic₆₇⋊C₈	—	2	4	no	—
59	Dic₆₇⋊C₈	—	2	4	no	—
60	#14523954	—	2	2	no	—
61	#14523953	—	2	2	no	—
62	#14523953	D₄	1	2	yes	—
63	#14523953	D₄	1	2	yes	—
64	#14523953	Q₈	1	4	yes	complemented
65	C₈×Dic₆₇	—	2	2	no	—
66	C₈×Dic₆₇	—	2	2	no	—
67	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
71	C₆₇⋊#95	—	2	2	no	—
72	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#117×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#108×C₆₇	—	2	2	no	—
78	#107×C₆₇	C₂×C₄	1	2	yes	—
79	#106×C₆₇	C₂×C₄	1	2	yes	—
80	#97×C₆₇	—	2	2	no	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	Q₈	1	4	yes	complemented
83	C₂×Dic₁₃₄	—	2	4	no	—
84	C₂²×Dic₆₇	—	2	1	no	—
85	C₂²×Dic₆₇	#41	1	2	yes	—
86	#11761612	—	2	2	no	—
87	#11761612	—	2	2	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	1	no	—
90	C₂×C₆₇⋊C₈	—	2	2	no	—
91	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
94	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
95	C₂×C₆₇⋊C₈	#41	1	2	yes	—
96	C₄⋊Dic₆₇	—	2	4	no	—
97	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
98	Dic₆₇⋊C₄	—	2	4	no	—
99	Dic₆₇⋊C₄	—	2	4	no	—
100	C₂²⋊Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	—	2	2	no	—
102	C₄×Dic₆₇	—	2	1	no	—
103	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
106	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
109	M₁₆×C₆₇	—	2	2	no	—
110	M₁₆×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₆₇	#41	1	2	yes	—
120	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂²⋊C₄×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	1	no	—
124	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
125	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₆₇	—	2	4	no	—
136	C₂×Dic₆₇	—	2	2	no	—
137	C₂×Dic₆₇	—	4	2	no	—
138	C₂×Dic₆₇	—	4	2	no	—
139	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
140	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
141	Dic₁₃₄	—	8	4	no	—
142	C₈×C₆₇	—	4	2	no	—
143	C₈×C₆₇	C₄×Q₈	1	2	yes	—
144	C₈×C₆₇	C₄×Q₈	1	2	yes	—
145	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
146	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
147	C₆₇⋊C₈	—	2	2	no	—
148	C₆₇⋊C₈	—	4	2	no	—
149	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
150	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₆₇	#533	1	1	yes	characteristic
154	C₂²×C₆₇	#533	1	1	yes	characteristic
155	C₄×C₆₇	—	4	2	no	—
156	C₄×C₆₇	—	4	4	no	—
157	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	C₄×C₆₇	#532	1	1	yes	characteristic
161	Dic₆₇	—	2	2	no	—
162	Dic₆₇	—	2	2	no	—
163	Dic₆₇	—	4	2	no	—
164	Dic₆₇	—	4	4	no	—
165	#21733	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	—	2	1	no	—
167	C₂×C₆₇	—	2	1	no	—
168	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
169	C₂×C₆₇	#2957	1	2	yes	—
170	#3160	—	67	1	no	—
171	#2961	—	67	2	no	—
172	#1995	—	67	2	no	—
173	#1946	—	67	2	no	—
174	#1945	—	67	1	no	—
175	#1945	—	67	1	no	—
176	#1848	—	67	1	no	—
177	#1773	—	67	2	no	—
178	#1772	—	67	1	no	—
179	#1772	D₆₇	1	1	yes	characteristic
180	#1726	—	67	1	no	—
181	C₆₇	#21733	1	1	yes	characteristic, complemented, Sylow
182	#533	—	67	2	no	—
183	#500	—	67	2	no	—
184	#499	—	67	2	no	—
185	#499	—	67	2	no	—
186	#498	—	67	2	no	—
187	#446	—	134	4	no	—
188	#446	—	134	4	no	—
189	C₂×#108	—	67	2	no	—
190	C₂×#108	—	67	1	no	—
191	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
192	#427	—	67	1	no	—
193	#427	—	67	2	no	—
194	#427	C₂×D₆₇	1	1	yes	characteristic
195	#424	—	67	2	no	—
196	#424	—	67	2	no	—
197	#423	—	67	2	no	—
198	#405	—	67	2	no	—
199	#405	C₂×D₆₇	1	2	yes	—
200	C₂×#97	—	67	1	no	—
201	C₂×#97	—	67	1	no	—
202	C₂×#97	—	67	1	no	—
203	C₂×C₄×C₈	—	67	1	no	—
204	C₂×C₄×C₈	—	67	1	no	—
205	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
206	#376	—	67	1	no	—
207	#339	—	67	2	no	—
208	#339	—	67	2	no	—
209	#339	—	67	2	no	—
210	#339	C₂×D₆₇	1	2	yes	—
211	C₂×M₁₆	—	67	2	no	—
