Subgroups — #34081823

Order 19328.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752931	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₃₀₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752571	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752570	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752561	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752508	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752505	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752490	C₂	1	1	yes	maximal, characteristic, complemented
9	#23752425	C₂	1	1	yes	maximal, characteristic
10	#23752415	C₂	1	1	yes	maximal, characteristic, complemented
11	#23752341	C₂	1	1	yes	maximal, characteristic, complemented
12	#23752313	C₂	1	1	yes	maximal, characteristic, complemented
13	#23752312	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
15	#463×C₁₅₁	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₃₀₂	—	2	1	no	—
17	C₂×#14558992	C₂²	1	1	yes	characteristic
18	#18012739	C₂²	1	1	yes	characteristic
19	#18012738	C₂²	1	1	yes	characteristic
20	#18012738	C₂²	1	1	yes	characteristic
21	#18012737	C₂²	1	1	yes	characteristic
22	#18012732	—	2	1	no	—
23	#18012732	—	2	1	no	—
24	#18012726	C₂²	1	1	yes	characteristic
25	#18012718	—	2	1	no	—
26	#18012716	—	2	1	no	—
27	#18012716	—	2	1	no	—
28	#18012716	—	2	1	no	—
29	Q₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₀₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₃₀₂	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
35	#18012695	—	2	1	no	—
36	#18012673	C₂²	1	1	yes	characteristic
37	#18012673	C₂²	1	1	yes	characteristic
38	#18012672	C₂²	1	1	yes	characteristic
39	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
41	#18012664	C₂²	1	1	yes	characteristic
42	#18012664	C₂²	1	1	yes	characteristic
43	#18012663	C₂²	1	1	yes	characteristic
44	#18012663	C₂²	1	1	yes	characteristic
45	#18012662	C₂²	1	1	yes	characteristic
46	#18012660	C₂²	1	1	yes	characteristic
47	#18012660	C₂²	1	1	yes	characteristic
48	#18012660	C₂²	1	1	yes	characteristic
49	#18012650	C₂²	1	1	yes	characteristic
50	#18012649	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₁₅₁	C₂²	1	1	yes	characteristic
54	#124×C₁₅₁	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
57	#103×C₁₅₁	C₂²	1	1	yes	characteristic
58	#102×C₁₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₀₂	—	2	1	no	—
60	C₂×Dic₃₀₂	—	2	1	no	—
61	C₂×Dic₃₀₂	—	2	1	no	—
62	C₂×Dic₃₀₂	—	4	4	no	—
63	C₂×Dic₃₀₂	D₄	1	1	yes	characteristic
64	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₅₁	—	2	1	no	—
66	C₂²×Dic₁₅₁	—	2	1	no	—
67	C₂²×Dic₁₅₁	—	2	1	no	—
68	#14558992	—	2	2	no	—
69	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₅₁	—	2	1	no	—
72	C₄⋊Dic₁₅₁	—	2	1	no	—
73	C₄⋊Dic₁₅₁	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
77	Dic₁₅₁⋊C₄	—	2	1	no	—
78	Dic₁₅₁⋊C₄	—	2	1	no	—
79	Dic₁₅₁⋊C₄	—	2	1	no	—
80	Dic₁₅₁⋊C₄	—	2	1	no	—
81	Dic₁₅₁⋊C₄	—	2	1	no	—
82	Dic₁₅₁⋊C₄	D₄	1	1	yes	characteristic
83	Dic₁₅₁⋊C₄	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₅₁	—	2	1	no	—
85	C₂²⋊Dic₁₅₁	—	2	1	no	—
86	C₂²⋊Dic₁₅₁	—	2	1	no	—
87	C₂²⋊Dic₁₅₁	—	2	1	no	—
88	C₂²⋊Dic₁₅₁	—	2	1	no	—
89	C₄×Dic₁₅₁	—	2	1	no	—
90	C₄×Dic₁₅₁	—	2	1	no	—
