Subgroups — #30677846

Order 16768.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553997	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553968	C₂	1	1	yes	maximal, characteristic
3	#22553665	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553655	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553618	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553554	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553539	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553533	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553475	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553470	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553406	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₁₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553305	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#16862169	C₂	1	1	yes	maximal, characteristic, complemented
15	#480×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523512	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₆₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₂₄	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₂₄	C₂²	1	1	yes	characteristic
20	C₂×#14523501	C₂²	1	1	yes	characteristic
21	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
23	#16862241	—	2	1	no	—
24	#16862237	C₂²	1	1	yes	characteristic
25	#16862236	—	2	1	no	—
26	#16862233	C₂²	1	1	yes	characteristic
27	#16862232	—	2	1	no	—
28	#16862229	—	2	1	no	—
29	#16862226	C₂²	1	1	yes	characteristic
30	#16862226	C₂²	1	1	yes	characteristic
31	#16862225	C₂²	1	1	yes	characteristic
32	#16862223	—	2	1	no	—
33	#16862218	—	2	1	no	—
34	#16862211	C₂²	1	1	yes	characteristic
35	#16862211	C₂²	1	1	yes	characteristic
36	D₄×Dic₁₃₁	—	2	1	no	—
37	C₄×C₂²⋊D₁₃₁	C₂²	1	1	yes	characteristic
38	#16862205	—	2	1	no	—
39	#16862204	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
42	#16862171	C₂²	1	1	yes	characteristic
43	#16862171	C₂²	1	1	yes	characteristic
44	#16862169	C₂²	1	1	yes	characteristic
45	#16862169	C₂²	1	1	yes	characteristic
46	#16862169	C₂²	1	1	yes	characteristic
47	#16862169	C₂²	1	1	yes	characteristic
48	#16862169	C₂²	1	1	yes	characteristic
49	#16862167	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	#121×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
54	#107×C₁₃₁	C₂²	1	1	yes	characteristic
55	#107×C₁₃₁	C₂²	1	1	yes	characteristic
56	#102×C₁₃₁	C₂²	1	1	yes	characteristic
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523512	—	2	2	no	—
60	#14523512	—	4	1	no	—
61	C₂×Dic₂₆₂	—	2	2	no	—
62	C₂×Dic₂₆₂	—	2	1	no	—
63	C₂×Dic₂₆₂	—	4	1	no	—
64	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
66	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
68	C₂×C₂²⋊D₁₃₁	D₄	1	1	yes	characteristic
69	C₂²×Dic₁₃₁	—	2	1	no	—
70	C₂²×Dic₁₃₁	—	2	1	no	—
71	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
72	Dic₅₂₄	—	2	2	no	—
73	Dic₅₂₄	—	4	1	no	—
74	#14523501	—	4	1	no	—
75	C₂×C₁₃₁⋊C₈	—	4	1	no	—
76	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×D₁₃₁	—	4	1	no	—
79	C₄⋊Dic₁₃₁	—	2	1	no	—
80	C₄⋊Dic₁₃₁	—	2	1	no	—
81	C₄⋊Dic₁₃₁	—	2	1	no	—
82	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
83	Dic₁₃₁⋊C₄	—	2	1	no	—
84	Dic₁₃₁⋊C₄	—	2	1	no	—
85	Dic₁₃₁⋊C₄	—	2	1	no	—
86	Dic₁₃₁⋊C₄	—	2	1	no	—
87	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₃₁	—	2	1	no	—
89	C₂²⋊Dic₁₃₁	—	2	1	no	—
90	C₂²⋊Dic₁₃₁	—	2	1	no	—
