Subgroups — #30755674

Order 17152.

562 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863743	C₂	1	1	yes	maximal, characteristic
2	#22561170	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561015	C₂	1	1	yes	maximal, characteristic
4	#22561015	C₂	1	1	yes	maximal, characteristic
5	#22561011	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560337	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524043	C₂²	1	1	yes	characteristic
9	C₂×#14524040	C₂²	1	1	yes	characteristic
10	#16863743	—	2	2	no	—
11	#16863743	C₄	1	4	yes	complemented
12	C₂×#14523968	—	2	2	no	—
13	C₂×#14523965	—	2	2	no	—
14	C₂×#14523965	—	2	2	no	—
15	C₂×#14523965	—	2	2	no	—
16	#16863266	C₂²	1	1	yes	characteristic
17	#16863266	C₂²	1	1	yes	characteristic
18	#16863159	—	2	2	no	—
19	#16863159	—	2	2	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#322×C₆₇	—	2	2	no	—
24	C₂²×Dic₁₃₄	—	2	1	no	—
25	C₂²×D₂₆₈	—	2	1	no	—
26	C₂²×D₂₆₈	—	2	1	no	—
27	C₂²×D₂₆₈	—	2	1	no	—
28	#14524043	—	2	1	no	—
29	#14524043	—	2	1	no	—
30	#14524043	—	2	1	no	—
31	#14524043	C₂×C₄	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	C₂×C₄⋊Dic₆₇	—	2	1	no	—
40	C₂×C₄⋊Dic₆₇	—	2	1	no	—
41	C₂×C₄⋊Dic₆₇	—	2	1	no	—
42	C₂×C₄⋊Dic₆₇	—	2	1	no	—
43	C₂×C₄⋊Dic₆₇	—	2	1	no	—
44	#14523968	—	4	4	no	—
45	#14523965	—	2	8	no	—
46	#14523965	—	4	4	no	—
47	#14523965	—	4	4	no	—
48	C₂²×C₈×C₆₇	D₄	1	2	yes	—
49	C₂²×C₈×C₆₇	D₄	1	2	yes	—
50	C₂²×C₈×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₆₇	—	2	1	no	—
53	C₄×C₈×C₆₇	—	2	1	no	—
54	C₄×C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₆₇	D₄	1	2	yes	—
58	C₂³×D₆₇	—	4	1	no	—
59	C₂×Dic₁₃₄	—	2	2	no	—
60	C₂×Dic₁₃₄	—	4	2	no	—
61	C₂×Dic₁₃₄	—	4	2	no	—
62	C₂×Dic₁₃₄	—	4	1	no	—
63	C₂×D₂₆₈	—	2	2	no	—
64	C₂×D₂₆₈	—	2	2	no	—
65	C₂×D₂₆₈	—	2	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	2	no	—
71	C₂×D₂₆₈	—	4	2	no	—
72	C₂×D₂₆₈	—	4	1	no	—
73	C₂×D₂₆₈	—	4	1	no	—
74	C₂×D₂₆₈	—	4	1	no	—
75	C₂²×Dic₆₇	—	4	1	no	—
76	C₂²×Dic₆₇	—	4	1	no	—
77	C₂²×Dic₆₇	—	4	1	no	—
78	C₄⋊Dic₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	—	2	2	no	—
80	C₄⋊Dic₆₇	—	4	1	no	—
81	C₄⋊Dic₆₇	—	4	1	no	—
82	C₄⋊Dic₆₇	—	4	2	no	—
83	C₄⋊Dic₆₇	—	4	2	no	—
84	C₄⋊Dic₆₇	—	4	2	no	—
85	C₄⋊Dic₆₇	—	4	2	no	—
86	C₄⋊Dic₆₇	—	4	2	no	—
87	C₄⋊Dic₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	#41	1	1	yes	characteristic
103	C₄²×C₆₇	#41	1	1	yes	characteristic
104	C₂³×C₆₇	#127	1	1	yes	characteristic
105	C₂²×D₆₇	—	4	2	no	—
106	C₂²×D₆₇	—	8	1	no	—
107	C₂²×D₆₇	—	8	1	no	—
108	C₂²×D₆₇	—	8	1	no	—
109	C₂²×D₆₇	—	8	1	no	—
110	C₂²×D₆₇	—	8	1	no	—
111	C₂²×D₆₇	—	8	1	no	—
112	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#135	1	2	yes	—
119	C₂×C₄×C₆₇	#135	1	2	yes	—
120	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₆₇	#137	1	2	yes	—
124	C₂×Dic₆₇	—	4	2	no	—
125	C₂×Dic₆₇	—	8	1	no	—
126	C₂×Dic₆₇	—	8	1	no	—
127	C₂×Dic₆₇	—	8	1	no	—
128	C₂×Dic₆₇	—	8	1	no	—
129	C₂×Dic₆₇	—	8	1	no	—
130	C₂×Dic₆₇	—	8	1	no	—
131	C₂×Dic₆₇	—	8	1	no	—
132	C₂×Dic₆₇	—	8	1	no	—
133	D₂₆₈	—	4	4	no	—
134	D₂₆₈	—	4	4	no	—
135	D₂₆₈	—	4	4	no	—
