Subgroups — #30675908

Order 16768.

410 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	C₂²⋊Q₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#22553654	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553653	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553650	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553649	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553646	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553644	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553641	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	#22553319	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553319	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553274	C₂	1	1	yes	maximal, characteristic
15	#485×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₃₁	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	#16862226	C₂²	1	1	yes	characteristic
21	#16862225	C₂²	1	1	yes	characteristic
22	#16862224	—	2	1	no	—
23	#16862223	—	2	1	no	—
24	#16862222	C₂²	1	1	yes	characteristic
25	#16862205	C₂²	1	1	yes	characteristic
26	#16862204	C₂²	1	1	yes	characteristic
27	#16862204	C₂²	1	1	yes	characteristic
28	#16862203	—	2	1	no	—
29	#16862202	—	2	1	no	—
30	C₄⋊C₄×D₁₃₁	—	2	1	no	—
31	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×D₁₃₁	—	2	1	no	—
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	#16862171	C₂²	1	1	yes	characteristic, complemented
44	#16862171	C₂²	1	1	yes	characteristic, complemented
45	#16862170	C₂²	1	1	yes	characteristic, complemented
46	#16862170	C₂²	1	1	yes	characteristic, complemented
47	#16862169	C₂²	1	1	yes	characteristic, complemented
48	#16862169	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	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	Q₈×D₁₃₁	—	2	2	no	—
63	Q₈×D₁₃₁	—	2	2	no	—
64	C₂²×Dic₁₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃₁	—	2	2	no	—
66	C₂×C₄×D₁₃₁	—	2	2	no	—
67	C₂×C₄×D₁₃₁	—	2	1	no	—
68	C₂×C₄×D₁₃₁	—	2	1	no	—
69	C₂×C₄×D₁₃₁	—	2	1	no	—
70	C₂×C₄×D₁₃₁	—	2	1	no	—
71	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
75	#14523496	—	2	2	no	—
76	#14523496	—	2	2	no	—
77	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₃₁	—	2	1	no	—
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	Dic₁₃₁⋊C₄	—	2	1	no	—
87	Dic₁₃₁⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₁₃₁	—	2	1	no	—
89	C₂²⋊Dic₁₃₁	—	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	(C₂×D₁₃₁)⋊C₄	—	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	2	no	—
110	C₂²×D₁₃₁	—	4	1	no	—
111	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
113	Dic₂₆₂	—	2	1	no	—
114	Dic₂₆₂	—	2	1	no	—
115	Dic₂₆₂	—	4	2	no	—
116	C₂×Dic₁₃₁	—	2	2	no	—
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	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₃₁	—	2	2	no	—
124	C₄×D₁₃₁	—	2	1	no	—
125	C₄×D₁₃₁	—	2	1	no	—
126	C₄×D₁₃₁	—	2	2	no	—
127	C₄×D₁₃₁	—	4	2	no	—
128	C₄×D₁₃₁	—	4	2	no	—
129	C₄×D₁₃₁	—	4	2	no	—
130	C₄×D₁₃₁	—	4	2	no	—
131	C₄×D₁₃₁	#41	1	2	yes	—
132	C₄×D₁₃₁	#41	1	2	yes	—
