Subgroups — #30676237

Order 16768.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553931	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553794	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553641	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553629	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553603	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553597	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553524	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553473	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553456	C₂	1	1	yes	maximal, characteristic
10	#22553421	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553419	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553397	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553372	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523514	C₂²	1	1	yes	characteristic
17	C₂×#14523500	C₂²	1	1	yes	characteristic
18	C₂×#14523498	C₂²	1	1	yes	characteristic
19	#16862246	C₂²	1	1	yes	characteristic
20	#16862239	—	2	1	no	—
21	#16862232	—	2	1	no	—
22	#16862230	—	2	1	no	—
23	#16862223	—	2	1	no	—
24	#16862222	C₂²	1	1	yes	characteristic
25	#16862217	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
28	C₄×D₅₂₄	—	2	1	no	—
29	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
31	#16862208	—	2	1	no	—
32	#16862208	C₂²	1	1	yes	characteristic
33	#16862203	—	2	1	no	—
34	#16862203	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₃₁	—	2	1	no	—
36	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
37	#16862178	C₂²	1	1	yes	characteristic
38	#16862178	C₂²	1	1	yes	characteristic
39	#16862177	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#16862170	C₂²	1	1	yes	characteristic
43	#16862170	C₂²	1	1	yes	characteristic
44	#16862170	C₂²	1	1	yes	characteristic
45	#16862168	C₂²	1	1	yes	characteristic
46	#16862168	C₂²	1	1	yes	characteristic
47	#16862167	C₂²	1	1	yes	characteristic
48	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#16862156	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
54	#124×C₁₃₁	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	#102×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523514	—	2	2	no	—
60	#14523514	—	2	2	no	—
61	C₂×Dic₂₆₂	—	2	1	no	—
62	C₂×D₅₂₄	—	2	1	no	—
63	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
65	C₂²×Dic₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	—	2	1	no	—
68	C₂×C₄×D₁₃₁	—	2	1	no	—
69	C₂×C₄×D₁₃₁	—	2	1	no	—
70	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
71	#14523500	—	2	2	no	—
72	#14523498	—	2	2	no	—
73	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₁	—	2	1	no	—
76	C₄⋊Dic₁₃₁	—	2	1	no	—
77	C₄⋊Dic₁₃₁	—	2	1	no	—
78	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
81	Dic₁₃₁⋊C₄	—	2	1	no	—
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₃₁	—	2	1	no	—
86	C₂²⋊Dic₁₃₁	—	2	1	no	—
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	D₄	1	1	yes	characteristic
91	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₁₃₁	—	2	1	no	—
93	C₄×Dic₁₃₁	—	2	1	no	—
94	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₃₁	—	2	2	no	—
99	SD₁₆×C₁₃₁	—	2	2	no	—
100	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₁	—	2	1	no	—
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
105	C₂²⋊C₄×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	—	8	1	no	—
