Subgroups — #30678643

Order 16768.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554039	C₂	1	1	yes	maximal, characteristic, complemented
2	#22554031	C₂	1	1	yes	maximal, characteristic
3	#22553665	C₂	1	1	yes	maximal, characteristic
4	#22553661	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553621	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553616	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553590	C₂	1	1	yes	maximal, characteristic
8	#22553533	C₂	1	1	yes	maximal, characteristic
9	#22553531	C₂	1	1	yes	maximal, characteristic
10	#22553521	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Dic₅₂₄	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553404	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553403	C₂	1	1	yes	maximal, characteristic
15	#488×C₁₃₁	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₅₂₄	C₂²	1	1	yes	characteristic
17	C₂×#14523503	C₂²	1	1	yes	characteristic
18	C₂×#14523501	C₂²	1	1	yes	characteristic
19	#16862245	—	2	1	no	—
20	#16862245	C₂²	1	1	yes	characteristic
21	#16862244	C₂²	1	1	yes	characteristic
22	#16862239	—	2	1	no	—
23	#16862239	—	2	1	no	—
24	#16862237	C₂²	1	1	yes	characteristic
25	#16862236	—	2	1	no	—
26	#16862236	—	2	1	no	—
27	#16862233	C₂²	1	1	yes	characteristic
28	#16862223	—	2	1	no	—
29	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
33	#16862211	—	2	1	no	—
34	#16862211	C₂²	1	1	yes	characteristic
35	#16862211	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
37	#16862202	—	2	1	no	—
38	#16862180	C₂²	1	1	yes	characteristic
39	#16862179	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
44	#16862171	C₂²	1	1	yes	characteristic
45	#16862171	C₂²	1	1	yes	characteristic
46	#16862170	C₂²	1	1	yes	characteristic
47	#16862169	C₂²	1	1	yes	characteristic
48	#16862169	C₂²	1	1	yes	characteristic
49	#16862167	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	#125×C₁₃₁	C₂²	1	1	yes	characteristic
53	#124×C₁₃₁	C₂²	1	1	yes	characteristic
54	#103×C₁₃₁	C₂²	1	1	yes	characteristic
55	#103×C₁₃₁	C₂²	1	1	yes	characteristic
56	#102×C₁₃₁	C₂²	1	1	yes	characteristic
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₃₁	—	2	1	no	—
64	Dic₅₂₄	—	2	2	no	—
65	#14523503	—	2	2	no	—
66	#14523501	—	2	2	no	—
67	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₃₁	—	2	1	no	—
70	C₄⋊Dic₁₃₁	—	2	1	no	—
71	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
74	Dic₁₃₁⋊C₄	—	2	1	no	—
75	Dic₁₃₁⋊C₄	—	2	1	no	—
76	Dic₁₃₁⋊C₄	—	2	1	no	—
77	Dic₁₃₁⋊C₄	—	2	1	no	—
78	Dic₁₃₁⋊C₄	—	2	1	no	—
79	Dic₁₃₁⋊C₄	—	2	1	no	—
80	Dic₁₃₁⋊C₄	—	2	1	no	—
81	Dic₁₃₁⋊C₄	—	2	1	no	—
82	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
83	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₃₁	—	2	1	no	—
85	C₂²⋊Dic₁₃₁	—	2	1	no	—
86	C₂²⋊Dic₁₃₁	—	2	1	no	—
87	C₄×Dic₁₃₁	—	2	1	no	—
88	C₄×Dic₁₃₁	—	2	1	no	—
89	C₄×Dic₁₃₁	—	2	1	no	—
90	C₄×Dic₁₃₁	—	2	1	no	—
91	C₄×Dic₁₃₁	—	2	1	no	—
92	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₃₁	—	2	1	no	—
