Subgroups — #30676141

Order 16768.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553927	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553784	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553655	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553635	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553600	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553591	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553526	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553525	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553464	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553444	C₂	1	1	yes	maximal, characteristic
11	#22553422	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553412	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×D₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#14523502	C₂²	1	1	yes	characteristic
18	C₂×#14523501	C₂²	1	1	yes	characteristic
19	C₂×#14523498	C₂²	1	1	yes	characteristic
20	#16862245	C₂²	1	1	yes	characteristic, complemented
21	#16862239	—	2	1	no	—
22	#16862233	C₂²	1	1	yes	characteristic, complemented
23	#16862232	—	2	1	no	—
24	#16862229	—	2	1	no	—
25	#16862226	C₂²	1	1	yes	characteristic
26	#16862225	—	2	1	no	—
27	#16862217	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
31	#16862208	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
33	(C₂²⋊D₁₃₁)⋊C₄	—	2	1	no	—
34	#16862204	C₂²	1	1	yes	characteristic
35	#16862203	C₂²	1	1	yes	characteristic
36	C₂×(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
37	C₂²⋊C₄×D₁₃₁	—	2	1	no	—
38	#16862179	C₂²	1	1	yes	characteristic, complemented
39	#16862178	C₂²	1	1	yes	characteristic, complemented
40	#16862177	C₂²	1	1	yes	characteristic, complemented
41	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#16862171	C₂²	1	1	yes	characteristic, complemented
44	#16862169	C₂²	1	1	yes	characteristic, complemented
45	#16862168	C₂²	1	1	yes	characteristic
46	#16862167	C₂²	1	1	yes	characteristic
47	#16862167	C₂²	1	1	yes	characteristic, complemented
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, complemented
52	C₂×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
55	#107×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
56	#102×C₁₃₁	C₂²	1	1	yes	characteristic
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₃₁	—	2	1	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
64	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
65	D₄×D₁₃₁	—	2	2	no	—
66	D₄×D₁₃₁	—	2	2	no	—
67	C₂²×Dic₁₃₁	—	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	#14523502	—	2	2	no	—
72	#14523501	—	2	2	no	—
73	#14523498	—	2	2	no	—
74	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
82	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃₁	—	2	1	no	—
84	C₂²⋊Dic₁₃₁	—	2	1	no	—
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	—	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×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₁	—	2	1	no	—
99	C₂²×C₄×C₁₃₁	—	2	1	no	—
100	D₈×C₁₃₁	—	2	2	no	—
101	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
106	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
107	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₁	—	2	1	no	—
109	C₂²×D₁₃₁	—	2	4	no	—
110	C₂²×D₁₃₁	—	8	1	no	—
111	C₂²×D₁₃₁	—	8	1	no	—
112	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
113	Dic₂₆₂	—	2	1	no	—
114	Dic₂₆₂	—	2	1	no	—
115	Dic₂₆₂	—	8	1	no	—
116	D₅₂₄	—	2	1	no	—
117	D₅₂₄	#41	1	2	yes	—
118	C₂²⋊D₁₃₁	—	2	4	no	—
119	C₂²⋊D₁₃₁	—	8	1	no	—
120	C₂×Dic₁₃₁	—	2	1	no	—
121	C₂×Dic₁₃₁	—	2	1	no	—
