Subgroups — #30676136

Order 16768.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553849	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553646	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553632	C₂	1	1	yes	maximal, characteristic
5	#22553622	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553621	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553537	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553536	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553466	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553452	C₂	1	1	yes	maximal, characteristic
11	#22553417	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553414	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553410	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553367	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×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	C₂×#14523502	C₂²	1	1	yes	characteristic
19	C₂×#14523500	C₂²	1	1	yes	characteristic
20	#16862241	—	2	1	no	—
21	#16862237	C₂²	1	1	yes	characteristic
22	#16862235	C₂²	1	1	yes	characteristic
23	#16862231	—	2	1	no	—
24	#16862222	C₂²	1	1	yes	characteristic
25	#16862221	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₆₂	—	2	1	no	—
28	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
29	#16862211	—	2	1	no	—
30	#16862211	C₂²	1	1	yes	characteristic
31	C₄×D₅₂₄	—	2	1	no	—
32	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
33	#16862203	—	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	#16862180	C₂²	1	1	yes	characteristic
39	#16862179	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	#16862170	C₂²	1	1	yes	characteristic
46	#16862169	C₂²	1	1	yes	characteristic
47	#16862168	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	#16862157	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
55	#106×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	#97×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
61	Q₈×D₁₃₁	—	2	2	no	—
62	Q₈×D₁₃₁	—	2	2	no	—
63	C₂×D₅₂₄	—	2	1	no	—
64	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	—	2	1	no	—
68	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
71	#14523503	—	2	2	no	—
72	#14523502	—	2	2	no	—
73	#14523500	—	2	2	no	—
74	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₃₁	—	2	1	no	—
77	C₄⋊Dic₁₃₁	—	2	1	no	—
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	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₁)⋊C₄	—	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	C₄×Dic₁₃₁	—	2	1	no	—
91	C₄×Dic₁₃₁	—	2	1	no	—
92	C₄×Dic₁₃₁	—	2	1	no	—
93	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
95	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₃₁	—	2	2	no	—
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	—	2	1	no	—
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₄²×C₁₃₁	—	2	1	no	—
106	C₄²×C₁₃₁	—	2	1	no	—
107	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₃₁	—	2	1	no	—
109	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
110	Dic₂₆₂	—	2	4	no	—
111	Dic₂₆₂	—	2	1	no	—
112	Dic₂₆₂	#41	1	2	yes	—
113	D₅₂₄	—	2	1	no	—
