Subgroups — #30677954

Order 16768.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553971	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553966	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553672	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553648	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553626	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553608	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553558	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553548	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553528	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553459	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553453	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553414	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553314	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#16862171	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523513	C₂²	1	1	yes	characteristic
17	C₂×#14523512	—	2	1	no	—
18	C₂²×Dic₂₆₂	C₂²	1	1	yes	characteristic
19	C₂×#14523502	C₂²	1	1	yes	characteristic
20	C₂×#14523502	C₂²	1	1	yes	characteristic
21	C₂×D₁₀₄₈	C₂²	1	1	yes	characteristic
22	#16862246	C₂²	1	1	yes	characteristic
23	#16862234	C₂²	1	1	yes	characteristic
24	#16862232	—	2	1	no	—
25	#16862230	—	2	1	no	—
26	#16862230	—	2	1	no	—
27	#16862226	—	2	1	no	—
28	#16862225	C₂²	1	1	yes	characteristic
29	#16862224	—	2	1	no	—
30	#16862220	C₂²	1	1	yes	characteristic
31	#16862220	C₂²	1	1	yes	characteristic
32	#16862220	C₂²	1	1	yes	characteristic
33	#16862218	—	2	1	no	—
34	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
35	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₁₃₁)⋊C₄	—	2	1	no	—
37	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
39	#16862180	C₂²	1	1	yes	characteristic
40	#16862180	C₂²	1	1	yes	characteristic
41	#16862180	C₂²	1	1	yes	characteristic
42	#16862171	C₂²	1	1	yes	characteristic
43	#16862171	C₂²	1	1	yes	characteristic
44	#16862171	C₂²	1	1	yes	characteristic
45	#16862171	C₂²	1	1	yes	characteristic
46	#16862171	C₂²	1	1	yes	characteristic
47	#16862171	C₂²	1	1	yes	characteristic
48	#16862168	C₂²	1	1	yes	characteristic
49	#16862168	C₂²	1	1	yes	characteristic
50	#16862168	C₂²	1	1	yes	characteristic
51	#16862168	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
58	#97×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523513	—	2	2	no	—
60	#14523513	—	4	1	no	—
61	#14523512	—	4	4	no	—
62	C₂×Dic₂₆₂	—	2	2	no	—
63	C₂×Dic₂₆₂	—	2	1	no	—
64	C₂×Dic₂₆₂	—	2	1	no	—
65	C₂×Dic₂₆₂	—	4	1	no	—
66	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
68	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
69	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
71	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
72	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
73	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
