Subgroups — #30863999

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570532	C₂	1	1	yes	maximal, characteristic
2	#22569149	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568928	C₂	1	1	yes	maximal, characteristic
4	#22561190	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561041	C₂	1	1	yes	maximal, characteristic
6	#22560920	C₂	1	2	yes	maximal, complemented
7	#22560913	C₂	1	2	yes	maximal, complemented
8	#22560347	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560053	C₂	1	2	yes	maximal, complemented
10	#22560046	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	C₂²	1	2	yes	—
13	#16863953	C₂²	1	1	yes	characteristic
14	#16863946	C₂²	1	1	yes	characteristic
15	#16863751	C₄	1	4	yes	complemented
16	#16863747	C₄	1	2	yes	complemented
17	#16863743	C₄	1	2	yes	complemented
18	#16863524	—	2	4	no	—
19	#16863469	—	2	4	no	—
20	#16863422	C₂²	1	2	yes	—
21	#16863420	C₂²	1	1	yes	characteristic
22	#16863420	C₂²	1	1	yes	characteristic
23	C₂×#14523971	C₂²	1	2	yes	—
24	C₂×#14523971	C₂²	1	2	yes	—
25	#16863393	C₂²	1	2	yes	—
26	#16863393	C₂²	1	2	yes	—
27	#16863362	C₂²	1	2	yes	—
28	#16863351	C₂²	1	2	yes	—
29	#16863161	C₂²	1	2	yes	—
30	#16863161	C₂²	1	2	yes	—
31	#16863156	C₂²	1	2	yes	—
32	#16863156	C₂²	1	2	yes	—
33	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#376×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#11761628	C₂³	1	2	yes	—
41	#14524049	—	2	4	no	—
42	#14524049	—	2	4	no	—
43	C₂×#11761612	C₂³	1	2	yes	—
44	C₂×#11761612	C₂³	1	2	yes	—
45	C₂×#11761611	—	2	2	no	—
46	C₂×C₈×D₆₇	—	2	2	no	—
47	#14524043	C₂×C₄	1	1	yes	characteristic
48	#14524043	C₂×C₄	1	2	yes	—
49	#14524040	C₂×C₄	1	1	yes	characteristic
50	#14524035	C₂×C₄	1	2	yes	—
51	#14524031	C₂×C₄	1	2	yes	—
52	#14524022	—	2	2	no	—
53	#14524017	—	2	2	no	—
54	C₄×Dic₁₃₄	—	2	2	no	—
55	C₄×D₂₆₈	—	2	2	no	—
56	#14524002	C₂³	1	2	yes	—
57	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
58	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
59	Dic₆₇⋊C₈	—	2	4	no	—
60	#14523971	—	2	4	no	—
61	#14523971	—	2	4	no	—
62	#14523968	C₂×C₄	1	4	yes	—
63	#14523968	C₂×C₄	1	4	yes	—
64	#14523965	C₂×C₄	1	4	yes	—
65	#14523965	C₂×C₄	1	4	yes	—
66	(C₂×D₆₇)⋊C₈	—	2	4	no	—
67	#14523953	—	2	2	no	—
68	C₈×Dic₆₇	—	2	2	no	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#97×C₆₇	—	2	2	no	—
77	#97×C₆₇	—	2	2	no	—
78	#97×C₆₇	D₄	1	4	yes	—
79	C₄×C₈×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	—	2	2	no	—
81	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
83	#95×C₆₇	—	2	2	no	—
84	#11761628	—	2	2	no	—
85	#11761628	—	2	2	no	—
86	#11761628	—	2	2	no	—
87	#11761628	—	2	2	no	—
88	C₂×Dic₁₃₄	—	2	2	no	—
89	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
90	C₂×D₂₆₈	—	2	2	no	—
91	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
92	C₂×C₂²⋊D₆₇	—	2	2	no	—
93	C₂×C₄×D₆₇	—	2	2	no	—
94	#11761612	—	2	2	no	—
95	#11761612	—	2	2	no	—
96	#11761612	—	2	2	no	—
97	#11761612	—	2	2	no	—
98	#11761611	—	4	4	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₈×D₆₇	—	4	4	no	—
102	C₄⋊Dic₆₇	—	2	2	no	—
103	C₄⋊Dic₆₇	—	2	2	no	—
104	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
105	Dic₆₇⋊C₄	—	2	4	no	—
106	C₂²⋊Dic₆₇	—	2	2	no	—
107	(C₂×D₆₇)⋊C₄	—	2	4	no	—
108	C₄×Dic₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	#41	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₆₇	#41	1	2	yes	—
124	C₂×C₈×C₆₇	#41	1	2	yes	—
