Subgroups — #30755696

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863770	C₂	1	1	yes	maximal, characteristic
2	#22561171	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561015	C₂	1	1	yes	maximal, characteristic
4	#22561014	C₂	1	1	yes	maximal, characteristic
5	#22560368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761615	—	2	2	no	—
9	C₂×#14524040	C₂²	1	1	yes	characteristic
10	#16863770	—	2	2	no	—
11	#16863770	C₄	1	4	yes	complemented
12	C₂×#14523976	—	2	2	no	—
13	C₂×#14523965	—	2	2	no	—
14	C₂×#14523963	—	2	2	no	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	#16863266	C₂²	1	1	yes	characteristic
18	#16863158	—	2	2	no	—
19	#16863118	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#384×C₆₇	C₂²	1	1	yes	characteristic
23	#340×C₆₇	—	2	2	no	—
24	C₂²×D₂₆₈	—	2	1	no	—
25	C₂²×D₂₆₈	—	2	1	no	—
26	C₂²×D₂₆₈	—	2	1	no	—
27	C₂×#11761615	—	2	8	no	—
28	C₂×#11761615	—	4	4	no	—
29	C₂×#11761615	—	4	4	no	—
30	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
31	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
32	#14524040	—	2	1	no	—
33	#14524040	—	2	1	no	—
34	#14524040	—	2	1	no	—
35	#14524040	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₆₇	—	2	1	no	—
37	C₂×C₄⋊Dic₆₇	—	2	1	no	—
38	C₂×C₄⋊Dic₆₇	—	2	1	no	—
39	#14523976	—	4	4	no	—
40	#14523965	—	4	4	no	—
41	#14523963	—	4	4	no	—
42	C₄×C₆₇⋊C₈	—	2	1	no	—
43	C₄×C₆₇⋊C₈	—	2	1	no	—
44	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
45	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
46	C₂²×Q₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	D₄	1	2	yes	—
48	C₄⋊Q₈×C₆₇	—	2	1	no	—
49	C₄⋊Q₈×C₆₇	—	2	1	no	—
50	C₄⋊Q₈×C₆₇	—	2	1	no	—
51	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	C₂³×D₆₇	—	4	1	no	—
61	C₂×D₂₆₈	—	2	2	no	—
62	C₂×D₂₆₈	—	2	2	no	—
63	C₂×D₂₆₈	—	2	2	no	—
64	C₂×D₂₆₈	—	4	2	no	—
65	C₂×D₂₆₈	—	4	2	no	—
66	C₂×D₂₆₈	—	4	2	no	—
67	C₂×D₂₆₈	—	4	2	no	—
68	C₂×D₂₆₈	—	4	2	no	—
69	C₂×D₂₆₈	—	4	2	no	—
70	C₂×D₂₆₈	—	4	1	no	—
71	C₂×D₂₆₈	—	4	1	no	—
72	C₂×D₂₆₈	—	4	1	no	—
73	C₂²×Dic₆₇	—	4	1	no	—
74	#11761615	—	4	16	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	4	yes	—
79	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
80	C₂×C₆₇⋊C₈	SD₁₆	1	4	yes	—
81	C₄⋊Dic₆₇	—	4	2	no	—
82	C₄⋊Dic₆₇	—	4	2	no	—
83	C₄⋊Dic₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	4	1	no	—
86	C₂×Q₈×C₆₇	—	4	2	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	4	1	no	—
99	C₄⋊C₄×C₆₇	—	4	1	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₄⋊C₄×C₆₇	—	4	2	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₇	#41	1	1	yes	characteristic
106	C₄²×C₆₇	#41	1	1	yes	characteristic
107	C₂³×C₆₇	#127	1	1	yes	characteristic
108	C₂²×D₆₇	—	4	2	no	—
109	C₂²×D₆₇	—	8	1	no	—
110	C₂²×D₆₇	—	8	1	no	—
111	C₂²×D₆₇	—	8	1	no	—
112	C₂²×D₆₇	—	8	1	no	—
113	C₂²×D₆₇	—	8	1	no	—
114	C₂²×D₆₇	—	8	1	no	—
115	Q₈×C₆₇	—	4	4	no	—
116	Q₈×C₆₇	—	8	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	8	1	no	—
119	C₂×C₄×C₆₇	—	8	1	no	—
120	C₂×C₄×C₆₇	—	8	1	no	—
121	C₂×C₄×C₆₇	—	8	1	no	—
122	C₂×C₄×C₆₇	—	8	1	no	—
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₆₇	#137	1	2	yes	—
134	C₂×C₄×C₆₇	#137	1	2	yes	—
