Subgroups — #34081756

Order 19328.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#23752918	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752565	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752561	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752517	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752516	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752500	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752426	C₂	1	1	yes	maximal, characteristic
9	#23752424	C₂	1	1	yes	maximal, characteristic
10	#23752416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₃₀₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#14558993	C₂	1	1	yes	maximal, characteristic, complemented
13	#23752297	C₂	1	1	yes	maximal, characteristic, complemented
14	#23752292	C₂	1	1	yes	maximal, characteristic
15	#488×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14558993	C₂²	1	1	yes	characteristic
17	C₂×#14558991	C₂²	1	1	yes	characteristic
18	C₂×#14558989	C₂²	1	1	yes	characteristic
19	#18012738	C₂²	1	1	yes	characteristic
20	#18012737	C₂²	1	1	yes	characteristic
21	#18012732	—	2	1	no	—
22	#18012732	—	2	1	no	—
23	#18012730	C₂²	1	1	yes	characteristic
24	#18012726	C₂²	1	1	yes	characteristic
25	#18012719	—	2	1	no	—
26	#18012717	—	2	1	no	—
27	#18012717	—	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	C₄×D₆₀₄	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
34	#18012701	C₂²	1	1	yes	characteristic
35	#18012701	C₂²	1	1	yes	characteristic
36	#18012695	—	2	1	no	—
37	C₄⋊C₄×D₁₅₁	—	2	1	no	—
38	#18012672	C₂²	1	1	yes	characteristic
39	#18012671	C₂²	1	1	yes	characteristic
40	#18012670	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
44	#18012664	C₂²	1	1	yes	characteristic
45	#18012663	C₂²	1	1	yes	characteristic
46	#18012662	C₂²	1	1	yes	characteristic
47	#18012662	C₂²	1	1	yes	characteristic
48	#18012661	C₂²	1	1	yes	characteristic
49	#18012660	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
52	#125×C₁₅₁	C₂²	1	1	yes	characteristic
53	#124×C₁₅₁	C₂²	1	1	yes	characteristic
54	#103×C₁₅₁	C₂²	1	1	yes	characteristic
55	#103×C₁₅₁	C₂²	1	1	yes	characteristic
56	#102×C₁₅₁	C₂²	1	1	yes	characteristic
57	#102×C₁₅₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₀₂	—	2	1	no	—
60	C₂×Dic₃₀₂	D₄	1	1	yes	characteristic
61	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
62	C₂×D₆₀₄	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₅₁	—	2	1	no	—
64	C₂×C₄×D₁₅₁	—	2	1	no	—
65	C₂×C₄×D₁₅₁	—	2	1	no	—
66	C₂×C₄×D₁₅₁	—	2	1	no	—
67	#14558993	—	2	2	no	—
68	#14558991	—	2	2	no	—
69	#14558989	—	2	2	no	—
70	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₅₁	—	2	1	no	—
73	C₄⋊Dic₁₅₁	—	2	1	no	—
74	C₄⋊Dic₁₅₁	D₄	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	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
90	C₄×Dic₁₅₁	—	2	1	no	—
91	C₄×Dic₁₅₁	—	2	1	no	—
92	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₅₁	—	2	1	no	—
99	C₄⋊C₄×C₁₅₁	—	2	1	no	—
100	C₄⋊C₄×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	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