125	C₄²×C₆₇	—	2	1	no	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂×D₄	1	2	yes	—
130	C₄²×C₆₇	#41	1	2	yes	—
131	C₄²×C₆₇	#41	1	2	yes	—
132	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₆₇	—	4	2	no	—
148	C₂²⋊D₆₇	—	4	2	no	—
149	C₂×Dic₆₇	—	2	4	no	—
150	C₂×Dic₆₇	—	2	2	no	—
151	D₂₆₈	—	2	2	no	—
152	D₂₆₈	—	2	2	no	—
153	D₂₆₈	—	8	2	no	—
154	C₄×D₆₇	—	4	2	no	—
155	C₄×D₆₇	—	4	2	no	—
156	Dic₁₃₄	—	2	2	no	—
157	Dic₁₃₄	—	2	2	no	—
158	Dic₁₃₄	—	8	2	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	—	2	2	no	—
161	C₈×C₆₇	—	2	2	no	—
162	C₈×C₆₇	#124	1	4	yes	—
163	C₆₇⋊C₈	—	4	2	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₂²×C₆₇	—	2	2	no	—
166	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₆₇	#529	1	1	yes	characteristic
168	C₂²×C₆₇	#529	1	1	yes	characteristic
169	C₂×D₆₇	—	4	4	no	—
170	C₂×D₆₇	—	4	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	4	2	no	—
174	C₄×C₆₇	—	4	2	no	—
175	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
176	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
177	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
178	C₄×C₆₇	#532	1	1	yes	characteristic
179	Dic₆₇	—	4	4	no	—
180	Dic₆₇	—	4	2	no	—
181	#21659	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	—	2	1	no	—
183	C₂×C₆₇	—	2	1	no	—
184	C₂×C₆₇	C₄×#124	1	1	yes	characteristic
185	C₂×C₆₇	#2957	1	2	yes	—
186	D₆₇	—	8	2	no	—
187	#3131	—	67	1	no	—
188	#2961	—	67	1	no	—
189	#2932	—	67	1	no	—
190	#1995	—	67	1	no	—
191	#1963	—	67	1	no	—
192	#1938	—	67	2	no	—
193	#1937	—	67	2	no	—
194	#1837	—	67	1	no	—
195	#1765	—	67	2	no	—
196	#1764	—	67	2	no	—
197	#1726	D₆₇	1	1	yes	characteristic, complemented
198	C₆₇	#21659	1	1	yes	characteristic, complemented, Sylow
199	C₂×#131	—	67	2	no	—
200	#533	—	67	1	no	—
201	#532	—	67	1	no	—
202	#488	—	67	4	no	—
203	#487	—	67	2	no	—
204	#486	—	67	2	no	—
205	#446	—	134	4	no	—
206	#435	—	134	4	no	—
207	#424	—	67	2	no	—
208	#423	—	67	1	no	—
209	#423	—	67	1	no	—
210	C₂×#104	—	67	2	no	—
211	C₂×#104	—	67	2	no	—
212	#416	—	67	2	no	—
213	#416	—	67	2	no	—
214	#408	—	67	2	no	—
215	#405	—	67	2	no	—
216	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
217	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
218	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
219	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
222	#376	C₂×D₆₇	1	1	yes	characteristic
223	#341	—	67	2	no	—
224	#341	—	67	2	no	—
225	#339	—	67	2	no	—
226	#339	—	67	2	no	—
227	C₂×#41	—	67	2	no	—
228	#131	—	134	4	no	—
229	#131	—	134	4	no	—
230	C₂×M₁₆	—	67	2	no	—
231	C₂×M₁₆	—	67	2	no	—
232	C₂×M₁₆	—	134	2	no	—
233	C₂²×C₈	—	134	2	no	—
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
236	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
238	C₄⋊Q₈	—	67	2	no	—
239	C₄⋊Q₈	—	67	1	no	—
240	#127	—	67	1	no	—
241	#125	—	67	2	no	—
242	#124	—	67	2	no	—
243	C₂²⋊Q₈	—	134	2	no	—
244	#121	—	134	2	no	—
245	C₄×Q₈	—	134	2	no	—
246	C₄×D₄	—	134	2	no	—
247	#117	—	67	2	no	—
248	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
251	C₄⋊C₈	—	134	4	no	—
252	C₄⋊C₈	—	134	2	no	—
253	C₄⋊C₈	—	134	2	no	—
254	#104	—	134	4	no	—
255	#104	—	134	4	no	—
256	#103	—	67	4	no	—
257	#103	—	67	4	no	—
258	#102	—	67	4	no	—
259	#102	—	67	4	no	—
260	C₂²⋊C₈	—	134	4	no	—
261	#97	—	2	2	no	—
262	#97	—	2	2	no	—
263	#97	—	134	2	no	—
264	#97	D₂₆₈	1	4	yes	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	—	134	2	no	—
268	C₄×C₈	C₄×D₆₇	1	2	yes	—
269	C₄×C₈	C₄×D₆₇	1	2	yes	—
