Subgroups — #31013511

Order 17152.

528 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573183	C₂	1	1	yes	maximal, characteristic
2	#22573092	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562434	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562431	C₂	1	2	yes	maximal, complemented
5	#22562355	C₂	1	2	yes	maximal, complemented
6	#22562250	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562248	C₂	1	2	yes	maximal, complemented
8	#22562111	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562060	C₂	1	2	yes	maximal, complemented
10	C₄×#14523987	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864126	—	2	1	no	—
13	#16864067	—	2	1	no	—
14	C₂×#14523988	C₂²	1	2	yes	—
15	C₂×#14523987	C₂²	1	1	yes	characteristic
16	#16863800	C₂²	1	2	yes	—
17	#16863773	C₂²	1	1	yes	characteristic
18	#16863771	C₂²	1	2	yes	—
19	#16863690	—	2	2	no	—
20	#16863688	—	2	1	no	—
21	#16863591	—	2	1	no	—
22	#16863590	—	2	2	no	—
23	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
25	#16863490	C₂²	1	2	yes	—
26	#16863478	C₂²	1	2	yes	—
27	#16863223	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	#16863199	C₂²	1	1	yes	characteristic
30	#16863198	C₂²	1	2	yes	—
31	#16863197	C₂²	1	1	yes	characteristic
32	#16863196	C₂²	1	2	yes	—
33	#16863196	C₂²	1	2	yes	—
34	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₆₇	C₂²	1	2	yes	—
36	#500×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
38	#367×C₆₇	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	#357×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761628	—	4	1	no	—
42	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
43	C₂×D₅₃₆	—	2	1	no	—
44	C₂×D₅₃₆	C₂³	1	2	yes	—
45	C₂×#11761613	—	2	2	no	—
46	#14524042	—	2	2	no	—
47	#14524042	—	4	1	no	—
48	#14524041	—	4	2	no	—
49	#14524040	—	2	1	no	—
50	#14524039	—	2	2	no	—
51	#14524017	—	4	1	no	—
52	#14524016	—	4	2	no	—
53	C₄×Dic₁₃₄	—	2	1	no	—
54	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
55	#14524008	—	2	2	no	—
56	#14524008	—	4	1	no	—
57	#14524005	—	2	2	no	—
58	#14523988	—	2	4	no	—
59	#14523987	—	2	2	no	—
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	#14523977	C₂³	1	1	yes	characteristic
63	#14523976	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
65	Dic₆₇⋊C₈	D₄	1	2	yes	—
66	#14523968	—	2	1	no	—
67	#14523966	—	2	2	no	—
68	#14523963	—	2	2	no	—
69	C₈×Dic₆₇	C₂³	1	2	yes	—
70	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₇	—	2	1	no	—
72	C₄×D₄×C₆₇	—	2	1	no	—
73	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	—	2	1	no	—
75	#107×C₆₇	C₂³	1	2	yes	—
76	#107×C₆₇	C₂³	1	1	yes	characteristic
77	#102×C₆₇	—	2	1	no	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#11761628	—	8	4	no	—
80	C₂×Dic₁₃₄	—	2	1	no	—
81	C₂×Dic₁₃₄	—	4	2	no	—
82	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
83	C₂×D₂₆₈	—	2	2	no	—
84	C₂×D₂₆₈	—	4	1	no	—
85	C₂×D₂₆₈	—	4	1	no	—
86	C₂×C₂²⋊D₆₇	—	4	2	no	—
87	C₂×C₂²⋊D₆₇	—	4	2	no	—
88	C₂×C₄×D₆₇	—	4	2	no	—
89	Dic₂₆₈	—	2	1	no	—
90	Dic₂₆₈	#41	1	2	yes	complemented
91	D₅₃₆	—	2	2	no	—
92	D₅₃₆	—	2	2	no	—
93	D₅₃₆	—	8	1	no	—
94	#11761613	—	8	2	no	—
95	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₄⋊Dic₆₇	—	2	1	no	—
97	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
98	Dic₆₇⋊C₄	—	4	1	no	—
99	Dic₆₇⋊C₄	—	4	2	no	—
100	Dic₆₇⋊C₄	—	4	2	no	—
101	Dic₆₇⋊C₄	D₈	1	2	yes	—
102	(C₂×D₆₇)⋊C₄	—	4	2	no	—
103	C₄×Dic₆₇	—	4	1	no	—
104	C₄×Dic₆₇	—	4	2	no	—
