Subgroups — #30676181

Order 16768.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553920	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553794	C₂	1	1	yes	maximal, characteristic
3	#22553649	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553641	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553617	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553615	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553520	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553516	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553477	C₂	1	1	yes	maximal, characteristic
10	#22553473	C₂	1	1	yes	maximal, characteristic
11	#22553417	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553410	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#14523502	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553371	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523514	C₂²	1	1	yes	characteristic
17	C₂×#14523503	C₂²	1	1	yes	characteristic
18	C₂×#14523502	C₂²	1	1	yes	characteristic
19	C₂×#14523500	C₂²	1	1	yes	characteristic
20	#16862244	C₂²	1	1	yes	characteristic
21	#16862241	—	2	1	no	—
22	#16862240	—	2	1	no	—
23	#16862238	—	2	1	no	—
24	#16862237	C₂²	1	1	yes	characteristic
25	#16862236	—	2	1	no	—
26	#16862226	C₂²	1	1	yes	characteristic
27	#16862222	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
30	C₄×D₅₂₄	—	2	1	no	—
31	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
32	#16862208	—	2	1	no	—
33	#16862208	C₂²	1	1	yes	characteristic
34	#16862203	—	2	1	no	—
35	#16862203	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₁	—	2	1	no	—
37	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
38	#16862179	C₂²	1	1	yes	characteristic
39	#16862178	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
43	#16862171	C₂²	1	1	yes	characteristic
44	#16862170	C₂²	1	1	yes	characteristic
45	#16862170	C₂²	1	1	yes	characteristic
46	#16862169	C₂²	1	1	yes	characteristic
47	#16862168	C₂²	1	1	yes	characteristic
48	#16862166	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#16862157	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
55	#107×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523514	—	2	2	no	—
60	#14523514	—	2	2	no	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×D₅₂₄	—	2	1	no	—
63	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₃₁	—	2	1	no	—
65	C₂×C₄×D₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	—	2	1	no	—
68	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
69	#14523503	—	2	2	no	—
70	#14523502	—	2	2	no	—
71	#14523500	—	2	2	no	—
72	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₁	—	2	1	no	—
75	C₄⋊Dic₁₃₁	—	2	1	no	—
76	C₄⋊Dic₁₃₁	—	2	1	no	—
77	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
79	Dic₁₃₁⋊C₄	—	2	1	no	—
80	Dic₁₃₁⋊C₄	—	2	1	no	—
81	Dic₁₃₁⋊C₄	—	2	1	no	—
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	—	2	1	no	—
84	Dic₁₃₁⋊C₄	—	2	1	no	—
85	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
86	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
87	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
92	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₁₃₁	—	2	1	no	—
94	C₄×Dic₁₃₁	—	2	1	no	—
95	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₃₁	—	2	2	no	—
