Subgroups — #30676236

Order 16768.

384 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	#22553934	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553794	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553641	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553632	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553601	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553595	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553536	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553530	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553473	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553460	C₂	1	1	yes	maximal, characteristic
11	#22553425	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14523500	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#14523500	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553374	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523514	C₂²	1	1	yes	characteristic
17	C₂×#14523500	C₂²	1	1	yes	characteristic
18	C₂×#14523500	C₂²	1	1	yes	characteristic
19	C₂×D₁₀₄₈	C₂²	1	1	yes	characteristic
20	#16862245	C₂²	1	1	yes	characteristic
21	#16862240	—	2	1	no	—
22	#16862235	C₂²	1	1	yes	characteristic
23	#16862234	—	2	1	no	—
24	#16862231	—	2	1	no	—
25	#16862224	—	2	1	no	—
26	#16862222	C₂²	1	1	yes	characteristic
27	#16862221	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
29	C₄×D₅₂₄	—	2	1	no	—
30	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
31	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
32	#16862208	—	2	1	no	—
33	#16862208	C₂²	1	1	yes	characteristic
34	#16862203	—	2	1	no	—
35	#16862203	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₁	—	2	1	no	—
37	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
38	#16862178	C₂²	1	1	yes	characteristic
39	#16862177	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
43	#16862170	C₂²	1	1	yes	characteristic
44	#16862170	C₂²	1	1	yes	characteristic
45	#16862168	C₂²	1	1	yes	characteristic
46	#16862168	C₂²	1	1	yes	characteristic
47	#16862166	C₂²	1	1	yes	characteristic
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	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×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	#97×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523514	—	2	2	no	—
60	#14523514	—	2	2	no	—
61	C₂×Dic₂₆₂	—	2	1	no	—
62	C₂×D₅₂₄	—	2	1	no	—
63	C₂×D₅₂₄	—	2	1	no	—
64	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
65	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
66	C₂×C₄×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	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
72	#14523500	—	2	2	no	—
73	#14523500	—	2	2	no	—
74	D₁₀₄₈	—	2	2	no	—
75	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₁	—	2	1	no	—
78	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
80	Dic₁₃₁⋊C₄	—	2	1	no	—
81	Dic₁₃₁⋊C₄	—	2	1	no	—
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
85	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₁)⋊C₄	—	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	D₄	1	1	yes	characteristic
93	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
94	C₄×Dic₁₃₁	—	2	1	no	—