74	C₂²×Dic₁₃₁	—	2	1	no	—
75	C₂²×Dic₁₃₁	—	2	1	no	—
76	C₂²×Dic₁₃₁	—	2	1	no	—
77	C₂×C₄×D₁₃₁	—	2	1	no	—
78	C₂×C₄×D₁₃₁	—	2	1	no	—
79	#14523502	—	2	2	no	—
80	#14523502	—	4	1	no	—
81	D₁₀₄₈	—	4	1	no	—
82	C₄⋊Dic₁₃₁	—	2	1	no	—
83	C₄⋊Dic₁₃₁	—	2	1	no	—
84	C₄⋊Dic₁₃₁	—	2	1	no	—
85	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
89	Dic₁₃₁⋊C₄	—	2	1	no	—
90	Dic₁₃₁⋊C₄	—	2	1	no	—
91	C₂²⋊Dic₁₃₁	—	2	1	no	—
92	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
93	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
94	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
95	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
96	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
97	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
98	C₄×Dic₁₃₁	—	2	1	no	—
99	C₂×D₄×C₁₃₁	D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
102	M₁₆×C₁₃₁	—	4	1	no	—
103	C₂×C₈×C₁₃₁	—	4	1	no	—
104	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
109	C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
110	C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
111	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
112	C₂²×D₁₃₁	—	2	1	no	—
113	C₂²×D₁₃₁	—	2	1	no	—
114	Dic₂₆₂	—	2	1	no	—
115	Dic₂₆₂	—	2	1	no	—
116	Dic₂₆₂	—	2	1	no	—
117	Dic₂₆₂	—	4	2	no	—
118	Dic₂₆₂	—	4	2	no	—
119	Dic₂₆₂	#41	1	2	yes	complemented
120	D₅₂₄	—	2	1	no	—
121	D₅₂₄	—	2	1	no	—
122	D₅₂₄	—	2	1	no	—
123	D₅₂₄	—	2	1	no	—
124	C₂²⋊D₁₃₁	—	4	2	no	—
125	C₂²⋊D₁₃₁	—	4	2	no	—
126	C₂²⋊D₁₃₁	—	4	2	no	—
127	C₂²⋊D₁₃₁	—	4	2	no	—
128	C₂×Dic₁₃₁	—	2	1	no	—
129	C₂×Dic₁₃₁	—	2	1	no	—
130	C₂×Dic₁₃₁	—	2	1	no	—
131	C₂×Dic₁₃₁	—	2	1	no	—
132	C₂×Dic₁₃₁	—	2	1	no	—
133	C₂×Dic₁₃₁	—	2	1	no	—
134	C₂×Dic₁₃₁	—	4	2	no	—
135	C₂×Dic₁₃₁	—	4	2	no	—
136	C₂×Dic₁₃₁	—	4	2	no	—
137	C₄×D₁₃₁	—	4	2	no	—
138	C₄×D₁₃₁	—	4	2	no	—
139	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
141	D₄×C₁₃₁	—	2	2	no	—
142	C₂×C₄×C₁₃₁	—	2	1	no	—
143	C₂×C₄×C₁₃₁	—	2	1	no	—
144	C₂×C₄×C₁₃₁	—	2	1	no	—
145	C₂×C₄×C₁₃₁	—	2	1	no	—
146	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
151	C₈×C₁₃₁	—	2	1	no	—
152	C₈×C₁₃₁	—	2	1	no	—
153	C₈×C₁₃₁	—	2	1	no	—
154	C₈×C₁₃₁	—	2	1	no	—
155	C₂²×C₁₃₁	—	2	1	no	—
156	C₂²×C₁₃₁	—	2	1	no	—
157	C₂²×C₁₃₁	—	2	1	no	—
158	C₂²×C₁₃₁	—	2	1	no	—
159	C₂²×C₁₃₁	—	2	1	no	—
160	C₂²×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
161	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₁₃₁	—	4	1	no	—
163	C₂×D₁₃₁	—	4	1	no	—
164	C₂×D₁₃₁	—	4	1	no	—
165	C₂×D₁₃₁	—	4	1	no	—
166	C₂×D₁₃₁	—	4	1	no	—
167	C₂×D₁₃₁	—	4	1	no	—
168	C₄×C₁₃₁	—	2	1	no	—
169	C₄×C₁₃₁	—	2	1	no	—
170	C₄×C₁₃₁	—	2	1	no	—
171	C₄×C₁₃₁	—	2	1	no	—
172	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
173	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
174	Dic₁₃₁	—	2	2	no	—
175	Dic₁₃₁	—	4	1	no	—
176	Dic₁₃₁	—	4	1	no	—
177	Dic₁₃₁	—	4	1	no	—
178	Dic₁₃₁	—	4	1	no	—
179	Dic₁₃₁	—	4	1	no	—
180	C₂×C₁₃₁	—	2	2	no	—