105	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	—	2	1	no	—
108	C₂×Q₈×C₆₇	—	4	1	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	4	1	no	—
111	D₈×C₆₇	—	2	1	no	—
112	D₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₂²⋊C₄×C₆₇	—	4	1	no	—
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₁₆×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	4	1	no	—
122	C₂²×D₆₇	—	4	2	no	—
123	Q₈×C₆₇	—	16	1	no	—
124	D₄×C₆₇	—	4	1	no	—
125	D₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	8	2	no	—
129	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₆₇	—	8	4	no	—
133	C₂²⋊D₆₇	—	16	2	no	—
134	C₂×Dic₆₇	—	4	1	no	—
135	C₂×Dic₆₇	—	4	1	no	—
136	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
137	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
138	D₂₆₈	—	4	2	no	—
139	D₂₆₈	—	4	2	no	—
140	D₂₆₈	—	8	2	no	—
141	D₂₆₈	—	16	1	no	—
142	C₄×D₆₇	—	8	4	no	—
143	Dic₁₃₄	—	2	2	no	—
144	Dic₁₃₄	—	4	1	no	—
145	Dic₁₃₄	—	16	2	no	—
146	Dic₁₃₄	SD₃₂	1	4	yes	complemented
147	C₈×C₆₇	—	2	1	no	—
148	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
150	C₆₇⋊C₈	—	2	2	no	—
151	C₂²×C₆₇	—	8	1	no	—
152	C₂²×C₆₇	—	8	1	no	—
153	C₂²×C₆₇	—	8	1	no	—
154	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₆₇	—	8	2	no	—
156	C₂×D₆₇	—	8	2	no	—
157	C₂×D₆₇	—	8	2	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	8	1	no	—
160	C₄×C₆₇	—	8	2	no	—
161	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
162	C₄×C₆₇	#548	1	1	yes	characteristic
163	C₄×C₆₇	#577	1	1	yes	characteristic
164	Dic₆₇	—	2	2	no	—
165	Dic₆₇	—	4	2	no	—
166	Dic₆₇	—	8	1	no	—
167	Dic₆₇	C₂×SD₃₂	1	4	yes	—
168	#33929	—	67	1	no	maximal, Sylow
169	C₂×C₆₇	—	16	1	no	—
170	C₂×C₆₇	#3358	1	1	yes	characteristic
171	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
172	C₂×C₆₇	#3415	1	1	yes	characteristic
173	D₆₇	—	16	2	no	—
174	#3391	—	67	1	no	—
175	Q₈×Q₁₆	—	67	1	no	—
176	#2240	—	67	1	no	—
177	#2240	—	67	2	no	—
178	#2226	—	67	2	no	—
179	#2226	D₆₇	1	1	yes	characteristic, complemented
180	#2207	—	67	1	no	—
181	#2207	—	67	2	no	—
182	Q₈×C₁₆	—	67	1	no	—
183	C₄×SD₃₂	—	67	1	no	—
184	C₄×SD₃₂	—	67	2	no	—
185	C₆₇	#33929	1	1	yes	characteristic, complemented, Sylow
186	Q₈²	—	134	1	no	—
187	#550	—	134	1	no	—
188	C₂×SD₃₂	—	67	1	no	—
189	C₂×SD₃₂	—	67	2	no	—
190	#500	—	67	2	no	—
191	#500	C₂×D₆₇	1	1	yes	characteristic
192	#493	—	67	2	no	—
193	#493	—	67	1	no	—
194	#477	—	134	2	no	—
195	#477	—	134	1	no	—
196	#459	—	134	2	no	—
197	#459	—	134	1	no	—
198	C₈×Q₈	—	67	1	no	—
199	C₄×Q₁₆	—	67	2	no	—
200	C₄×Q₁₆	—	67	1	no	—
201	C₄×D₈	—	67	2	no	—
202	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
203	#367	—	67	1	no	—
204	#367	C₂×D₆₇	1	2	yes	—
205	C₄⋊C₁₆	—	67	2	no	—
206	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
207	#358	—	67	1	no	—
208	#357	—	67	1	no	—
209	#357	—	67	2	no	—
210	#357	—	67	2	no	—
211	#357	—	67	2	no	—
212	#357	C₂×D₆₇	1	2	yes	—
213	C₄×C₁₆	—	67	2	no	—
214	C₄×C₁₆	—	67	1	no	—
215	C₂×#41	—	268	1	no	—
216	C₂×Q₁₆	—	67	1	no	—
217	C₂×D₈	—	67	2	no	—
218	C₂×D₈	—	134	2	no	—
219	C₂×D₈	—	134	1	no	—
220	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	268	1	no	—
223	C₄⋊Q₈	—	268	2	no	—
224	C₄⋊Q₈	—	134	2	no	—
225	#127	—	134	2	no	—
226	#127	—	134	1	no	—
227	#121	—	268	2	no	—
228	#121	—	268	1	no	—
229	C₄×Q₈	—	268	1	no	—
230	C₄×Q₈	—	67	1	no	—
231	C₄×Q₈	—	134	2	no	—
232	C₄×Q₈	—	134	1	no	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	—	134	2	no	—
235	SD₃₂	—	134	4	no	—
236	SD₃₂	—	134	2	no	—
