Subgroups — #30838981

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571332	C₂	1	1	yes	maximal, characteristic
2	#22571136	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863747	C₂	1	1	yes	maximal, characteristic, complemented
4	#22568922	C₂	1	1	yes	maximal, characteristic
5	#22559757	C₂	1	4	yes	maximal, complemented
6	#22559633	C₂	1	4	yes	maximal, complemented
7	#22558893	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558880	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524049	—	2	1	no	—
11	#16863946	C₂²	1	1	yes	characteristic
12	C₂×#14524043	C₂²	1	1	yes	characteristic
13	C₂×#14524043	C₂²	1	1	yes	characteristic
14	#16863801	—	2	1	no	—
15	#16863769	—	2	1	no	—
16	#16863747	—	2	1	no	—
17	#16863747	—	2	1	no	—
18	#16863747	—	2	1	no	—
19	#16863747	C₂²	1	2	yes	—
20	#16863747	C₂²	1	2	yes	—
21	#16863743	—	2	1	no	—
22	#16863710	—	2	2	no	—
23	#16863677	C₂²	1	4	yes	—
24	#16863602	C₂²	1	4	yes	—
25	#16863537	—	2	2	no	—
26	#16863475	—	2	1	no	—
27	#16863475	C₂²	1	2	yes	—
28	C₈×D₂₆₈	—	2	1	no	—
29	C₈×D₂₆₈	C₂²	1	2	yes	—
30	#16863420	C₂²	1	1	yes	characteristic
31	C₂×#14523968	—	2	1	no	—
32	C₂×#14523968	—	2	1	no	—
33	#16863362	—	2	1	no	—
34	#16863143	C₂²	1	4	yes	—
35	#16863141	C₂²	1	4	yes	—
36	#431×C₆₇	C₂²	1	1	yes	characteristic
37	#431×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#335×C₆₇	C₂²	1	2	yes	—
40	#334×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761628	—	2	1	no	—
42	C₂²×Dic₁₃₄	—	2	1	no	—
43	C₂²×Dic₁₃₄	—	2	1	no	—
44	C₂×#11761617	—	2	4	no	—
45	#14524049	—	4	4	no	—
46	C₂×#11761612	—	2	1	no	—
47	C₂×#11761611	—	2	2	no	—
48	C₂×C₈×D₆₇	—	2	2	no	—
49	#14524043	—	2	1	no	—
50	#14524043	—	2	1	no	—
51	#14524043	—	2	1	no	—
52	#14524043	—	2	1	no	—
53	#14524043	—	2	1	no	—
54	#14524043	—	2	1	no	—
55	#14524043	D₄	1	1	yes	characteristic
56	#14524043	C₂³	1	2	yes	—
57	#14524043	C₂³	1	2	yes	—
58	#14524040	D₄	1	1	yes	characteristic
59	#14524031	—	2	1	no	—
60	#14524017	—	2	1	no	—
61	#14524017	D₄	1	2	yes	—
62	C₄×D₂₆₈	—	2	1	no	—
63	C₄×D₂₆₈	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₆₇	—	2	1	no	—
65	C₂×C₄⋊Dic₆₇	—	2	1	no	—
66	C₂×C₄⋊Dic₆₇	—	2	1	no	—
67	C₂×C₄⋊Dic₆₇	—	2	1	no	—
68	#14523977	—	2	4	no	—
69	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
71	#14523968	—	2	4	no	—
72	#14523968	—	2	2	no	—
73	#14523968	—	2	2	no	—
74	#14523968	—	2	2	no	—
75	#14523968	—	2	2	no	—
76	#14523965	—	2	4	no	—
77	(C₂×D₆₇)⋊C₈	—	2	2	no	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
83	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₆₇	—	2	1	no	—
85	C₂²⋊C₈×C₆₇	—	2	1	no	—
86	C₄×C₈×C₆₇	—	2	1	no	—
87	C₄×C₈×C₆₇	—	2	1	no	—
88	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	#11761628	—	4	2	no	—
93	#11761628	—	4	2	no	—
94	C₂×Dic₁₃₄	—	2	1	no	—
