Subgroups — #30675998

Order 16768.

384 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553866	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#22553649	C₂	1	1	yes	maximal, characteristic
4	#22553646	C₂	1	1	yes	maximal, characteristic
5	#22553606	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553605	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553521	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553520	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553469	C₂	1	1	yes	maximal, characteristic
10	#22553466	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₅₂₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14523503	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#14523503	C₂	1	1	yes	maximal, characteristic, complemented
14	#22553363	C₂	1	1	yes	maximal, characteristic
15	C₄×Q₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₂₄	C₂²	1	1	yes	characteristic
18	C₂×#14523503	C₂²	1	1	yes	characteristic
19	C₂×#14523503	C₂²	1	1	yes	characteristic
20	#16862245	C₂²	1	1	yes	characteristic
21	#16862244	C₂²	1	1	yes	characteristic
22	#16862236	—	2	1	no	—
23	#16862226	C₂²	1	1	yes	characteristic
24	#16862224	—	2	1	no	—
25	#16862222	C₂²	1	1	yes	characteristic
26	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₆₂	—	2	1	no	—
28	C₄×Dic₂₆₂	—	2	1	no	—
29	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
31	#16862211	—	2	1	no	—
32	#16862211	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	#16862179	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	#16862171	C₂²	1	1	yes	characteristic
45	#16862170	C₂²	1	1	yes	characteristic
46	#16862170	C₂²	1	1	yes	characteristic
47	#16862169	C₂²	1	1	yes	characteristic
48	#16862169	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×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	#107×C₁₃₁	C₂²	1	1	yes	characteristic
56	#103×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
63	Q₈×D₁₃₁	—	2	2	no	—
64	Q₈×D₁₃₁	—	2	2	no	—
65	C₂×C₄×D₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
70	Dic₅₂₄	—	2	2	no	—
71	#14523503	—	2	2	no	—
72	#14523503	—	2	2	no	—
73	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₁	—	2	1	no	—
76	C₄⋊Dic₁₃₁	—	2	1	no	—
77	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
79	Dic₁₃₁⋊C₄	—	2	1	no	—
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	Dic₁₃₁⋊C₄	—	2	1	no	—
86	Dic₁₃₁⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
89	C₄×Dic₁₃₁	—	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	C₂×D₄	1	1	yes	characteristic
109	Dic₂₆₂	—	2	4	no	—
110	Dic₂₆₂	—	2	1	no	—
111	Dic₂₆₂	—	2	1	no	—
112	Dic₂₆₂	—	2	1	no	—
113	Dic₂₆₂	—	4	2	no	—
114	Dic₂₆₂	#41	1	2	yes	—
115	C₂×Dic₁₃₁	—	2	1	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	1	no	—
124	C₄×D₁₃₁	—	2	1	no	—
125	C₄×D₁₃₁	—	4	1	no	—
126	C₄×D₁₃₁	—	4	2	no	—