111	D₅₂₄	—	2	4	no	—
112	D₅₂₄	—	2	1	no	—
113	D₅₂₄	#41	1	2	yes	—
114	C₂²⋊D₁₃₁	—	8	1	no	—
115	C₂×Dic₁₃₁	—	2	1	no	—
116	C₂×Dic₁₃₁	—	2	1	no	—
117	C₂×Dic₁₃₁	—	2	1	no	—
118	C₂×Dic₁₃₁	—	8	1	no	—
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₃₁	—	2	4	no	—
123	C₄×D₁₃₁	—	2	1	no	—
124	C₄×D₁₃₁	—	2	1	no	—
125	C₄×D₁₃₁	—	8	1	no	—
126	C₄×D₁₃₁	—	8	1	no	—
127	C₄×D₁₃₁	—	8	1	no	—
128	C₁₃₁⋊C₈	—	2	1	no	—
129	C₁₃₁⋊C₈	—	2	1	no	—
130	C₂³×C₁₃₁	—	2	1	no	—
131	Q₈×C₁₃₁	—	2	1	no	—
132	Q₈×C₁₃₁	—	2	1	no	—
133	Q₈×C₁₃₁	—	2	1	no	—
134	Q₈×C₁₃₁	#41	1	2	yes	—
135	D₄×C₁₃₁	—	2	1	no	—
136	D₄×C₁₃₁	—	2	1	no	—
137	C₂×C₄×C₁₃₁	—	2	1	no	—
138	C₂×C₄×C₁₃₁	—	2	1	no	—
139	C₂×C₄×C₁₃₁	—	2	1	no	—
140	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₁	—	2	1	no	—
144	C₈×C₁₃₁	—	2	1	no	—
145	C₂²×C₁₃₁	—	4	1	no	—
146	C₂²×C₁₃₁	—	4	1	no	—
147	C₂²×C₁₃₁	—	4	1	no	—
148	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₃₁	—	2	1	no	—
150	C₂×D₁₃₁	—	2	2	no	—
151	C₂×D₁₃₁	—	2	1	no	—
152	C₂×D₁₃₁	—	2	1	no	—
153	C₂×D₁₃₁	—	4	1	no	—
154	C₂×D₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	—	2	2	no	—
156	C₄×C₁₃₁	—	2	1	no	—
157	C₄×C₁₃₁	—	2	1	no	—
158	C₄×C₁₃₁	—	4	1	no	—
159	C₄×C₁₃₁	—	4	1	no	—
160	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
162	Dic₁₃₁	—	2	1	no	—
163	Dic₁₃₁	—	2	1	no	—
164	Dic₁₃₁	—	2	1	no	—
165	Dic₁₃₁	—	4	1	no	—
166	Dic₁₃₁	—	4	1	no	—
167	Dic₁₃₁	—	4	1	no	—
168	C₂×C₁₃₁	—	8	1	no	—
169	C₂×C₁₃₁	#525	1	1	yes	characteristic
170	C₂×C₁₃₁	#577	1	1	yes	characteristic
171	C₂×C₁₃₁	#577	1	1	yes	characteristic
172	D₁₃₁	—	4	1	no	—
173	D₁₃₁	—	4	2	no	—
174	C₁₃₁	#3125	1	1	yes	characteristic, complemented, Sylow
175	#3125	—	131	1	no	maximal, Sylow
176	#542	—	131	1	no	—
177	#520	—	131	1	no	—
178	#483	—	131	1	no	—
179	#481	—	131	1	no	—
180	#476	—	131	1	no	—
181	#475	—	131	1	no	—
182	#464	D₁₃₁	1	1	yes	characteristic, complemented
183	#463	—	131	1	no	—
184	#452	—	131	1	no	—
185	#449	—	131	1	no	—
186	#442	—	131	1	no	—
187	#441	—	131	1	no	—
188	C₄×SD₁₆	—	131	1	no	—
189	C₄×SD₁₆	—	131	1	no	—
190	#433	—	131	1	no	—
191	C₂×#41	—	131	1	no	—
192	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
193	C₂×SD₁₆	—	131	1	no	—
194	C₂×SD₁₆	C₂×D₁₃₁	1	1	yes	characteristic
195	C₂×D₈	—	131	1	no	—
196	C₄⋊Q₈	—	131	1	no	—
197	#126	—	262	1	no	—
198	#124	—	262	1	no	—
199	#124	C₂×D₁₃₁	1	1	yes	characteristic
200	#123	—	262	1	no	—
201	C₂²⋊Q₈	—	262	1	no	—
202	C₂²⋊Q₈	—	131	1	no	—
203	#121	—	131	1	no	—
204	C₄×Q₈	—	131	1	no	—
205	C₄×Q₈	—	131	1	no	—
206	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
207	C₄×D₄	—	262	1	no	—
208	C₄×D₄	—	262	1	no	—
209	C₄×D₄	—	131	1	no	—
210	C₄×D₄	—	131	1	no	—
211	C₄×D₄	—	131	1	no	—
212	#117	—	262	1	no	—
213	#117	—	131	1	no	—
214	C₂×C₄⋊C₄	—	262	1	no	—
215	C₂×C₄⋊C₄	—	131	1	no	—
216	#106	—	131	1	no	—
217	#106	—	131	1	no	—
218	#106	—	131	1	no	—
219	#106	—	131	1	no	—
220	C₄⋊C₈	—	131	1	no	—
221	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
222	#103	—	131	1	no	—
223	#103	—	131	1	no	—
224	#103	—	131	1	no	—
225	#103	C₂×D₁₃₁	1	1	yes	characteristic
226	#102	—	131	1	no	—
227	#102	—	131	1	no	—
228	#102	—	131	1	no	—
229	#102	C₂×D₁₃₁	1	1	yes	characteristic
230	C₂²⋊C₈	—	131	1	no	—
231	C₂²⋊C₈	—	131	1	no	—
232	#97	—	131	1	no	—
233	C₄×C₈	—	131	1	no	—
234	#41	—	262	2	no	—
235	#41	—	262	2	no	—
236	C₂×Q₈	—	262	1	no	—
237	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