122	C₂×Dic₁₃₁	—	2	1	no	—
123	C₂×Dic₁₃₁	—	8	1	no	—
124	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₃₁	—	2	1	no	—
128	C₄×D₁₃₁	—	2	1	no	—
129	C₄×D₁₃₁	—	8	1	no	—
130	C₄×D₁₃₁	—	8	1	no	—
131	C₁₃₁⋊C₈	—	2	1	no	—
132	C₁₃₁⋊C₈	—	2	1	no	—
133	C₂³×C₁₃₁	—	2	1	no	—
134	C₂³×C₁₃₁	—	2	1	no	—
135	D₄×C₁₃₁	—	2	1	no	—
136	D₄×C₁₃₁	—	2	1	no	—
137	D₄×C₁₃₁	—	2	1	no	—
138	D₄×C₁₃₁	#41	1	2	yes	—
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	—	8	1	no	—
143	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
146	C₈×C₁₃₁	—	2	1	no	—
147	C₈×C₁₃₁	—	2	1	no	—
148	C₂²×C₁₃₁	—	2	2	no	—
149	C₂²×C₁₃₁	—	4	1	no	—
150	C₂²×C₁₃₁	—	4	1	no	—
151	C₂²×C₁₃₁	—	4	1	no	—
152	C₂²×C₁₃₁	—	4	1	no	—
153	C₂²×C₁₃₁	—	4	1	no	—
154	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₃₁	—	2	1	no	—
156	C₂×D₁₃₁	—	2	2	no	—
157	C₂×D₁₃₁	—	2	1	no	—
158	C₂×D₁₃₁	—	2	1	no	—
159	C₂×D₁₃₁	—	4	1	no	—
160	C₂×D₁₃₁	—	4	1	no	—
161	C₂×D₁₃₁	—	8	4	no	—
162	C₄×C₁₃₁	—	2	1	no	—
163	C₄×C₁₃₁	—	2	1	no	—
164	C₄×C₁₃₁	—	4	1	no	—
165	C₄×C₁₃₁	—	4	1	no	—
166	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
167	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
168	Dic₁₃₁	—	2	1	no	—
169	Dic₁₃₁	—	2	1	no	—
170	Dic₁₃₁	—	2	1	no	—
171	Dic₁₃₁	—	4	1	no	—
172	Dic₁₃₁	—	4	1	no	—
173	Dic₁₃₁	—	4	1	no	—
174	C₂×C₁₃₁	—	4	2	no	—
175	C₂×C₁₃₁	—	8	1	no	—
176	C₂×C₁₃₁	#525	1	1	yes	characteristic
177	C₂×C₁₃₁	#576	1	1	yes	characteristic
178	C₂×C₁₃₁	#577	1	1	yes	characteristic
179	D₁₃₁	—	4	1	no	—
180	D₁₃₁	—	4	2	no	—
181	C₁₃₁	#3118	1	1	yes	characteristic, complemented, Sylow
182	#3118	—	131	1	no	maximal, Sylow
183	#542	—	131	1	no	—
184	#518	—	131	1	no	—
185	#485	—	131	1	no	—
186	#482	—	131	1	no	—
187	#476	—	131	1	no	—
188	#474	—	131	1	no	—
189	#463	—	131	1	no	—
190	#463	—	131	1	no	—
191	#450	—	131	1	no	—
192	#447	—	131	1	no	—
193	#442	—	131	1	no	—
194	#440	—	131	1	no	—
195	C₄×SD₁₆	—	131	1	no	—
196	C₄×D₈	D₁₃₁	1	1	yes	characteristic, complemented
197	#433	—	131	1	no	—
198	C₂²×D₄	—	131	1	no	—
199	C₂×SD₁₆	—	131	1	no	—
200	C₂×SD₁₆	—	131	1	no	—
201	C₂×D₈	—	131	1	no	—
202	C₂×D₈	C₂×D₁₃₁	1	1	yes	characteristic
203	C₄⋊Q₈	—	131	1	no	—
204	#126	—	262	1	no	—
205	#124	—	262	1	no	—
206	#124	—	131	1	no	—
207	#123	—	262	1	no	—
208	C₂²⋊Q₈	—	262	1	no	—
209	C₂²⋊Q₈	—	131	1	no	—
210	#121	—	131	1	no	—
211	C₄×Q₈	—	131	1	no	—
212	C₄×D₄	—	262	1	no	—
213	C₄×D₄	—	262	1	no	—
214	C₄×D₄	—	131	1	no	—
215	C₄×D₄	—	131	1	no	—
216	C₄×D₄	—	131	1	no	—
217	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
218	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic, complemented
219	#117	—	131	1	no	—
220	#117	—	131	1	no	—
221	C₂×C₂²⋊C₄	—	262	1	no	—
222	C₂×C₂²⋊C₄	—	262	1	no	—
223	#107	—	131	1	no	—
224	#107	C₂×D₁₃₁	1	1	yes	characteristic, complemented
225	#106	—	131	1	no	—
226	#106	—	131	1	no	—
227	C₄⋊C₈	—	131	1	no	—
228	C₄⋊C₈	—	131	1	no	—
229	#103	—	131	1	no	—
230	#103	—	131	1	no	—
231	#102	—	131	1	no	—
232	#102	—	131	1	no	—
233	#102	—	131	1	no	—
234	#102	—	131	1	no	—
235	#102	C₂×D₁₃₁	1	1	yes	characteristic
236	#102	C₂×D₁₃₁	1	1	yes	characteristic, complemented
237	C₂²⋊C₈	—	131	1	no	—
238	C₂²⋊C₈	—	131	1	no	—
239	#97	—	131	1	no	—
240	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
241	C₂⁴	—	262	1	no	—
242	C₂×Q₈	—	262	1	no	—
243	C₂×Q₈	—	131	1	no	—
244	C₂×D₄	—	262	2	no	—
245	C₂×D₄	—	262	2	no	—
246	C₂×D₄	—	262	1	no	—
247	C₂×D₄	—	262	1	no	—
248	C₂×D₄	—	131	1	no	—
249	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
250	C₂×D₄	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	C₂²×C₄	—	131	1	no	—
257	SD₁₆	—	262	2	no	—
