Subgroups — #30677939

Order 16768.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554008	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553967	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553681	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553654	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553643	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553613	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553558	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553541	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553518	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553459	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553459	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553425	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553314	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553309	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×#14523513	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	#16862246	C₂²	1	1	yes	characteristic
22	#16862241	—	2	1	no	—
23	#16862238	C₂²	1	1	yes	characteristic
24	#16862228	—	2	1	no	—
25	#16862228	—	2	1	no	—
26	#16862228	—	2	1	no	—
27	#16862225	C₂²	1	1	yes	characteristic
28	#16862225	C₂²	1	1	yes	characteristic
29	#16862220	C₂²	1	1	yes	characteristic
30	#16862220	C₂²	1	1	yes	characteristic
31	#16862218	—	2	1	no	—
32	#16862217	—	2	1	no	—
33	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
34	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₁₃₁)⋊C₄	—	2	1	no	—
36	#16862205	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₁₃₁	—	2	1	no	—
39	#16862180	C₂²	1	1	yes	characteristic
40	#16862178	C₂²	1	1	yes	characteristic
41	#16862178	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	—	2	2	no	—
61	#14523513	—	4	1	no	—
62	#14523513	—	4	1	no	—
63	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
65	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
66	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
68	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
69	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
70	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
71	C₂²×Dic₁₃₁	—	2	1	no	—
72	C₂×C₄×D₁₃₁	—	2	1	no	—
73	C₂×C₄×D₁₃₁	—	2	1	no	—
74	C₂×C₄×D₁₃₁	—	2	1	no	—
75	C₂×C₄×D₁₃₁	—	2	1	no	—
76	#14523502	—	4	1	no	—
77	#14523502	—	4	1	no	—
78	D₁₀₄₈	—	2	2	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	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
86	Dic₁₃₁⋊C₄	—	2	1	no	—
87	Dic₁₃₁⋊C₄	—	2	1	no	—
88	Dic₁₃₁⋊C₄	—	2	1	no	—
89	Dic₁₃₁⋊C₄	—	2	1	no	—
90	C₂²⋊Dic₁₃₁	—	2	1	no	—
91	C₂²⋊Dic₁₃₁	—	2	1	no	—
92	C₂²⋊Dic₁₃₁	—	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	C₄×Dic₁₃₁	—	2	1	no	—
98	C₂×D₄×C₁₃₁	D₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
101	M₁₆×C₁₃₁	—	4	1	no	—
102	C₂×C₈×C₁₃₁	—	4	1	no	—
103	C₂×C₈×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	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
109	C₂²⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
110	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
111	C₂²×D₁₃₁	—	2	1	no	—
112	C₂²×D₁₃₁	—	2	1	no	—
113	Dic₂₆₂	—	2	1	no	—
114	Dic₂₆₂	—	2	1	no	—
115	Dic₂₆₂	—	2	1	no	—
116	Dic₂₆₂	—	2	1	no	—
117	D₅₂₄	—	2	1	no	—
118	D₅₂₄	—	2	1	no	—
119	D₅₂₄	—	2	1	no	—
120	D₅₂₄	#41	1	2	yes	—
121	C₂²⋊D₁₃₁	—	4	2	no	—
