Subgroups — #30677830

Order 16768.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554020	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₂²⋊D₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#22553661	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553648	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553646	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553605	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553551	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553550	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553520	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553466	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₁₆×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₁₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic
13	C₂×#16862170	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#451×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#14523503	C₂²	1	1	yes	characteristic
18	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
20	#16862245	C₂²	1	1	yes	characteristic
21	#16862244	—	2	1	no	—
22	#16862244	C₂²	1	1	yes	characteristic
23	#16862236	—	2	1	no	—
24	#16862226	—	2	1	no	—
25	#16862225	—	2	1	no	—
26	#16862225	C₂²	1	1	yes	characteristic
27	#16862223	—	2	1	no	—
28	#16862222	—	2	1	no	—
29	#16862222	C₂²	1	1	yes	characteristic
30	#16862222	C₂²	1	1	yes	characteristic
31	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
32	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
34	C₄×C₂²⋊D₁₃₁	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₁₃₁)⋊C₄	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
38	#16862180	C₂²	1	1	yes	characteristic
39	#16862180	C₂²	1	1	yes	characteristic
40	#16862179	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	#16862170	C₂²	1	1	yes	characteristic
45	#16862170	C₂²	1	1	yes	characteristic
46	#16862170	C₂²	1	1	yes	characteristic
47	#16862170	C₂²	1	1	yes	characteristic
48	#16862170	C₂²	1	1	yes	characteristic
49	#16862169	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	Q₈×D₁₃₁	—	2	2	no	—
63	Q₈×D₁₃₁	—	4	1	no	—
64	C₂×C₂²⋊D₁₃₁	D₄	1	1	yes	characteristic
65	C₂²×Dic₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	—	2	1	no	—
68	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
69	#14523503	—	4	1	no	—
70	C₂×C₁₃₁⋊C₈	—	4	1	no	—
71	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₁₃₁	—	4	1	no	—
74	C₄⋊Dic₁₃₁	—	2	1	no	—
75	C₄⋊Dic₁₃₁	—	2	1	no	—
76	C₄⋊Dic₁₃₁	—	2	1	no	—
77	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
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	—	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	D₄	1	1	yes	characteristic
90	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
91	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
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×Q₈×C₁₃₁	—	2	1	no	—
97	C₂×Q₈×C₁₃₁	—	2	2	no	—
98	C₂×Q₈×C₁₃₁	—	4	1	no	—
99	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₁	—	2	1	no	—
103	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	Q₁₆×C₁₃₁	—	2	2	no	—
105	Q₁₆×C₁₃₁	—	4	1	no	—