114	D₅₂₄	—	2	1	no	—
115	D₅₂₄	—	4	2	no	—
116	C₂×Dic₁₃₁	—	2	1	no	—
117	C₂×Dic₁₃₁	—	2	1	no	—
118	C₂×Dic₁₃₁	—	2	1	no	—
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₃₁	—	2	4	no	—
123	C₄×D₁₃₁	—	2	2	no	—
124	C₄×D₁₃₁	—	2	1	no	—
125	C₄×D₁₃₁	—	2	1	no	—
126	C₄×D₁₃₁	—	4	1	no	—
127	C₄×D₁₃₁	—	4	2	no	—
128	C₄×D₁₃₁	—	4	2	no	—
129	C₁₃₁⋊C₈	—	2	1	no	—
130	C₁₃₁⋊C₈	—	2	1	no	—
131	Q₈×C₁₃₁	—	2	1	no	—
132	Q₈×C₁₃₁	—	2	1	no	—
133	Q₈×C₁₃₁	—	2	1	no	—
134	Q₈×C₁₃₁	#41	1	2	yes	—
135	C₂×C₄×C₁₃₁	—	2	1	no	—
136	C₂×C₄×C₁₃₁	—	2	1	no	—
137	C₂×C₄×C₁₃₁	—	2	1	no	—
138	C₂×C₄×C₁₃₁	—	2	1	no	—
139	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₃₁	—	2	1	no	—
143	C₈×C₁₃₁	—	2	1	no	—
144	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₃₁	—	2	1	no	—
146	C₂×D₁₃₁	—	2	1	no	—
147	C₂×D₁₃₁	—	4	1	no	—
148	C₂×D₁₃₁	—	4	1	no	—
149	C₂×D₁₃₁	—	4	1	no	—
150	C₂×D₁₃₁	#137	1	2	yes	—
151	C₄×C₁₃₁	—	2	2	no	—
152	C₄×C₁₃₁	—	2	1	no	—
153	C₄×C₁₃₁	—	2	1	no	—
154	C₄×C₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	—	4	1	no	—
156	C₄×C₁₃₁	—	4	1	no	—
157	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
159	Dic₁₃₁	—	2	2	no	—
160	Dic₁₃₁	—	2	1	no	—
161	Dic₁₃₁	—	2	1	no	—
162	Dic₁₃₁	—	2	1	no	—
163	Dic₁₃₁	—	4	1	no	—
164	Dic₁₃₁	—	4	1	no	—
165	C₂×C₁₃₁	#525	1	1	yes	characteristic
166	C₂×C₁₃₁	C₂×#137	1	1	yes	characteristic
167	C₂×C₁₃₁	#575	1	1	yes	characteristic
168	D₁₃₁	—	2	2	no	—
169	D₁₃₁	—	8	1	no	—
170	C₁₃₁	#3117	1	1	yes	characteristic, complemented, Sylow
171	#3117	—	131	1	no	maximal, Sylow
172	#529	—	131	1	no	—
173	C₂×C₄×Q₈	—	131	1	no	—
174	#484	—	131	1	no	—
175	#481	—	131	1	no	—
176	#479	—	131	1	no	—
177	#479	—	131	1	no	—
178	#464	—	131	1	no	—
179	#464	—	131	1	no	—
180	#451	—	131	1	no	—
181	#448	—	131	1	no	—
182	#441	—	131	1	no	—
183	#441	D₁₃₁	1	1	yes	characteristic, complemented
184	#440	—	131	1	no	—
185	#440	—	131	1	no	—
186	#432	—	131	1	no	—
187	C₂²×Q₈	—	131	1	no	—
188	C₂×Q₁₆	—	131	1	no	—
189	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
190	C₂×SD₁₆	—	131	1	no	—
191	C₂×SD₁₆	—	131	1	no	—
192	#126	—	262	1	no	—
193	#125	—	131	1	no	—
194	#124	—	131	1	no	—
195	#123	—	262	1	no	—
196	C₂²⋊Q₈	—	131	1	no	—
197	#121	—	131	1	no	—
198	C₄×Q₈	—	262	1	no	—
199	C₄×Q₈	—	262	1	no	—
200	C₄×Q₈	—	131	1	no	—
201	C₄×Q₈	—	131	1	no	—
202	C₄×Q₈	—	131	1	no	—
203	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
204	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
205	C₄×D₄	—	262	1	no	—
206	C₄×D₄	—	131	1	no	—
207	#117	—	262	1	no	—
208	C₂×C₄⋊C₄	—	262	1	no	—
209	C₂×C₄⋊C₄	—	131	1	no	—
210	C₂×C₄²	—	262	1	no	—
211	C₂×C₄²	—	131	1	no	—
212	#107	—	131	1	no	—
213	#107	—	131	1	no	—
214	#106	—	131	1	no	—
215	#106	C₂×D₁₃₁	1	1	yes	characteristic
216	C₄⋊C₈	—	131	1	no	—
217	C₄⋊C₈	—	131	1	no	—
218	#103	—	131	1	no	—
219	#103	—	131	1	no	—
220	#103	—	131	1	no	—
221	#103	—	131	1	no	—
222	#103	C₂×D₁₃₁	1	1	yes	characteristic
223	#103	C₂×D₁₃₁	1	1	yes	characteristic
224	#102	—	131	1	no	—
225	#102	—	131	1	no	—
226	C₂²⋊C₈	—	131	1	no	—
227	C₂²⋊C₈	—	131	1	no	—
228	#97	—	131	1	no	—
229	#97	C₂×D₁₃₁	1	1	yes	characteristic
230	C₂×Q₈	—	262	1	no	—
231	C₂×Q₈	—	262	2	no	—
232	C₂×Q₈	—	262	2	no	—
