Subgroups — #30675486

Order 16768.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862278	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₁₆×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553887	C₂	1	1	yes	maximal, characteristic
4	#22553788	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553537	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553536	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553535	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553534	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553521	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553520	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553519	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553337	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553319	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553307	C₂	1	1	yes	maximal, characteristic
15	#462×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523514	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
19	#16862278	—	2	4	no	—
20	C₂×Dic₅₂₄	C₂²	1	1	yes	characteristic
21	C₂×#14523503	C₂²	1	1	yes	characteristic
22	C₂×#14523503	C₂²	1	1	yes	characteristic
23	C₂×#14523503	C₂²	1	1	yes	characteristic
24	Q₁₆×D₁₃₁	—	2	4	no	—
25	C₂×#14523502	C₂²	1	1	yes	characteristic
26	C₂×#14523500	C₂²	1	1	yes	characteristic
27	C₂×#14523496	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
29	#16862244	C₂²	1	1	yes	characteristic
30	#16862235	C₂²	1	1	yes	characteristic
31	#16862234	C₂²	1	1	yes	characteristic
32	#16862231	—	2	1	no	—
33	#16862231	—	2	1	no	—
34	#16862224	—	2	1	no	—
35	#16862224	—	2	1	no	—
36	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₆₂	—	2	1	no	—
38	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
39	#16862211	—	2	1	no	—
40	#16862211	C₂²	1	1	yes	characteristic
41	#16862204	—	2	1	no	—
42	#16862204	C₂²	1	1	yes	characteristic
43	#16862203	—	2	1	no	—
44	#16862203	C₂²	1	1	yes	characteristic
45	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
46	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
48	#16862171	C₂²	1	1	yes	characteristic
49	#16862170	C₂²	1	1	yes	characteristic
50	#16862170	C₂²	1	1	yes	characteristic
51	#16862169	C₂²	1	1	yes	characteristic
52	#16862168	C₂²	1	1	yes	characteristic
53	#16862166	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
55	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	#103×C₁₃₁	C₂²	1	1	yes	characteristic
60	#103×C₁₃₁	C₂²	1	1	yes	characteristic
61	#14523514	—	2	2	no	—
62	#14523514	—	2	2	no	—
63	C₂×Dic₂₆₂	—	2	1	no	—
64	C₂×Dic₂₆₂	—	2	1	no	—
65	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
66	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
67	Q₈×D₁₃₁	—	2	2	no	—
68	Q₈×D₁₃₁	—	2	2	no	—
69	Q₈×D₁₃₁	—	2	2	no	—
70	Q₈×D₁₃₁	D₄	1	4	yes	—
71	C₂×D₅₂₄	—	2	1	no	—