95	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₃₁	—	2	2	no	—
99	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₁	—	2	1	no	—
102	C₄⋊C₄×C₁₃₁	—	2	1	no	—
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₄²×C₁₃₁	—	2	1	no	—
106	C₄²×C₁₃₁	—	2	1	no	—
107	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₁	—	2	1	no	—
109	C₂²×D₁₃₁	—	2	1	no	—
110	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
111	Dic₂₆₂	—	8	1	no	—
112	D₅₂₄	—	2	4	no	—
113	D₅₂₄	—	2	1	no	—
114	D₅₂₄	—	2	1	no	—
115	D₅₂₄	—	2	1	no	—
116	D₅₂₄	—	8	1	no	—
117	D₅₂₄	#41	1	2	yes	—
118	C₂×Dic₁₃₁	—	2	1	no	—
119	C₂×Dic₁₃₁	—	2	1	no	—
120	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₃₁	—	2	4	no	—
124	C₄×D₁₃₁	—	2	1	no	—
125	C₄×D₁₃₁	—	2	1	no	—
126	C₄×D₁₃₁	—	8	1	no	—
127	C₄×D₁₃₁	—	8	1	no	—
128	C₄×D₁₃₁	—	8	1	no	—
129	C₄×D₁₃₁	—	8	1	no	—
130	C₁₃₁⋊C₈	—	2	1	no	—
131	C₁₃₁⋊C₈	—	2	1	no	—
132	Q₈×C₁₃₁	—	2	1	no	—
133	Q₈×C₁₃₁	—	2	1	no	—
134	Q₈×C₁₃₁	—	2	1	no	—
135	Q₈×C₁₃₁	#41	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₃₁	—	2	1	no	—
144	C₈×C₁₃₁	—	2	1	no	—
145	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₃₁	—	2	1	no	—
147	C₂×D₁₃₁	—	2	2	no	—
148	C₂×D₁₃₁	—	2	1	no	—
149	C₂×D₁₃₁	—	2	1	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	2	no	—
156	C₄×C₁₃₁	—	2	1	no	—
157	C₄×C₁₃₁	—	2	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	—	2	1	no	—
165	Dic₁₃₁	—	2	1	no	—
166	Dic₁₃₁	—	4	1	no	—
167	Dic₁₃₁	—	4	1	no	—
168	C₂×C₁₃₁	#525	1	1	yes	characteristic
169	C₂×C₁₃₁	#577	1	1	yes	characteristic
170	C₂×C₁₃₁	#577	1	1	yes	characteristic
171	D₁₃₁	—	4	1	no	—
172	D₁₃₁	—	4	2	no	—
173	D₁₃₁	—	8	1	no	—
174	C₁₃₁	#3125	1	1	yes	characteristic, complemented, Sylow
175	#3125	—	131	1	no	maximal, Sylow
176	#542	—	131	1	no	—
177	#520	—	131	1	no	—
178	#483	—	131	1	no	—
179	#481	—	131	1	no	—
180	#476	—	131	1	no	—
181	#475	—	131	1	no	—
182	#464	—	131	1	no	—
183	#463	—	131	1	no	—
184	#452	—	131	1	no	—
185	#449	—	131	1	no	—
186	#442	—	131	1	no	—
187	#441	D₁₃₁	1	1	yes	characteristic, complemented
188	C₄×SD₁₆	—	131	1	no	—
189	C₄×SD₁₆	—	131	1	no	—
190	#433	—	131	1	no	—
191	C₂×#41	—	131	1	no	—
192	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
193	C₂×SD₁₆	—	131	1	no	—
194	C₂×SD₁₆	—	131	1	no	—
195	C₂×D₈	—	131	1	no	—
196	C₄⋊Q₈	—	131	1	no	—
197	#126	—	262	1	no	—
198	#124	—	262	1	no	—
199	#124	—	131	1	no	—
200	#123	—	262	1	no	—
201	C₂²⋊Q₈	—	262	1	no	—
202	C₂²⋊Q₈	—	131	1	no	—
203	#121	—	131	1	no	—
204	C₄×Q₈	—	131	1	no	—
205	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
206	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
207	C₄×D₄	—	262	1	no	—
208	C₄×D₄	—	262	1	no	—
209	C₄×D₄	—	131	1	no	—
210	C₄×D₄	—	131	1	no	—
211	C₄×D₄	—	131	1	no	—
212	#117	—	262	1	no	—
213	#117	—	131	1	no	—
214	C₂×C₄⋊C₄	—	262	1	no	—
215	C₂×C₄⋊C₄	—	131	1	no	—
216	#106	—	131	1	no	—
217	#106	—	131	1	no	—
218	#106	—	131	1	no	—
219	#106	C₂×D₁₃₁	1	1	yes	characteristic
220	C₄⋊C₈	—	131	1	no	—
221	C₄⋊C₈	—	131	1	no	—
222	#103	—	131	1	no	—
223	#103	—	131	1	no	—
224	#103	C₂×D₁₃₁	1	1	yes	characteristic
225	#103	C₂×D₁₃₁	1	1	yes	characteristic
226	#102	—	131	1	no	—
227	#102	—	131	1	no	—
228	#102	—	131	1	no	—
229	#102	—	131	1	no	—
230	C₂²⋊C₈	—	131	1	no	—