122	C₂²⋊D₁₃₁	—	4	2	no	—
123	C₂²⋊D₁₃₁	—	4	2	no	—
124	C₂²⋊D₁₃₁	—	8	1	no	—
125	C₂×Dic₁₃₁	—	2	1	no	—
126	C₂×Dic₁₃₁	—	2	1	no	—
127	C₂×Dic₁₃₁	—	2	1	no	—
128	C₂×Dic₁₃₁	—	2	1	no	—
129	C₂×Dic₁₃₁	—	2	1	no	—
130	C₂×Dic₁₃₁	—	2	1	no	—
131	C₂×Dic₁₃₁	—	8	1	no	—
132	C₄×D₁₃₁	—	4	2	no	—
133	C₄×D₁₃₁	—	4	2	no	—
134	C₄×D₁₃₁	—	4	2	no	—
135	C₄×D₁₃₁	—	8	1	no	—
136	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
138	D₄×C₁₃₁	—	4	1	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	C₂×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
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	C₈×C₁₃₁	—	2	1	no	—
151	C₈×C₁₃₁	—	2	1	no	—
152	C₂²×C₁₃₁	—	2	1	no	—
153	C₂²×C₁₃₁	—	2	1	no	—
154	C₂²×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	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₁₃₁	—	2	2	no	—
160	C₂×D₁₃₁	—	4	1	no	—
161	C₂×D₁₃₁	—	4	1	no	—
162	C₂×D₁₃₁	—	4	1	no	—
163	C₂×D₁₃₁	—	4	1	no	—
164	C₂×D₁₃₁	—	4	1	no	—
165	C₄×C₁₃₁	—	2	1	no	—
166	C₄×C₁₃₁	—	2	1	no	—
167	C₄×C₁₃₁	—	2	1	no	—
168	C₄×C₁₃₁	—	2	1	no	—
169	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
170	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₁₃₁	—	4	1	no	—
172	Dic₁₃₁	—	4	1	no	—
173	Dic₁₃₁	—	4	1	no	—
174	Dic₁₃₁	—	4	1	no	—
175	Dic₁₃₁	—	4	1	no	—
176	Dic₁₃₁	—	4	1	no	—
177	C₂×C₁₃₁	—	4	1	no	—
178	C₂×C₁₃₁	—	4	1	no	—
179	C₂×C₁₃₁	#546	1	1	yes	characteristic
180	C₂×C₁₃₁	#575	1	1	yes	characteristic
181	C₂×C₁₃₁	#577	1	1	yes	characteristic
182	D₁₃₁	—	4	2	no	—
183	D₁₃₁	—	8	1	no	—
184	C₁₃₁	#3307	1	1	yes	characteristic, complemented, Sylow
185	#3307	—	131	1	no	maximal, Sylow
186	#553	—	131	1	no	—
187	#547	—	131	1	no	—
188	#491	—	131	1	no	—
189	#485	—	131	1	no	—
190	#483	—	131	1	no	—
191	#478	—	131	1	no	—
192	#468	—	131	1	no	—
193	#465	—	131	1	no	—
194	#461	—	131	1	no	—
195	#449	—	131	1	no	—
196	#449	—	131	1	no	—
197	#442	—	131	1	no	—
198	#435	D₁₃₁	1	1	yes	characteristic, complemented
199	#417	—	131	1	no	—
200	#416	—	131	1	no	—
201	C₂×#41	—	131	1	no	—
202	C₂×#41	—	131	1	no	—
203	C₂×SD₁₆	—	131	1	no	—
204	C₂×SD₁₆	—	131	1	no	—
205	C₂×D₈	—	131	1	no	—
206	C₂×M₁₆	C₂×D₁₃₁	1	1	yes	characteristic
207	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
208	C₄⋊Q₈	—	131	1	no	—
209	#126	—	262	1	no	—
210	#125	—	131	1	no	—
211	#123	—	262	1	no	—
212	#123	—	262	1	no	—
213	#123	—	262	1	no	—
214	C₂²⋊Q₈	—	131	1	no	—
215	C₂²⋊Q₈	—	131	1	no	—
216	#121	—	262	1	no	—
217	#121	—	262	1	no	—
218	#121	—	131	1	no	—
219	#121	—	131	1	no	—
220	C₄×D₄	—	262	1	no	—
221	C₄×D₄	—	131	1	no	—
222	C₄×D₄	—	131	1	no	—
223	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
224	#117	—	131	1	no	—
225	C₂×C₄⋊C₄	—	262	1	no	—
226	C₂×C₄⋊C₄	—	131	1	no	—
227	#107	—	131	1	no	—
228	#106	—	131	1	no	—
229	#106	—	131	1	no	—
230	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
231	#103	—	131	1	no	—
232	#103	—	131	1	no	—
233	#103	—	131	1	no	—
234	#103	—	131	1	no	—
235	#103	—	131	1	no	—
236	#103	—	131	1	no	—
237	#102	—	131	1	no	—
238	#102	—	131	1	no	—
239	#102	—	131	1	no	—
240	#102	—	131	1	no	—
241	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
242	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
243	#97	C₂×D₁₃₁	1	1	yes	characteristic
244	#41	—	262	2	no	—
245	#41	—	262	2	no	—
246	#41	—	524	1	no	—
247	#41	—	524	1	no	—
248	C₂×Q₈	—	131	1	no	—
249	C₂×Q₈	—	131	1	no	—
250	C₂×D₄	—	262	1	no	—
251	C₂×D₄	—	262	1	no	—
252	C₂×D₄	—	262	1	no	—
