Subgroups — #30675923

Order 16768.

386 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	#22553885	C₂	1	1	yes	maximal, characteristic
2	#22553793	C₂	1	1	yes	maximal, characteristic
3	#22553656	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553654	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553651	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553650	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553647	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553646	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553644	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553351	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553347	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553319	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553277	C₂	1	1	yes	maximal, characteristic
15	#483×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523513	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×#14523496	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
20	#16862245	—	2	1	no	—
21	#16862232	—	2	1	no	—
22	#16862231	—	2	1	no	—
23	#16862228	—	2	1	no	—
24	#16862226	C₂²	1	1	yes	characteristic
25	#16862225	C₂²	1	1	yes	characteristic
26	#16862222	C₂²	1	1	yes	characteristic
27	#16862211	—	2	1	no	—
28	#16862208	—	2	1	no	—
29	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
30	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
31	#16862205	C₂²	1	1	yes	characteristic
32	#16862204	C₂²	1	1	yes	characteristic
33	#16862204	C₂²	1	1	yes	characteristic
34	#16862204	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
37	#16862180	C₂²	1	1	yes	characteristic
38	#16862179	C₂²	1	1	yes	characteristic
39	#16862179	C₂²	1	1	yes	characteristic
40	#16862179	C₂²	1	1	yes	characteristic
41	#16862178	C₂²	1	1	yes	characteristic
42	#16862177	C₂²	1	1	yes	characteristic
43	#16862171	C₂²	1	1	yes	characteristic
44	#16862171	C₂²	1	1	yes	characteristic
45	#16862170	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	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
54	#106×C₁₃₁	C₂²	1	1	yes	characteristic
55	#106×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523513	—	2	2	no	—
60	#14523513	—	2	2	no	—
61	C₂×Dic₂₆₂	—	2	1	no	—
62	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
63	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
64	Q₈×D₁₃₁	—	2	2	no	—
65	Q₈×D₁₃₁	—	2	2	no	—
66	C₂×D₅₂₄	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃₁	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₁	—	2	1	no	—
70	C₂×C₄×D₁₃₁	—	2	1	no	—
71	C₂×C₄×D₁₃₁	—	2	1	no	—
72	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
74	#14523496	—	2	2	no	—
75	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×D₁₃₁	—	2	2	no	—
78	C₄⋊Dic₁₃₁	—	2	1	no	—
79	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	—	2	1	no	—
84	Dic₁₃₁⋊C₄	—	2	1	no	—
85	Dic₁₃₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₁₃₁	—	2	1	no	—
87	C₂²⋊Dic₁₃₁	—	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	—	2	1	no	—
92	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
93	C₄×Dic₁₃₁	—	2	1	no	—
94	C₄×Dic₁₃₁	—	2	1	no	—
95	C₄×Dic₁₃₁	—	2	1	no	—
96	C₄×Dic₁₃₁	—	2	1	no	—
97	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₁	—	2	1	no	—
99	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