136	D₂₆₈	—	8	2	no	—
137	D₂₆₈	—	8	2	no	—
138	D₂₆₈	—	8	2	no	—
139	Dic₁₃₄	—	4	4	no	—
140	Dic₁₃₄	—	8	2	no	—
141	C₈×C₆₇	—	2	4	no	—
142	C₈×C₆₇	—	4	2	no	—
143	C₈×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	#390	1	1	yes	characteristic
145	C₂²×C₆₇	#495	1	1	yes	characteristic
146	C₂²×C₆₇	#495	1	1	yes	characteristic
147	C₂²×C₆₇	#496	1	1	yes	characteristic
148	C₂²×C₆₇	#496	1	1	yes	characteristic
149	C₂²×C₆₇	#496	1	1	yes	characteristic
150	C₂²×C₆₇	#497	1	1	yes	characteristic
151	C₂×D₆₇	—	8	2	no	—
152	C₂×D₆₇	—	8	2	no	—
153	C₂×D₆₇	—	8	2	no	—
154	C₂×D₆₇	—	16	1	no	—
155	C₂×D₆₇	—	16	1	no	—
156	C₂×D₆₇	—	16	1	no	—
157	C₂×D₆₇	—	16	1	no	—
158	C₄×C₆₇	#416	1	2	yes	—
159	C₄×C₆₇	#440	1	2	yes	—
160	C₄×C₆₇	#442	1	2	yes	—
161	C₄×C₆₇	#468	1	2	yes	—
162	C₄×C₆₇	#469	1	2	yes	—
163	C₄×C₆₇	#488	1	2	yes	—
164	Dic₆₇	—	8	2	no	—
165	Dic₆₇	—	16	1	no	—
166	Dic₆₇	—	16	1	no	—
167	#9838	—	67	1	no	maximal, Sylow
168	C₂×C₆₇	#1717	1	1	yes	characteristic
169	C₂×C₆₇	#1717	1	1	yes	characteristic
170	C₂×C₆₇	#1719	1	1	yes	characteristic
171	C₂×C₆₇	#1720	1	1	yes	characteristic
172	C₂×C₆₇	#1958	1	1	yes	characteristic
173	C₂×C₆₇	#1972	1	1	yes	characteristic
174	C₂×C₆₇	#1973	1	1	yes	characteristic
175	D₆₇	—	16	2	no	—
176	C₂×#486	—	67	1	no	—
177	#1990	—	67	1	no	—
178	#1957	—	67	1	no	—
179	#1957	—	67	1	no	—
180	#1956	—	67	1	no	—
181	#1834	—	67	1	no	—
182	C₂×#322	D₆₇	1	1	yes	characteristic, complemented
183	C₆₇	#9838	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄⋊Q₈	—	67	1	no	—
185	C₂×#127	—	67	1	no	—
186	#486	—	67	4	no	—
187	#486	—	134	2	no	—
188	C₂×#103	—	134	2	no	—
189	C₂×#102	—	134	2	no	—
190	C₂×#102	—	134	2	no	—
191	C₂×#102	—	134	2	no	—
192	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
195	#384	—	67	1	no	—
196	#384	—	67	1	no	—
197	#340	—	134	2	no	—
198	#340	—	134	2	no	—
199	#322	—	2	2	no	—
200	C₂²×Q₈	—	134	1	no	—
201	C₂²×D₄	—	134	1	no	—
202	C₂²×D₄	—	134	1	no	—
203	C₂²×D₄	—	134	1	no	—
204	C₂²×C₈	D₂₆₈	1	2	yes	—
205	C₂²×C₈	D₂₆₈	1	2	yes	—
206	C₂²×C₈	D₂₆₈	1	2	yes	—
207	C₄⋊Q₈	—	67	2	no	—
208	C₄⋊Q₈	—	134	1	no	—
209	C₄⋊Q₈	—	134	1	no	—
210	C₄⋊Q₈	—	134	1	no	—
211	#127	—	67	2	no	—
212	#127	—	134	1	no	—
213	#127	—	134	1	no	—
214	#127	—	134	1	no	—
215	C₂×C₄⋊C₄	—	134	1	no	—
216	C₂×C₄⋊C₄	—	134	1	no	—
217	C₂×C₄⋊C₄	—	134	1	no	—
218	C₂×C₄⋊C₄	—	134	1	no	—
219	C₂×C₄⋊C₄	—	134	1	no	—
220	C₂×C₄⋊C₄	—	134	1	no	—
221	C₂×C₄⋊C₄	—	134	1	no	—
222	C₂×C₄⋊C₄	—	134	1	no	—
223	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
224	#103	—	268	4	no	—
225	#102	—	268	4	no	—
226	#102	—	268	4	no	—
227	#102	—	134	8	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	C₄×D₆₇	1	2	yes	—
233	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
234	C₂⁴	—	268	1	no	—
235	C₂×Q₈	—	268	1	no	—
236	C₂×Q₈	—	268	2	no	—
237	C₂×Q₈	—	268	2	no	—
238	C₂×Q₈	—	134	2	no	—
239	C₂×D₄	—	268	1	no	—
240	C₂×D₄	—	268	1	no	—
241	C₂×D₄	—	268	1	no	—
242	C₂×D₄	—	268	2	no	—
243	C₂×D₄	—	268	2	no	—
244	C₂×D₄	—	268	2	no	—
245	C₂×D₄	—	268	2	no	—
246	C₂×D₄	—	268	2	no	—
247	C₂×D₄	—	268	2	no	—