95	C₂×Dic₁₃₄	—	2	2	no	—
96	C₂×Dic₁₃₄	—	2	2	no	—
97	C₂×Dic₁₃₄	—	4	2	no	—
98	C₂×Dic₁₃₄	—	4	2	no	—
99	C₂×Dic₁₃₄	—	4	1	no	—
100	C₂×Dic₁₃₄	—	4	1	no	—
101	C₂×Dic₁₃₄	—	8	2	no	—
102	C₂×D₂₆₈	—	2	1	no	—
103	C₂×D₂₆₈	—	2	1	no	—
104	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
105	C₂×C₂²⋊D₆₇	—	2	2	no	—
106	C₂²×Dic₆₇	—	4	1	no	—
107	C₂²×Dic₆₇	—	4	1	no	—
108	C₂×C₄×D₆₇	—	2	2	no	—
109	#11761617	—	8	4	no	—
110	#11761612	—	4	2	no	—
111	#11761611	—	4	4	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₈×D₆₇	—	4	4	no	—
114	C₄⋊Dic₆₇	—	2	1	no	—
115	C₄⋊Dic₆₇	—	2	2	no	—
116	C₄⋊Dic₆₇	—	2	2	no	—
117	C₄⋊Dic₆₇	—	2	2	no	—
118	C₄⋊Dic₆₇	—	2	2	no	—
119	C₄⋊Dic₆₇	—	4	1	no	—
120	C₄⋊Dic₆₇	—	4	1	no	—
121	C₄⋊Dic₆₇	—	4	1	no	—
122	C₄⋊Dic₆₇	—	4	1	no	—
123	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
124	(C₂×D₆₇)⋊C₄	—	2	2	no	—
125	(C₂×D₆₇)⋊C₄	—	2	2	no	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₆₇	#41	1	1	yes	characteristic
140	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
141	C₂³×C₆₇	#121	1	1	yes	characteristic
142	C₂²×D₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#135	1	2	yes	—
149	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₂²⋊D₆₇	—	4	4	no	—
154	C₂×Dic₆₇	—	2	2	no	—
155	C₂×Dic₆₇	—	4	2	no	—
156	C₂×Dic₆₇	—	4	2	no	—
157	C₂×Dic₆₇	—	8	2	no	—
158	C₂×Dic₆₇	—	8	2	no	—
159	D₂₆₈	—	2	4	no	—
160	D₂₆₈	—	4	2	no	—
161	D₂₆₈	—	4	2	no	—
162	C₄×D₆₇	—	4	2	no	—
163	C₄×D₆₇	—	4	2	no	—
164	Dic₁₃₄	—	4	2	no	—
165	Dic₁₃₄	—	4	2	no	—
166	Dic₁₃₄	—	4	2	no	—
167	Dic₁₃₄	—	4	4	no	—
168	Dic₁₃₄	—	8	2	no	—
169	Dic₁₃₄	—	8	2	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	2	1	no	—
173	C₈×C₆₇	—	2	1	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₈×C₆₇	—	4	2	no	—
176	C₆₇⋊C₈	—	4	2	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	#463	1	2	yes	—
180	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
181	C₂×D₆₇	—	4	2	no	—
182	C₂×D₆₇	—	4	2	no	—
183	C₂×D₆₇	—	4	2	no	—
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	#463	1	2	yes	—
187	C₄×C₆₇	#473	1	2	yes	—
188	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
189	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₄×C₆₇	#577	1	2	yes	—
191	Dic₆₇	—	4	2	no	—
192	Dic₆₇	—	8	2	no	—
193	Dic₆₇	—	8	2	no	—
194	#19197	—	67	1	no	maximal, Sylow
195	C₂×C₆₇	—	2	2	no	—
196	C₂×C₆₇	C₂×#463	1	1	yes	characteristic
197	C₂×C₆₇	#3057	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
199	D₆₇	—	8	2	no	—
200	#3226	—	67	1	no	—
201	#3206	—	67	1	no	—
202	C₂×#487	—	67	1	no	—
203	#2931	—	67	1	no	—
204	#1699	—	67	4	no	—
205	#1679	—	67	4	no	—
206	#1564	D₆₇	1	1	yes	characteristic, complemented
207	#1549	—	67	1	no	—
208	#1546	—	67	1	no	—
209	C₆₇	#19197	1	1	yes	characteristic, complemented, Sylow
210	C₂×#131	—	134	1	no	—