104	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
108	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
110	Dic₂₆₂	—	2	2	no	—
111	Dic₂₆₂	—	2	1	no	—
112	Dic₂₆₂	—	2	1	no	—
113	Dic₂₆₂	—	4	2	no	—
114	D₅₂₄	—	2	2	no	—
115	C₂²⋊D₁₃₁	—	2	2	no	—
116	C₂²⋊D₁₃₁	—	2	2	no	—
117	C₂×Dic₁₃₁	—	2	1	no	—
118	C₂×Dic₁₃₁	—	2	1	no	—
119	C₂×Dic₁₃₁	—	2	1	no	—
120	C₂×Dic₁₃₁	—	2	1	no	—
121	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₃₁	—	2	1	no	—
124	C₄×D₁₃₁	—	2	1	no	—
125	C₄×D₁₃₁	—	4	2	no	—
126	C₄×D₁₃₁	—	8	1	no	—
127	C₄×D₁₃₁	—	8	1	no	—
128	C₄×D₁₃₁	#41	1	2	yes	—
129	C₄×D₁₃₁	#41	1	2	yes	—
130	C₁₃₁⋊C₈	—	2	1	no	—
131	C₁₃₁⋊C₈	—	2	1	no	—
132	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
133	Q₈×C₁₃₁	—	2	1	no	—
134	Q₈×C₁₃₁	—	2	1	no	—
135	C₂×C₄×C₁₃₁	—	2	1	no	—
136	C₂×C₄×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	—	2	1	no	—
141	C₂×C₄×C₁₃₁	—	8	1	no	—
142	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₃₁	—	2	1	no	—
145	C₈×C₁₃₁	—	2	1	no	—
146	C₂²×C₁₃₁	—	2	1	no	—
147	C₂²×C₁₃₁	—	2	1	no	—
148	C₂²×C₁₃₁	—	2	1	no	—
149	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₃₁	—	2	1	no	—
151	C₂×D₁₃₁	—	2	1	no	—
152	C₂×D₁₃₁	—	2	1	no	—
153	C₂×D₁₃₁	—	2	1	no	—
154	C₂×D₁₃₁	—	2	1	no	—
155	C₂×D₁₃₁	#123	1	2	yes	—
156	C₄×C₁₃₁	—	2	1	no	—
157	C₄×C₁₃₁	—	4	1	no	—
158	C₄×C₁₃₁	—	4	1	no	—
159	C₄×C₁₃₁	—	4	1	no	—
160	C₄×C₁₃₁	—	4	1	no	—
161	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
163	Dic₁₃₁	—	2	1	no	—
164	Dic₁₃₁	—	4	1	no	—
165	Dic₁₃₁	—	4	1	no	—
166	Dic₁₃₁	—	4	1	no	—
167	Dic₁₃₁	—	4	1	no	—
168	Dic₁₃₁	#123	1	2	yes	—
169	C₂×C₁₃₁	—	4	1	no	—
170	C₂×C₁₃₁	C₂×#123	1	1	yes	characteristic
171	C₂×C₁₃₁	#577	1	1	yes	characteristic
172	C₂×C₁₃₁	#578	1	1	yes	characteristic
173	D₁₃₁	—	2	2	no	—
174	D₁₃₁	—	4	1	no	—
175	C₁₃₁	#3097	1	1	yes	characteristic, complemented, Sylow
176	#3097	—	131	1	no	maximal, Sylow
177	#536	—	131	1	no	—
178	#520	—	131	1	no	—
179	#485	—	131	1	no	—
180	#485	—	131	1	no	—
181	#485	—	131	1	no	—
182	#485	—	131	1	no	—
183	#484	—	131	1	no	—
184	#484	—	131	1	no	—
185	#483	—	131	1	no	—
186	#483	D₁₃₁	1	1	yes	characteristic, complemented
187	#428	—	131	1	no	—
188	#427	—	131	1	no	—
189	#419	—	131	1	no	—
190	C₂×#103	—	131	1	no	—
191	#408	—	131	1	no	—
192	C₂×#41	—	131	1	no	—
193	C₂²×Q₈	—	131	1	no	—
194	C₂×M₁₆	—	131	1	no	—
195	C₂²×C₈	—	131	1	no	—
196	C₄⋊Q₈	—	262	1	no	—
197	#124	—	262	1	no	—
198	#123	—	262	1	no	—
199	#123	—	262	1	no	—
200	C₂²⋊Q₈	—	131	1	no	—
201	C₂²⋊Q₈	—	131	1	no	—
202	C₂²⋊Q₈	—	131	1	no	—
203	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
204	C₄×Q₈	—	262	1	no	—
205	C₄×D₄	—	262	1	no	—
206	C₄×D₄	—	262	1	no	—
207	C₄×D₄	—	262	1	no	—
208	#117	—	131	1	no	—
209	#117	—	131	1	no	—
210	#117	—	131	1	no	—
211	#117	—	131	1	no	—
212	C₂×C₄⋊C₄	—	131	1	no	—
213	C₂×C₄⋊C₄	—	131	1	no	—
214	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
215	#107	—	131	1	no	—
216	#107	—	131	1	no	—
217	#107	—	131	1	no	—
218	#107	—	131	1	no	—
219	#106	—	131	1	no	—
220	#106	—	131	1	no	—
221	#106	C₂×D₁₃₁	1	1	yes	characteristic
222	#106	C₂×D₁₃₁	1	1	yes	characteristic
223	#103	—	131	1	no	—
224	#103	—	131	1	no	—
225	#103	—	131	1	no	—
226	#103	—	131	1	no	—
227	#103	—	131	1	no	—
228	#103	—	131	1	no	—
229	#103	C₂×D₁₃₁	1	1	yes	characteristic
230	#103	C₂×D₁₃₁	1	1	yes	characteristic
231	C₂²⋊C₈	—	131	1	no	—
232	C₂²⋊C₈	—	131	1	no	—