105	C₂²×D₁₅₁	—	2	1	no	—
106	Dic₃₀₂	—	2	2	no	—
107	Dic₃₀₂	—	2	1	no	—
108	Dic₃₀₂	—	2	1	no	—
109	Dic₃₀₂	—	8	1	no	—
110	D₆₀₄	—	2	1	no	—
111	D₆₀₄	#41	1	2	yes	—
112	C₂×Dic₁₅₁	—	2	1	no	—
113	C₂×Dic₁₅₁	—	2	1	no	—
114	C₂×Dic₁₅₁	—	2	1	no	—
115	C₂×Dic₁₅₁	—	4	2	no	—
116	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
120	C₄×D₁₅₁	—	4	2	no	—
121	C₄×D₁₅₁	—	8	1	no	—
122	C₄×D₁₅₁	—	8	1	no	—
123	C₁₅₁⋊C₈	—	2	1	no	—
124	C₁₅₁⋊C₈	—	2	1	no	—
125	C₂³×C₁₅₁	—	2	1	no	—
126	Q₈×C₁₅₁	—	2	1	no	—
127	Q₈×C₁₅₁	—	2	1	no	—
128	D₄×C₁₅₁	—	2	1	no	—
129	D₄×C₁₅₁	—	2	1	no	—
130	C₂×C₄×C₁₅₁	—	2	1	no	—
131	C₂×C₄×C₁₅₁	—	2	1	no	—
132	C₂×C₄×C₁₅₁	—	2	1	no	—
133	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₅₁	—	2	1	no	—
137	C₈×C₁₅₁	—	2	1	no	—
138	C₂²×C₁₅₁	—	4	1	no	—
139	C₂²×C₁₅₁	—	4	1	no	—
140	C₂²×C₁₅₁	—	4	1	no	—
141	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₅₁	—	2	2	no	—
143	C₂×D₁₅₁	—	4	1	no	—
144	C₂×D₁₅₁	—	4	1	no	—
145	C₄×C₁₅₁	—	2	1	no	—
146	C₄×C₁₅₁	—	2	1	no	—
147	C₄×C₁₅₁	—	4	1	no	—
148	C₄×C₁₅₁	—	4	1	no	—
149	C₄×C₁₅₁	—	4	1	no	—
150	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₅₁	—	2	1	no	—
153	Dic₁₅₁	—	2	1	no	—
154	Dic₁₅₁	—	2	1	no	—
155	Dic₁₅₁	—	4	1	no	—
156	Dic₁₅₁	—	4	1	no	—
157	Dic₁₅₁	—	4	1	no	—
158	Dic₁₅₁	#135	1	2	yes	—
159	C₂×C₁₅₁	—	8	1	no	—
160	C₂×C₁₅₁	#548	1	1	yes	characteristic
161	C₂×C₁₅₁	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₅₁	#577	1	1	yes	characteristic
163	D₁₅₁	—	4	2	no	—
164	C₁₅₁	#3360	1	1	yes	characteristic, complemented, Sylow
165	#3360	—	151	1	no	maximal, Sylow
166	#556	—	151	1	no	—
167	#554	—	151	1	no	—
168	#488	—	151	1	no	—
169	#488	D₁₅₁	1	1	yes	characteristic, complemented
170	#487	—	151	1	no	—
171	#479	—	151	1	no	—
172	#479	—	151	1	no	—
173	#476	—	151	1	no	—
174	#463	—	151	1	no	—
175	#463	—	151	1	no	—
176	#461	—	151	1	no	—
177	C₈×Q₈	—	151	1	no	—
178	C₄×SD₁₆	—	151	1	no	—
179	C₄×SD₁₆	—	151	1	no	—
180	C₄×D₈	—	151	1	no	—
181	C₂×SD₁₆	—	151	1	no	—
182	C₂×SD₁₆	—	151	1	no	—
183	C₂×D₈	—	151	1	no	—
184	C₄⋊Q₈	—	151	1	no	—
185	C₄⋊Q₈	—	151	1	no	—
186	#126	—	302	1	no	—
187	#126	—	302	1	no	—
188	#125	—	151	1	no	—
189	#125	C₂×D₁₅₁	1	1	yes	characteristic
190	#124	—	151	1	no	—
191	#124	C₂×D₁₅₁	1	1	yes	characteristic
192	C₂²⋊Q₈	—	302	1	no	—
193	C₂²⋊Q₈	—	302	1	no	—
194	C₂²⋊Q₈	—	302	1	no	—
195	C₂²⋊Q₈	—	302	1	no	—
196	C₄×Q₈	—	151	1	no	—
197	C₄×Q₈	—	151	1	no	—
198	C₄×Q₈	—	151	1	no	—
199	C₄×D₄	—	151	1	no	—
200	C₄×D₄	—	151	1	no	—
201	C₄×D₄	—	151	1	no	—
202	C₄×D₄	—	151	1	no	—
203	#117	—	302	1	no	—
204	C₂×C₄⋊C₄	—	302	1	no	—
205	#107	—	151	1	no	—
206	#106	—	151	1	no	—
207	#106	—	151	1	no	—
208	C₄⋊C₈	—	151	1	no	—
209	C₄⋊C₈	—	151	1	no	—
210	C₄⋊C₈	—	151	1	no	—
211	#103	—	151	1	no	—
212	#103	—	151	1	no	—
213	#103	—	151	1	no	—
214	#103	—	151	1	no	—
215	#103	C₂×D₁₅₁	1	1	yes	characteristic
216	#103	C₂×D₁₅₁	1	1	yes	characteristic
217	#102	—	151	1	no	—
218	#102	—	151	1	no	—
219	#102	C₂×D₁₅₁	1	1	yes	characteristic
220	#102	C₂×D₁₅₁	1	1	yes	characteristic
221	C₄×C₈	—	151	1	no	—
222	C₄×C₈	—	151	1	no	—
223	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
