Subgroups — #30675981

Order 16768.

396 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	#22553846	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14523513	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553654	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553637	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553620	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553613	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553535	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553528	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553476	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553459	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₅₂₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14523502	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#14523502	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₁₀₄₈	C₂	1	1	yes	maximal, characteristic, complemented
15	#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523513	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₂₄	C₂²	1	1	yes	characteristic
18	C₂×#14523502	C₂²	1	1	yes	characteristic
19	C₂×#14523502	C₂²	1	1	yes	characteristic
20	C₂×D₁₀₄₈	C₂²	1	1	yes	characteristic
21	#16862240	—	2	1	no	—
22	#16862238	C₂²	1	1	yes	characteristic
23	#16862234	C₂²	1	1	yes	characteristic
24	#16862229	—	2	1	no	—
25	#16862225	C₂²	1	1	yes	characteristic
26	#16862220	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
30	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
31	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
32	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
34	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
35	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
36	#16862205	C₂²	1	1	yes	characteristic
37	#16862203	—	2	1	no	—
38	#16862203	—	2	1	no	—
39	C₄²×D₁₃₁	—	2	1	no	—
40	#16862180	C₂²	1	1	yes	characteristic
41	#16862180	C₂²	1	1	yes	characteristic
42	#16862178	C₂²	1	1	yes	characteristic
43	#16862178	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×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
57	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523513	—	2	2	no	—
60	#14523513	—	2	2	no	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
63	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
64	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
66	C₂×C₂²⋊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	—	2	1	no	—
71	Dic₅₂₄	—	2	2	no	—
72	#14523502	—	2	2	no	—
73	#14523502	—	2	2	no	—
74	D₁₀₄₈	—	2	2	no	—
75	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	—	2	1	no	—
84	Dic₁₃₁⋊C₄	—	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	—	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	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
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	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
105	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
106	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
107	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₁	—	2	1	no	—
109	C₂²×D₁₃₁	—	2	1	no	—
110	Dic₂₆₂	—	2	1	no	—
111	Dic₂₆₂	—	2	1	no	—
112	Dic₂₆₂	—	2	1	no	—
113	Dic₂₆₂	#41	1	2	yes	—
114	D₅₂₄	—	2	1	no	—