135	C₂×Dic₆₇	—	8	1	no	—
136	C₂×Dic₆₇	—	8	1	no	—
137	C₂×Dic₆₇	—	8	1	no	—
138	D₂₆₈	—	4	4	no	—
139	D₂₆₈	—	4	4	no	—
140	D₂₆₈	—	4	4	no	—
141	D₂₆₈	—	8	2	no	—
142	D₂₆₈	—	8	2	no	—
143	D₂₆₈	—	8	2	no	—
144	C₈×C₆₇	—	4	2	no	—
145	C₆₇⋊C₈	—	4	2	no	—
146	C₆₇⋊C₈	#102	1	8	yes	complemented
147	C₂²×C₆₇	#390	1	1	yes	characteristic
148	C₂²×C₆₇	#492	1	1	yes	characteristic
149	C₂²×C₆₇	#493	1	1	yes	characteristic
150	C₂²×C₆₇	#496	1	1	yes	characteristic
151	C₂²×C₆₇	#496	1	1	yes	characteristic
152	C₂²×C₆₇	#497	1	1	yes	characteristic
153	C₂²×C₆₇	#497	1	1	yes	characteristic
154	C₂×D₆₇	—	8	2	no	—
155	C₂×D₆₇	—	8	2	no	—
156	C₂×D₆₇	—	8	2	no	—
157	C₂×D₆₇	—	16	1	no	—
158	C₂×D₆₇	—	16	1	no	—
159	C₂×D₆₇	—	16	1	no	—
160	C₂×D₆₇	—	16	1	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	—	16	1	no	—
163	C₄×C₆₇	C₂×#102	1	2	yes	—
164	C₄×C₆₇	#440	1	2	yes	—
165	C₄×C₆₇	#440	1	2	yes	—
166	C₄×C₆₇	#468	1	2	yes	—
167	C₄×C₆₇	#468	1	2	yes	—
168	C₄×C₆₇	#486	1	2	yes	—
169	Dic₆₇	—	16	1	no	—
170	#9842	—	67	1	no	maximal, Sylow
171	C₂×C₆₇	#1714	1	1	yes	characteristic
172	C₂×C₆₇	#1714	1	1	yes	characteristic
173	C₂×C₆₇	#1721	1	1	yes	characteristic
174	C₂×C₆₇	#1722	1	1	yes	characteristic
175	C₂×C₆₇	#1957	1	1	yes	characteristic
176	C₂×C₆₇	#1973	1	1	yes	characteristic
177	C₂×C₆₇	#1973	1	1	yes	characteristic
178	D₆₇	—	16	2	no	—
179	C₂×#492	—	67	1	no	—
180	#1990	—	67	1	no	—
181	#1990	D₆₇	1	1	yes	characteristic, complemented
182	#1957	—	67	1	no	—
183	#1957	—	67	1	no	—
184	#1845	—	67	1	no	—
185	C₂×#322	—	67	1	no	—
186	C₆₇	#9842	1	1	yes	characteristic, complemented, Sylow
187	C₂²×SD₁₆	—	134	2	no	—
188	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
189	C₂×#127	—	67	1	no	—
190	#492	—	67	4	no	—
191	#492	—	134	2	no	—
192	C₂×#107	—	134	2	no	—
193	C₂×#102	—	134	2	no	—
194	C₂×#102	—	134	2	no	—
195	C₂×C₄×C₈	—	67	1	no	—
196	C₂×C₄×C₈	—	67	1	no	—
197	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
198	#384	—	67	1	no	—
199	#384	C₂×D₆₇	1	1	yes	characteristic
200	#340	—	2	2	no	—
201	#340	—	134	2	no	—
202	#322	—	134	2	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	134	1	no	—
205	C₂²×D₄	—	134	1	no	—
206	C₂²×D₄	—	134	1	no	—
207	C₂×SD₁₆	—	268	4	no	—
208	C₂×SD₁₆	—	268	4	no	—
209	C₂×SD₁₆	—	134	8	no	—
210	C₂²×C₈	—	67	2	no	—
211	C₂²×C₈	—	67	2	no	—
212	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	C₄×D₆₇	1	2	yes	—
217	#127	—	67	2	no	—
218	#127	—	134	1	no	—
219	#127	—	134	1	no	—
220	#127	—	134	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	134	1	no	—
227	C₂×C₄⋊C₄	—	134	1	no	—
228	C₂×C₄⋊C₄	—	134	1	no	—
229	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
230	#107	—	268	4	no	—
231	#102	—	268	4	no	—
232	#102	—	268	4	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	67	2	no	—
236	C₄×C₈	—	67	2	no	—
237	C₄×C₈	—	134	1	no	—
238	C₄×C₈	—	134	1	no	—
239	C₂⁴	—	268	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×D₄	—	268	1	no	—
245	C₂×D₄	—	268	1	no	—
246	C₂×D₄	—	268	1	no	—
247	C₂×D₄	—	268	2	no	—
248	C₂×D₄	—	268	2	no	—
249	C₂×D₄	—	268	2	no	—
250	C₂×D₄	—	268	2	no	—
251	C₂×D₄	—	268	2	no	—
252	C₂×D₄	—	268	2	no	—
253	C₂×D₄	—	134	2	no	—
254	C₂×D₄	—	134	2	no	—
