Subgroups — #34078022

Order 19328.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#23752977	C₂	1	2	yes	maximal, complemented
3	#23752332	C₂	1	2	yes	maximal, complemented
4	#23752331	C₂	1	2	yes	maximal, complemented
5	#23752330	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752329	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752328	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#23752227	C₂	1	2	yes	maximal, complemented
10	#23752166	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₅₁	C₂²	1	2	yes	—
13	#18012762	C₄	1	1	yes	characteristic, complemented
14	#18012761	C₂²	1	1	yes	characteristic
15	#18012760	C₄	1	2	yes	complemented
16	#18012759	C₂²	1	2	yes	—
17	#18012758	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₅₁	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₁₅₁	C₄	1	2	yes	complemented
20	D₈×D₁₅₁	C₄	1	1	yes	characteristic, complemented
21	#18012747	—	2	2	no	—
22	#18012747	C₂²	1	2	yes	complemented
23	#18012746	C₂²	1	2	yes	complemented
24	#18012745	—	2	2	no	—
25	#18012745	C₂²	1	2	yes	complemented
26	C₂×#14558987	—	2	2	no	—
27	C₂×#14558987	C₂²	1	1	yes	characteristic
28	M₁₆×D₁₅₁	—	2	2	no	—
29	M₁₆×D₁₅₁	C₂²	1	2	yes	—
30	C₂×C₈×D₁₅₁	C₂²	1	1	yes	characteristic
31	#18012696	C₂²	1	1	yes	characteristic
32	#18012675	C₂²	1	1	yes	characteristic, complemented
33	#18012674	C₂²	1	2	yes	complemented
34	#18012667	C₂²	1	2	yes	complemented
35	#18012666	C₂²	1	2	yes	complemented
36	#18012665	C₂²	1	2	yes	complemented
37	#18012650	C₂²	1	1	yes	characteristic, complemented
38	#18012649	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic, complemented
40	#135×C₁₅₁	C₂²	1	1	yes	characteristic
41	#131×C₁₅₁	C₂²	1	2	yes	complemented
42	#108×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
43	#104×C₁₅₁	C₂²	1	2	yes	complemented
44	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
45	#14559005	C₂×C₄	1	2	yes	—
46	#14559004	—	2	2	no	—
47	#14559004	C₂³	1	2	yes	—
48	#14559003	—	2	2	no	—
49	#14559003	C₂×C₄	1	2	yes	—
50	Q₈×D₁₅₁	C₂×C₄	1	2	yes	—
51	D₄×D₁₅₁	—	2	2	no	—
52	D₄×D₁₅₁	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₅₁	—	2	2	no	—
54	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
55	Dic₆₀₄	C₂×C₄	1	1	yes	characteristic
56	#14558994	C₂×C₄	1	2	yes	—
57	#14558993	C₂×C₄	1	2	yes	—
58	#14558992	C₂×C₄	1	2	yes	—
59	#14558991	C₂×C₄	1	2	yes	—
60	D₁₂₀₈	C₂×C₄	1	1	yes	characteristic
61	#14558989	C₂×C₄	1	2	yes	—
62	#14558988	—	2	2	no	—
63	#14558988	C₂³	1	2	yes	—
64	#14558987	—	2	2	no	—
65	#14558987	—	2	2	no	—
66	#14558987	—	2	2	no	—
67	#14558987	D₄	1	1	yes	characteristic
68	#14558987	D₄	1	1	yes	characteristic
69	#14558987	D₄	1	1	yes	characteristic
70	#14558987	D₄	1	1	yes	characteristic
71	C₂×C₁₅₁⋊C₈	—	2	2	no	—
72	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₁₅₁	—	2	2	no	—
75	C₈×D₁₅₁	—	2	2	no	—
76	C₈×D₁₅₁	—	2	2	no	—
77	C₈×D₁₅₁	C₂×C₄	1	1	yes	characteristic
78	C₈×D₁₅₁	C₂×C₄	1	1	yes	characteristic
79	C₈×D₁₅₁	C₂×C₄	1	1	yes	characteristic
80	C₈×D₁₅₁	C₂×C₄	1	1	yes	characteristic
81	Dic₁₅₁⋊C₄	—	2	1	no	—