270	#95	—	2	2	no	—
271	#41	—	134	2	no	—
272	#41	—	134	2	no	—
273	#41	—	134	2	no	—
274	#41	—	134	2	no	—
275	C₂×Q₈	—	67	2	no	—
276	C₂×Q₈	—	134	2	no	—
277	C₂×D₄	—	67	2	no	—
278	C₂×D₄	—	134	2	no	—
279	C₂×D₄	—	134	2	no	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	134	2	no	—
283	C₂²×C₄	#11761628	1	2	yes	—
284	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
285	M₁₆	—	268	4	no	—
286	M₁₆	—	134	2	no	—
287	M₁₆	—	134	2	no	—
288	M₁₆	—	134	2	no	—
289	M₁₆	—	134	2	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	268	4	no	—
297	C₂×C₈	—	134	2	no	—
298	C₂×C₈	—	134	2	no	—
299	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
300	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
301	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
302	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
303	C₂×C₈	#11761628	1	2	yes	—
304	C₂×C₈	#11761628	1	2	yes	—
305	C₄⋊C₄	—	67	4	no	—
306	C₄⋊C₄	—	134	2	no	—
307	C₄⋊C₄	—	134	4	no	—
308	C₄⋊C₄	—	134	2	no	—
309	C₂²⋊C₄	—	134	2	no	—
310	C₂²⋊C₄	—	134	4	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	134	2	no	—
314	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
315	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
316	C₄²	C₂×D₂₆₈	1	2	yes	—
317	C₄²	#11761628	1	2	yes	—
318	C₄²	#11761628	1	2	yes	—
319	C₂³	—	134	2	no	—
320	C₂³	C₂×#11761628	1	1	yes	characteristic
321	Q₈	—	536	2	no	—
322	Q₈	—	134	2	no	—
323	Q₈	—	134	2	no	—
324	D₄	—	268	2	no	—
325	D₄	—	268	2	no	—
326	D₄	—	536	2	no	—
327	D₄	—	134	2	no	—
328	D₄	—	134	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	268	2	no	—
335	C₂×C₄	—	268	2	no	—
336	C₂×C₄	—	134	4	no	—
337	C₂×C₄	—	134	2	no	—
338	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
339	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
340	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
342	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
343	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
345	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	268	2	no	—
350	C₈	—	268	2	no	—
351	C₈	#14524031	1	4	yes	—
352	C₂²	—	2	2	no	—
353	C₂²	—	268	4	no	—
354	C₂²	—	268	2	no	—
355	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
356	C₂²	#16863930	1	1	yes	characteristic
357	C₂²	#16863930	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	268	4	no	—
363	C₄	—	268	2	no	—
364	C₄	C₄×#11761628	1	1	yes	characteristic
365	C₄	C₄×#11761628	1	1	yes	characteristic
366	C₄	C₂×#14524031	1	1	yes	characteristic
367	C₄	#16863946	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	536	2	no	—
371	C₂	C₄×#14524031	1	1	yes	characteristic
372	C₂	#22569127	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#21659
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726

Complements

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

N	H ≅ G/N	type
2. #22569149	C₂	semidirect
4. #22561190	C₂	semidirect
6. #22560920	C₂	semidirect
7. #22560913	C₂	semidirect
8. #22560347	C₂	semidirect
9. #22560053	C₂	semidirect
10. #22560046	C₂	semidirect
11. C₆₇×#1726	C₂	semidirect
15. #16863751	C₄	semidirect
16. #16863747	C₄	semidirect
17. #16863743	C₄	semidirect
197. #1726	D₆₇	semidirect
198. C₆₇	#21659	semidirect

Maximal subgroups

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

subgroup	index
#22570532	2
#22569149	2
#22568928	2
#22561190	2
#22561041	2
#22560920	2
#22560913	2
#22560347	2
#22560053	2
#22560046	2
C₆₇×#1726	2
#21659	67