212	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
213	C₂²×C₈	—	67	1	no	—
214	C₂²×C₈	—	67	1	no	—
215	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₄⋊Q₈	—	67	4	no	—
218	#125	—	67	4	no	—
219	C₂²⋊Q₈	—	134	4	no	—
220	C₄×Q₈	—	134	4	no	—
221	#117	—	67	2	no	—
222	#117	C₂²×D₆₇	1	2	yes	—
223	C₂×C₄²	—	67	1	no	—
224	C₂×C₄²	—	67	1	no	—
225	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
226	#108	—	2	2	no	—
227	#108	—	134	4	no	—
228	#108	—	134	2	no	—
229	#107	—	67	2	no	—
230	#107	—	67	4	no	—
231	#107	C₄×D₆₇	1	2	yes	—
232	#106	—	67	2	no	—
233	#106	—	67	4	no	—
234	#106	C₄×D₆₇	1	2	yes	—
235	C₄⋊C₈	—	134	4	no	—
236	C₄⋊C₈	—	134	4	no	—
237	C₄⋊C₈	—	134	4	no	—
238	#97	—	2	2	no	—
239	#97	—	67	2	no	—
240	#97	—	67	2	no	—
241	#97	—	67	4	no	—
242	#97	—	134	2	no	—
243	#97	—	134	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	67	1	no	—
246	C₄×C₈	—	67	1	no	—
247	C₄×C₈	—	67	1	no	—
248	C₄×C₈	—	67	1	no	—
249	C₄×C₈	—	134	2	no	—
250	C₄×C₈	—	134	2	no	—
251	C₄×C₈	Dic₁₃₄	1	4	yes	complemented
252	#95	—	134	2	no	—
253	C₂×Q₈	—	134	4	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	67	2	no	—
256	C₂²×C₄	—	134	1	no	—
257	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	134	2	no	—
261	M₁₆	—	134	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	67	1	no	—
264	C₂×C₈	—	67	1	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	67	2	no	—
267	C₂×C₈	—	67	2	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	—	134	1	no	—
270	C₂×C₈	—	134	1	no	—
271	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
273	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
274	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
275	C₂×C₈	Q₈×D₆₇	1	2	yes	—
276	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
277	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
278	C₂×C₈	#11761628	1	2	yes	—
279	C₄⋊C₄	—	67	4	no	—
280	C₄⋊C₄	—	134	4	no	—
281	C₄⋊C₄	—	134	4	no	—
282	C₄⋊C₄	—	134	4	no	—
283	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
284	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	134	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	67	1	no	—
289	C₄²	—	67	1	no	—
290	C₄²	—	67	2	no	—
291	C₄²	—	67	2	no	—
292	C₄²	—	134	2	no	—
293	C₄²	—	134	1	no	—
294	C₄²	C₂×Dic₁₃₄	1	2	yes	—
295	C₂³	#41×D₆₇	1	1	yes	characteristic
296	Q₈	—	536	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	67	2	no	—
303	C₂×C₄	—	67	2	no	—
304	C₂×C₄	—	134	4	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
310	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
312	C₈	—	4	2	no	—
313	C₈	—	268	2	no	—
314	C₈	—	67	2	no	—
315	C₈	—	67	2	no	—
316	C₈	—	134	2	no	—
317	C₈	C₄×Dic₁₃₄	1	2	yes	—
318	C₈	C₄×Dic₁₃₄	1	2	yes	—
319	C₈	#14524042	1	2	yes	—
320	C₈	#14524042	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
323	C₂²	#16863949	1	1	yes	characteristic
324	C₂²	#16863950	1	1	yes	characteristic
325	C₄	—	4	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	268	2	no	—
328	C₄	—	268	4	no	—
329	C₄	—	134	2	no	—
330	C₄	—	134	2	no	—
331	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
332	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
333	C₄	C₂×#14524042	1	1	yes	characteristic
334	C₄	#16863943	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	C₄×#14524042	1	1	yes	characteristic
338	C₂	#22569121	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#21733
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
13. #16863800	C₄	semidirect
14. #16863797	C₄	semidirect
15. #16863796	C₄	semidirect
16. #16863793	C₄	semidirect
64. #14523953	Q₈	semidirect
82. C₄×C₈×C₆₇	Q₈	semidirect
181. C₆₇	#21733	semidirect
251. C₄×C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570721	2
#22569147	2
#22561188	2
#22560961	2
#22560958	2
#22560955	2
#22560379	2
#22560094	2
#22560088	2
#22559895	2
C₆₇×#1772	2
#21733	67