115	D₅₂₄	—	2	1	no	—
116	D₅₂₄	—	2	1	no	—
117	D₅₂₄	#41	1	2	yes	—
118	C₂²⋊D₁₃₁	—	2	4	no	—
119	C₂²⋊D₁₃₁	—	4	2	no	—
120	C₂×Dic₁₃₁	—	2	1	no	—
121	C₂×Dic₁₃₁	—	2	1	no	—
122	C₂×Dic₁₃₁	—	2	1	no	—
123	C₂×Dic₁₃₁	—	2	1	no	—
124	C₂×Dic₁₃₁	—	2	1	no	—
125	C₂×Dic₁₃₁	—	2	1	no	—
126	C₄×D₁₃₁	—	2	4	no	—
127	C₄×D₁₃₁	—	4	2	no	—
128	C₄×D₁₃₁	—	4	2	no	—
129	C₄×D₁₃₁	—	4	2	no	—
130	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₁	—	2	2	no	—
132	C₂×C₄×C₁₃₁	—	2	1	no	—
133	C₂×C₄×C₁₃₁	—	2	1	no	—
134	C₂×C₄×C₁₃₁	—	4	1	no	—
135	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
141	C₈×C₁₃₁	—	2	1	no	—
142	C₈×C₁₃₁	—	2	1	no	—
143	C₈×C₁₃₁	—	2	1	no	—
144	C₈×C₁₃₁	—	2	1	no	—
145	C₂²×C₁₃₁	—	2	1	no	—
146	C₂²×C₁₃₁	—	2	1	no	—
147	C₂²×C₁₃₁	—	2	1	no	—
148	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₃₁	—	2	2	no	—
150	C₂×D₁₃₁	—	4	1	no	—
151	C₂×D₁₃₁	—	4	1	no	—
152	C₂×D₁₃₁	—	4	1	no	—
153	C₂×D₁₃₁	—	4	1	no	—
154	C₂×D₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	—	2	1	no	—
156	C₄×C₁₃₁	—	2	1	no	—
157	C₄×C₁₃₁	—	2	1	no	—
158	C₄×C₁₃₁	—	2	1	no	—
159	C₄×C₁₃₁	#135	1	2	yes	—
160	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
162	Dic₁₃₁	—	2	2	no	—
163	Dic₁₃₁	—	4	1	no	—
164	Dic₁₃₁	—	4	1	no	—
165	Dic₁₃₁	—	4	1	no	—
166	Dic₁₃₁	—	4	1	no	—
167	Dic₁₃₁	—	4	1	no	—
168	C₂×C₁₃₁	—	4	1	no	—
169	C₂×C₁₃₁	#525	1	1	yes	characteristic
170	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
171	C₂×C₁₃₁	#575	1	1	yes	characteristic
172	D₁₃₁	—	4	2	no	—
173	D₁₃₁	—	8	1	no	—
174	C₁₃₁	#3102	1	1	yes	characteristic, complemented, Sylow
175	#3102	—	131	1	no	maximal, Sylow
176	#529	—	131	1	no	—
177	C₄×#41	—	131	1	no	—
178	#485	—	131	1	no	—
179	#482	—	131	1	no	—
180	#479	—	131	1	no	—
181	#478	—	131	1	no	—
182	#464	—	131	1	no	—
183	#463	—	131	1	no	—
184	#452	—	131	1	no	—
185	#449	—	131	1	no	—
186	C₄×Q₁₆	—	131	1	no	—
187	C₄×SD₁₆	—	131	1	no	—
188	C₄×SD₁₆	—	131	1	no	—
189	C₄×D₈	—	131	1	no	—
190	#431	D₁₃₁	1	1	yes	characteristic, complemented
191	C₂×#41	—	131	1	no	—
192	C₂×Q₁₆	—	131	1	no	—
193	C₂×SD₁₆	—	131	1	no	—
194	C₂×SD₁₆	—	131	1	no	—
195	C₂×D₈	—	131	1	no	—
196	#126	—	262	1	no	—
197	#125	—	131	1	no	—
198	#124	—	131	1	no	—
199	#123	—	262	1	no	—
200	C₂²⋊Q₈	—	131	1	no	—
201	#121	—	131	1	no	—
202	C₄×Q₈	—	131	1	no	—
203	C₄×Q₈	—	131	1	no	—
204	C₄×Q₈	—	131	1	no	—
205	C₄×D₄	—	262	1	no	—
206	C₄×D₄	—	262	1	no	—
207	C₄×D₄	—	262	1	no	—
208	C₄×D₄	—	131	1	no	—
209	C₄×D₄	—	131	1	no	—
210	C₄×D₄	—	131	1	no	—
211	#117	—	262	1	no	—
212	#117	—	262	1	no	—
213	#117	—	131	1	no	—
214	C₂×C₄²	—	262	1	no	—
215	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
216	#107	—	131	1	no	—
217	#107	—	131	1	no	—
218	#106	—	131	1	no	—
219	#106	—	131	1	no	—
220	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
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	—	131	1	no	—
226	#102	—	131	1	no	—
227	#102	—	131	1	no	—
228	#102	—	131	1	no	—
229	#102	—	131	1	no	—
230	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
231	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
232	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
233	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
234	#41	—	262	2	no	—
235	#41	—	262	2	no	—
236	C₂×Q₈	—	131	1	no	—
237	C₂×Q₈	—	131	1	no	—
238	C₂×D₄	—	262	1	no	—
239	C₂×D₄	—	262	1	no	—