99	C₄⋊C₄×C₁₃₁	—	2	1	no	—
100	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
104	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
105	Dic₂₆₂	—	2	1	no	—
106	Dic₂₆₂	—	2	2	no	—
107	Dic₂₆₂	—	2	1	no	—
108	Dic₂₆₂	—	2	1	no	—
109	Dic₂₆₂	—	4	2	no	—
110	Dic₂₆₂	#41	1	2	yes	—
111	C₂×Dic₁₃₁	—	2	1	no	—
112	C₂×Dic₁₃₁	—	2	1	no	—
113	C₂×Dic₁₃₁	—	2	1	no	—
114	C₂×Dic₁₃₁	—	2	1	no	—
115	C₂×Dic₁₃₁	—	4	2	no	—
116	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₁₃₁⋊C₈	—	2	1	no	—
121	C₁₃₁⋊C₈	—	2	1	no	—
122	C₂³×C₁₃₁	—	2	1	no	—
123	Q₈×C₁₃₁	—	2	1	no	—
124	Q₈×C₁₃₁	—	2	1	no	—
125	D₄×C₁₃₁	—	2	1	no	—
126	D₄×C₁₃₁	—	2	1	no	—
127	C₂×C₄×C₁₃₁	—	2	1	no	—
128	C₂×C₄×C₁₃₁	—	2	1	no	—
129	C₂×C₄×C₁₃₁	—	2	1	no	—
130	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₃₁	—	2	1	no	—
134	C₈×C₁₃₁	—	2	1	no	—
135	C₂²×C₁₃₁	—	4	1	no	—
136	C₂²×C₁₃₁	—	4	1	no	—
137	C₂²×C₁₃₁	—	4	1	no	—
138	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₃₁	—	2	1	no	—
140	C₄×C₁₃₁	—	2	1	no	—
141	C₄×C₁₃₁	—	4	1	no	—
142	C₄×C₁₃₁	—	4	1	no	—
143	C₄×C₁₃₁	—	4	1	no	—
144	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
145	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₁₃₁	—	2	2	no	—
147	Dic₁₃₁	—	2	1	no	—
148	Dic₁₃₁	—	2	1	no	—
149	Dic₁₃₁	—	2	1	no	—
150	Dic₁₃₁	—	4	1	no	—
151	Dic₁₃₁	—	4	1	no	—
152	Dic₁₃₁	—	4	1	no	—
153	Dic₁₃₁	#135	1	2	yes	—
154	C₂×C₁₃₁	—	8	1	no	—
155	C₂×C₁₃₁	#548	1	1	yes	characteristic
156	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
157	C₂×C₁₃₁	#578	1	1	yes	characteristic
158	C₁₃₁	#3364	1	1	yes	characteristic, complemented, Sylow
159	#3364	—	131	1	no	maximal, Sylow
160	#557	—	131	1	no	—
161	#556	—	131	1	no	—
162	#488	—	131	1	no	—
163	#488	D₁₃₁	1	1	yes	characteristic
164	#487	—	131	1	no	—
165	#479	—	131	1	no	—
166	#479	—	131	1	no	—
167	#474	—	131	1	no	—
168	#464	—	131	1	no	—
169	#464	—	131	1	no	—
170	#462	—	131	1	no	—
171	C₈×Q₈	—	131	1	no	—
172	C₄×Q₁₆	—	131	1	no	—
173	C₄×Q₁₆	—	131	1	no	—
174	C₄×SD₁₆	—	131	1	no	—
175	C₂×Q₁₆	—	131	1	no	—
176	C₂×Q₁₆	—	131	1	no	—
177	C₂×SD₁₆	—	131	1	no	—
178	C₄⋊Q₈	—	262	1	no	—
179	C₄⋊Q₈	—	131	1	no	—
180	C₄⋊Q₈	—	131	1	no	—
181	#126	—	262	1	no	—
182	#126	—	262	1	no	—
183	#125	—	262	1	no	—
184	#125	—	262	1	no	—
185	#125	—	131	1	no	—
186	#125	C₂×D₁₃₁	1	1	yes	characteristic
187	#124	—	131	1	no	—
188	#124	C₂×D₁₃₁	1	1	yes	characteristic
189	C₂²⋊Q₈	—	262	1	no	—
190	C₄×Q₈	—	262	1	no	—
191	C₄×Q₈	—	131	1	no	—
192	C₄×Q₈	—	131	1	no	—
193	C₄×Q₈	—	131	1	no	—
194	C₄×Q₈	—	131	1	no	—
195	C₄×Q₈	—	131	1	no	—
196	C₄×Q₈	—	131	1	no	—
197	C₄×D₄	—	131	1	no	—
198	#117	—	262	1	no	—
199	#107	—	131	1	no	—
200	#107	—	131	1	no	—
201	#106	—	131	1	no	—
202	C₄⋊C₈	—	131	1	no	—
203	C₄⋊C₈	—	131	1	no	—
204	C₄⋊C₈	—	131	1	no	—
205	#103	—	131	1	no	—
206	#103	—	131	1	no	—
207	#103	—	131	1	no	—
208	#103	—	131	1	no	—
209	#103	—	131	1	no	—
210	#103	C₂×D₁₃₁	1	1	yes	characteristic
211	#103	C₂×D₁₃₁	1	1	yes	characteristic
212	#102	—	131	1	no	—