72	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₃₁	—	2	1	no	—
74	C₂×C₄×D₁₃₁	—	2	1	no	—
75	C₂×C₄×D₁₃₁	—	2	1	no	—
76	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
77	Dic₅₂₄	—	2	2	no	—
78	#14523503	—	2	2	no	—
79	#14523503	—	2	2	no	—
80	#14523503	—	2	2	no	—
81	#14523502	—	2	2	no	—
82	#14523500	—	2	2	no	—
83	#14523496	—	2	2	no	—
84	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
85	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
86	C₈×D₁₃₁	—	2	2	no	—
87	C₄⋊Dic₁₃₁	—	2	1	no	—
88	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
89	Dic₁₃₁⋊C₄	—	2	1	no	—
90	Dic₁₃₁⋊C₄	—	2	1	no	—
91	Dic₁₃₁⋊C₄	—	2	1	no	—
92	Dic₁₃₁⋊C₄	—	2	1	no	—
93	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
94	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
95	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
96	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
97	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
98	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
99	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
100	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
101	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₁₃₁	—	2	1	no	—
103	C₄×Dic₁₃₁	—	2	1	no	—
104	C₄×Dic₁₃₁	—	2	1	no	—
105	C₄×Dic₁₃₁	—	2	1	no	—
106	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
110	Q₁₆×C₁₃₁	—	2	2	no	—
111	Q₁₆×C₁₃₁	—	2	2	no	—
112	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₁₃₁	—	2	1	no	—
115	C₄⋊C₄×C₁₃₁	—	2	1	no	—
116	C₄⋊C₄×C₁₃₁	—	2	1	no	—
117	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
118	C₄²×C₁₃₁	—	2	1	no	—
119	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
120	C₂²×D₁₃₁	—	2	1	no	—
121	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
122	Dic₂₆₂	—	2	1	no	—
123	Dic₂₆₂	—	2	4	no	—
124	Dic₂₆₂	—	2	1	no	—
125	Dic₂₆₂	—	2	1	no	—
126	Dic₂₆₂	—	4	2	no	—
127	Dic₂₆₂	C₂×D₄	1	2	yes	—
128	D₅₂₄	—	2	1	no	—
129	D₅₂₄	—	2	1	no	—
130	D₅₂₄	—	4	2	no	—
131	C₂×Dic₁₃₁	—	2	1	no	—
132	C₂×Dic₁₃₁	—	2	1	no	—
133	C₂×Dic₁₃₁	—	2	1	no	—
134	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄×D₁₃₁	—	2	4	no	—
138	C₄×D₁₃₁	—	4	2	no	—
139	C₄×D₁₃₁	—	4	2	no	—
140	C₄×D₁₃₁	C₂×D₄	1	2	yes	—
141	C₄×D₁₃₁	#41	1	2	yes	—
142	C₁₃₁⋊C₈	—	2	1	no	—
143	C₁₃₁⋊C₈	—	2	1	no	—
144	Q₈×C₁₃₁	—	2	1	no	—
145	Q₈×C₁₃₁	—	2	1	no	—
146	Q₈×C₁₃₁	—	2	1	no	—
147	Q₈×C₁₃₁	—	2	1	no	—
148	Q₈×C₁₃₁	—	2	1	no	—
149	Q₈×C₁₃₁	C₂×D₄	1	2	yes	—
150	C₂×C₄×C₁₃₁	—	2	1	no	—
151	C₂×C₄×C₁₃₁	—	2	1	no	—
152	C₂×C₄×C₁₃₁	—	2	1	no	—
153	C₂×C₄×C₁₃₁	—	2	1	no	—
154	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
157	C₈×C₁₃₁	—	2	1	no	—
158	C₈×C₁₃₁	—	2	1	no	—
159	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₃₁	—	2	1	no	—
161	C₂×D₁₃₁	—	2	1	no	—
162	C₂×D₁₃₁	—	4	1	no	—
163	C₂×D₁₃₁	—	4	1	no	—
164	C₂×D₁₃₁	—	4	1	no	—
165	C₂×D₁₃₁	#121	1	2	yes	—
166	C₄×C₁₃₁	—	2	2	no	—
167	C₄×C₁₃₁	—	2	1	no	—
168	C₄×C₁₃₁	—	2	1	no	—