181	C₂×C₁₃₁	—	4	1	no	—
182	C₂×C₁₃₁	#546	1	1	yes	characteristic
183	C₂×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₁₃₁	#578	1	1	yes	characteristic
185	D₁₃₁	—	8	1	no	—
186	D₁₃₁	—	8	1	no	—
187	C₁₃₁	#3308	1	1	yes	characteristic, complemented, Sylow
188	#3308	—	131	1	no	maximal, Sylow
189	#548	—	131	1	no	—
190	#547	—	131	1	no	—
191	#489	—	131	1	no	—
192	#484	—	131	1	no	—
193	#480	—	131	1	no	—
194	#477	—	131	1	no	—
195	#468	—	131	1	no	—
196	#466	—	131	1	no	—
197	#463	—	131	1	no	—
198	#449	—	131	1	no	—
199	#448	—	131	1	no	—
200	#440	—	131	1	no	—
201	#435	D₁₃₁	1	1	yes	characteristic, complemented
202	#417	—	131	1	no	—
203	C₂×#103	—	131	1	no	—
204	C₂×#41	—	262	1	no	—
205	C₂×#41	—	131	1	no	—
206	C₂²×Q₈	—	131	1	no	—
207	C₂×SD₁₆	—	131	1	no	—
208	C₂×SD₁₆	—	131	1	no	—
209	C₂×D₈	—	131	1	no	—
210	C₂×M₁₆	C₂×D₁₃₁	1	1	yes	characteristic
211	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
212	C₄⋊Q₈	—	131	1	no	—
213	#124	—	262	1	no	—
214	#124	—	131	1	no	—
215	#123	—	262	1	no	—
216	#123	—	262	1	no	—
217	C₂²⋊Q₈	—	262	1	no	—
218	C₂²⋊Q₈	—	262	1	no	—
219	C₂²⋊Q₈	—	131	1	no	—
220	#121	—	262	1	no	—
221	#121	—	131	1	no	—
222	#121	—	131	1	no	—
223	#121	—	131	1	no	—
224	C₄×Q₈	—	131	1	no	—
225	C₄×D₄	—	262	1	no	—
226	C₄×D₄	—	131	1	no	—
227	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
228	C₂×C₄⋊C₄	—	131	1	no	—
229	C₂×C₄⋊C₄	—	131	1	no	—
230	#107	—	131	1	no	—
231	#107	—	131	1	no	—
232	#107	—	131	1	no	—
233	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
234	#103	—	131	1	no	—
235	#103	—	131	1	no	—
236	#103	—	131	1	no	—
237	#103	—	131	1	no	—
238	#103	—	131	1	no	—
239	#103	—	131	1	no	—
240	#102	—	131	1	no	—
241	#102	—	131	1	no	—
242	#102	—	131	1	no	—
243	#102	—	131	1	no	—
244	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
245	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
246	#97	C₂×D₁₃₁	1	1	yes	characteristic
247	#41	—	262	2	no	—
248	#41	—	524	1	no	—
249	#41	—	524	4	no	—
250	C₂×Q₈	—	262	2	no	—
251	C₂×Q₈	—	262	1	no	—
252	C₂×Q₈	—	262	1	no	—
253	C₂×Q₈	—	524	1	no	—
254	C₂×Q₈	—	131	1	no	—
255	C₂×Q₈	—	131	1	no	—
256	C₂×D₄	—	262	1	no	—
257	C₂×D₄	—	262	1	no	—
258	C₂×D₄	—	262	1	no	—
259	C₂×D₄	—	262	1	no	—
260	C₂×D₄	—	131	1	no	—
261	C₂×D₄	—	131	1	no	—
262	C₂×D₄	D₅₂₄	1	1	yes	characteristic
263	C₂²×C₄	—	262	1	no	—
264	C₂²×C₄	—	262	1	no	—
265	C₂²×C₄	—	262	1	no	—
266	C₂²×C₄	—	262	1	no	—
267	C₂²×C₄	—	262	1	no	—
268	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
269	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
270	SD₁₆	—	262	2	no	—
271	SD₁₆	—	524	1	no	—
272	D₈	—	524	1	no	—
273	M₁₆	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
278	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
279	C₄⋊C₄	—	262	1	no	—
280	C₄⋊C₄	—	262	1	no	—
281	C₄⋊C₄	—	262	1	no	—
282	C₄⋊C₄	—	262	1	no	—
283	C₄⋊C₄	—	262	1	no	—
284	C₄⋊C₄	—	131	1	no	—
285	C₄⋊C₄	—	131	1	no	—
286	C₄⋊C₄	—	131	1	no	—
287	C₄⋊C₄	—	131	1	no	—