91	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₅₁	—	2	1	no	—
97	SD₁₆×C₁₅₁	—	2	2	no	—
98	D₈×C₁₅₁	—	2	2	no	—
99	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
103	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
104	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
105	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
106	Dic₃₀₂	—	2	2	no	—
107	Dic₃₀₂	—	2	1	no	—
108	Dic₃₀₂	—	2	1	no	—
109	Dic₃₀₂	—	4	2	no	—
110	Dic₃₀₂	—	4	2	no	—
111	Dic₃₀₂	—	4	2	no	—
112	C₂×Dic₁₅₁	—	2	1	no	—
113	C₂×Dic₁₅₁	—	2	1	no	—
114	C₂×Dic₁₅₁	—	2	1	no	—
115	C₂×Dic₁₅₁	—	2	1	no	—
116	C₂×Dic₁₅₁	—	4	2	no	—
117	C₂×Dic₁₅₁	—	4	2	no	—
118	C₂×Dic₁₅₁	—	4	2	no	—
119	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
123	C₁₅₁⋊C₈	—	2	1	no	—
124	C₁₅₁⋊C₈	—	2	1	no	—
125	C₂³×C₁₅₁	—	2	1	no	—
126	C₂³×C₁₅₁	—	2	1	no	—
127	Q₈×C₁₅₁	—	2	1	no	—
128	Q₈×C₁₅₁	—	2	1	no	—
129	D₄×C₁₅₁	—	2	1	no	—
130	D₄×C₁₅₁	—	2	1	no	—
131	D₄×C₁₅₁	—	2	1	no	—
132	D₄×C₁₅₁	#41	1	2	yes	—
133	C₂×C₄×C₁₅₁	—	2	1	no	—
134	C₂×C₄×C₁₅₁	—	2	1	no	—
135	C₂×C₄×C₁₅₁	—	4	2	no	—
136	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₅₁	—	2	1	no	—
140	C₈×C₁₅₁	—	2	1	no	—
141	C₂²×C₁₅₁	—	2	2	no	—
142	C₂²×C₁₅₁	—	4	1	no	—
143	C₂²×C₁₅₁	—	4	1	no	—
144	C₂²×C₁₅₁	—	4	1	no	—
145	C₂²×C₁₅₁	—	4	1	no	—
146	C₂²×C₁₅₁	—	4	1	no	—
147	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₅₁	—	2	1	no	—
149	C₄×C₁₅₁	—	2	1	no	—
150	C₄×C₁₅₁	—	4	1	no	—
151	C₄×C₁₅₁	—	4	1	no	—
152	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
153	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₅₁	—	2	1	no	—
155	Dic₁₅₁	—	2	1	no	—
156	Dic₁₅₁	—	2	1	no	—
157	Dic₁₅₁	—	4	1	no	—
158	Dic₁₅₁	—	4	1	no	—
159	Dic₁₅₁	—	4	1	no	—
160	Dic₁₅₁	—	4	1	no	—
161	Dic₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₁₅₁	—	4	2	no	—
163	C₂×C₁₅₁	—	8	1	no	—
164	C₂×C₁₅₁	#548	1	1	yes	characteristic
165	C₂×C₁₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₁₅₁	#578	1	1	yes	characteristic
167	C₁₅₁	#3367	1	1	yes	characteristic, complemented, Sylow
168	#3367	—	151	1	no	maximal, Sylow
169	#556	—	151	1	no	—
170	D₄×Q₈	—	151	1	no	—
171	#489	—	151	1	no	—
172	#489	—	151	1	no	—
173	#487	—	151	1	no	—
174	#477	—	151	1	no	—
175	#477	—	151	1	no	—
176	#474	—	151	1	no	—
177	#463	—	151	1	no	—
178	#463	D₁₅₁	1	1	yes	characteristic
179	#461	—	151	1	no	—
180	#446	—	151	1	no	—
181	#440	—	151	1	no	—
182	#440	—	151	1	no	—
183	C₄×D₈	—	151	1	no	—
184	C₂²×Q₈	—	302	1	no	—
185	C₂×SD₁₆	—	151	1	no	—
186	C₂×SD₁₆	C₂×D₁₅₁	1	1	yes	characteristic
187	C₂×D₈	C₂×D₁₅₁	1	1	yes	characteristic
188	C₄⋊Q₈	—	151	1	no	—
189	C₄⋊Q₈	—	151	1	no	—
190	C₄⋊Q₈	—	151	1	no	—
191	C₄⋊Q₈	—	151	1	no	—
192	#126	—	302	1	no	—
193	#126	—	302	1	no	—
194	#124	—	151	1	no	—
195	#124	C₂×D₁₅₁	1	1	yes	characteristic
196	C₂²⋊Q₈	—	302	1	no	—
197	C₂²⋊Q₈	—	302	1	no	—
198	C₂²⋊Q₈	—	302	1	no	—
199	C₂²⋊Q₈	—	302	1	no	—
200	C₄×Q₈	—	151	1	no	—
201	C₄×Q₈	—	151	1	no	—
202	C₄×Q₈	—	151	1	no	—
203	C₄×D₄	—	151	1	no	—
204	C₄×D₄	—	151	1	no	—
205	C₄×D₄	—	151	1	no	—
206	C₄×D₄	C₂×D₁₅₁	1	1	yes	characteristic
207	#117	—	302	1	no	—
208	#107	—	151	1	no	—
209	#107	—	151	1	no	—
210	#107	—	151	1	no	—
211	C₄⋊C₈	—	151	1	no	—
212	C₄⋊C₈	—	151	1	no	—
213	C₄⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
214	#103	—	151	1	no	—
215	#103	—	151	1	no	—
216	#103	—	151	1	no	—