248	C₂×D₄	—	134	2	no	—
249	C₂×D₄	—	134	2	no	—
250	C₂×D₄	—	134	2	no	—
251	C₂²×C₄	—	268	1	no	—
252	C₂²×C₄	—	268	1	no	—
253	C₂²×C₄	—	268	1	no	—
254	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
266	C₄⋊C₄	—	268	1	no	—
267	C₄⋊C₄	—	268	1	no	—
268	C₄⋊C₄	—	268	2	no	—
269	C₄⋊C₄	—	268	2	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	134	2	no	—
275	C₄⋊C₄	—	134	2	no	—
276	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
277	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
278	C₄²	#11761628	1	1	yes	characteristic
279	C₄²	#11761628	1	1	yes	characteristic
280	C₂³	—	268	2	no	—
281	C₂³	—	536	1	no	—
282	C₂³	—	536	1	no	—
283	C₂³	—	536	1	no	—
284	C₂³	—	536	1	no	—
285	C₂³	—	536	1	no	—
286	C₂³	—	536	1	no	—
287	C₂³	#14524040	1	1	yes	characteristic
288	Q₈	—	268	4	no	—
289	Q₈	—	536	2	no	—
290	D₄	—	268	4	no	—
291	D₄	—	268	4	no	—
292	D₄	—	268	4	no	—
293	D₄	—	536	2	no	—
294	D₄	—	536	2	no	—
295	D₄	—	536	2	no	—
296	C₂×C₄	—	268	2	no	—
297	C₂×C₄	—	536	1	no	—
298	C₂×C₄	—	536	1	no	—
299	C₂×C₄	—	536	1	no	—
300	C₂×C₄	—	536	1	no	—
301	C₂×C₄	—	536	1	no	—
302	C₂×C₄	—	536	1	no	—
303	C₂×C₄	—	536	1	no	—
304	C₂×C₄	—	536	1	no	—
305	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
307	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
308	C₂×C₄	#14524017	1	1	yes	characteristic
309	C₂×C₄	#14524017	1	1	yes	characteristic
310	C₂×C₄	#14524031	1	1	yes	characteristic
311	C₂×C₄	#14524065	1	2	yes	—
312	C₂×C₄	#14524065	1	2	yes	—
313	C₂×C₄	#14524071	1	2	yes	—
314	C₂×C₄	#14524071	1	2	yes	—
315	C₂×C₄	#14524071	1	2	yes	—
316	C₂×C₄	#14524076	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	2	no	—
320	C₂²	—	536	2	no	—
321	C₂²	—	536	2	no	—
322	C₂²	—	536	2	no	—
323	C₂²	—	1072	1	no	—
324	C₂²	—	1072	1	no	—
325	C₂²	—	1072	1	no	—
326	C₂²	—	1072	1	no	—
327	C₂²	#16863298	1	1	yes	characteristic
328	C₂²	#16863780	1	1	yes	characteristic
329	C₂²	#16863780	1	1	yes	characteristic
330	C₂²	#16863785	1	1	yes	characteristic
331	C₂²	#16863785	1	1	yes	characteristic
332	C₂²	#16863785	1	1	yes	characteristic
333	C₂²	#16863790	1	1	yes	characteristic
334	C₄	—	536	2	no	—
335	C₄	—	1072	1	no	—
336	C₄	—	1072	1	no	—
337	C₄	#16863393	1	2	yes	—
338	C₄	#16863498	1	2	yes	—
339	C₄	#16863509	1	2	yes	—
340	C₄	#16863642	1	2	yes	—
341	C₄	#16863649	1	2	yes	—
342	C₄	#16863751	1	2	yes	—
343	C₂	—	1072	2	no	—
344	C₂	#22559848	1	1	yes	characteristic
345	C₂	#22559848	1	1	yes	characteristic
346	C₂	#22559862	1	1	yes	characteristic
347	C₂	#22559869	1	1	yes	characteristic
348	C₂	#22561020	1	1	yes	characteristic
349	C₂	#22561085	1	1	yes	characteristic
350	C₂	#22561090	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#9838
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #22561170	C₂	semidirect
5. #22561011	C₂	semidirect
6. #22560337	C₂	semidirect
7. C₂×#322×C₆₇	C₂	semidirect
11. #16863743	C₄	semidirect
182. C₂×#322	D₆₇	semidirect
183. C₆₇	#9838	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863743	2
#22561170	2
#22561015	2
#22561011	2
#22560337	2
C₂×#322×C₆₇	2
#9838	67