Subgroups — #30675893

Order 16768.

414 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553825	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553652	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553651	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553650	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553639	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553638	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553637	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553636	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553353	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553351	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553312	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553312	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553274	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523514	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×#14523496	C₂²	1	1	yes	characteristic
19	C₂×#14523496	C₂²	1	1	yes	characteristic
20	#16862231	—	2	1	no	—
21	#16862222	C₂²	1	1	yes	characteristic
22	#16862221	C₂²	1	1	yes	characteristic
23	#16862220	C₂²	1	1	yes	characteristic
24	#16862216	—	2	1	no	—
25	#16862208	—	2	1	no	—
26	(C₂²⋊D₁₃₁)⋊C₄	—	2	1	no	—
27	#16862205	C₂²	1	1	yes	characteristic
28	#16862204	C₂²	1	1	yes	characteristic
29	#16862204	C₂²	1	1	yes	characteristic
30	#16862204	C₂²	1	1	yes	characteristic
31	#16862203	—	2	1	no	—
32	#16862202	—	2	1	no	—
33	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₁₃₁	—	2	1	no	—
36	C₄²×D₁₃₁	—	2	1	no	—
37	#16862180	C₂²	1	1	yes	characteristic, complemented
38	#16862179	C₂²	1	1	yes	characteristic, complemented
39	#16862179	C₂²	1	1	yes	characteristic, complemented
40	#16862178	C₂²	1	1	yes	characteristic, complemented
41	#16862177	C₂²	1	1	yes	characteristic, complemented
42	#16862177	C₂²	1	1	yes	characteristic, complemented
43	#16862170	C₂²	1	1	yes	characteristic, complemented
44	#16862170	C₂²	1	1	yes	characteristic, complemented
45	#16862168	C₂²	1	1	yes	characteristic, complemented
46	#16862168	C₂²	1	1	yes	characteristic, complemented
47	#16862166	C₂²	1	1	yes	characteristic, complemented
48	#16862166	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
53	#117×C₁₃₁	C₂²	1	1	yes	characteristic
54	#107×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
55	#106×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂³×D₁₃₁	D₄	1	1	yes	characteristic
60	#14523514	—	2	2	no	—
61	#14523514	—	2	2	no	—
62	C₂×D₅₂₄	—	2	1	no	—
63	C₂×D₅₂₄	—	2	1	no	—
64	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
66	C₂²×Dic₁₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃₁	—	2	2	no	—
68	C₂×C₄×D₁₃₁	—	2	2	no	—
69	C₂×C₄×D₁₃₁	—	2	1	no	—
70	C₂×C₄×D₁₃₁	—	2	1	no	—
71	C₂×C₄×D₁₃₁	—	2	1	no	—
72	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
76	#14523496	—	2	2	no	—
77	#14523496	—	2	2	no	—
78	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
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	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	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
106	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
107	C₄²×C₁₃₁	—	2	1	no	—
108	C₂²×D₁₃₁	—	2	2	no	—
109	C₂²×D₁₃₁	—	2	1	no	—
110	C₂²×D₁₃₁	—	4	1	no	—
111	C₂²×D₁₃₁	—	4	1	no	—
112	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
114	D₅₂₄	—	2	1	no	—
115	D₅₂₄	—	2	1	no	—
116	D₅₂₄	—	4	2	no	—
117	D₅₂₄	—	4	2	no	—
118	C₂²⋊D₁₃₁	—	4	2	no	—
119	C₂×Dic₁₃₁	—	2	2	no	—
120	C₂×Dic₁₃₁	—	2	1	no	—
121	C₂×Dic₁₃₁	—	2	1	no	—