217	#103	—	151	1	no	—
218	#103	—	151	1	no	—
219	#103	C₂×D₁₅₁	1	1	yes	characteristic
220	#102	—	151	1	no	—
221	#102	—	151	1	no	—
222	#102	—	151	1	no	—
223	#102	C₂×D₁₅₁	1	1	yes	characteristic
224	#97	—	151	1	no	—
225	#97	—	151	1	no	—
226	C₄×C₈	—	151	1	no	—
227	C₂×Q₈	—	302	1	no	—
228	C₂×Q₈	—	302	1	no	—
229	C₂×Q₈	—	302	1	no	—
230	C₂×Q₈	—	604	4	no	—
231	C₂×Q₈	—	151	1	no	—
232	C₂×Q₈	—	151	1	no	—
233	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	302	1	no	—
238	C₂²×C₄	—	302	1	no	—
239	C₂²×C₄	—	302	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	302	2	no	—
242	D₈	—	2	2	no	—
243	C₂×C₈	—	151	1	no	—
244	C₂×C₈	—	151	1	no	—
245	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
247	C₄⋊C₄	—	302	1	no	—
248	C₄⋊C₄	—	302	1	no	—
249	C₄⋊C₄	—	302	1	no	—
250	C₄⋊C₄	—	302	1	no	—
251	C₄⋊C₄	—	302	1	no	—
252	C₄⋊C₄	—	302	1	no	—
253	C₄⋊C₄	—	302	1	no	—
254	C₄⋊C₄	—	151	1	no	—
255	C₄⋊C₄	—	151	1	no	—
256	C₄⋊C₄	—	151	1	no	—
257	C₄⋊C₄	—	151	1	no	—
258	C₄⋊C₄	—	151	1	no	—
259	C₄⋊C₄	—	151	1	no	—
260	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	302	1	no	—
265	C₂²⋊C₄	—	302	1	no	—
266	C₂²⋊C₄	—	302	1	no	—
267	C₂²⋊C₄	—	302	1	no	—
268	C₂²⋊C₄	—	302	1	no	—
269	C₄²	—	302	1	no	—
270	C₄²	—	302	1	no	—
271	C₄²	—	151	1	no	—
272	C₄²	—	151	1	no	—
273	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	302	2	no	—
279	Q₈	—	302	1	no	—
280	Q₈	—	302	1	no	—
281	Q₈	—	604	2	no	—
282	Q₈	—	604	2	no	—
283	Q₈	—	604	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#14559003	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	302	1	no	—
292	C₂×C₄	—	302	1	no	—
293	C₂×C₄	—	302	1	no	—
294	C₂×C₄	—	302	1	no	—
295	C₂×C₄	—	604	2	no	—
296	C₂×C₄	—	604	2	no	—
297	C₂×C₄	—	604	2	no	—
298	C₂×C₄	—	151	1	no	—
299	C₂×C₄	—	151	1	no	—
300	C₂×C₄	—	151	1	no	—
301	C₂×C₄	—	151	1	no	—
302	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
303	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	302	1	no	—
308	C₈	—	302	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	302	1	no	—
321	C₄	—	302	1	no	—
322	C₄	—	302	1	no	—
323	C₄	—	604	1	no	—
324	C₄	—	604	1	no	—
325	C₄	—	604	1	no	—
326	C₄	—	604	1	no	—
327	C₄	—	151	2	no	—
328	C₄	C₂×#14559003	1	1	yes	characteristic
329	C₄	#18012786	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#23752868	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₁₅₁	1	1	yes	characteristic
334	C₂	#23753026	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#463×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×D₈×C₁₅₁	C₂²
hypercenter	#463	D₁₅₁
hypocenter	C₁₅₁	#3367
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #23752931	C₂	semidirect
2. D₄×Dic₃₀₂	C₂	semidirect
3. #23752571	C₂	semidirect
4. #23752570	C₂	semidirect
5. #23752561	C₂	semidirect
6. #23752508	C₂	semidirect
7. #23752505	C₂	semidirect
8. #23752490	C₂	semidirect
10. #23752415	C₂	semidirect
11. #23752341	C₂	semidirect
12. #23752313	C₂	semidirect
13. #23752312	C₂	semidirect
167. C₁₅₁	#3367	semidirect

Maximal subgroups

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

subgroup	index
#23752931	2
D₄×Dic₃₀₂	2
#23752571	2
#23752570	2
#23752561	2
#23752508	2
#23752505	2
#23752490	2
#23752425	2
#23752415	2
#23752341	2
#23752313	2
#23752312	2
D₈×Dic₁₅₁	2
#463×C₁₅₁	2
#3367	151