106	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₁₃₁	—	2	1	no	—
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	—	4	2	no	—
115	Dic₂₆₂	—	8	1	no	—
116	C₂²⋊D₁₃₁	Q₁₆	1	4	yes	complemented
117	C₂×Dic₁₃₁	—	2	1	no	—
118	C₂×Dic₁₃₁	—	2	1	no	—
119	C₂×Dic₁₃₁	—	2	1	no	—
120	C₂×Dic₁₃₁	—	2	1	no	—
121	C₂×Dic₁₃₁	—	8	1	no	—
122	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₃₁	—	2	1	no	—
126	C₄×D₁₃₁	—	2	1	no	—
127	C₄×D₁₃₁	—	4	2	no	—
128	C₄×D₁₃₁	—	8	1	no	—
129	C₁₃₁⋊C₈	—	2	1	no	—
130	C₁₃₁⋊C₈	—	2	1	no	—
131	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
132	Q₈×C₁₃₁	—	2	1	no	—
133	Q₈×C₁₃₁	—	2	1	no	—
134	Q₈×C₁₃₁	—	2	1	no	—
135	Q₈×C₁₃₁	—	2	1	no	—
136	Q₈×C₁₃₁	—	2	1	no	—
137	Q₈×C₁₃₁	—	4	2	no	—
138	Q₈×C₁₃₁	#41	1	2	yes	—
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	—	2	1	no	—
145	C₂×C₄×C₁₃₁	—	4	2	no	—
146	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
148	C₈×C₁₃₁	—	2	1	no	—
149	C₈×C₁₃₁	—	2	1	no	—
150	C₂²×C₁₃₁	—	2	1	no	—
151	C₂²×C₁₃₁	—	2	1	no	—
152	C₂²×C₁₃₁	C₂×Q₁₆	1	2	yes	—
153	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₃₁	—	2	1	no	—
155	C₂×D₁₃₁	—	2	1	no	—
156	C₂×D₁₃₁	C₂×Q₁₆	1	2	yes	—
157	C₄×C₁₃₁	—	2	1	no	—
158	C₄×C₁₃₁	—	2	2	no	—
159	C₄×C₁₃₁	—	4	1	no	—
160	C₄×C₁₃₁	—	4	1	no	—
161	C₄×C₁₃₁	—	4	1	no	—
162	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₁₃₁	—	2	1	no	—
165	Dic₁₃₁	—	2	1	no	—
166	Dic₁₃₁	—	4	1	no	—
167	Dic₁₃₁	—	4	1	no	—
168	Dic₁₃₁	—	4	1	no	—
169	Dic₁₃₁	—	4	1	no	—
170	Dic₁₃₁	C₂×Q₁₆	1	2	yes	—
171	C₂×C₁₃₁	—	2	2	no	—
172	C₂×C₁₃₁	#546	1	1	yes	characteristic
173	C₂×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂×C₁₃₁	#577	1	1	yes	characteristic
175	D₁₃₁	—	2	2	no	—
176	C₁₃₁	#3299	1	1	yes	characteristic, complemented, Sylow
177	#3299	—	131	1	no	maximal, Sylow
178	#554	—	131	1	no	—
179	D₄×Q₈	—	131	1	no	—
180	#487	—	131	1	no	—
181	#484	—	131	1	no	—
182	#484	—	131	1	no	—
183	#477	—	131	1	no	—
184	#467	—	131	1	no	—
185	#467	—	131	1	no	—
186	#462	—	131	1	no	—
187	#451	—	131	1	no	—
188	#451	D₁₃₁	1	1	yes	characteristic, complemented
189	C₄×Q₁₆	—	131	1	no	—
190	C₈×D₄	—	131	1	no	—
191	C₂×#103	—	131	1	no	—
192	C₂×#103	—	131	1	no	—
193	C₂²×Q₈	—	131	1	no	—
194	C₂²×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
195	C₂×Q₁₆	—	131	1	no	—
196	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
197	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
198	C₂²×C₈	—	131	1	no	—
199	C₂²×C₈	—	131	1	no	—
200	C₄⋊Q₈	—	262	1	no	—
201	C₄⋊Q₈	—	131	1	no	—
202	C₄⋊Q₈	—	131	1	no	—
203	#125	—	262	1	no	—
204	C₂²⋊Q₈	—	262	1	no	—
205	C₂²⋊Q₈	—	262	1	no	—
206	C₂²⋊Q₈	—	262	1	no	—
207	C₂²⋊Q₈	—	262	1	no	—
208	C₂²⋊Q₈	—	131	1	no	—
209	C₂²⋊Q₈	—	131	1	no	—
210	C₂²⋊Q₈	—	131	1	no	—
211	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
212	C₄×Q₈	—	131	1	no	—
213	C₄×Q₈	—	131	1	no	—
214	C₄×D₄	—	262	1	no	—
215	C₄×D₄	—	262	1	no	—
216	C₄×D₄	—	131	1	no	—
217	C₂×C₄⋊C₄	—	131	1	no	—
218	C₂×C₄⋊C₄	—	131	1	no	—
219	#107	—	131	1	no	—
220	#107	—	131	1	no	—
221	#107	—	131	1	no	—
222	C₄⋊C₈	—	131	1	no	—
223	#103	—	131	1	no	—
224	#103	—	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	C₂×D₁₃₁	1	1	yes	characteristic
