Subgroups — #30847433

Order 17152.

638 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	#22573184	C₂	1	1	yes	maximal, characteristic
2	#22573077	C₂	1	1	yes	maximal, characteristic, complemented
3	#22572549	C₂	1	2	yes	maximal, complemented
4	#22559801	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559743	C₂	1	2	yes	maximal, complemented
6	#22559613	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559569	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559338	C₂	1	2	yes	maximal, complemented
9	#22559284	C₂	1	2	yes	maximal, complemented
10	#22559095	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864106	C₂²	1	2	yes	—
13	#16864068	C₂²	1	2	yes	—
14	#16863801	C₂²	1	1	yes	characteristic
15	#16863798	—	2	1	no	—
16	#16863773	—	2	1	no	—
17	#16863773	C₂²	1	1	yes	characteristic
18	#16863769	C₂²	1	2	yes	—
19	#16863743	C₂²	1	2	yes	—
20	#16863711	—	2	2	no	—
21	#16863697	—	2	2	no	—
22	#16863691	C₂²	1	2	yes	—
23	#16863687	C₂²	1	2	yes	—
24	#16863645	—	2	2	no	—
25	#16863596	C₂²	1	2	yes	—
26	#16863592	—	2	2	no	—
27	#16863592	C₂²	1	2	yes	—
28	#16863557	—	2	2	no	—
29	C₄×D₅₃₆	—	2	1	no	—
30	C₄×D₅₃₆	—	2	1	no	—
31	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
32	#16863475	C₂²	1	2	yes	—
33	#16863396	—	2	2	no	—
34	#16863149	C₂²	1	1	yes	characteristic
35	#16863144	C₂²	1	2	yes	—
36	#16863139	C₂²	1	2	yes	—
37	#16863125	C₂²	1	2	yes	—
38	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#11761629	—	2	2	no	—
46	C₂×#11761617	—	4	2	no	—
47	C₂×#11761615	—	2	2	no	—
48	C₂×D₅₃₆	—	2	2	no	—
49	C₂×D₅₃₆	—	2	1	no	—
50	C₂×D₅₃₆	—	2	1	no	—
51	C₂×D₅₃₆	D₄	1	1	yes	characteristic
52	C₂×#11761611	—	2	2	no	—
53	#14524043	C₂³	1	2	yes	—
54	#14524040	—	2	2	no	—
55	#14524040	C₂³	1	2	yes	—
56	#14524035	—	2	2	no	—
57	#14524021	—	4	2	no	—
58	#14524019	—	2	2	no	—
59	#14524018	—	2	2	no	—
60	#14524018	—	4	2	no	—
61	C₄×D₂₆₈	—	2	2	no	—
62	C₄×D₂₆₈	—	2	2	no	—
63	C₄×D₂₆₈	C₂³	1	2	yes	—
64	C₄⋊C₄×D₆₇	—	2	2	no	—
65	#14523977	—	2	1	no	—
66	#14523977	—	2	1	no	—
67	#14523977	D₄	1	1	yes	characteristic
68	#14523976	—	2	2	no	—
69	#14523975	—	2	2	no	—
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
71	#14523967	—	2	2	no	—
72	#14523965	—	2	2	no	—
73	#14523965	—	2	2	no	—
74	#14523965	—	2	2	no	—
75	#14523965	—	2	2	no	—
76	#14523965	D₄	1	2	yes	—
77	#14523963	—	2	2	no	—
78	(C₂×D₆₇)⋊C₈	—	2	2	no	—
79	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	#107×C₆₇	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₆₇	—	2	1	no	—
84	C₄⋊C₈×C₆₇	—	2	1	no	—
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#103×C₆₇	D₄	1	1	yes	characteristic
88	#103×C₆₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₆₇	—	2	1	no	—
90	C₄×C₈×C₆₇	—	2	1	no	—
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	#11761629	—	4	4	no	—