255	C₂×D₄	—	134	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	268	1	no	—
259	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
262	SD₁₆	—	268	16	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	67	4	no	—
267	C₂×C₈	—	67	4	no	—
268	C₂×C₈	—	67	4	no	—
269	C₂×C₈	—	134	2	no	—
270	C₂×C₈	—	134	2	no	—
271	C₂×C₈	—	134	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	268	2	no	—
280	C₄⋊C₄	—	268	2	no	—
281	C₄⋊C₄	—	268	2	no	—
282	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
283	C₄²	D₄×D₆₇	1	1	yes	characteristic
284	C₄²	#11761627	1	1	yes	characteristic
285	C₄²	#11761628	1	1	yes	characteristic
286	C₂³	—	268	2	no	—
287	C₂³	—	536	1	no	—
288	C₂³	—	536	1	no	—
289	C₂³	—	536	1	no	—
290	C₂³	—	536	1	no	—
291	C₂³	—	536	1	no	—
292	C₂³	—	536	1	no	—
293	C₂³	#14524039	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	268	4	no	—
297	D₄	—	268	4	no	—
298	D₄	—	268	4	no	—
299	D₄	—	536	2	no	—
300	D₄	—	536	2	no	—
301	D₄	—	536	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	536	1	no	—
309	C₂×C₄	—	536	1	no	—
310	C₂×C₄	—	536	1	no	—
311	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
312	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
313	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
314	C₂×C₄	#14524015	1	1	yes	characteristic
315	C₂×C₄	#14524016	1	1	yes	characteristic
316	C₂×C₄	#14524029	1	1	yes	characteristic
317	C₂×C₄	D₈×D₆₇	1	2	yes	—
318	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
319	C₂×C₄	#14524068	1	2	yes	—
320	C₂×C₄	#14524070	1	2	yes	—
321	C₂×C₄	#14524074	1	2	yes	—
322	C₂×C₄	#14524075	1	2	yes	—
323	C₈	—	4	2	no	—
324	C₈	—	268	2	no	—
325	C₈	—	67	8	no	—
326	C₂²	—	536	2	no	—
327	C₂²	—	536	2	no	—
328	C₂²	—	536	2	no	—
329	C₂²	—	1072	1	no	—
330	C₂²	—	1072	1	no	—
331	C₂²	—	1072	1	no	—
332	C₂²	—	1072	1	no	—
333	C₂²	#16863296	1	1	yes	characteristic
334	C₂²	#16863767	1	1	yes	characteristic
335	C₂²	#16863771	1	1	yes	characteristic
336	C₂²	#16863782	1	1	yes	characteristic
337	C₂²	#16863786	1	1	yes	characteristic
338	C₂²	#16863788	1	1	yes	characteristic
339	C₂²	#16863789	1	1	yes	characteristic
340	C₄	—	8	2	no	—
341	C₄	—	16	1	no	—
342	C₄	—	1072	1	no	—
343	C₄	#102×D₆₇	1	2	yes	—
344	C₄	#16863494	1	2	yes	—
345	C₄	#16863499	1	2	yes	—
346	C₄	#16863644	1	2	yes	—
347	C₄	#16863645	1	2	yes	—
348	C₄	#16863741	1	2	yes	—
349	C₂	—	1072	2	no	—
350	C₂	#22559833	1	1	yes	characteristic
351	C₂	#22559834	1	1	yes	characteristic
352	C₂	#22559877	1	1	yes	characteristic
353	C₂	#22559879	1	1	yes	characteristic
354	C₂	#22561013	1	1	yes	characteristic
355	C₂	#22561087	1	1	yes	characteristic
356	C₂	#22561093	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1990	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1990	D₆₇
hypocenter	C₆₇	#9842
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. #22561171	C₂	semidirect
5. #22560368	C₂	semidirect
6. C₂×#16863118	C₂	semidirect
7. C₆₇×#1990	C₂	semidirect
11. #16863770	C₄	semidirect
146. C₆₇⋊C₈	#102	semidirect
181. #1990	D₆₇	semidirect
186. C₆₇	#9842	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863770	2
#22561171	2
#22561015	2
#22561014	2
#22560368	2
C₂×#16863118	2
C₆₇×#1990	2
#9842	67