127	C₄×D₁₃₁	Q₁₆	1	4	yes	complemented
128	C₁₃₁⋊C₈	—	2	1	no	—
129	C₁₃₁⋊C₈	—	2	1	no	—
130	Q₈×C₁₃₁	—	2	1	no	—
131	Q₈×C₁₃₁	—	2	1	no	—
132	Q₈×C₁₃₁	—	2	1	no	—
133	Q₈×C₁₃₁	#41	1	2	yes	—
134	C₂×C₄×C₁₃₁	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
141	C₈×C₁₃₁	—	2	1	no	—
142	C₈×C₁₃₁	—	2	1	no	—
143	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₃₁	—	2	1	no	—
145	C₂×D₁₃₁	—	2	1	no	—
146	C₂×D₁₃₁	C₂×Q₁₆	1	2	yes	—
147	C₄×C₁₃₁	—	2	2	no	—
148	C₄×C₁₃₁	—	2	1	no	—
149	C₄×C₁₃₁	—	4	1	no	—
150	C₄×C₁₃₁	—	4	1	no	—
151	C₄×C₁₃₁	—	4	1	no	—
152	C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
153	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
155	Dic₁₃₁	—	2	2	no	—
156	Dic₁₃₁	—	2	1	no	—
157	Dic₁₃₁	—	2	1	no	—
158	Dic₁₃₁	—	4	1	no	—
159	Dic₁₃₁	—	4	1	no	—
160	Dic₁₃₁	—	4	1	no	—
161	Dic₁₃₁	C₂×Q₁₆	1	2	yes	—
162	C₂×C₁₃₁	#525	1	1	yes	characteristic
163	C₂×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
164	C₂×C₁₃₁	#575	1	1	yes	characteristic
165	D₁₃₁	—	2	2	no	—
166	C₁₃₁	#3104	1	1	yes	characteristic, complemented, Sylow
167	#3104	—	131	1	no	maximal, Sylow
168	#533	—	131	1	no	—
169	C₂×C₄×Q₈	—	131	1	no	—
170	#484	—	131	1	no	—
171	#484	—	131	1	no	—
172	#477	—	131	1	no	—
173	#477	—	131	1	no	—
174	#462	—	131	1	no	—
175	#462	—	131	1	no	—
176	#451	—	131	1	no	—
177	#451	—	131	1	no	—
178	C₄×Q₁₆	—	131	1	no	—
179	C₄×Q₁₆	—	131	1	no	—
180	C₄×Q₁₆	—	131	1	no	—
181	C₄×Q₁₆	D₁₃₁	1	1	yes	characteristic, complemented
182	#431	—	131	1	no	—
183	C₂²×Q₈	—	131	1	no	—
184	C₂×Q₁₆	—	131	1	no	—
185	C₂×Q₁₆	—	131	1	no	—
186	C₂×Q₁₆	—	131	1	no	—
187	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
188	C₄⋊Q₈	—	131	1	no	—
189	C₄⋊Q₈	—	131	1	no	—
190	#125	—	262	1	no	—
191	C₂²⋊Q₈	—	262	1	no	—
192	C₂²⋊Q₈	—	131	1	no	—
193	C₂²⋊Q₈	—	131	1	no	—
194	C₄×Q₈	—	262	1	no	—
195	C₄×Q₈	—	262	1	no	—
196	C₄×Q₈	—	262	1	no	—
197	C₄×Q₈	—	131	1	no	—
198	C₄×Q₈	—	131	1	no	—
199	C₄×Q₈	—	131	1	no	—
200	C₄×Q₈	—	131	1	no	—
201	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
202	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
203	#117	—	262	1	no	—
204	C₂×C₄⋊C₄	—	262	1	no	—
205	C₂×C₄⋊C₄	—	131	1	no	—
206	C₂×C₄²	—	262	1	no	—
207	C₂×C₄²	—	131	1	no	—
208	#107	—	131	1	no	—
209	#107	—	131	1	no	—
210	#107	—	131	1	no	—
211	#107	C₂×D₁₃₁	1	1	yes	characteristic
212	C₄⋊C₈	—	131	1	no	—
213	C₄⋊C₈	—	131	1	no	—
214	#103	—	131	1	no	—
215	#103	—	131	1	no	—
216	#103	—	131	1	no	—
217	#103	—	131	1	no	—
218	#103	—	131	1	no	—
219	#103	—	131	1	no	—
220	#103	C₂×D₁₃₁	1	1	yes	characteristic
221	#103	C₂×D₁₃₁	1	1	yes	characteristic
222	C₂²⋊C₈	—	131	1	no	—
223	C₂²⋊C₈	—	131	1	no	—
224	C₄×C₈	—	131	1	no	—
225	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
226	C₂×Q₈	—	262	1	no	—
227	C₂×Q₈	—	262	1	no	—
228	C₂×Q₈	—	262	2	no	—
229	C₂×Q₈	—	262	2	no	—
230	C₂×Q₈	—	131	1	no	—
231	C₂×Q₈	—	131	1	no	—
232	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