237	C₂×C₁₆	—	67	1	no	—
238	C₂×C₁₆	—	67	1	no	—
239	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
240	#107	—	2	1	no	—
241	#107	—	67	1	no	—
242	#107	—	134	2	no	—
243	#107	C₂²×D₆₇	1	2	yes	—
244	#107	C₂²×D₆₇	1	1	yes	characteristic
245	C₄⋊C₈	—	67	2	no	—
246	C₄⋊C₈	—	67	1	no	—
247	#103	—	134	2	no	—
248	#103	—	134	1	no	—
249	#102	—	2	1	no	—
250	#102	—	134	2	no	—
251	C₄×C₈	—	67	2	no	—
252	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
253	#41	—	536	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	268	2	no	—
256	C₂×Q₈	—	67	1	no	—
257	C₂×Q₈	—	134	1	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	268	2	no	—
260	C₂×D₄	—	268	2	no	—
261	C₂×D₄	—	268	1	no	—
262	C₂×D₄	—	268	1	no	—
263	C₂×D₄	—	134	2	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	268	2	no	—
266	Q₁₆	—	67	2	no	—
267	Q₁₆	—	134	1	no	—
268	D₈	—	2	1	no	—
269	D₈	—	2	1	no	—
270	D₈	—	536	1	no	—
271	D₈	—	536	2	no	—
272	D₈	—	134	2	no	—
273	D₈	—	134	2	no	—
274	C₂×C₈	—	67	2	no	—
275	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	268	2	no	—
281	C₄⋊C₄	—	268	2	no	—
282	C₄⋊C₄	—	268	1	no	—
283	C₄⋊C₄	—	67	2	no	—
284	C₄⋊C₄	—	67	1	no	—
285	C₄⋊C₄	—	134	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	268	2	no	—
288	C₄²	—	268	2	no	—
289	C₄²	—	268	1	no	—
290	C₄²	—	67	2	no	—
291	C₄²	D₄×D₆₇	1	1	yes	characteristic
292	C₁₆	—	2	2	no	—
293	C₁₆	—	134	1	no	—
294	C₁₆	—	134	1	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	268	2	no	—
297	Q₈	—	268	1	no	—
298	Q₈	—	16	1	no	—
299	Q₈	—	1072	2	no	—
300	Q₈	—	67	4	no	—
301	Q₈	—	134	2	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	1	no	—
304	D₄	—	268	2	no	—
305	D₄	—	268	2	no	—
306	D₄	—	536	4	no	—
307	D₄	—	536	2	no	—
308	D₄	—	1072	1	no	—
309	D₄	—	1072	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	268	1	no	—
314	C₂×C₄	—	268	1	no	—
315	C₂×C₄	—	536	4	no	—
316	C₂×C₄	—	67	2	no	—
317	C₂×C₄	—	67	2	no	—
318	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
319	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	134	2	no	—
323	C₈	#41×D₆₇	1	1	yes	characteristic
324	C₈	#14524091	1	1	yes	characteristic
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	536	2	no	—
329	C₂²	—	536	2	no	—
330	C₂²	—	536	2	no	—
331	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	268	2	no	—
336	C₄	—	536	1	no	—
337	C₄	—	67	4	no	—
338	C₄	—	134	2	no	—
339	C₄	SD₃₂×D₆₇	1	2	yes	—
340	C₄	#16864056	1	1	yes	characteristic
341	C₄	#16864205	1	1	yes	characteristic
342	C₂	—	16	1	no	—
343	C₂	—	1072	2	no	—
344	C₂	#22572843	1	1	yes	characteristic
345	C₂	C₂×SD₃₂×D₆₇	1	1	yes	characteristic
346	C₂	#22573349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2226	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22573183	C₂
hypercenter	#2226	D₆₇
hypocenter	C₆₇	#33929
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
2. #22573092	C₂	semidirect
3. #22562434	C₂	semidirect
4. #22562431	C₂	semidirect
5. #22562355	C₂	semidirect
6. #22562250	C₂	semidirect
7. #22562248	C₂	semidirect
8. #22562111	C₂	semidirect
9. #22562060	C₂	semidirect
10. C₄×#14523987	C₂	semidirect
11. C₆₇×#2226	C₂	semidirect
90. Dic₂₆₈	#41	semidirect
146. Dic₁₃₄	SD₃₂	semidirect
179. #2226	D₆₇	semidirect
185. C₆₇	#33929	semidirect

Maximal subgroups

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

subgroup	index
#22573183	2
#22573092	2
#22562434	2
#22562431	2
#22562355	2
#22562250	2
#22562248	2
#22562111	2
#22562060	2
C₄×#14523987	2
C₆₇×#2226	2
#33929	67