213	#102	C₂×D₁₃₁	1	1	yes	characteristic
214	#102	C₂×D₁₃₁	1	1	yes	characteristic
215	C₄×C₈	—	131	1	no	—
216	C₄×C₈	—	131	1	no	—
217	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
218	C₂×Q₈	—	262	1	no	—
219	C₂×Q₈	—	131	1	no	—
220	C₂×Q₈	—	131	1	no	—
221	C₂×Q₈	—	131	1	no	—
222	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
223	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
224	C₂²×C₄	—	262	1	no	—
225	Q₁₆	—	262	2	no	—
226	Q₁₆	—	262	2	no	—
227	SD₁₆	—	262	2	no	—
228	C₂×C₈	—	131	1	no	—
229	C₂×C₈	—	131	1	no	—
230	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	262	1	no	—
235	C₄⋊C₄	—	262	1	no	—
236	C₄⋊C₄	—	262	1	no	—
237	C₄⋊C₄	—	262	1	no	—
238	C₄⋊C₄	—	262	1	no	—
239	C₄⋊C₄	—	262	1	no	—
240	C₄⋊C₄	—	262	1	no	—
241	C₄⋊C₄	—	262	1	no	—
242	C₄⋊C₄	—	262	1	no	—
243	C₄⋊C₄	—	262	1	no	—
244	C₄⋊C₄	—	131	1	no	—
245	C₄⋊C₄	—	131	1	no	—
246	C₄⋊C₄	—	131	1	no	—
247	C₄⋊C₄	—	131	1	no	—
248	C₄⋊C₄	—	131	1	no	—
249	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	262	1	no	—
254	C₂²⋊C₄	—	262	1	no	—
255	C₂²⋊C₄	—	262	1	no	—
256	C₄²	—	262	1	no	—
257	C₄²	—	262	1	no	—
258	C₄²	—	262	1	no	—
259	C₄²	—	262	1	no	—
260	C₄²	—	262	1	no	—
261	C₄²	—	131	1	no	—
262	C₄²	—	131	1	no	—
263	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	262	2	no	—
268	Q₈	—	262	1	no	—
269	Q₈	—	262	1	no	—
270	Q₈	—	262	1	no	—
271	Q₈	—	524	2	no	—
272	Q₈	—	131	2	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	262	1	no	—
279	C₂×C₄	—	262	1	no	—
280	C₂×C₄	—	262	1	no	—
281	C₂×C₄	—	262	1	no	—
282	C₂×C₄	—	524	2	no	—
283	C₂×C₄	—	131	1	no	—
284	C₂×C₄	—	131	1	no	—
285	C₂×C₄	—	131	1	no	—
286	C₂×C₄	—	131	1	no	—
287	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
288	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	262	1	no	—
293	C₈	—	262	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	262	1	no	—
304	C₄	—	262	1	no	—
305	C₄	—	262	2	no	—
306	C₄	—	262	1	no	—
307	C₄	—	524	1	no	—
308	C₄	—	524	1	no	—
309	C₄	—	524	1	no	—
310	C₄	—	131	2	no	—
311	C₄	#41×D₁₃₁	1	1	yes	characteristic
312	C₄	#16862293	1	1	yes	characteristic
313	C₂	—	8	1	no	—
314	C₂	#22553970	1	1	yes	characteristic
315	C₂	#135×D₁₃₁	1	1	yes	characteristic
316	C₂	#22554126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#488×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×D₄×C₁₃₁	C₂³
hypercenter	#488	D₁₃₁
hypocenter	C₁₃₁	#3364
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #22554039	C₂	semidirect
4. #22553661	C₂	semidirect
5. #22553621	C₂	semidirect
6. #22553616	C₂	semidirect
10. #22553521	C₂	semidirect
11. C₈×Dic₂₆₂	C₂	semidirect
12. C₄×Dic₅₂₄	C₂	semidirect
13. #22553404	C₂	semidirect
158. C₁₃₁	#3364	semidirect

Maximal subgroups

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

subgroup	index
#22554039	2
#22554031	2
#22553665	2
#22553661	2
#22553621	2
#22553616	2
#22553590	2
#22553533	2
#22553531	2
#22553521	2
C₈×Dic₂₆₂	2
C₄×Dic₅₂₄	2
#22553404	2
#22553403	2
#488×C₁₃₁	2
#3364	131