91	C₂²⋊Dic₁₃₁	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
96	C₄×Dic₁₃₁	—	2	1	no	—
97	C₄×Dic₁₃₁	—	2	1	no	—
98	C₄×Dic₁₃₁	—	2	1	no	—
99	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₃₁	—	2	1	no	—
101	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₁	—	2	1	no	—
103	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₃₁	—	2	1	no	—
107	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
111	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
112	Dic₂₆₂	—	2	1	no	—
113	Dic₂₆₂	—	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	#41	1	2	yes	—
119	C₂²⋊D₁₃₁	—	2	2	no	—
120	C₂²⋊D₁₃₁	—	4	2	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	—	4	2	no	—
126	C₂×Dic₁₃₁	—	4	2	no	—
127	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄×D₁₃₁	—	2	1	no	—
131	C₄×D₁₃₁	—	2	1	no	—
132	C₁₃₁⋊C₈	—	2	1	no	—
133	C₁₃₁⋊C₈	—	2	1	no	—
134	C₂³×C₁₃₁	—	2	1	no	—
135	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
136	D₄×C₁₃₁	—	2	1	no	—
137	D₄×C₁₃₁	—	2	1	no	—
138	D₄×C₁₃₁	—	4	2	no	—
139	C₂×C₄×C₁₃₁	—	2	1	no	—
140	C₂×C₄×C₁₃₁	—	2	1	no	—
141	C₂×C₄×C₁₃₁	—	2	1	no	—
142	C₂×C₄×C₁₃₁	—	2	1	no	—
143	C₂×C₄×C₁₃₁	—	2	1	no	—
144	C₂×C₄×C₁₃₁	—	4	2	no	—
145	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
147	C₈×C₁₃₁	—	2	1	no	—
148	C₈×C₁₃₁	—	2	1	no	—
149	C₂²×C₁₃₁	—	2	1	no	—
150	C₂²×C₁₃₁	—	2	1	no	—
151	C₂²×C₁₃₁	—	4	1	no	—
152	C₂²×C₁₃₁	—	4	1	no	—
153	C₂²×C₁₃₁	—	4	1	no	—
154	C₂²×C₁₃₁	#135	1	2	yes	—
155	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₃₁	—	2	1	no	—
157	C₂×D₁₃₁	—	2	1	no	—
158	C₂×D₁₃₁	—	2	1	no	—
159	C₄×C₁₃₁	—	2	1	no	—
160	C₄×C₁₃₁	—	4	1	no	—
161	C₄×C₁₃₁	—	4	1	no	—
162	C₄×C₁₃₁	—	4	1	no	—
163	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₁₃₁	—	2	1	no	—
166	Dic₁₃₁	—	2	2	no	—
167	Dic₁₃₁	—	2	1	no	—
168	Dic₁₃₁	—	2	1	no	—
169	Dic₁₃₁	—	4	1	no	—
170	Dic₁₃₁	—	4	1	no	—
171	Dic₁₃₁	—	4	1	no	—
172	C₂×C₁₃₁	—	2	2	no	—
173	C₂×C₁₃₁	—	8	1	no	—
174	C₂×C₁₃₁	#546	1	1	yes	characteristic
175	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
176	C₂×C₁₃₁	#578	1	1	yes	characteristic
177	D₁₃₁	—	4	1	no	—
178	C₁₃₁	#3300	1	1	yes	characteristic, complemented, Sylow
179	#3300	—	131	1	no	maximal, Sylow
180	#552	—	131	1	no	—
181	#548	—	131	1	no	—
182	#488	—	131	1	no	—
183	#485	—	131	1	no	—
184	#480	D₁₃₁	1	1	yes	characteristic, complemented
185	#479	—	131	1	no	—
186	#467	—	131	1	no	—
187	#465	—	131	1	no	—
188	#464	—	131	1	no	—
189	#452	—	131	1	no	—
190	#451	—	131	1	no	—
191	C₄×Q₁₆	—	131	1	no	—
192	C₈×D₄	—	131	1	no	—
193	#416	—	131	1	no	—
194	C₂×#103	—	131	1	no	—
195	C₂×#41	—	131	1	no	—
196	C₂²×Q₈	—	131	1	no	—
197	C₂×Q₁₆	—	131	1	no	—
198	C₂×Q₁₆	—	131	1	no	—
199	C₂×SD₁₆	—	131	1	no	—
200	C₂²×C₈	—	131	1	no	—
201	C₂²×C₈	—	131	1	no	—
202	#126	—	262	1	no	—
203	#125	—	262	1	no	—
204	#125	—	131	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	C₂²⋊Q₈	—	131	1	no	—
211	C₂²⋊Q₈	—	131	1	no	—
212	#121	—	262	1	no	—
213	#121	C₂×D₁₃₁	1	1	yes	characteristic
214	C₄×Q₈	—	131	1	no	—
215	C₄×Q₈	—	131	1	no	—
216	C₄×D₄	—	262	1	no	—
217	C₄×D₄	—	131	1	no	—
218	#117	—	262	1	no	—
219	#117	—	131	1	no	—
220	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
221	#107	C₂×D₁₃₁	1	1	yes	characteristic
222	#107	C₂×D₁₃₁	1	1	yes	characteristic
223	#106	—	131	1	no	—
224	C₄⋊C₈	—	131	1	no	—
225	#103	—	131	1	no	—
226	#103	—	131	1	no	—
227	#103	—	131	1	no	—
228	#103	—	131	1	no	—
229	#103	—	131	1	no	—
230	#103	—	131	1	no	—
231	#103	—	131	1	no	—
232	#102	—	131	1	no	—