238	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
239	C₂×D₄	—	262	1	no	—
240	C₂×D₄	—	262	1	no	—
241	C₂×D₄	—	131	1	no	—
242	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
243	C₂²×C₄	—	262	1	no	—
244	C₂²×C₄	—	262	1	no	—
245	C₂²×C₄	—	262	1	no	—
246	C₂²×C₄	—	262	1	no	—
247	C₂²×C₄	—	262	1	no	—
248	C₂²×C₄	—	131	1	no	—
249	Q₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	262	2	no	—
252	D₈	—	262	2	no	—
253	C₂×C₈	—	131	1	no	—
254	C₂×C₈	—	131	1	no	—
255	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	262	1	no	—
259	C₄⋊C₄	—	262	1	no	—
260	C₄⋊C₄	—	262	1	no	—
261	C₄⋊C₄	—	262	1	no	—
262	C₄⋊C₄	—	262	1	no	—
263	C₄⋊C₄	—	131	1	no	—
264	C₄⋊C₄	—	131	1	no	—
265	C₄⋊C₄	—	131	1	no	—
266	C₄⋊C₄	—	131	1	no	—
267	C₄⋊C₄	—	131	1	no	—
268	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	262	1	no	—
272	C₂²⋊C₄	—	262	1	no	—
273	C₂²⋊C₄	—	262	1	no	—
274	C₂²⋊C₄	—	262	1	no	—
275	C₂²⋊C₄	—	262	1	no	—
276	C₂²⋊C₄	—	131	1	no	—
277	C₂²⋊C₄	—	131	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	262	1	no	—
280	C₄²	—	262	1	no	—
281	C₄²	—	131	1	no	—
282	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	262	1	no	—
285	C₂³	—	131	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	1048	1	no	—
290	Q₈	#14523512	1	2	yes	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	262	4	no	—
294	D₄	—	262	1	no	—
295	D₄	—	1048	1	no	—
296	D₄	—	131	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	262	4	no	—
301	C₂×C₄	—	262	1	no	—
302	C₂×C₄	—	262	1	no	—
303	C₂×C₄	—	262	1	no	—
304	C₂×C₄	—	262	1	no	—
305	C₂×C₄	—	262	1	no	—
306	C₂×C₄	—	1048	1	no	—
307	C₂×C₄	—	1048	1	no	—
308	C₂×C₄	—	1048	1	no	—
309	C₂×C₄	—	1048	1	no	—
310	C₂×C₄	—	131	1	no	—
311	C₂×C₄	—	131	1	no	—
312	C₂×C₄	—	131	1	no	—
313	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
314	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	262	1	no	—
319	C₈	—	262	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	262	1	no	—
324	C₂²	—	262	1	no	—
325	C₂²	—	262	1	no	—
326	C₂²	—	262	2	no	—
327	C₂²	—	524	1	no	—
328	C₂²	—	524	1	no	—
329	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	262	1	no	—
336	C₄	—	262	1	no	—
337	C₄	—	262	1	no	—
338	C₄	—	524	1	no	—
339	C₄	—	524	1	no	—
340	C₄	—	524	1	no	—
341	C₄	#41×D₁₃₁	1	1	yes	characteristic
342	C₄	C₂×#14523512	1	1	yes	characteristic
343	C₂	—	8	1	no	—
344	C₂	—	524	1	no	—
345	C₂	—	524	2	no	—
346	C₂	#22553822	1	1	yes	characteristic
347	C₂	#22554119	1	1	yes	characteristic
348	C₂	#22554121	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#464×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553456	C₂
hypercenter	#464	D₁₃₁
hypocenter	C₁₃₁	#3125
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #22553931	C₂	semidirect
2. #22553794	C₂	semidirect
3. #22553641	C₂	semidirect
4. #22553629	C₂	semidirect
5. #22553603	C₂	semidirect
6. #22553597	C₂	semidirect
7. #22553524	C₂	semidirect
8. #22553473	C₂	semidirect
10. #22553421	C₂	semidirect
11. #22553419	C₂	semidirect
12. SD₁₆×Dic₁₃₁	C₂	semidirect
13. #22553397	C₂	semidirect
14. #22553372	C₂	semidirect
15. #464×C₁₃₁	C₂	semidirect
174. C₁₃₁	#3125	semidirect
182. #464	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553931	2
#22553794	2
#22553641	2
#22553629	2
#22553603	2
#22553597	2
#22553524	2
#22553473	2
#22553456	2
#22553421	2
#22553419	2
SD₁₆×Dic₁₃₁	2
#22553397	2
#22553372	2
#464×C₁₃₁	2
#3125	131