240	C₂×D₄	—	131	1	no	—
241	C₂×D₄	—	131	1	no	—
242	C₂²×C₄	—	262	1	no	—
243	C₂²×C₄	—	262	1	no	—
244	C₂²×C₄	—	262	1	no	—
245	C₂²×C₄	—	262	1	no	—
246	C₂²×C₄	D₅₂₄	1	1	yes	characteristic
247	C₂²×C₄	D₅₂₄	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
249	Q₁₆	—	262	2	no	—
250	SD₁₆	—	262	2	no	—
251	SD₁₆	—	262	2	no	—
252	D₈	—	262	2	no	—
253	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
257	C₄⋊C₄	—	262	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	C₂²⋊C₄	—	262	1	no	—
268	C₂²⋊C₄	—	262	1	no	—
269	C₂²⋊C₄	—	262	1	no	—
270	C₂²⋊C₄	—	262	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	C₄²	—	262	1	no	—
276	C₄²	—	262	1	no	—
277	C₄²	—	262	1	no	—
278	C₄²	—	262	1	no	—
279	C₄²	D₅₂₄	1	1	yes	characteristic
280	C₄²	D₅₂₄	1	1	yes	characteristic
281	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
282	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
283	C₂³	—	262	1	no	—
284	C₂³	—	262	1	no	—
285	C₂³	C₂×D₅₂₄	1	1	yes	characteristic
286	Q₈	—	262	1	no	—
287	Q₈	—	262	1	no	—
288	Q₈	—	262	1	no	—
289	Q₈	—	131	2	no	—
290	D₄	—	262	4	no	—
291	D₄	—	262	1	no	—
292	D₄	—	262	1	no	—
293	D₄	—	262	1	no	—
294	D₄	—	524	2	no	—
295	D₄	—	131	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	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	—	262	1	no	—
307	C₂×C₄	—	524	2	no	—
308	C₂×C₄	—	524	2	no	—
309	C₂×C₄	—	524	2	no	—
310	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
311	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
312	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
313	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
314	C₂×C₄	C₂×D₅₂₄	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	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	262	2	no	—
324	C₂²	—	524	1	no	—
325	C₂²	—	524	1	no	—
326	C₂²	—	524	1	no	—
327	C₂²	—	524	1	no	—
328	C₂²	—	524	1	no	—
329	C₂²	C₂²×D₅₂₄	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	262	2	no	—
335	C₄	—	524	1	no	—
336	C₄	—	524	1	no	—
337	C₄	—	524	1	no	—
338	C₄	—	524	1	no	—
339	C₄	—	524	1	no	—
340	C₄	#16862268	1	2	yes	—
341	C₄	#41×D₁₃₁	1	1	yes	characteristic
342	C₄	#41×D₁₃₁	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	524	2	no	—
345	C₂	—	1048	1	no	—
346	C₂	#22553824	1	1	yes	characteristic
347	C₂	C₂×#16862268	1	1	yes	characteristic
348	C₂	#22554112	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₂₄
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#431×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553459	C₂
hypercenter	#431	D₁₃₁
hypocenter	C₁₃₁	#3102
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
1. #22553846	C₂	semidirect
2. C₄×#14523513	C₂	semidirect
3. #22553654	C₂	semidirect
4. #22553637	C₂	semidirect
5. #22553620	C₂	semidirect
6. #22553613	C₂	semidirect
7. #22553535	C₂	semidirect
8. #22553528	C₂	semidirect
9. #22553476	C₂	semidirect
11. C₄×Dic₅₂₄	C₂	semidirect
12. C₄×#14523502	C₂	semidirect
13. C₄×#14523502	C₂	semidirect
14. C₄×D₁₀₄₈	C₂	semidirect
15. #431×C₁₃₁	C₂	semidirect
174. C₁₃₁	#3102	semidirect
190. #431	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553846	2
C₄×#14523513	2
#22553654	2
#22553637	2
#22553620	2
#22553613	2
#22553535	2
#22553528	2
#22553476	2
#22553459	2
C₄×Dic₅₂₄	2
C₄×#14523502	2
C₄×D₁₀₄₈	2
#431×C₁₃₁	2
#3102	131