211	#532	—	67	1	no	—
212	C₂×C₄⋊Q₈	—	67	1	no	—
213	C₂×C₄⋊Q₈	—	67	1	no	—
214	#500	—	134	1	no	—
215	#492	—	134	1	no	—
216	#487	—	67	2	no	—
217	#487	—	67	2	no	—
218	#487	—	134	1	no	—
219	#487	—	134	1	no	—
220	#487	—	134	1	no	—
221	#486	—	134	1	no	—
222	#480	—	134	2	no	—
223	#474	—	67	4	no	—
224	#461	—	67	4	no	—
225	#448	—	134	2	no	—
226	#436	—	67	2	no	—
227	#436	—	134	1	no	—
228	C₈×D₄	—	67	2	no	—
229	C₈×D₄	—	134	1	no	—
230	#431	C₂×D₆₇	1	1	yes	characteristic
231	#431	C₂×D₆₇	1	1	yes	characteristic
232	#423	—	67	1	no	—
233	C₂×#103	—	134	1	no	—
234	C₂×#103	—	134	1	no	—
235	#408	—	134	1	no	—
236	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
237	#335	C₂×D₆₇	1	2	yes	—
238	#334	C₂×D₆₇	1	2	yes	—
239	#332	—	67	4	no	—
240	#332	—	67	4	no	—
241	C₂×#41	—	134	1	no	—
242	C₂²×Q₈	—	134	1	no	—
243	C₂²×Q₈	—	134	1	no	—
244	C₂×SD₁₆	—	134	4	no	—
245	#131	—	268	4	no	—
246	C₂×M₁₆	—	134	1	no	—
247	C₂×M₁₆	—	134	2	no	—
248	C₂²×C₈	—	134	2	no	—
249	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
250	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
251	C₄⋊Q₈	—	67	1	no	—
252	C₄⋊Q₈	—	67	2	no	—
253	C₄⋊Q₈	—	67	2	no	—
254	C₄⋊Q₈	—	134	1	no	—
255	C₄⋊Q₈	—	134	1	no	—
256	C₄⋊Q₈	—	134	1	no	—
257	C₄⋊Q₈	—	134	1	no	—
258	C₄⋊Q₈	—	134	1	no	—
259	C₄⋊Q₈	—	134	1	no	—
260	#127	—	67	1	no	—
261	#124	—	134	1	no	—
262	#121	—	67	2	no	—
263	#121	—	134	1	no	—
264	C₄×D₄	—	67	2	no	—
265	C₄×D₄	—	134	1	no	—
266	C₂×C₄⋊C₄	—	134	1	no	—
267	C₂×C₄⋊C₄	—	134	1	no	—
268	C₂×C₄⋊C₄	—	134	1	no	—
269	C₂×C₄⋊C₄	—	134	1	no	—
270	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
271	#107	—	134	4	no	—
272	C₄⋊C₈	—	67	2	no	—
273	C₄⋊C₈	—	134	1	no	—
274	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
275	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
276	#103	—	134	4	no	—
277	#103	—	134	2	no	—
278	#103	—	134	2	no	—
279	#103	—	134	2	no	—
280	#103	—	134	2	no	—
281	#102	—	134	4	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	134	2	no	—
285	C₂²⋊C₈	—	134	2	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	D₂₆₈	1	1	yes	characteristic
289	C₄×C₈	D₂₆₈	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
292	#41	—	268	2	no	—
293	#41	—	268	2	no	—
294	C₂×Q₈	—	268	1	no	—
295	C₂×Q₈	—	268	1	no	—
296	C₂×Q₈	—	268	2	no	—
297	C₂×Q₈	—	268	2	no	—
298	C₂×Q₈	—	536	2	no	—
299	C₂×Q₈	—	134	2	no	—
300	C₂×Q₈	—	134	2	no	—
301	C₂×Q₈	—	134	1	no	—
302	C₂×D₄	—	67	2	no	—
303	C₂×D₄	—	134	2	no	—
304	C₂×D₄	—	134	1	no	—
305	C₂×D₄	—	134	1	no	—
306	C₂²×C₄	—	268	1	no	—
307	C₂²×C₄	—	268	1	no	—
308	C₂²×C₄	—	134	2	no	—
309	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
311	C₂²×C₄	#11761629	1	1	yes	characteristic
312	SD₁₆	—	536	4	no	—
313	M₁₆	—	268	4	no	—
314	M₁₆	—	268	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	268	4	no	—