169	C₄×C₁₃₁	—	4	1	no	—
170	C₄×C₁₃₁	—	4	1	no	—
171	C₄×C₁₃₁	—	4	1	no	—
172	C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
174	Dic₁₃₁	—	2	2	no	—
175	Dic₁₃₁	—	2	1	no	—
176	Dic₁₃₁	—	2	1	no	—
177	Dic₁₃₁	—	4	1	no	—
178	Dic₁₃₁	—	4	1	no	—
179	Dic₁₃₁	#121	1	2	yes	—
180	C₂×C₁₃₁	C₂×#121	1	1	yes	characteristic
181	C₂×C₁₃₁	#578	1	1	yes	characteristic
182	C₂×C₁₃₁	#578	1	1	yes	characteristic
183	D₁₃₁	—	2	2	no	—
184	D₁₃₁	—	8	1	no	—
185	C₁₃₁	#3044	1	1	yes	characteristic, complemented, Sylow
186	#3044	—	131	1	no	maximal, Sylow
187	C₂×#137	—	131	1	no	—
188	C₂²×Q₁₆	—	131	1	no	—
189	#536	—	131	1	no	—
190	#519	—	131	1	no	—
191	#464	—	131	1	no	—
192	#464	—	131	1	no	—
193	#464	—	131	1	no	—
194	#464	—	131	1	no	—
195	#462	—	131	1	no	—
196	#462	—	131	1	no	—
197	#462	—	131	1	no	—
198	#462	D₁₃₁	1	1	yes	characteristic, complemented
199	#424	—	131	1	no	—
200	#419	—	131	1	no	—
201	#416	—	131	1	no	—
202	C₂×#41	—	131	1	no	—
203	C₂²×Q₈	—	131	1	no	—
204	C₂²×Q₈	—	131	1	no	—
205	#137	—	262	4	no	—
206	C₂×Q₁₆	—	262	4	no	—
207	C₂×Q₁₆	—	131	1	no	—
208	C₂×Q₁₆	—	131	1	no	—
209	C₂×Q₁₆	—	131	1	no	—
210	C₂×Q₁₆	—	131	1	no	—
211	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
212	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
213	C₂×SD₁₆	—	131	1	no	—
214	C₂×SD₁₆	—	131	1	no	—
215	C₂×M₁₆	—	131	1	no	—
216	C₂²×C₈	—	131	1	no	—
217	C₄⋊Q₈	—	131	1	no	—
218	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
219	#124	—	131	1	no	—
220	#124	—	131	1	no	—
221	#123	—	262	1	no	—
222	#123	—	262	1	no	—
223	C₂²⋊Q₈	—	262	1	no	—
224	C₂²⋊Q₈	—	262	1	no	—
225	C₄×Q₈	—	262	1	no	—
226	C₄×Q₈	—	262	1	no	—
227	C₄×Q₈	—	131	1	no	—
228	C₄×Q₈	—	131	1	no	—
229	C₄×Q₈	—	131	1	no	—
230	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
231	#117	—	262	1	no	—
232	#117	—	262	1	no	—
233	#117	—	131	1	no	—
234	#117	—	131	1	no	—
235	C₄⋊C₈	—	131	1	no	—
236	C₄⋊C₈	—	131	1	no	—
237	C₄⋊C₈	—	131	1	no	—
238	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
239	#103	—	131	1	no	—
240	#103	—	131	1	no	—
241	#103	—	131	1	no	—
242	#103	—	131	1	no	—
243	#103	C₂×D₁₃₁	1	1	yes	characteristic
244	#103	C₂×D₁₃₁	1	1	yes	characteristic
245	#102	—	131	1	no	—
246	#102	—	131	1	no	—
247	#41	—	262	2	no	—
248	#41	—	262	2	no	—
249	C₂×Q₈	—	262	1	no	—
250	C₂×Q₈	—	262	1	no	—
251	C₂×Q₈	—	262	2	no	—
252	C₂×Q₈	—	262	2	no	—
253	C₂×Q₈	—	262	2	no	—
254	C₂×Q₈	—	131	4	no	—
255	C₂×Q₈	—	131	1	no	—
256	C₂×Q₈	—	131	1	no	—
257	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
260	C₂×D₄	—	262	1	no	—
261	C₂×D₄	—	131	1	no	—
262	C₂²×C₄	—	262	1	no	—
263	C₂²×C₄	—	262	1	no	—
264	C₂²×C₄	—	262	1	no	—
265	C₂²×C₄	—	131	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	262	2	no	—
269	Q₁₆	—	262	2	no	—
270	Q₁₆	—	262	2	no	—
271	Q₁₆	—	262	2	no	—
272	SD₁₆	—	262	2	no	—
273	SD₁₆	—	262	2	no	—
274	M₁₆	—	262	2	no	—
275	C₂×C₈	—	262	2	no	—
276	C₂×C₈	—	131	1	no	—