233	#102	C₂×D₁₃₁	1	1	yes	characteristic
234	#102	C₂×D₁₃₁	1	1	yes	characteristic
235	C₂²⋊C₈	—	131	1	no	—
236	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
237	C₄×C₈	—	131	1	no	—
238	#41	—	262	2	no	—
239	#41	—	524	1	no	—
240	C₂×Q₈	—	262	2	no	—
241	C₂×Q₈	—	262	1	no	—
242	C₂×Q₈	—	524	1	no	—
243	C₂×Q₈	—	131	1	no	—
244	C₂×Q₈	—	131	1	no	—
245	C₂×Q₈	—	131	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	262	1	no	—
248	C₂×D₄	—	131	1	no	—
249	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	262	1	no	—
252	C₂²×C₄	—	262	1	no	—
253	C₂²×C₄	—	131	1	no	—
254	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
255	Q₁₆	—	262	2	no	—
256	Q₁₆	—	524	1	no	—
257	SD₁₆	—	524	1	no	—
258	C₂×C₈	—	524	1	no	—
259	C₂×C₈	—	524	1	no	—
260	C₂×C₈	—	131	1	no	—
261	C₂×C₈	—	131	1	no	—
262	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	262	1	no	—
266	C₄⋊C₄	—	262	1	no	—
267	C₄⋊C₄	—	262	1	no	—
268	C₄⋊C₄	—	262	1	no	—
269	C₄⋊C₄	—	262	1	no	—
270	C₄⋊C₄	—	262	1	no	—
271	C₄⋊C₄	—	262	1	no	—
272	C₄⋊C₄	—	131	1	no	—
273	C₄⋊C₄	—	131	1	no	—
274	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	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	—	262	1	no	—
289	C₄²	—	131	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	131	1	no	—
292	C₂³	D₄×D₁₃₁	1	1	yes	characteristic
293	Q₈	—	262	1	no	—
294	Q₈	—	262	1	no	—
295	Q₈	—	262	1	no	—
296	Q₈	—	262	1	no	—
297	Q₈	—	262	1	no	—
298	Q₈	—	524	2	no	—
299	Q₈	—	131	2	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	262	2	no	—
304	D₄	—	524	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	2	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	—	262	1	no	—
316	C₂×C₄	—	262	1	no	—
317	C₂×C₄	—	524	2	no	—
318	C₂×C₄	—	524	2	no	—
319	C₂×C₄	—	131	1	no	—
320	C₂×C₄	—	131	1	no	—
321	C₂×C₄	—	131	1	no	—
322	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	262	1	no	—
327	C₈	—	262	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	262	1	no	—
334	C₂²	—	262	1	no	—
335	C₂²	—	262	1	no	—
336	C₂²	#16862265	1	2	yes	—
337	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	262	1	no	—
343	C₄	—	262	1	no	—
344	C₄	—	262	2	no	—
345	C₄	—	262	1	no	—
346	C₄	—	524	1	no	—
347	C₄	—	524	1	no	—
348	C₄	—	524	1	no	—
349	C₄	#41×D₁₃₁	1	1	yes	characteristic
350	C₄	#16862292	1	1	yes	characteristic
351	C₂	—	2	2	no	—
352	C₂	—	8	1	no	—
353	C₂	—	524	1	no	—
354	C₂	#22553954	1	1	yes	characteristic
355	C₂	C₂×#16862265	1	1	yes	characteristic
356	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	#480×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553968	C₂
hypercenter	#480	D₁₃₁
hypocenter	C₁₃₁	#3300
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #480

Complements

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

N	H ≅ G/N	type
1. #22553997	C₂	semidirect
3. #22553665	C₂	semidirect
4. #22553655	C₂	semidirect
5. #22553618	C₂	semidirect
6. #22553554	C₂	semidirect
7. #22553539	C₂	semidirect
8. #22553533	C₂	semidirect
9. #22553475	C₂	semidirect
10. #22553470	C₂	semidirect
11. #22553406	C₂	semidirect
12. (C₂²⋊D₁₃₁)⋊C₈	C₂	semidirect
13. #22553305	C₂	semidirect
14. C₂×#16862169	C₂	semidirect
15. #480×C₁₃₁	C₂	semidirect
178. C₁₃₁	#3300	semidirect
184. #480	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553997	2
#22553968	2
#22553665	2
#22553655	2
#22553618	2
#22553554	2
#22553539	2
#22553533	2
#22553475	2
#22553470	2
#22553406	2
(C₂²⋊D₁₃₁)⋊C₈	2
#22553305	2
C₂×#16862169	2
#480×C₁₃₁	2
#3300	131