Subgroups — #30675990

Order 16768.

428 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553859	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553632	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553629	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553596	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553595	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553511	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553510	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553452	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553449	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#14523502	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14523500	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#14523501	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553363	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523513	—	2	1	no	—
17	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×#14523502	C₂²	1	1	yes	characteristic
19	C₂×#14523501	C₂²	1	1	yes	characteristic
20	C₂×#14523500	C₂²	1	1	yes	characteristic
21	#16862245	C₂²	1	1	yes	characteristic
22	#16862242	C₂²	1	1	yes	characteristic
23	#16862232	—	2	1	no	—
24	#16862221	C₂²	1	1	yes	characteristic
25	#16862219	—	2	1	no	—
26	#16862217	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆₂	—	2	1	no	—
29	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
30	#16862211	—	2	1	no	—
31	#16862211	C₂²	1	1	yes	characteristic
32	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
34	C₄⋊C₄×D₁₃₁	—	2	1	no	—
35	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
36	C₄²×D₁₃₁	—	2	1	no	—
37	C₄²×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	#16862171	C₂²	1	1	yes	characteristic
44	#16862170	C₂²	1	1	yes	characteristic
45	#16862169	C₂²	1	1	yes	characteristic
46	#16862168	C₂²	1	1	yes	characteristic
47	#16862167	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×SD₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
55	#106×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523513	—	4	4	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	—	2	1	no	—
62	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
63	Q₈×D₁₃₁	—	2	2	no	—
64	Q₈×D₁₃₁	—	2	2	no	—
65	C₂×D₅₂₄	—	2	1	no	—
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×C₄×D₁₃₁	—	2	1	no	—
71	C₂×C₄×D₁₃₁	—	2	1	no	—
72	C₂×C₄×D₁₃₁	—	2	1	no	—
73	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
76	#14523502	—	2	2	no	—
77	#14523501	—	2	2	no	—
78	#14523500	—	2	2	no	—
79	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₃₁	—	2	1	no	—
82	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
84	Dic₁₃₁⋊C₄	—	2	1	no	—
85	Dic₁₃₁⋊C₄	—	2	1	no	—
86	Dic₁₃₁⋊C₄	—	2	1	no	—
87	Dic₁₃₁⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₁₃₁	—	2	1	no	—
89	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
91	C₄×Dic₁₃₁	—	2	1	no	—
92	C₄×Dic₁₃₁	—	2	1	no	—
93	C₄×Dic₁₃₁	—	2	1	no	—
94	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
95	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
96	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₃₁	—	2	1	no	—
100	SD₁₆×C₁₃₁	—	2	2	no	—
101	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₁	—	2	1	no	—
104	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
107	C₄²×C₁₃₁	—	2	1	no	—