232	#103	C₂×D₁₃₁	1	1	yes	characteristic
233	C₂²⋊C₈	—	131	1	no	—
234	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
235	C₄×C₈	—	131	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	262	1	no	—
240	C₂×Q₈	—	262	1	no	—
241	C₂×Q₈	—	262	2	no	—
242	C₂×Q₈	—	524	1	no	—
243	C₂×Q₈	—	131	1	no	—
244	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
247	C₂×D₄	—	131	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	262	1	no	—
250	C₂²×C₄	—	262	1	no	—
251	C₂²×C₄	—	262	1	no	—
252	C₂²×C₄	—	131	1	no	—
253	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	524	1	no	—
257	C₂×C₈	—	524	1	no	—
258	C₂×C₈	—	524	1	no	—
259	C₂×C₈	—	131	1	no	—
260	C₂×C₈	—	131	1	no	—
261	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	262	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	—	262	1	no	—
273	C₄⋊C₄	—	262	1	no	—
274	C₄⋊C₄	—	131	1	no	—
275	C₄⋊C₄	—	131	1	no	—
276	C₄⋊C₄	—	131	1	no	—
277	C₄⋊C₄	—	131	1	no	—
278	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	262	1	no	—
281	C₂²⋊C₄	—	262	1	no	—
282	C₂²⋊C₄	—	262	1	no	—
283	C₂²⋊C₄	—	131	1	no	—
284	C₂²⋊C₄	—	131	1	no	—
285	C₄²	—	262	1	no	—
286	C₄²	—	262	1	no	—
287	C₄²	—	131	1	no	—
288	C₂³	—	131	1	no	—
289	C₂³	D₄×D₁₃₁	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	262	1	no	—
297	Q₈	—	262	1	no	—
298	Q₈	—	524	2	no	—
299	Q₈	—	1048	1	no	—
300	Q₈	#14523512	1	2	yes	—
301	D₄	—	131	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	262	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	—	524	2	no	—
316	C₂×C₄	—	1048	1	no	—
317	C₂×C₄	—	1048	1	no	—
318	C₂×C₄	—	131	1	no	—
319	C₂×C₄	—	131	1	no	—
320	C₂×C₄	—	131	1	no	—
321	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	262	1	no	—
326	C₈	—	262	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	262	1	no	—
330	C₂²	—	262	1	no	—
331	C₂²	—	131	2	no	—
332	C₂²	Q₁₆×D₁₃₁	1	2	yes	—
333	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	262	1	no	—
340	C₄	—	262	1	no	—
341	C₄	—	524	1	no	—
342	C₄	—	524	1	no	—
343	C₄	—	524	1	no	—
344	C₄	—	524	1	no	—
345	C₄	—	131	2	no	—
346	C₄	C₂×#14523512	1	1	yes	characteristic
347	C₄	#16862291	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	262	2	no	—
350	C₂	#22553949	1	1	yes	characteristic
351	C₂	C₂×Q₁₆×D₁₃₁	1	1	yes	characteristic
352	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	#451×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×C₂²⋊D₁₃₁	D₄
hypercenter	#451	D₁₃₁
hypocenter	C₁₃₁	#3299
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
3. #22553661	C₂	semidirect
4. #22553648	C₂	semidirect
5. #22553646	C₂	semidirect
6. #22553605	C₂	semidirect
7. #22553551	C₂	semidirect
8. #22553550	C₂	semidirect
9. #22553520	C₂	semidirect
10. #22553466	C₂	semidirect
11. Q₁₆×Dic₁₃₁	C₂	semidirect
13. C₂×#16862170	C₂	semidirect
14. #103×D₁₃₁	C₂	semidirect
15. #451×C₁₃₁	C₂	semidirect
116. C₂²⋊D₁₃₁	Q₁₆	semidirect
176. C₁₃₁	#3299	semidirect
188. #451	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22554020	2
Q₈×C₂²⋊D₁₃₁	2
#22553661	2
#22553648	2
#22553646	2
#22553605	2
#22553551	2
#22553550	2
#22553520	2
#22553466	2
Q₁₆×Dic₁₃₁	2
(C₂²⋊D₁₃₁)⋊C₈	2
C₂×#16862170	2
#103×D₁₃₁	2
#451×C₁₃₁	2
#3299	131