122	C₂×Dic₁₃₁	—	2	1	no	—
123	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₃₁	—	2	2	no	—
126	C₄×D₁₃₁	—	2	1	no	—
127	C₄×D₁₃₁	—	2	1	no	—
128	C₄×D₁₃₁	—	2	2	no	—
129	C₄×D₁₃₁	—	4	2	no	—
130	C₄×D₁₃₁	—	4	2	no	—
131	C₄×D₁₃₁	—	4	2	no	—
132	C₄×D₁₃₁	#41	1	2	yes	—
133	C₄×D₁₃₁	#41	1	2	yes	—
134	C₁₃₁⋊C₈	—	2	1	no	—
135	C₁₃₁⋊C₈	—	2	1	no	—
136	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
137	Q₈×C₁₃₁	—	2	1	no	—
138	Q₈×C₁₃₁	—	2	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	—	2	1	no	—
144	C₂×C₄×C₁₃₁	—	2	1	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	—	2	1	no	—
152	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₃₁	—	2	1	no	—
154	C₂×D₁₃₁	—	2	1	no	—
155	C₂×D₁₃₁	—	2	1	no	—
156	C₂×D₁₃₁	—	2	1	no	—
157	C₂×D₁₃₁	—	2	1	no	—
158	C₂×D₁₃₁	—	4	4	no	—
159	C₂×D₁₃₁	—	4	1	no	—
160	C₂×D₁₃₁	—	4	1	no	—
161	C₂×D₁₃₁	—	4	1	no	—
162	C₂×D₁₃₁	#123	1	2	yes	—
163	C₄×C₁₃₁	—	2	1	no	—
164	C₄×C₁₃₁	—	4	1	no	—
165	C₄×C₁₃₁	—	4	1	no	—
166	C₄×C₁₃₁	—	4	1	no	—
167	C₄×C₁₃₁	—	4	1	no	—
168	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
169	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
170	Dic₁₃₁	—	2	1	no	—
171	Dic₁₃₁	—	4	1	no	—
172	Dic₁₃₁	—	4	1	no	—
173	Dic₁₃₁	—	4	1	no	—
174	Dic₁₃₁	#123	1	2	yes	—
175	C₂×C₁₃₁	—	4	1	no	—
176	C₂×C₁₃₁	C₂×#123	1	1	yes	characteristic
177	C₂×C₁₃₁	#575	1	1	yes	characteristic
178	C₂×C₁₃₁	#575	1	1	yes	characteristic
179	D₁₃₁	—	2	2	no	—
180	D₁₃₁	—	4	1	no	—
181	D₁₃₁	—	8	1	no	—
182	C₁₃₁	#3093	1	1	yes	characteristic, complemented, Sylow
183	#3093	—	131	1	no	maximal, Sylow
184	#525	—	131	1	no	—
185	C₂×#117	—	131	1	no	—
186	#485	—	131	1	no	—
187	#485	—	131	1	no	—
188	#485	—	131	1	no	—
189	#485	D₁₃₁	1	1	yes	characteristic, complemented
190	#482	—	131	1	no	—
191	#482	—	131	1	no	—
192	#482	—	131	1	no	—
193	#482	—	131	1	no	—
194	#428	—	131	1	no	—
195	#428	—	131	1	no	—
196	#417	—	131	1	no	—
197	#417	—	131	1	no	—
198	#407	—	131	1	no	—
199	C₂×#41	—	131	1	no	—
200	C₂³×C₄	—	131	1	no	—
201	C₂×M₁₆	—	131	1	no	—
202	C₂×M₁₆	—	131	1	no	—
203	#123	—	262	1	no	—
204	C₂²⋊Q₈	—	131	1	no	—
205	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
206	#121	—	131	1	no	—
207	#121	—	131	1	no	—
208	#120	—	262	1	no	—
209	C₄×D₄	—	262	1	no	—
210	C₄×D₄	—	262	1	no	—
211	#117	—	262	1	no	—
212	#117	—	262	1	no	—
213	#117	—	131	1	no	—
214	#117	—	131	1	no	—
215	#117	—	131	1	no	—
216	#117	—	131	1	no	—
217	#117	C₂×D₁₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	131	1	no	—
219	C₂×C₄⋊C₄	—	131	1	no	—
220	C₂×C₂²⋊C₄	—	262	1	no	—
221	C₂×C₄²	—	262	1	no	—
222	#107	—	131	1	no	—
223	#107	—	131	1	no	—
224	#107	—	131	1	no	—
225	#107	C₂×D₁₃₁	1	1	yes	characteristic, complemented
226	#106	—	131	1	no	—
227	#106	—	131	1	no	—
228	#106	—	131	1	no	—
229	#106	C₂×D₁₃₁	1	1	yes	characteristic, complemented
230	#103	—	131	1	no	—
231	#103	—	131	1	no	—
232	#103	C₂×D₁₃₁	1	1	yes	characteristic, complemented
233	#103	C₂×D₁₃₁	1	1	yes	characteristic, complemented
234	#102	—	131	1	no	—
235	#102	—	131	1	no	—
236	#102	—	131	1	no	—
237	#102	—	131	1	no	—
238	C₂²⋊C₈	—	131	1	no	—
239	C₂²⋊C₈	—	131	1	no	—
240	C₂²⋊C₈	—	131	1	no	—
241	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
242	C₂⁴	—	131	1	no	—
243	#41	—	262	2	no	—
244	#41	—	262	2	no	—
245	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
246	C₂×D₄	—	262	1	no	—
247	C₂×D₄	—	262	1	no	—
248	C₂×D₄	—	262	1	no	—
249	C₂×D₄	—	131	1	no	—
250	C₂²×C₄	—	262	2	no	—
251	C₂²×C₄	—	262	2	no	—
252	C₂²×C₄	—	262	1	no	—
253	C₂²×C₄	—	262	1	no	—
254	C₂²×C₄	—	262	1	no	—
255	C₂²×C₄	—	131	1	no	—
256	C₂²×C₄	—	131	1	no	—
257	C₂²×C₄	—	131	1	no	—