277	C₂×C₈	—	131	1	no	—
278	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
279	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	262	1	no	—
284	C₄⋊C₄	—	262	1	no	—
285	C₄⋊C₄	—	262	1	no	—
286	C₄⋊C₄	—	262	1	no	—
287	C₄⋊C₄	—	262	1	no	—
288	C₄⋊C₄	—	131	1	no	—
289	C₄⋊C₄	—	131	1	no	—
290	C₄⋊C₄	—	131	1	no	—
291	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
292	C₂²⋊C₄	—	262	1	no	—
293	C₂²⋊C₄	—	262	1	no	—
294	C₂²⋊C₄	—	262	1	no	—
295	C₂²⋊C₄	—	262	1	no	—
296	C₂²⋊C₄	—	262	1	no	—
297	C₂²⋊C₄	—	131	1	no	—
298	C₂²⋊C₄	—	131	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	262	1	no	—
301	C₄²	—	262	1	no	—
302	C₄²	—	262	1	no	—
303	C₄²	—	262	1	no	—
304	C₄²	—	131	1	no	—
305	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
306	C₂³	—	262	1	no	—
307	C₂³	—	131	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	2	1	no	—
313	Q₈	—	262	4	no	—
314	Q₈	—	262	1	no	—
315	Q₈	—	262	1	no	—
316	Q₈	—	262	1	no	—
317	Q₈	—	524	2	no	—
318	Q₈	—	131	2	no	—
319	Q₈	D₄×D₁₃₁	1	2	yes	—
320	D₄	—	262	1	no	—
321	D₄	—	262	1	no	—
322	D₄	—	524	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	262	4	no	—
328	C₂×C₄	—	262	1	no	—
329	C₂×C₄	—	262	1	no	—
330	C₂×C₄	—	262	1	no	—
331	C₂×C₄	—	524	2	no	—
332	C₂×C₄	—	524	2	no	—
333	C₂×C₄	—	131	1	no	—
334	C₂×C₄	—	131	1	no	—
335	C₂×C₄	—	131	1	no	—
336	C₂×C₄	—	131	2	no	—
337	C₂×C₄	—	131	2	no	—
338	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
339	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	262	1	no	—
344	C₈	—	262	1	no	—
345	C₂²	—	262	1	no	—
346	C₂²	—	262	1	no	—
347	C₂²	—	524	1	no	—
348	C₂²	—	524	1	no	—
349	C₂²	—	524	1	no	—
350	C₂²	—	131	2	no	—
351	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	262	1	no	—
359	C₄	—	262	1	no	—
360	C₄	—	262	2	no	—
361	C₄	—	524	1	no	—
362	C₄	—	524	1	no	—
363	C₄	—	131	2	no	—
364	C₄	C₂×D₄×D₁₃₁	1	1	yes	characteristic
365	C₄	#41×D₁₃₁	1	1	yes	characteristic
366	C₂	—	262	2	no	—
367	C₂	—	1048	1	no	—
368	C₂	#121×D₁₃₁	1	1	yes	characteristic
369	C₂	#22554125	1	1	yes	characteristic
370	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	#462×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×#14523514	C₂²
hypercenter	#462	D₁₃₁
hypocenter	C₁₃₁	#3044
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₁₃₁	C₂	semidirect
4. #22553788	C₂	semidirect
5. #22553537	C₂	semidirect
6. #22553536	C₂	semidirect
7. #22553535	C₂	semidirect
8. #22553534	C₂	semidirect
9. #22553521	C₂	semidirect
10. #22553520	C₂	semidirect
11. #22553519	C₂	semidirect
12. #22553337	C₂	semidirect
13. #22553319	C₂	semidirect
15. #462×C₁₃₁	C₂	semidirect
185. C₁₃₁	#3044	semidirect
198. #462	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16862278	2
C₂×Q₁₆×D₁₃₁	2
#22553887	2
#22553788	2
#22553537	2
#22553536	2
#22553535	2
#22553534	2
#22553521	2
#22553520	2
#22553519	2
#22553337	2
#22553319	2
#22553307	2
#462×C₁₃₁	2
#3044	131