234	C₂²×C₄	—	262	1	no	—
235	C₂²×C₄	—	262	1	no	—
236	C₂²×C₄	—	131	1	no	—
237	C₂²×C₄	—	131	1	no	—
238	C₂²×C₄	—	131	1	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	262	2	no	—
241	Q₁₆	—	262	2	no	—
242	Q₁₆	—	262	2	no	—
243	C₂×C₈	—	131	1	no	—
244	C₂×C₈	—	131	1	no	—
245	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	262	1	no	—
250	C₄⋊C₄	—	262	1	no	—
251	C₄⋊C₄	—	262	1	no	—
252	C₄⋊C₄	—	262	1	no	—
253	C₄⋊C₄	—	262	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	—	131	1	no	—
260	C₄⋊C₄	—	131	1	no	—
261	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	262	1	no	—
264	C₂²⋊C₄	—	262	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	262	1	no	—
268	C₄²	—	262	1	no	—
269	C₄²	—	262	1	no	—
270	C₄²	—	262	1	no	—
271	C₄²	—	131	1	no	—
272	C₄²	—	131	1	no	—
273	C₄²	—	131	1	no	—
274	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
275	C₂³	—	131	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	262	4	no	—
280	Q₈	—	262	1	no	—
281	Q₈	—	262	1	no	—
282	Q₈	—	262	1	no	—
283	Q₈	—	524	2	no	—
284	Q₈	—	131	2	no	—
285	Q₈	#14523513	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	262	4	no	—
291	C₂×C₄	—	262	1	no	—
292	C₂×C₄	—	262	1	no	—
293	C₂×C₄	—	262	1	no	—
294	C₂×C₄	—	262	1	no	—
295	C₂×C₄	—	262	1	no	—
296	C₂×C₄	—	262	1	no	—
297	C₂×C₄	—	524	1	no	—
298	C₂×C₄	—	524	2	no	—
299	C₂×C₄	—	131	1	no	—
300	C₂×C₄	—	131	1	no	—
301	C₂×C₄	—	131	1	no	—
302	C₂×C₄	—	131	4	no	—
303	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
304	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	262	1	no	—
309	C₈	—	262	1	no	—
310	C₂²	—	262	1	no	—
311	C₂²	—	262	1	no	—
312	C₂²	—	131	2	no	—
313	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	262	2	no	—
320	C₄	—	262	1	no	—
321	C₄	—	262	1	no	—
322	C₄	—	524	1	no	—
323	C₄	—	524	1	no	—
324	C₄	—	524	1	no	—
325	C₄	—	131	2	no	—
326	C₄	Q₁₆×D₁₃₁	1	2	yes	—
327	C₄	#41×D₁₃₁	1	1	yes	characteristic
328	C₄	C₂×#14523513	1	1	yes	characteristic
329	C₂	—	262	2	no	—
330	C₂	#22553821	1	1	yes	characteristic
331	C₂	C₂×Q₁₆×D₁₃₁	1	1	yes	characteristic
332	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×Q₁₆×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂²×D₁₃₁	C₂×D₄
hypercenter	C₄×Q₁₆	D₁₃₁
hypocenter	C₁₃₁	#3104
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
5. #22553606	C₂	semidirect
6. #22553605	C₂	semidirect
7. #22553521	C₂	semidirect
8. #22553520	C₂	semidirect
11. C₄×Dic₅₂₄	C₂	semidirect
12. C₄×#14523503	C₂	semidirect
13. C₄×#14523503	C₂	semidirect
15. C₄×Q₁₆×C₁₃₁	C₂	semidirect
127. C₄×D₁₃₁	Q₁₆	semidirect
166. C₁₃₁	#3104	semidirect
181. C₄×Q₁₆	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553866	2
C₄×Q₈×D₁₃₁	2
#22553649	2
#22553646	2
#22553606	2
#22553605	2
#22553521	2
#22553520	2
#22553469	2
#22553466	2
C₄×Dic₅₂₄	2
C₄×#14523503	2
#22553363	2
C₄×Q₁₆×C₁₃₁	2
#3104	131