99	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₁	—	2	1	no	—
102	C₄⋊C₄×C₁₃₁	—	2	1	no	—
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₄²×C₁₃₁	—	2	1	no	—
106	C₄²×C₁₃₁	—	2	1	no	—
107	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₁	—	2	1	no	—
109	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
110	Dic₂₆₂	—	2	1	no	—
111	Dic₂₆₂	—	2	1	no	—
112	D₅₂₄	—	2	4	no	—
113	D₅₂₄	—	2	1	no	—
114	D₅₂₄	#41	1	2	yes	—
115	C₂×Dic₁₃₁	—	2	1	no	—
116	C₂×Dic₁₃₁	—	2	1	no	—
117	C₂×Dic₁₃₁	—	2	1	no	—
118	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
121	C₄×D₁₃₁	—	2	4	no	—
122	C₄×D₁₃₁	—	2	1	no	—
123	C₄×D₁₃₁	—	2	1	no	—
124	C₄×D₁₃₁	—	8	1	no	—
125	C₄×D₁₃₁	—	8	1	no	—
126	C₄×D₁₃₁	—	8	1	no	—
127	C₁₃₁⋊C₈	—	2	1	no	—
128	C₁₃₁⋊C₈	—	2	1	no	—
129	Q₈×C₁₃₁	—	2	1	no	—
130	Q₈×C₁₃₁	—	2	1	no	—
131	Q₈×C₁₃₁	—	2	1	no	—
132	Q₈×C₁₃₁	#41	1	2	yes	—
133	C₂×C₄×C₁₃₁	—	2	1	no	—
134	C₂×C₄×C₁₃₁	—	2	1	no	—
135	C₂×C₄×C₁₃₁	—	2	1	no	—
136	C₂×C₄×C₁₃₁	—	2	1	no	—
137	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
140	C₈×C₁₃₁	—	2	1	no	—
141	C₈×C₁₃₁	—	2	1	no	—
142	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₃₁	—	2	1	no	—
144	C₂×D₁₃₁	—	2	2	no	—
145	C₂×D₁₃₁	—	2	1	no	—
146	C₂×D₁₃₁	—	2	1	no	—
147	C₂×D₁₃₁	—	4	1	no	—
148	C₂×D₁₃₁	—	4	1	no	—
149	C₄×C₁₃₁	—	2	2	no	—
150	C₄×C₁₃₁	—	2	1	no	—
151	C₄×C₁₃₁	—	2	1	no	—
152	C₄×C₁₃₁	—	4	1	no	—
153	C₄×C₁₃₁	—	4	1	no	—
154	C₄×C₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
157	Dic₁₃₁	—	2	1	no	—
158	Dic₁₃₁	—	2	1	no	—
159	Dic₁₃₁	—	2	1	no	—
160	Dic₁₃₁	—	4	1	no	—
161	Dic₁₃₁	—	4	1	no	—
162	Dic₁₃₁	—	4	1	no	—
163	C₂×C₁₃₁	#525	1	1	yes	characteristic
164	C₂×C₁₃₁	#577	1	1	yes	characteristic
165	C₂×C₁₃₁	#578	1	1	yes	characteristic
166	D₁₃₁	—	4	1	no	—
167	D₁₃₁	—	4	2	no	—
168	C₁₃₁	#3120	1	1	yes	characteristic, complemented, Sylow
169	#3120	—	131	1	no	maximal, Sylow
170	#541	—	131	1	no	—
171	#520	—	131	1	no	—
172	#484	—	131	1	no	—
173	#483	—	131	1	no	—
174	#479	—	131	1	no	—
175	#478	—	131	1	no	—
176	#462	—	131	1	no	—
177	#461	—	131	1	no	—
178	#452	—	131	1	no	—
179	#452	—	131	1	no	—
180	#441	—	131	1	no	—
181	#440	—	131	1	no	—
182	C₄×Q₁₆	D₁₃₁	1	1	yes	characteristic, complemented
183	C₄×SD₁₆	—	131	1	no	—
184	#433	—	131	1	no	—
185	C₂×#41	—	131	1	no	—
186	C₂×Q₁₆	—	131	1	no	—
187	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
188	C₂×SD₁₆	—	131	1	no	—
189	C₂×SD₁₆	—	131	1	no	—
190	C₄⋊Q₈	—	131	1	no	—
191	#126	—	262	1	no	—
192	#126	—	262	1	no	—
193	#125	—	262	1	no	—
194	#125	—	262	1	no	—
195	#125	—	131	1	no	—
196	C₂²⋊Q₈	—	131	1	no	—
197	C₂²⋊Q₈	—	131	1	no	—
198	C₄×Q₈	—	131	1	no	—
199	C₄×Q₈	—	131	1	no	—
200	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
201	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
202	C₄×D₄	—	262	1	no	—
203	C₄×D₄	—	262	1	no	—
204	C₄×D₄	—	131	1	no	—
205	C₄×D₄	—	131	1	no	—
206	#117	—	262	1	no	—
207	#117	—	131	1	no	—
208	C₂×C₄⋊C₄	—	262	1	no	—
209	C₂×C₄⋊C₄	—	131	1	no	—
210	#107	—	131	1	no	—
211	#107	C₂×D₁₃₁	1	1	yes	characteristic
212	#106	—	131	1	no	—
213	#106	—	131	1	no	—
214	C₄⋊C₈	—	131	1	no	—
215	C₄⋊C₈	—	131	1	no	—
216	#103	—	131	1	no	—
217	#103	—	131	1	no	—
218	#103	—	131	1	no	—
219	#103	—	131	1	no	—
220	#103	C₂×D₁₃₁	1	1	yes	characteristic
221	#103	C₂×D₁₃₁	1	1	yes	characteristic
222	#102	—	131	1	no	—
223	#102	—	131	1	no	—
224	C₂²⋊C₈	—	131	1	no	—
225	C₂²⋊C₈	—	131	1	no	—