258	C₂²×C₄	—	131	1	no	—
259	C₂²×C₄	—	131	1	no	—
260	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
261	M₁₆	—	262	2	no	—
262	M₁₆	—	262	2	no	—
263	C₂×C₈	—	131	1	no	—
264	C₂×C₈	—	131	1	no	—
265	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	262	1	no	—
269	C₄⋊C₄	—	262	1	no	—
270	C₄⋊C₄	—	262	1	no	—
271	C₄⋊C₄	—	131	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	—	2	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	—	262	1	no	—
285	C₂²⋊C₄	—	262	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	262	1	no	—
288	C₄²	—	262	1	no	—
289	C₄²	—	262	1	no	—
290	C₄²	—	262	1	no	—
291	C₂³	—	262	2	no	—
292	C₂³	—	262	1	no	—
293	C₂³	—	524	1	no	—
294	C₂³	—	524	1	no	—
295	C₂³	—	131	1	no	—
296	C₂³	—	131	1	no	—
297	C₂³	D₄×D₁₃₁	1	1	yes	characteristic
298	Q₈	—	2	1	no	—
299	Q₈	—	2	1	no	—
300	D₄	—	262	1	no	—
301	D₄	—	262	1	no	—
302	D₄	—	524	2	no	—
303	D₄	—	524	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	—	2	1	no	—
311	C₂×C₄	—	262	2	no	—
312	C₂×C₄	—	262	2	no	—
313	C₂×C₄	—	262	2	no	—
314	C₂×C₄	—	262	1	no	—
315	C₂×C₄	—	262	1	no	—
316	C₂×C₄	—	262	1	no	—
317	C₂×C₄	—	262	1	no	—
318	C₂×C₄	—	262	1	no	—
319	C₂×C₄	—	524	2	no	—
320	C₂×C₄	—	524	2	no	—
321	C₂×C₄	—	524	2	no	—
322	C₂×C₄	—	131	1	no	—
323	C₂×C₄	—	131	1	no	—
324	C₂×C₄	—	131	2	no	—
325	C₂×C₄	—	131	2	no	—
326	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	262	1	no	—
331	C₈	—	262	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	262	1	no	—
336	C₂²	—	262	1	no	—
337	C₂²	—	262	1	no	—
338	C₂²	—	262	1	no	—
339	C₂²	—	262	1	no	—
340	C₂²	—	524	4	no	—
341	C₂²	—	524	1	no	—
342	C₂²	—	524	1	no	—
343	C₂²	—	524	1	no	—
344	C₂²	—	131	2	no	—
345	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	262	1	no	—
352	C₄	—	524	1	no	—
353	C₄	—	524	1	no	—
354	C₄	—	524	1	no	—
355	C₄	—	131	2	no	—
356	C₄	#41×D₁₃₁	1	1	yes	characteristic
357	C₄	#41×D₁₃₁	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	262	2	no	—
360	C₂	—	524	1	no	—
361	C₂	—	1048	1	no	—
362	C₂	#123×D₁₃₁	1	1	yes	characteristic
363	C₂	#22554109	1	1	yes	characteristic
364	C₂	#22554111	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#485×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553825	C₂
hypercenter	#485	D₁₃₁
hypocenter	C₁₃₁	#3093
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
2. #117×D₁₃₁	C₂	semidirect
3. #22553652	C₂	semidirect
4. #22553651	C₂	semidirect
5. #22553650	C₂	semidirect
6. #22553639	C₂	semidirect
7. #22553638	C₂	semidirect
8. #22553637	C₂	semidirect
9. #22553636	C₂	semidirect
10. #22553353	C₂	semidirect
11. #22553351	C₂	semidirect
12. #22553312	C₂	semidirect
13. #22553312	C₂	semidirect
14. #22553274	C₂	semidirect
15. #485×C₁₃₁	C₂	semidirect
37. #16862180	C₂²	semidirect
38. #16862179	C₂²	semidirect
39. #16862179	C₂²	semidirect
40. #16862178	C₂²	semidirect
41. #16862177	C₂²	semidirect
42. #16862177	C₂²	semidirect
43. #16862170	C₂²	semidirect
44. #16862170	C₂²	semidirect
45. #16862168	C₂²	semidirect
46. #16862168	C₂²	semidirect
47. #16862166	C₂²	semidirect
48. #16862166	C₂²	semidirect
54. #107×C₁₃₁	C₂²	semidirect
55. #106×C₁₃₁	C₂²	semidirect
56. #103×C₁₃₁	C₂²	semidirect
57. #103×C₁₃₁	C₂²	semidirect
182. C₁₃₁	#3093	semidirect
189. #485	D₁₃₁	semidirect
225. #107	C₂×D₁₃₁	semidirect
229. #106	C₂×D₁₃₁	semidirect
232. #103	C₂×D₁₃₁	semidirect
233. #103	C₂×D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553825	2
#117×D₁₃₁	2
#22553652	2
#22553651	2
#22553650	2
#22553639	2
#22553638	2
#22553637	2
#22553636	2
#22553353	2
#22553351	2
#22553312	2
#22553274	2
#485×C₁₃₁	2
#3093	131