233	C₂²⋊C₈	—	131	1	no	—
234	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
235	#41	—	262	2	no	—
236	#41	—	262	2	no	—
237	C₂×Q₈	—	262	1	no	—
238	C₂×Q₈	—	262	2	no	—
239	C₂×Q₈	—	262	2	no	—
240	C₂×Q₈	—	131	1	no	—
241	C₂×Q₈	—	131	1	no	—
242	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
243	C₂×D₄	—	131	1	no	—
244	C₂×D₄	—	131	1	no	—
245	C₂×D₄	—	131	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	262	1	no	—
248	C₂²×C₄	—	262	1	no	—
249	C₂²×C₄	—	262	1	no	—
250	C₂²×C₄	—	131	1	no	—
251	C₂²×C₄	—	131	1	no	—
252	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
253	M₁₆	—	262	2	no	—
254	C₂×C₈	—	262	2	no	—
255	C₂×C₈	—	131	1	no	—
256	C₂×C₈	—	131	1	no	—
257	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	262	1	no	—
262	C₄⋊C₄	—	262	1	no	—
263	C₄⋊C₄	—	262	1	no	—
264	C₄⋊C₄	—	262	1	no	—
265	C₄⋊C₄	—	262	1	no	—
266	C₄⋊C₄	—	131	1	no	—
267	C₄⋊C₄	—	131	1	no	—
268	C₄⋊C₄	—	131	1	no	—
269	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	262	1	no	—
274	C₂²⋊C₄	—	262	1	no	—
275	C₂²⋊C₄	—	262	1	no	—
276	C₂²⋊C₄	—	262	1	no	—
277	C₂²⋊C₄	—	262	1	no	—
278	C₂²⋊C₄	—	262	1	no	—
279	C₂²⋊C₄	—	262	1	no	—
280	C₄²	—	262	1	no	—
281	C₄²	—	262	1	no	—
282	C₄²	—	262	1	no	—
283	C₄²	—	262	1	no	—
284	C₂³	—	131	1	no	—
285	C₂³	—	131	1	no	—
286	C₂³	D₄×D₁₃₁	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	262	2	no	—
290	Q₈	—	262	1	no	—
291	Q₈	—	262	1	no	—
292	Q₈	—	524	2	no	—
293	D₄	—	262	2	no	—
294	D₄	—	262	2	no	—
295	D₄	—	262	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	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	—	262	1	no	—
307	C₂×C₄	—	262	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	524	2	no	—
310	C₂×C₄	—	1048	1	no	—
311	C₂×C₄	—	1048	1	no	—
312	C₂×C₄	—	131	1	no	—
313	C₂×C₄	—	131	1	no	—
314	C₂×C₄	—	131	2	no	—
315	C₂×C₄	—	131	2	no	—
316	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	262	1	no	—
321	C₈	—	262	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	262	1	no	—
326	C₂²	—	262	1	no	—
327	C₂²	—	262	1	no	—
328	C₂²	—	262	1	no	—
329	C₂²	—	262	1	no	—
330	C₂²	—	131	2	no	—
331	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	262	1	no	—
338	C₄	—	524	1	no	—
339	C₄	—	524	1	no	—
340	C₄	—	524	1	no	—
341	C₄	—	524	1	no	—
342	C₄	—	131	2	no	—
343	C₄	#41×D₁₃₁	1	1	yes	characteristic
344	C₄	#41×D₁₃₁	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	262	2	no	—
347	C₂	—	524	1	no	—
348	C₂	#123×D₁₃₁	1	1	yes	characteristic
349	C₂	#22554118	1	1	yes	characteristic
350	C₂	#22554128	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#483×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×#14523513	C₂²
hypercenter	#483	D₁₃₁
hypocenter	C₁₃₁	#3097
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
3. #22553656	C₂	semidirect
4. #22553654	C₂	semidirect
5. #22553651	C₂	semidirect
6. #22553650	C₂	semidirect
7. #22553647	C₂	semidirect
8. #22553646	C₂	semidirect
9. #22553644	C₂	semidirect
10. #22553351	C₂	semidirect
11. #22553347	C₂	semidirect
12. #22553319	C₂	semidirect
13. #103×D₁₃₁	C₂	semidirect
15. #483×C₁₃₁	C₂	semidirect
175. C₁₃₁	#3097	semidirect
186. #483	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553885	2
#22553793	2
#22553656	2
#22553654	2
#22553651	2
#22553650	2
#22553647	2
#22553646	2
#22553644	2
#22553351	2
#22553347	2
#22553319	2
#103×D₁₃₁	2
#22553277	2
#483×C₁₃₁	2
#3097	131