231	C₂²⋊C₈	—	131	1	no	—
232	#97	C₂×D₁₃₁	1	1	yes	characteristic
233	C₄×C₈	—	131	1	no	—
234	#41	—	262	2	no	—
235	#41	—	262	2	no	—
236	C₂×Q₈	—	262	1	no	—
237	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
238	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
239	C₂×D₄	—	262	1	no	—
240	C₂×D₄	—	262	1	no	—
241	C₂×D₄	—	131	1	no	—
242	C₂×D₄	—	131	1	no	—
243	C₂²×C₄	—	262	1	no	—
244	C₂²×C₄	—	262	1	no	—
245	C₂²×C₄	—	262	1	no	—
246	C₂²×C₄	—	262	1	no	—
247	C₂²×C₄	—	262	1	no	—
248	C₂²×C₄	—	131	1	no	—
249	Q₁₆	—	2	2	no	—
250	SD₁₆	—	262	2	no	—
251	SD₁₆	—	262	2	no	—
252	D₈	—	262	2	no	—
253	C₂×C₈	—	131	1	no	—
254	C₂×C₈	—	131	1	no	—
255	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	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	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
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	C₂²⋊C₄	—	262	1	no	—
276	C₂²⋊C₄	—	131	1	no	—
277	C₂²⋊C₄	—	131	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	262	1	no	—
281	C₄²	—	131	1	no	—
282	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
283	C₂³	—	262	1	no	—
284	C₂³	—	262	1	no	—
285	C₂³	—	131	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	1048	1	no	—
290	Q₈	#14523513	1	2	yes	—
291	D₄	—	262	4	no	—
292	D₄	—	262	1	no	—
293	D₄	—	262	1	no	—
294	D₄	—	262	1	no	—
295	D₄	—	1048	1	no	—
296	D₄	—	131	2	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	—	262	4	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	—	1048	1	no	—
307	C₂×C₄	—	1048	1	no	—
308	C₂×C₄	—	1048	1	no	—
309	C₂×C₄	—	1048	1	no	—
310	C₂×C₄	—	131	1	no	—
311	C₂×C₄	—	131	1	no	—
312	C₂×C₄	—	131	1	no	—
313	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
314	C₂×C₄	D₄×D₁₃₁	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	—	262	1	no	—
319	C₈	—	262	1	no	—
320	C₂²	—	262	1	no	—
321	C₂²	—	262	1	no	—
322	C₂²	—	262	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	C₂×D₄×D₁₃₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	262	1	no	—
337	C₄	—	262	1	no	—
338	C₄	—	262	1	no	—
339	C₄	—	524	1	no	—
340	C₄	—	524	1	no	—
341	C₄	#41×D₁₃₁	1	1	yes	characteristic
342	C₄	C₂×#14523513	1	1	yes	characteristic
343	C₂	—	524	1	no	—
344	C₂	—	524	2	no	—
345	C₂	—	1048	1	no	—
346	C₂	#22553821	1	1	yes	characteristic
347	C₂	#22554121	1	1	yes	characteristic
348	C₂	#22554121	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#441×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553460	C₂
hypercenter	#441	D₁₃₁
hypocenter	C₁₃₁	#3125
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. #22553934	C₂	semidirect
2. #22553794	C₂	semidirect
3. #22553641	C₂	semidirect
4. #22553632	C₂	semidirect
5. #22553601	C₂	semidirect
6. #22553595	C₂	semidirect
7. #22553536	C₂	semidirect
8. #22553530	C₂	semidirect
9. #22553473	C₂	semidirect
11. #22553425	C₂	semidirect
12. C₄×#14523500	C₂	semidirect
13. C₄×#14523500	C₂	semidirect
14. #22553374	C₂	semidirect
15. #441×C₁₃₁	C₂	semidirect
174. C₁₃₁	#3125	semidirect
187. #441	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553934	2
#22553794	2
#22553641	2
#22553632	2
#22553601	2
#22553595	2
#22553536	2
#22553530	2
#22553473	2
#22553460	2
#22553425	2
C₄×#14523500	2
#22553374	2
#441×C₁₃₁	2
#3125	131