95	#11761629	—	8	2	no	—
96	C₂×Dic₁₃₄	—	2	2	no	—
97	C₂×D₂₆₈	—	2	2	no	—
98	C₂×D₂₆₈	—	2	2	no	—
99	C₂×D₂₆₈	—	2	2	no	—
100	C₂×D₂₆₈	—	4	2	no	—
101	C₂×D₂₆₈	—	4	2	no	—
102	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
103	C₂×C₄×D₆₇	—	2	2	no	—
104	C₂×C₄×D₆₇	—	4	2	no	—
105	C₂×C₄×D₆₇	—	4	2	no	—
106	#11761617	—	8	4	no	—
107	#11761615	—	8	2	no	—
108	D₅₃₆	—	2	2	no	—
109	D₅₃₆	—	4	2	no	—
110	D₅₃₆	—	8	1	no	—
111	D₅₃₆	—	8	2	no	—
112	#11761611	—	8	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₄⋊Dic₆₇	—	2	2	no	—
115	C₄⋊Dic₆₇	—	2	2	no	—
116	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
117	Dic₆₇⋊C₄	—	4	2	no	—
118	Dic₆₇⋊C₄	—	4	2	no	—
119	(C₂×D₆₇)⋊C₄	—	2	2	no	—
120	(C₂×D₆₇)⋊C₄	—	4	2	no	—
121	(C₂×D₆₇)⋊C₄	—	4	2	no	—
122	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
124	Q₁₆×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₆₇	—	2	1	no	—
137	C₄²×C₆₇	—	2	1	no	—
138	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
139	C₂²×D₆₇	—	2	2	no	—
140	C₂²×D₆₇	—	4	2	no	—
141	Q₈×C₆₇	—	2	1	no	—
142	Q₈×C₆₇	—	2	1	no	—
143	Q₈×C₆₇	—	2	1	no	—
144	Q₈×C₆₇	#135	1	2	yes	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×Dic₆₇	—	2	2	no	—
153	C₂×Dic₆₇	—	4	2	no	—
154	D₂₆₈	—	2	2	no	—
155	D₂₆₈	—	2	2	no	—
156	D₂₆₈	—	4	2	no	—
157	D₂₆₈	—	4	2	no	—
158	D₂₆₈	—	4	2	no	—
159	D₂₆₈	—	4	2	no	—
160	D₂₆₈	—	8	4	no	—
161	D₂₆₈	—	8	2	no	—
162	D₂₆₈	—	16	2	no	—
163	C₄×D₆₇	—	4	2	no	—
164	C₄×D₆₇	—	4	2	no	—
165	C₄×D₆₇	—	8	4	no	—
166	C₄×D₆₇	—	16	2	no	—
167	Dic₁₃₄	—	8	2	no	—
168	C₈×C₆₇	—	2	2	no	—
169	C₈×C₆₇	—	2	1	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	4	1	no	—
173	C₈×C₆₇	—	4	1	no	—
174	C₈×C₆₇	#137	1	2	yes	—
175	C₆₇⋊C₈	—	4	2	no	—
176	C₂²×C₆₇	D₄²	1	1	yes	characteristic
177	C₂×D₆₇	—	4	2	no	—
178	C₂×D₆₇	—	4	2	no	—
179	C₂×D₆₇	—	4	2	no	—
180	C₂×D₆₇	—	8	2	no	—
181	C₂×D₆₇	—	8	2	no	—
182	C₂×D₆₇	—	8	2	no	—
183	C₄×C₆₇	—	2	2	no	—
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	4	1	no	—
186	C₄×C₆₇	—	4	1	no	—
187	C₄×C₆₇	—	4	1	no	—
188	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
189	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
190	C₄×C₆₇	#576	1	1	yes	characteristic
191	C₄×C₆₇	#577	1	1	yes	characteristic
192	Dic₆₇	—	4	2	no	—
193	Dic₆₇	—	8	2	no	—
194	#19909	—	67	1	no	maximal, Sylow
195	C₂×C₆₇	#3278	1	1	yes	characteristic
196	C₂×C₆₇	#3280	1	1	yes	characteristic
197	C₂×C₆₇	#3292	1	1	yes	characteristic
198	D₆₇	—	8	2	no	—
199	D₆₇	—	16	2	no	—
200	#3391	—	67	1	no	—
201	#3380	—	67	1	no	—
202	#3333	—	67	2	no	—
203	#1706	—	67	1	no	—
204	#1697	—	67	2	no	—
205	#1676	—	67	1	no	—
206	#1668	—	67	1	no	—
207	#1631	—	67	2	no	—
208	#1622	—	67	2	no	—
209	#1589	—	67	1	no	—
210	C₈×Q₁₆	D₆₇	1	1	yes	characteristic, complemented
211	C₆₇	#19909	1	1	yes	characteristic, complemented, Sylow