224	C₂×Q₈	—	302	1	no	—
225	C₂×Q₈	—	151	1	no	—
226	C₂×Q₈	—	151	1	no	—
227	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
228	C₂×D₄	—	151	1	no	—
229	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
230	C₂²×C₄	—	302	1	no	—
231	C₂²×C₄	—	302	1	no	—
232	C₂²×C₄	—	302	1	no	—
233	C₂²×C₄	—	302	1	no	—
234	SD₁₆	—	302	2	no	—
235	SD₁₆	—	302	2	no	—
236	D₈	—	302	2	no	—
237	C₂×C₈	—	151	1	no	—
238	C₂×C₈	—	151	1	no	—
239	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	302	1	no	—
244	C₄⋊C₄	—	302	1	no	—
245	C₄⋊C₄	—	302	1	no	—
246	C₄⋊C₄	—	302	1	no	—
247	C₄⋊C₄	—	302	1	no	—
248	C₄⋊C₄	—	302	1	no	—
249	C₄⋊C₄	—	302	1	no	—
250	C₄⋊C₄	—	151	1	no	—
251	C₄⋊C₄	—	151	1	no	—
252	C₄⋊C₄	—	151	1	no	—
253	C₄⋊C₄	—	151	1	no	—
254	C₄⋊C₄	—	151	1	no	—
255	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	302	1	no	—
260	C₂²⋊C₄	—	302	1	no	—
261	C₂²⋊C₄	—	302	1	no	—
262	C₂²⋊C₄	—	302	1	no	—
263	C₂²⋊C₄	—	302	1	no	—
264	C₂²⋊C₄	—	302	1	no	—
265	C₄²	—	302	1	no	—
266	C₄²	—	302	1	no	—
267	C₄²	—	151	1	no	—
268	C₄²	—	151	1	no	—
269	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	302	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	302	2	no	—
275	Q₈	—	302	1	no	—
276	Q₈	—	302	1	no	—
277	Q₈	—	1208	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	302	1	no	—
281	D₄	—	151	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	302	1	no	—
286	C₂×C₄	—	302	1	no	—
287	C₂×C₄	—	302	1	no	—
288	C₂×C₄	—	604	2	no	—
289	C₂×C₄	—	604	2	no	—
290	C₂×C₄	—	151	1	no	—
291	C₂×C₄	—	151	1	no	—
292	C₂×C₄	—	151	1	no	—
293	C₂×C₄	—	151	1	no	—
294	C₂×C₄	—	1208	1	no	—
295	C₂×C₄	—	1208	1	no	—
296	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	302	1	no	—
302	C₈	—	302	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	302	2	no	—
307	C₂²	—	604	1	no	—
308	C₂²	—	604	1	no	—
309	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	302	1	no	—
316	C₄	—	302	1	no	—
317	C₄	—	302	1	no	—
318	C₄	—	604	1	no	—
319	C₄	—	604	1	no	—
320	C₄	—	604	1	no	—
321	C₄	—	151	2	no	—
322	C₄	#41×D₁₅₁	1	1	yes	characteristic
323	C₄	#18012786	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	604	2	no	—
326	C₂	#23752870	1	1	yes	characteristic
327	C₂	#135×D₁₅₁	1	1	yes	characteristic
328	C₂	#23753019	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#488×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×#14558989	C₂²
hypercenter	#488	D₁₅₁
hypocenter	C₁₅₁	#3360
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #23752931	C₂	semidirect
2. #23752918	C₂	semidirect
3. #23752565	C₂	semidirect
4. #23752561	C₂	semidirect
5. #23752517	C₂	semidirect
6. #23752516	C₂	semidirect
7. #23752500	C₂	semidirect
10. #23752416	C₂	semidirect
11. C₈×Dic₃₀₂	C₂	semidirect
12. C₄×#14558993	C₂	semidirect
13. #23752297	C₂	semidirect
15. #488×C₁₅₁	C₂	semidirect
164. C₁₅₁	#3360	semidirect
169. #488	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752931	2
#23752918	2
#23752565	2
#23752561	2
#23752517	2
#23752516	2
#23752500	2
#23752426	2
#23752424	2
#23752416	2
C₈×Dic₃₀₂	2
C₄×#14558993	2
#23752297	2
#23752292	2
#488×C₁₅₁	2
#3360	151