288	C₄⋊C₄	D₅₂₄	1	1	yes	characteristic
289	C₂²⋊C₄	—	262	1	no	—
290	C₂²⋊C₄	—	262	1	no	—
291	C₂²⋊C₄	—	262	1	no	—
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	D₅₂₄	1	1	yes	characteristic
297	C₂²⋊C₄	D₅₂₄	1	1	yes	characteristic
298	C₄²	—	262	1	no	—
299	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
300	C₂³	—	262	1	no	—
301	C₂³	—	262	1	no	—
302	C₂³	C₂×D₅₂₄	1	1	yes	characteristic
303	C₂³	C₂×D₅₂₄	1	1	yes	characteristic
304	Q₈	—	262	1	no	—
305	Q₈	—	262	1	no	—
306	Q₈	—	262	1	no	—
307	Q₈	—	524	2	no	—
308	Q₈	—	524	2	no	—
309	Q₈	—	131	2	no	—
310	D₄	—	2	2	no	—
311	D₄	—	262	1	no	—
312	D₄	—	262	1	no	—
313	D₄	—	262	1	no	—
314	D₄	—	262	1	no	—
315	D₄	—	524	2	no	—
316	D₄	—	524	2	no	—
317	D₄	—	524	2	no	—
318	D₄	—	524	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	262	1	no	—
324	C₂×C₄	—	262	1	no	—
325	C₂×C₄	—	262	1	no	—
326	C₂×C₄	—	262	1	no	—
327	C₂×C₄	—	262	1	no	—
328	C₂×C₄	—	262	1	no	—
329	C₂×C₄	—	524	2	no	—
330	C₂×C₄	—	524	2	no	—
331	C₂×C₄	—	524	2	no	—
332	C₂×C₄	—	524	2	no	—
333	C₂×C₄	—	524	2	no	—
334	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
335	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
336	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
337	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	524	1	no	—
349	C₂²	—	524	1	no	—
350	C₂²	—	524	1	no	—
351	C₂²	—	524	1	no	—
352	C₂²	—	524	1	no	—
353	C₂²	—	524	1	no	—
354	C₂²	#16862274	1	2	yes	—
355	C₂²	C₂²×D₅₂₄	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	262	2	no	—
361	C₄	—	524	1	no	—
362	C₄	—	524	1	no	—
363	C₄	—	524	1	no	—
364	C₄	—	524	1	no	—
365	C₄	—	524	1	no	—
366	C₄	#41×D₁₃₁	1	1	yes	characteristic
367	C₄	#16862291	1	1	yes	characteristic
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	—	1048	1	no	—
371	C₂	—	1048	1	no	—
372	C₂	#22553957	1	1	yes	characteristic
373	C₂	C₂×#16862274	1	1	yes	characteristic
374	C₂	#22554129	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂₄
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#435×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
hypercenter	#435	D₁₃₁
hypocenter	C₁₃₁	#3308
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. #22553971	C₂	semidirect
2. #22553966	C₂	semidirect
3. #22553672	C₂	semidirect
4. #22553648	C₂	semidirect
5. #22553626	C₂	semidirect
6. #22553608	C₂	semidirect
7. #22553558	C₂	semidirect
8. #22553548	C₂	semidirect
9. #22553528	C₂	semidirect
10. #22553459	C₂	semidirect
11. #22553453	C₂	semidirect
12. #22553414	C₂	semidirect
13. #22553314	C₂	semidirect
14. C₂×#16862171	C₂	semidirect
15. #435×C₁₃₁	C₂	semidirect
119. Dic₂₆₂	#41	semidirect
187. C₁₃₁	#3308	semidirect
201. #435	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553971	2
#22553966	2
#22553672	2
#22553648	2
#22553626	2
#22553608	2
#22553558	2
#22553548	2
#22553528	2
#22553459	2
#22553453	2
#22553414	2
#22553314	2
C₂×#16862171	2
#435×C₁₃₁	2
#3308	131