226	#97	—	131	1	no	—
227	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
228	#41	—	262	2	no	—
229	#41	—	262	2	no	—
230	C₂×Q₈	—	131	1	no	—
231	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
233	C₂×D₄	—	262	1	no	—
234	C₂×D₄	—	131	1	no	—
235	C₂²×C₄	—	262	1	no	—
236	C₂²×C₄	—	262	1	no	—
237	C₂²×C₄	—	262	1	no	—
238	C₂²×C₄	—	262	1	no	—
239	C₂²×C₄	—	131	1	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	262	2	no	—
242	SD₁₆	—	262	2	no	—
243	SD₁₆	—	262	2	no	—
244	C₂×C₈	—	131	1	no	—
245	C₂×C₈	—	131	1	no	—
246	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	262	1	no	—
251	C₄⋊C₄	—	262	1	no	—
252	C₄⋊C₄	—	262	1	no	—
253	C₄⋊C₄	—	262	1	no	—
254	C₄⋊C₄	—	262	1	no	—
255	C₄⋊C₄	—	262	1	no	—
256	C₄⋊C₄	—	262	1	no	—
257	C₄⋊C₄	—	262	1	no	—
258	C₄⋊C₄	—	262	1	no	—
259	C₄⋊C₄	—	131	1	no	—
260	C₄⋊C₄	—	131	1	no	—
261	C₄⋊C₄	—	131	1	no	—
262	C₄⋊C₄	—	131	1	no	—
263	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	262	1	no	—
266	C₂²⋊C₄	—	262	1	no	—
267	C₂²⋊C₄	—	262	1	no	—
268	C₂²⋊C₄	—	262	1	no	—
269	C₂²⋊C₄	—	131	1	no	—
270	C₂²⋊C₄	—	131	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	262	1	no	—
274	C₄²	—	262	1	no	—
275	C₄²	—	131	1	no	—
276	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
277	C₂³	—	262	1	no	—
278	C₂³	—	131	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	262	1	no	—
283	Q₈	—	262	1	no	—
284	Q₈	#14523513	1	2	yes	—
285	D₄	—	262	4	no	—
286	D₄	—	262	1	no	—
287	D₄	—	131	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	262	4	no	—
293	C₂×C₄	—	262	1	no	—
294	C₂×C₄	—	262	1	no	—
295	C₂×C₄	—	262	1	no	—
296	C₂×C₄	—	262	1	no	—
297	C₂×C₄	—	262	1	no	—
298	C₂×C₄	—	1048	1	no	—
299	C₂×C₄	—	1048	1	no	—
300	C₂×C₄	—	1048	1	no	—
301	C₂×C₄	—	131	1	no	—
302	C₂×C₄	—	131	1	no	—
303	C₂×C₄	—	131	1	no	—
304	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
305	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	262	1	no	—
310	C₈	—	262	1	no	—
311	C₂²	—	262	1	no	—
312	C₂²	—	262	1	no	—
313	C₂²	—	262	1	no	—
314	C₂²	—	262	2	no	—
315	C₂²	—	524	1	no	—
316	C₂²	—	524	1	no	—
317	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	262	1	no	—
325	C₄	—	262	1	no	—
326	C₄	—	262	1	no	—
327	C₄	—	524	1	no	—
328	C₄	—	524	1	no	—
329	C₄	—	524	1	no	—
330	C₄	#41×D₁₃₁	1	1	yes	characteristic
331	C₄	C₂×#14523513	1	1	yes	characteristic
332	C₂	—	524	1	no	—
333	C₂	—	524	2	no	—
334	C₂	#22553821	1	1	yes	characteristic
335	C₂	#22554119	1	1	yes	characteristic
336	C₂	#22554125	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	C₄×Q₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×#14523514	C₂²
hypercenter	C₄×Q₁₆	D₁₃₁
hypocenter	C₁₃₁	#3120
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #22553920	C₂	semidirect
3. #22553649	C₂	semidirect
4. #22553641	C₂	semidirect
5. #22553617	C₂	semidirect
6. #22553615	C₂	semidirect
7. #22553520	C₂	semidirect
8. #22553516	C₂	semidirect
11. #22553417	C₂	semidirect
12. #22553410	C₂	semidirect
13. C₄×#14523502	C₂	semidirect
14. #22553371	C₂	semidirect
15. C₄×Q₁₆×C₁₃₁	C₂	semidirect
168. C₁₃₁	#3120	semidirect
182. C₄×Q₁₆	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553920	2
#22553794	2
#22553649	2
#22553641	2
#22553617	2
#22553615	2
#22553520	2
#22553516	2
#22553477	2
#22553473	2
#22553417	2
#22553410	2
C₄×#14523502	2
#22553371	2
C₄×Q₁₆×C₁₃₁	2
#3120	131