253	C₂×D₄	—	262	1	no	—
254	C₂×D₄	—	131	1	no	—
255	C₂×D₄	—	131	1	no	—
256	C₂×D₄	D₅₂₄	1	1	yes	characteristic
257	C₂²×C₄	—	262	1	no	—
258	C₂²×C₄	—	262	1	no	—
259	C₂²×C₄	—	262	1	no	—
260	C₂²×C₄	—	262	1	no	—
261	C₂²×C₄	—	262	1	no	—
262	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
264	SD₁₆	—	524	1	no	—
265	SD₁₆	—	524	1	no	—
266	D₈	—	262	2	no	—
267	M₁₆	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
273	C₄⋊C₄	—	262	1	no	—
274	C₄⋊C₄	—	262	1	no	—
275	C₄⋊C₄	—	262	1	no	—
276	C₄⋊C₄	—	262	1	no	—
277	C₄⋊C₄	—	262	1	no	—
278	C₄⋊C₄	—	262	1	no	—
279	C₄⋊C₄	—	262	1	no	—
280	C₄⋊C₄	—	131	1	no	—
281	C₄⋊C₄	—	131	1	no	—
282	C₄⋊C₄	—	131	1	no	—
283	C₄⋊C₄	—	131	1	no	—
284	C₄⋊C₄	D₅₂₄	1	1	yes	characteristic
285	C₂²⋊C₄	—	262	1	no	—
286	C₂²⋊C₄	—	262	1	no	—
287	C₂²⋊C₄	—	262	1	no	—
288	C₂²⋊C₄	—	262	1	no	—
289	C₂²⋊C₄	—	262	1	no	—
290	C₂²⋊C₄	—	262	1	no	—
291	C₂²⋊C₄	—	262	1	no	—
292	C₂²⋊C₄	D₅₂₄	1	1	yes	characteristic
293	C₂²⋊C₄	D₅₂₄	1	1	yes	characteristic
294	C₄²	—	262	1	no	—
295	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
296	C₂³	—	262	1	no	—
297	C₂³	—	262	1	no	—
298	C₂³	C₂×D₅₂₄	1	1	yes	characteristic
299	C₂³	C₂×D₅₂₄	1	1	yes	characteristic
300	Q₈	—	262	1	no	—
301	Q₈	—	262	1	no	—
302	Q₈	—	262	1	no	—
303	Q₈	—	262	1	no	—
304	D₄	—	4	1	no	—
305	D₄	—	262	1	no	—
306	D₄	—	262	1	no	—
307	D₄	—	262	1	no	—
308	D₄	—	524	2	no	—
309	D₄	—	524	2	no	—
310	D₄	—	524	2	no	—
311	D₄	—	1048	1	no	—
312	D₄	—	131	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	262	1	no	—
318	C₂×C₄	—	262	1	no	—
319	C₂×C₄	—	262	1	no	—
320	C₂×C₄	—	262	1	no	—
321	C₂×C₄	—	262	1	no	—
322	C₂×C₄	—	262	1	no	—
323	C₂×C₄	—	524	2	no	—
324	C₂×C₄	—	524	2	no	—
325	C₂×C₄	—	524	2	no	—
326	C₂×C₄	—	1048	1	no	—
327	C₂×C₄	—	1048	1	no	—
328	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
329	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
330	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
331	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	262	2	no	—
344	C₂²	—	524	1	no	—
345	C₂²	—	524	1	no	—
346	C₂²	—	524	1	no	—
347	C₂²	—	524	1	no	—
348	C₂²	—	524	1	no	—
349	C₂²	C₂²×D₅₂₄	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	524	1	no	—
355	C₄	—	524	1	no	—
356	C₄	—	524	1	no	—
357	C₄	—	524	1	no	—
358	C₄	—	524	1	no	—
359	C₄	—	524	1	no	—
360	C₄	#41×D₁₃₁	1	1	yes	characteristic
361	C₄	#16862291	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	4	1	no	—
364	C₂	—	524	2	no	—
365	C₂	—	1048	1	no	—
366	C₂	#22553957	1	1	yes	characteristic
367	C₂	#22554112	1	1	yes	characteristic
368	C₂	#22554124	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	#3307
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. #22554008	C₂	semidirect
2. #22553967	C₂	semidirect
3. #22553681	C₂	semidirect
4. #22553654	C₂	semidirect
5. #22553643	C₂	semidirect
6. #22553613	C₂	semidirect
7. #22553558	C₂	semidirect
8. #22553541	C₂	semidirect
9. #22553518	C₂	semidirect
10. #22553459	C₂	semidirect
11. #22553459	C₂	semidirect
12. #22553425	C₂	semidirect
13. #22553314	C₂	semidirect
14. #22553309	C₂	semidirect
15. #435×C₁₃₁	C₂	semidirect
184. C₁₃₁	#3307	semidirect
198. #435	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22554008	2
#22553967	2
#22553681	2
#22553654	2
#22553643	2
#22553613	2
#22553558	2
#22553541	2
#22553518	2
#22553459	2
#22553425	2
#22553314	2
#22553309	2
#435×C₁₃₁	2
#3307	131