212	#554	—	67	2	no	—
213	#550	—	67	2	no	—
214	#500	—	67	1	no	—
215	#499	—	134	1	no	—
216	#493	—	67	1	no	—
217	#493	—	134	1	no	—
218	#492	—	67	2	no	—
219	#486	—	67	2	no	—
220	#480	—	134	2	no	—
221	#478	—	134	2	no	—
222	#477	—	67	2	no	—
223	#476	—	67	2	no	—
224	#468	—	134	2	no	—
225	#460	—	67	2	no	—
226	#459	—	67	2	no	—
227	#459	—	134	2	no	—
228	#452	—	134	2	no	—
229	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
230	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
231	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
232	C₄×D₈	—	67	1	no	—
233	C₄×D₈	—	134	1	no	—
234	C₄×D₈	—	134	1	no	—
235	#436	—	67	2	no	—
236	#417	—	134	2	no	—
237	#335	—	67	1	no	—
238	#335	C₂×D₆₇	1	1	yes	characteristic
239	#333	—	67	2	no	—
240	#331	—	67	2	no	—
241	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
242	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
243	D₄⋊C₈	—	67	2	no	—
244	C₈²	C₂×D₆₇	1	1	yes	characteristic
245	C₂×#41	—	134	2	no	—
246	C₂×Q₁₆	D₂₆₈	1	1	yes	characteristic
247	C₂×SD₁₆	—	268	2	no	—
248	C₂×SD₁₆	—	134	2	no	—
249	C₂×D₈	—	67	1	no	—
250	C₂×D₈	—	134	2	no	—
251	C₂×D₈	—	134	1	no	—
252	C₂×D₈	—	134	1	no	—
253	C₂×M₁₆	—	134	2	no	—
254	C₄⋊Q₈	—	67	2	no	—
255	#127	—	67	2	no	—
256	#127	—	134	2	no	—
257	#125	—	134	2	no	—
258	C₂²⋊Q₈	—	268	2	no	—
259	C₂²⋊Q₈	—	134	2	no	—
260	#121	—	268	2	no	—
261	#121	—	134	2	no	—
262	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
263	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
264	C₄×D₄	—	67	2	no	—
265	C₄×D₄	—	134	2	no	—
266	C₄×D₄	—	134	2	no	—
267	C₂×C₄⋊C₄	—	134	2	no	—
268	#107	—	67	1	no	—
269	#107	—	134	1	no	—
270	#107	—	134	1	no	—
271	#107	—	134	2	no	—
272	#107	D₂₆₈	1	1	yes	characteristic
273	#106	—	134	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	67	2	no	—
277	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
278	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
279	#103	—	134	2	no	—
280	#103	D₂₆₈	1	1	yes	characteristic
281	#103	D₂₆₈	1	1	yes	characteristic
282	#102	—	67	2	no	—
283	#102	—	134	2	no	—
284	#102	—	134	2	no	—
285	#102	—	134	2	no	—
286	#102	—	134	2	no	—
287	#102	—	134	2	no	—
288	C₂²⋊C₈	—	134	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
294	#41	—	268	4	no	—
295	#41	—	536	2	no	—
296	C₂×Q₈	—	134	2	no	—
297	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
298	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
299	C₂×D₄	—	268	2	no	—
300	C₂×D₄	—	268	2	no	—
301	C₂×D₄	—	67	2	no	—
302	C₂×D₄	—	134	2	no	—
303	C₂×D₄	—	134	2	no	—
304	C₂×D₄	—	134	2	no	—
305	C₂²×C₄	—	268	2	no	—
306	C₂²×C₄	—	268	2	no	—
307	C₂²×C₄	—	134	2	no	—
308	Q₁₆	—	2	2	no	—
309	SD₁₆	—	536	4	no	—
310	SD₁₆	—	536	2	no	—
311	D₈	—	268	2	no	—
312	D₈	—	536	1	no	—
313	D₈	—	536	2	no	—