320	C₂×C₈	—	134	2	no	—
321	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
322	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
323	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
324	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
325	C₄⋊C₄	—	268	1	no	—
326	C₄⋊C₄	—	268	1	no	—
327	C₄⋊C₄	—	268	1	no	—
328	C₄⋊C₄	—	268	1	no	—
329	C₄⋊C₄	—	67	2	no	—
330	C₄⋊C₄	—	134	2	no	—
331	C₄⋊C₄	—	134	2	no	—
332	C₄⋊C₄	—	134	2	no	—
333	C₄⋊C₄	—	134	2	no	—
334	C₄⋊C₄	—	134	1	no	—
335	C₂²⋊C₄	—	134	2	no	—
336	C₂²⋊C₄	—	134	2	no	—
337	C₄²	D₄×D₆₇	1	1	yes	characteristic
338	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
339	C₄²	#11761629	1	1	yes	characteristic
340	C₄²	C₂³×D₆₇	1	1	yes	characteristic
341	C₂³	—	134	2	no	—
342	C₂³	#14524018	1	1	yes	characteristic
343	Q₈	—	268	2	no	—
344	Q₈	—	268	2	no	—
345	Q₈	—	268	2	no	—
346	Q₈	—	268	4	no	—
347	Q₈	—	536	2	no	—
348	Q₈	—	536	2	no	—
349	D₄	—	268	4	no	—
350	D₄	—	268	2	no	—
351	D₄	—	268	2	no	—
352	D₄	—	134	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	268	2	no	—
356	C₂×C₄	—	268	2	no	—
357	C₂×C₄	—	268	2	no	—
358	C₂×C₄	—	268	2	no	—
359	C₂×C₄	—	536	2	no	—
360	C₂×C₄	—	536	2	no	—
361	C₂×C₄	—	134	2	no	—
362	C₂×C₄	#14524018	1	1	yes	characteristic
363	C₂×C₄	#14524018	1	1	yes	characteristic
364	C₂×C₄	#14524018	1	1	yes	characteristic
365	C₂×C₄	#14524065	1	2	yes	—
366	C₂×C₄	#14524071	1	2	yes	—
367	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
369	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₈	—	268	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	268	2	no	—
380	C₂²	—	268	2	no	—
381	C₂²	—	268	2	no	—
382	C₂²	#16863612	1	2	yes	—
383	C₂²	C₂×#14524018	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	268	2	no	—
387	C₄	—	536	2	no	—
388	C₄	—	536	2	no	—
389	C₄	#16863612	1	2	yes	—
390	C₄	#16863672	1	2	yes	—
391	C₄	C₂×#14524065	1	1	yes	characteristic
392	C₄	C₂×#14524071	1	1	yes	characteristic
393	C₄	#16864203	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	536	2	no	—
396	C₂	C₂×#16863612	1	1	yes	characteristic
397	C₂	#22569849	1	1	yes	characteristic
398	C₂	C₂×#16863672	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	#16863946	C₂²
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#19197
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₂×#16863747	C₂	semidirect
5. #22559757	C₂	semidirect
6. #22559633	C₂	semidirect
7. #22558893	C₂	semidirect
8. #22558880	C₂	semidirect
9. C₆₇×#1564	C₂	semidirect
206. #1564	D₆₇	semidirect
209. C₆₇	#19197	semidirect

Maximal subgroups

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

subgroup	index
#22571332	2
#22571136	2
C₂×#16863747	2
#22568922	2
#22559757	2
#22559633	2
#22558893	2
#22558880	2
C₆₇×#1564	2
#19197	67