108	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
109	C₂²×D₁₃₁	—	2	1	no	—
110	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
111	Dic₂₆₂	—	2	4	no	—
112	Dic₂₆₂	—	2	1	no	—
113	Dic₂₆₂	—	4	2	no	—
114	Dic₂₆₂	#41	1	2	yes	complemented
115	D₅₂₄	—	2	1	no	—
116	D₅₂₄	—	2	1	no	—
117	D₅₂₄	—	4	2	no	—
118	C₂²⋊D₁₃₁	—	4	2	no	—
119	C₂²⋊D₁₃₁	—	4	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	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₃₁	—	2	4	no	—
128	C₄×D₁₃₁	—	2	1	no	—
129	C₄×D₁₃₁	—	2	1	no	—
130	C₄×D₁₃₁	—	4	1	no	—
131	C₄×D₁₃₁	—	4	2	no	—
132	C₄×D₁₃₁	—	4	2	no	—
133	C₄×D₁₃₁	SD₁₆	1	4	yes	complemented
134	C₁₃₁⋊C₈	—	2	1	no	—
135	C₁₃₁⋊C₈	—	2	1	no	—
136	C₂³×C₁₃₁	—	2	1	no	—
137	Q₈×C₁₃₁	—	2	1	no	—
138	Q₈×C₁₃₁	#41	1	2	yes	complemented
139	D₄×C₁₃₁	—	2	1	no	—
140	D₄×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	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	—	4	1	no	—
151	C₂²×C₁₃₁	—	4	1	no	—
152	C₂²×C₁₃₁	—	4	1	no	—
153	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₃₁	—	2	1	no	—
155	C₂×D₁₃₁	—	2	1	no	—
156	C₂×D₁₃₁	—	4	1	no	—
157	C₂×D₁₃₁	—	4	1	no	—
158	C₂×D₁₃₁	—	4	1	no	—
159	C₂×D₁₃₁	C₂×SD₁₆	1	2	yes	—
160	C₄×C₁₃₁	—	2	2	no	—
161	C₄×C₁₃₁	—	2	1	no	—
162	C₄×C₁₃₁	—	4	1	no	—
163	C₄×C₁₃₁	—	4	1	no	—
164	C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
165	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
167	Dic₁₃₁	—	2	2	no	—
168	Dic₁₃₁	—	2	1	no	—
169	Dic₁₃₁	—	2	1	no	—
170	Dic₁₃₁	—	4	1	no	—
171	Dic₁₃₁	—	4	1	no	—
172	Dic₁₃₁	C₂×SD₁₆	1	2	yes	—
173	C₂×C₁₃₁	—	8	1	no	—
174	C₂×C₁₃₁	#525	1	1	yes	characteristic
175	C₂×C₁₃₁	C₂²×SD₁₆	1	1	yes	characteristic
176	C₂×C₁₃₁	#575	1	1	yes	characteristic
177	D₁₃₁	—	2	2	no	—
178	D₁₃₁	—	8	1	no	—
179	C₁₃₁	#3103	1	1	yes	characteristic, complemented, Sylow
180	#3103	—	131	1	no	maximal, Sylow
181	#532	—	131	1	no	—
182	C₂×C₄×Q₈	—	131	1	no	—
183	#481	—	131	1	no	—
184	#481	—	131	1	no	—
185	#475	—	131	1	no	—
186	#475	—	131	1	no	—
187	#460	—	131	1	no	—
188	#460	—	131	1	no	—
189	#448	—	131	1	no	—
190	#448	—	131	1	no	—
191	C₄×SD₁₆	—	131	1	no	—
192	C₄×SD₁₆	—	131	1	no	—
193	C₄×SD₁₆	—	131	1	no	—
194	C₄×SD₁₆	D₁₃₁	1	1	yes	characteristic, complemented
195	#431	—	131	1	no	—
196	C₂×#41	—	262	1	no	—
197	C₂²×Q₈	—	131	1	no	—
198	C₂×SD₁₆	—	131	1	no	—
199	C₂×SD₁₆	—	131	1	no	—
200	C₂×SD₁₆	—	131	1	no	—
201	C₂×SD₁₆	C₂×D₁₃₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	131	1	no	—
203	#127	—	131	1	no	—
204	#124	—	262	1	no	—
205	#121	—	262	1	no	—
206	#121	—	131	1	no	—
207	#121	—	131	1	no	—
208	C₄×Q₈	—	262	1	no	—
209	C₄×Q₈	—	262	1	no	—
210	C₄×Q₈	—	131	1	no	—
211	C₄×Q₈	—	131	1	no	—
212	C₄×Q₈	—	131	1	no	—
213	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
214	C₄×D₄	—	262	1	no	—
215	C₄×D₄	—	131	1	no	—
216	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	262	1	no	—
218	C₂×C₄⋊C₄	—	131	1	no	—
219	C₂×C₄²	—	262	1	no	—
220	C₂×C₄²	—	131	1	no	—
221	#106	—	131	1	no	—
222	#106	—	131	1	no	—
223	#106	—	131	1	no	—
224	#106	C₂×D₁₃₁	1	1	yes	characteristic
225	C₄⋊C₈	—	131	1	no	—
226	C₄⋊C₈	—	131	1	no	—
227	#103	—	131	1	no	—
228	#103	—	131	1	no	—
229	#103	—	131	1	no	—
230	#103	C₂×D₁₃₁	1	1	yes	characteristic
231	#102	—	131	1	no	—
232	#102	—	131	1	no	—
233	#102	—	131	1	no	—
234	#102	C₂×D₁₃₁	1	1	yes	characteristic
235	C₂²⋊C₈	—	131	1	no	—