314	D₈	—	134	2	no	—
315	M₁₆	—	536	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	134	2	no	—
320	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
321	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
322	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
323	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	268	2	no	—
327	C₄⋊C₄	—	268	2	no	—
328	C₄⋊C₄	—	67	2	no	—
329	C₄⋊C₄	—	134	2	no	—
330	C₄⋊C₄	—	134	2	no	—
331	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
332	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
333	C₂²⋊C₄	—	268	2	no	—
334	C₂²⋊C₄	—	268	2	no	—
335	C₂²⋊C₄	—	134	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	C₂³×D₆₇	1	1	yes	characteristic
339	C₂³	—	268	2	no	—
340	C₂³	—	134	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	2	1	no	—
344	Q₈	—	536	2	no	—
345	Q₈	#14524065	1	2	yes	—
346	D₄	—	268	2	no	—
347	D₄	—	268	2	no	—
348	D₄	—	268	2	no	—
349	D₄	—	268	2	no	—
350	D₄	—	536	4	no	—
351	D₄	—	536	2	no	—
352	D₄	—	1072	2	no	—
353	D₄	—	134	2	no	—
354	D₄	—	134	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	268	2	no	—
360	C₂×C₄	—	268	2	no	—
361	C₂×C₄	—	268	2	no	—
362	C₂×C₄	—	536	4	no	—
363	C₂×C₄	—	1072	2	no	—
364	C₂×C₄	—	134	2	no	—
365	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
367	C₂×C₄	#14524089	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	268	2	no	—
375	C₈	#14524075	1	2	yes	—
376	C₂²	—	268	2	no	—
377	C₂²	—	268	2	no	—
378	C₂²	—	268	2	no	—
379	C₂²	—	536	2	no	—
380	C₂²	—	536	2	no	—
381	C₂²	—	536	2	no	—
382	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
383	C₄	—	2	2	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	268	2	no	—
389	C₄	—	536	2	no	—
390	C₄	C₂×#14524065	1	1	yes	characteristic
391	C₄	C₂×#14524075	1	1	yes	characteristic
392	C₄	#16864201	1	1	yes	characteristic
393	C₄	#16864203	1	1	yes	characteristic
394	C₂	—	536	2	no	—
395	C₂	—	1072	2	no	—
396	C₂	#22571876	1	1	yes	characteristic
397	C₂	#22571896	1	1	yes	characteristic
398	C₂	#22572009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
prime radical	#22573184	C₂
hypercenter	C₈×Q₁₆	D₆₇
hypocenter	C₆₇	#19909
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22573077	C₂	semidirect
3. #22572549	C₂	semidirect
4. #22559801	C₂	semidirect
5. #22559743	C₂	semidirect
6. #22559613	C₂	semidirect
7. #22559569	C₂	semidirect
8. #22559338	C₂	semidirect
9. #22559284	C₂	semidirect
10. #22559095	C₂	semidirect
11. C₈×Q₁₆×C₆₇	C₂	semidirect
210. C₈×Q₁₆	D₆₇	semidirect
211. C₆₇	#19909	semidirect

Maximal subgroups

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

subgroup	index
#22573184	2
#22573077	2
#22572549	2
#22559801	2
#22559743	2
#22559613	2
#22559569	2
#22559338	2
#22559284	2
#22559095	2
C₈×Q₁₆×C₆₇	2
#19909	67