233	C₂×Q₈	—	131	1	no	—
234	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
236	C₂×D₄	—	262	1	no	—
237	C₂×D₄	—	131	1	no	—
238	C₂²×C₄	—	262	1	no	—
239	C₂²×C₄	—	262	1	no	—
240	C₂²×C₄	—	262	1	no	—
241	C₂²×C₄	—	131	1	no	—
242	C₂²×C₄	—	131	1	no	—
243	C₂²×C₄	—	131	1	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	262	2	no	—
246	SD₁₆	—	262	2	no	—
247	SD₁₆	—	262	2	no	—
248	C₂×C₈	—	131	1	no	—
249	C₂×C₈	—	131	1	no	—
250	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	262	1	no	—
255	C₄⋊C₄	—	262	1	no	—
256	C₄⋊C₄	—	262	1	no	—
257	C₄⋊C₄	—	262	1	no	—
258	C₄⋊C₄	—	262	1	no	—
259	C₄⋊C₄	—	262	1	no	—
260	C₄⋊C₄	—	262	1	no	—
261	C₄⋊C₄	—	131	1	no	—
262	C₄⋊C₄	—	131	1	no	—
263	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	262	1	no	—
266	C₂²⋊C₄	—	262	1	no	—
267	C₂²⋊C₄	—	262	1	no	—
268	C₂²⋊C₄	—	262	1	no	—
269	C₂²⋊C₄	—	262	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	262	1	no	—
273	C₄²	—	262	1	no	—
274	C₄²	—	262	1	no	—
275	C₄²	—	131	1	no	—
276	C₄²	—	131	1	no	—
277	C₄²	—	131	1	no	—
278	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
279	C₂³	—	262	1	no	—
280	C₂³	—	131	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	262	4	no	—
285	Q₈	—	262	1	no	—
286	Q₈	—	131	2	no	—
287	Q₈	#14523513	1	2	yes	—
288	D₄	—	262	1	no	—
289	D₄	—	262	1	no	—
290	D₄	—	524	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	262	4	no	—
296	C₂×C₄	—	262	2	no	—
297	C₂×C₄	—	262	1	no	—
298	C₂×C₄	—	262	1	no	—
299	C₂×C₄	—	262	1	no	—
300	C₂×C₄	—	262	1	no	—
301	C₂×C₄	—	262	1	no	—
302	C₂×C₄	—	524	1	no	—
303	C₂×C₄	—	524	2	no	—
304	C₂×C₄	—	524	2	no	—
305	C₂×C₄	—	131	1	no	—
306	C₂×C₄	—	131	1	no	—
307	C₂×C₄	—	131	1	no	—
308	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	262	1	no	—
314	C₈	—	262	1	no	—
315	C₂²	—	262	1	no	—
316	C₂²	—	262	1	no	—
317	C₂²	—	524	1	no	—
318	C₂²	—	524	1	no	—
319	C₂²	—	524	1	no	—
320	C₂²	—	131	2	no	—
321	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	262	2	no	—
329	C₄	—	262	1	no	—
330	C₄	—	262	1	no	—
331	C₄	—	262	1	no	—
332	C₄	—	524	1	no	—
333	C₄	—	524	1	no	—
334	C₄	#41×D₁₃₁	1	1	yes	characteristic
335	C₄	C₂×#14523513	1	1	yes	characteristic
336	C₂	—	262	2	no	—
337	C₂	—	1048	1	no	—
338	C₂	#22553821	1	1	yes	characteristic
339	C₂	#137×D₁₃₁	1	1	yes	characteristic
340	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	#441×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#16862221	C₂²
hypercenter	#441	D₁₃₁
hypocenter	C₁₃₁	#3117
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₁₃₁	C₂	semidirect
3. #22553646	C₂	semidirect
5. #22553622	C₂	semidirect
6. #22553621	C₂	semidirect
7. #22553537	C₂	semidirect
8. #22553536	C₂	semidirect
9. #22553466	C₂	semidirect
11. #22553417	C₂	semidirect
12. #22553414	C₂	semidirect
13. #22553410	C₂	semidirect
14. #22553367	C₂	semidirect
15. #441×C₁₃₁	C₂	semidirect
170. C₁₃₁	#3117	semidirect
183. #441	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553849	2
C₄×Q₈×D₁₃₁	2
#22553646	2
#22553632	2
#22553622	2
#22553621	2
#22553537	2
#22553536	2
#22553466	2
#22553452	2
#22553417	2
#22553414	2
#22553410	2
#22553367	2
#441×C₁₃₁	2
#3117	131