258	SD₁₆	—	262	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	262	2	no	—
261	C₂×C₈	—	131	1	no	—
262	C₂×C₈	—	131	1	no	—
263	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
265	C₄⋊C₄	—	262	1	no	—
266	C₄⋊C₄	—	262	1	no	—
267	C₄⋊C₄	—	262	1	no	—
268	C₄⋊C₄	—	131	1	no	—
269	C₄⋊C₄	—	131	1	no	—
270	C₄⋊C₄	—	131	1	no	—
271	C₄⋊C₄	—	131	1	no	—
272	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	262	1	no	—
281	C₂²⋊C₄	—	262	1	no	—
282	C₂²⋊C₄	—	262	1	no	—
283	C₂²⋊C₄	—	262	1	no	—
284	C₂²⋊C₄	—	131	1	no	—
285	C₂²⋊C₄	—	131	1	no	—
286	C₄²	—	262	1	no	—
287	C₄²	—	262	1	no	—
288	C₄²	—	131	1	no	—
289	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	262	4	no	—
293	C₂³	—	262	1	no	—
294	C₂³	—	1048	1	no	—
295	C₂³	—	1048	1	no	—
296	C₂³	—	131	1	no	—
297	Q₈	—	262	1	no	—
298	Q₈	—	262	1	no	—
299	Q₈	—	1048	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	262	4	no	—
304	D₄	—	262	1	no	—
305	D₄	—	1048	1	no	—
306	D₄	—	131	2	no	—
307	D₄	#14523513	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	262	1	no	—
311	C₂×C₄	—	262	1	no	—
312	C₂×C₄	—	262	1	no	—
313	C₂×C₄	—	262	1	no	—
314	C₂×C₄	—	262	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	1048	1	no	—
318	C₂×C₄	—	1048	1	no	—
319	C₂×C₄	—	1048	1	no	—
320	C₂×C₄	—	131	1	no	—
321	C₂×C₄	—	131	1	no	—
322	C₂×C₄	—	131	1	no	—
323	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
324	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	262	1	no	—
329	C₈	—	262	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	262	1	no	—
337	C₂²	—	262	1	no	—
338	C₂²	—	262	1	no	—
339	C₂²	—	262	2	no	—
340	C₂²	—	524	1	no	—
341	C₂²	—	524	1	no	—
342	C₂²	—	1048	4	no	—
343	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	262	1	no	—
349	C₄	—	262	1	no	—
350	C₄	—	262	1	no	—
351	C₄	—	524	1	no	—
352	C₄	—	524	1	no	—
353	C₄	—	524	1	no	—
354	C₄	#41×D₁₃₁	1	1	yes	characteristic
355	C₄	C₂×#14523513	1	1	yes	characteristic
356	C₂	—	4	2	no	—
357	C₂	—	8	1	no	—
358	C₂	—	524	1	no	—
359	C₂	—	524	2	no	—
360	C₂	#22553821	1	1	yes	characteristic
361	C₂	#22554113	1	1	yes	characteristic
362	C₂	#22554119	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×D₈×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553444	C₂
hypercenter	C₄×D₈	D₁₃₁
hypocenter	C₁₃₁	#3118
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #22553927	C₂	semidirect
2. #22553784	C₂	semidirect
3. #22553655	C₂	semidirect
4. #22553635	C₂	semidirect
5. #22553600	C₂	semidirect
6. #22553591	C₂	semidirect
7. #22553526	C₂	semidirect
8. #22553525	C₂	semidirect
9. #22553464	C₂	semidirect
11. #22553422	C₂	semidirect
12. #22553412	C₂	semidirect
13. C₄×#14523502	C₂	semidirect
14. #22553371	C₂	semidirect
15. C₄×D₈×C₁₃₁	C₂	semidirect
20. #16862245	C₂²	semidirect
22. #16862233	C₂²	semidirect
28. C₄×Dic₂₆₂	C₂²	semidirect
38. #16862179	C₂²	semidirect
39. #16862178	C₂²	semidirect
40. #16862177	C₂²	semidirect
41. Dic₁₃₁⋊C₈	C₂²	semidirect
42. Dic₁₃₁⋊C₈	C₂²	semidirect
43. #16862171	C₂²	semidirect
44. #16862169	C₂²	semidirect
47. #16862167	C₂²	semidirect
51. #16862157	C₂²	semidirect
54. C₄×D₄×C₁₃₁	C₂²	semidirect
55. #107×C₁₃₁	C₂²	semidirect
57. #102×C₁₃₁	C₂²	semidirect
58. C₄×C₈×C₁₃₁	C₂²	semidirect
181. C₁₃₁	#3118	semidirect
196. C₄×D₈	D₁₃₁	semidirect
218. C₄×D₄	C₂×D₁₃₁	semidirect
224. #107	C₂×D₁₃₁	semidirect
236. #102	C₂×D₁₃₁	semidirect
240. C₄×C₈	C₂×D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553927	2
#22553784	2
#22553655	2
#22553635	2
#22553600	2
#22553591	2
#22553526	2
#22553525	2
#22553464	2
#22553444	2
#22553422	2
#22553412	2
C₄×#14523502	2
#22553371	2
C₄×D₈×C₁₃₁	2
#3118	131