236	C₂²⋊C₈	—	131	1	no	—
237	C₄×C₈	—	131	1	no	—
238	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
239	#41	—	524	4	no	—
240	C₂×Q₈	—	262	1	no	—
241	C₂×Q₈	—	262	1	no	—
242	C₂×Q₈	—	262	2	no	—
243	C₂×Q₈	—	262	2	no	—
244	C₂×Q₈	—	131	1	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	—	262	1	no	—
250	C₂×D₄	—	131	1	no	—
251	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	262	1	no	—
254	C₂²×C₄	—	262	1	no	—
255	C₂²×C₄	—	262	1	no	—
256	C₂²×C₄	—	131	1	no	—
257	C₂²×C₄	—	131	1	no	—
258	C₂²×C₄	—	131	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	262	2	no	—
261	SD₁₆	—	262	2	no	—
262	SD₁₆	—	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	—	262	1	no	—
272	C₄⋊C₄	—	262	1	no	—
273	C₄⋊C₄	—	131	1	no	—
274	C₄⋊C₄	—	131	1	no	—
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	C₄²	—	2	1	no	—
282	C₄²	—	262	1	no	—
283	C₄²	—	262	1	no	—
284	C₄²	—	262	1	no	—
285	C₄²	—	131	1	no	—
286	C₄²	—	131	1	no	—
287	C₄²	—	131	1	no	—
288	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	262	1	no	—
291	C₂³	—	131	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	262	4	no	—
294	Q₈	—	262	1	no	—
295	Q₈	—	524	2	no	—
296	Q₈	—	131	2	no	—
297	Q₈	#14523513	1	2	yes	complemented
298	D₄	—	2	1	no	—
299	D₄	—	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	D₄	—	524	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	262	4	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	—	524	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	—	131	4	no	—
323	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
324	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	262	1	no	—
329	C₈	—	262	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	—	524	1	no	—
336	C₂²	—	524	1	no	—
337	C₂²	—	524	1	no	—
338	C₂²	—	131	2	no	—
339	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	262	2	no	—
345	C₄	—	262	1	no	—
346	C₄	—	262	1	no	—
347	C₄	—	524	1	no	—
348	C₄	—	524	1	no	—
349	C₄	—	131	2	no	—
350	C₄	SD₁₆×D₁₃₁	1	2	yes	—
351	C₄	#41×D₁₃₁	1	1	yes	characteristic
352	C₄	C₂×#14523513	1	1	yes	characteristic
353	C₂	—	262	2	no	—
354	C₂	—	8	1	no	—
355	C₂	—	1048	1	no	—
356	C₂	#22553821	1	1	yes	characteristic
357	C₂	C₂×SD₁₆×D₁₃₁	1	1	yes	characteristic
358	C₂	#22554109	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	C₄×SD₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553859	C₂
hypercenter	C₄×SD₁₆	D₁₃₁
hypocenter	C₁₃₁	#3103
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₁₃₁	C₂	semidirect
3. #22553632	C₂	semidirect
4. #22553629	C₂	semidirect
5. #22553596	C₂	semidirect
6. #22553595	C₂	semidirect
7. #22553511	C₂	semidirect
8. #22553510	C₂	semidirect
9. #22553452	C₂	semidirect
10. #22553449	C₂	semidirect
11. C₄×#14523502	C₂	semidirect
12. C₄×#14523500	C₂	semidirect
13. C₄×#14523501	C₂	semidirect
14. #22553363	C₂	semidirect
15. C₄×SD₁₆×C₁₃₁	C₂	semidirect
114. Dic₂₆₂	#41	semidirect
133. C₄×D₁₃₁	SD₁₆	semidirect
138. Q₈×C₁₃₁	#41	semidirect
179. C₁₃₁	#3103	semidirect
194. C₄×SD₁₆	D₁₃₁	semidirect
297. Q₈	#14523513	semidirect

Maximal subgroups

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

subgroup	index
#22553859	2
C₄×Q₈×D₁₃₁	2
#22553632	2
#22553629	2
#22553596	2
#22553595	2
#22553511	2
#22553510	2
#22553452	2
#22553449	2
C₄×#14523502	2
C₄×#14523500	2
C₄×#14523501	2
#22553363	2
C₄×SD₁₆×C₁₃₁	2
#3103	131