133	C₁₃₁⋊C₈	—	2	1	no	—
134	C₁₃₁⋊C₈	—	2	1	no	—
135	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
136	Q₈×C₁₃₁	—	2	1	no	—
137	Q₈×C₁₃₁	—	2	1	no	—
138	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
146	C₈×C₁₃₁	—	2	1	no	—
147	C₈×C₁₃₁	—	2	1	no	—
148	C₂²×C₁₃₁	—	2	1	no	—
149	C₂²×C₁₃₁	—	2	1	no	—
150	C₂²×C₁₃₁	—	2	1	no	—
151	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₁₃₁	—	2	1	no	—
153	C₂×D₁₃₁	—	2	1	no	—
154	C₂×D₁₃₁	—	2	1	no	—
155	C₂×D₁₃₁	—	2	1	no	—
156	C₂×D₁₃₁	—	4	4	no	—
157	C₂×D₁₃₁	#123	1	2	yes	—
158	C₂×D₁₃₁	#137	1	2	yes	—
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	—	4	1	no	—
164	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
166	Dic₁₃₁	—	2	1	no	—
167	Dic₁₃₁	—	4	1	no	—
168	Dic₁₃₁	—	4	1	no	—
169	Dic₁₃₁	—	4	1	no	—
170	Dic₁₃₁	—	4	1	no	—
171	Dic₁₃₁	#123	1	2	yes	—
172	C₂×C₁₃₁	—	4	1	no	—
173	C₂×C₁₃₁	C₂×#123	1	1	yes	characteristic
174	C₂×C₁₃₁	C₂×#137	1	1	yes	characteristic
175	C₂×C₁₃₁	#575	1	1	yes	characteristic
176	D₁₃₁	—	2	2	no	—
177	D₁₃₁	—	2	2	no	—
178	C₁₃₁	#3095	1	1	yes	characteristic, complemented, Sylow
179	#3095	—	131	1	no	maximal, Sylow
180	C₂×C₂²⋊Q₈	—	131	1	no	—
181	C₂×#117	—	131	1	no	—
182	#485	—	131	1	no	—
183	#485	—	131	1	no	—
184	#485	—	131	1	no	—
185	#485	D₁₃₁	1	1	yes	characteristic, complemented
186	#484	—	131	1	no	—
187	#484	—	131	1	no	—
188	#483	—	131	1	no	—
189	#483	—	131	1	no	—
190	#428	—	131	1	no	—
191	#428	—	131	1	no	—
192	#419	—	131	1	no	—
193	#419	—	131	1	no	—
194	#407	—	131	1	no	—
195	C₂²×Q₈	—	131	1	no	—
196	C₂³×C₄	—	131	1	no	—
197	C₂×M₁₆	—	131	1	no	—
198	C₂×M₁₆	—	131	1	no	—
199	C₂²⋊Q₈	—	262	1	no	—
200	C₂²⋊Q₈	—	262	1	no	—
201	C₂²⋊Q₈	—	131	1	no	—
202	C₂²⋊Q₈	—	131	1	no	—
203	C₂²⋊Q₈	—	131	1	no	—
204	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
205	#117	—	262	1	no	—
206	#117	—	262	1	no	—
207	#117	—	131	1	no	—
208	#117	—	131	1	no	—
209	#117	—	131	1	no	—
210	#117	C₂×D₁₃₁	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	262	1	no	—
212	C₂×C₄⋊C₄	—	131	1	no	—
213	C₂×C₄⋊C₄	—	131	1	no	—
214	C₂×C₄⋊C₄	—	131	1	no	—
215	C₂×C₂²⋊C₄	—	262	1	no	—
216	C₂×C₂²⋊C₄	—	262	1	no	—
217	C₂×C₄²	—	262	1	no	—
218	#107	—	131	1	no	—
219	#107	—	131	1	no	—
220	#107	—	131	1	no	—
221	#107	C₂×D₁₃₁	1	1	yes	characteristic, complemented
222	#106	—	131	1	no	—
223	#106	—	131	1	no	—
224	#106	—	131	1	no	—
225	#106	C₂×D₁₃₁	1	1	yes	characteristic, complemented
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	#103	C₂×D₁₃₁	1	1	yes	characteristic, complemented
233	#103	C₂×D₁₃₁	1	1	yes	characteristic, complemented
234	C₂²⋊C₈	—	131	1	no	—
235	C₂²⋊C₈	—	131	1	no	—
236	C₂²⋊C₈	—	131	1	no	—
237	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
238	C₂⁴	—	131	1	no	—
239	C₂×Q₈	—	262	1	no	—
240	C₂×Q₈	—	262	2	no	—
241	C₂×Q₈	—	262	2	no	—
242	C₂×Q₈	—	131	1	no	—
243	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
244	C₂²×C₄	—	262	2	no	—
245	C₂²×C₄	—	262	2	no	—
246	C₂²×C₄	—	262	1	no	—
247	C₂²×C₄	—	262	1	no	—
248	C₂²×C₄	—	262	1	no	—
249	C₂²×C₄	—	262	1	no	—
250	C₂²×C₄	—	131	1	no	—
251	C₂²×C₄	—	131	1	no	—
252	C₂²×C₄	—	131	1	no	—
253	C₂²×C₄	—	131	1	no	—
254	C₂²×C₄	—	131	1	no	—
255	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
256	M₁₆	—	262	2	no	—
257	M₁₆	—	262	2	no	—
258	C₂×C₈	—	131	1	no	—
259	C₂×C₈	—	131	1	no	—
260	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	262	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	—	131	1	no	—
270	C₄⋊C₄	—	131	1	no	—
271	C₄⋊C₄	—	131	1	no	—
272	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	262	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	C₄²	—	2	1	no	—
284	C₄²	—	262	1	no	—
285	C₄²	—	262	1	no	—
286	C₄²	—	262	1	no	—
287	C₂³	—	262	2	no	—
288	C₂³	—	262	2	no	—
289	C₂³	—	524	1	no	—
290	C₂³	—	131	1	no	—
291	C₂³	—	131	1	no	—
292	C₂³	D₄×D₁₃₁	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	262	1	no	—
296	Q₈	—	262	1	no	—
297	Q₈	—	524	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	262	2	no	—
305	C₂×C₄	—	262	2	no	—
306	C₂×C₄	—	262	2	no	—
307	C₂×C₄	—	262	1	no	—
308	C₂×C₄	—	262	1	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	—	524	2	no	—
314	C₂×C₄	—	524	2	no	—
315	C₂×C₄	—	524	2	no	—
316	C₂×C₄	—	524	2	no	—
317	C₂×C₄	—	131	1	no	—
318	C₂×C₄	—	131	1	no	—
319	C₂×C₄	—	131	2	no	—
320	C₂×C₄	—	131	2	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	—	2	1	no	—
330	C₂²	—	262	1	no	—
331	C₂²	—	262	1	no	—
332	C₂²	—	262	1	no	—
333	C₂²	—	262	1	no	—
334	C₂²	—	524	4	no	—
335	C₂²	—	131	2	no	—
336	C₂²	—	131	2	no	—
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	—	4	1	no	—
343	C₄	—	262	1	no	—
344	C₄	—	524	1	no	—
345	C₄	—	524	1	no	—
346	C₄	—	524	1	no	—
347	C₄	—	524	1	no	—
348	C₄	—	131	2	no	—
349	C₄	#41×D₁₃₁	1	1	yes	characteristic
350	C₄	#41×D₁₃₁	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	262	2	no	—
353	C₂	—	262	2	no	—
354	C₂	#123×D₁₃₁	1	1	yes	characteristic
355	C₂	#137×D₁₃₁	1	1	yes	characteristic
356	C₂	#22554109	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	C₂³×D₁₃₁	D₄
hypercenter	#485	D₁₃₁
hypocenter	C₁₃₁	#3095
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
3. #22553654	C₂	semidirect
4. #22553653	C₂	semidirect
5. #22553650	C₂	semidirect
6. #22553649	C₂	semidirect
7. #22553646	C₂	semidirect
8. #22553644	C₂	semidirect
9. #22553641	C₂	semidirect
10. #22553353	C₂	semidirect
11. #22553351	C₂	semidirect
12. #22553319	C₂	semidirect
13. #22553319	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. #16862171	C₂²	semidirect
44. #16862171	C₂²	semidirect
45. #16862170	C₂²	semidirect
46. #16862170	C₂²	semidirect
47. #16862169	C₂²	semidirect
48. #16862169	C₂²	semidirect
54. #107×C₁₃₁	C₂²	semidirect
55. #106×C₁₃₁	C₂²	semidirect
56. #103×C₁₃₁	C₂²	semidirect
57. #103×C₁₃₁	C₂²	semidirect
178. C₁₃₁	#3095	semidirect
185. #485	D₁₃₁	semidirect
221. #107	C₂×D₁₃₁	semidirect
225. #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
C₂²⋊Q₈×D₁₃₁	2
#117×D₁₃₁	2
#22553654	2
#22553653	2
#22553650	2
#22553649	2
#22553646	2
#22553644	2
#22553641	2
#22553353	2
#22553351	2
#22553319	2
#22553274	2
#485×C₁₃₁	2
#3095	131