82	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
83	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
84	#41×C₁₅₁	C₂³	1	2	yes	—
85	Q₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
86	SD₁₆×C₁₅₁	C₂×C₄	1	2	yes	—
87	D₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
88	M₁₆×C₁₅₁	—	2	2	no	—
89	M₁₆×C₁₅₁	C₂³	1	2	yes	—
90	C₂×C₈×C₁₅₁	—	2	2	no	—
91	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
93	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
94	C₂²×D₁₅₁	—	2	2	no	—
95	C₂²×D₁₅₁	#41	1	1	yes	characteristic
96	Dic₃₀₂	—	2	2	no	—
97	Dic₃₀₂	C₂²×C₄	1	2	yes	—
98	D₆₀₄	—	2	2	no	—
99	D₆₀₄	C₂²×C₄	1	2	yes	—
100	C₂²⋊D₁₅₁	—	2	2	no	—
101	C₂²⋊D₁₅₁	—	2	2	no	—
102	C₂²⋊D₁₅₁	—	2	2	no	—
103	C₂×Dic₁₅₁	—	2	2	no	—
104	C₂×Dic₁₅₁	—	2	1	no	—
105	C₂×Dic₁₅₁	#41	1	1	yes	characteristic
106	C₄×D₁₅₁	—	2	2	no	—
107	C₄×D₁₅₁	—	2	2	no	—
108	C₄×D₁₅₁	—	2	2	no	—
109	C₄×D₁₅₁	C₂²×C₄	1	1	yes	characteristic
110	C₄×D₁₅₁	C₂²×C₄	1	1	yes	characteristic
111	C₄×D₁₅₁	C₂×D₄	1	1	yes	characteristic
112	C₄×D₁₅₁	C₂×D₄	1	1	yes	characteristic
113	C₁₅₁⋊C₈	—	2	2	no	—
114	C₁₅₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
115	C₁₅₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₁₅₁⋊C₈	C₂×D₄	1	1	yes	characteristic
117	C₁₅₁⋊C₈	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₁₅₁	C₂²×C₄	1	2	yes	—
119	D₄×C₁₅₁	—	2	2	no	—
120	D₄×C₁₅₁	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₁₅₁	—	2	2	no	—
122	C₂×C₄×C₁₅₁	—	2	1	no	—
123	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
124	C₈×C₁₅₁	—	2	2	no	—
125	C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
126	C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
127	C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
128	C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₁₅₁	—	2	2	no	—
130	C₂²×C₁₅₁	C₂×#41	1	1	yes	characteristic
131	C₂×D₁₅₁	—	2	2	no	—
132	C₂×D₁₅₁	—	4	1	no	—
133	C₂×D₁₅₁	—	8	2	no	—
134	C₂×D₁₅₁	C₄×D₄	1	1	yes	characteristic
135	C₂×D₁₅₁	C₄×D₄	1	1	yes	characteristic
136	C₄×C₁₅₁	—	2	2	no	—
137	C₄×C₁₅₁	—	4	1	no	—
138	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
140	Dic₁₅₁	—	2	2	no	—
141	Dic₁₅₁	—	4	1	no	—
142	Dic₁₅₁	C₄×D₄	1	1	yes	characteristic
143	Dic₁₅₁	C₄×D₄	1	1	yes	characteristic
144	C₂×C₁₅₁	—	2	1	no	—
145	C₂×C₁₅₁	—	4	2	no	—
146	C₂×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
147	D₁₅₁	—	2	1	no	—
148	D₁₅₁	—	2	1	no	—
149	D₁₅₁	—	4	2	no	—
150	C₁₅₁	#2957	1	1	yes	characteristic, complemented, Sylow
151	#2957	—	151	1	no	maximal, Sylow
152	C₂×#135	—	151	1	no	—
153	#568	—	151	2	no	—
154	#444	—	151	2	no	—
155	#444	—	151	2	no	—
156	#443	—	151	1	no	—
157	#443	—	151	1	no	—
158	#443	—	151	1	no	—
159	#443	D₁₅₁	1	1	yes	characteristic, complemented
160	C₂×#108	—	151	1	no	—
161	#421	—	151	2	no	—
162	#405	—	151	1	no	—
163	C₂×#41	—	151	2	no	—
164	#135	—	151	2	no	—
165	#135	—	151	1	no	—
166	#135	—	151	2	no	—
167	#135	—	151	1	no	—
168	#135	—	151	1	no	—
169	#135	C₂×D₁₅₁	1	1	yes	characteristic
170	C₂×Q₁₆	—	151	1	no	—
171	C₂×SD₁₆	—	151	2	no	—
172	C₂×D₈	—	151	1	no	—
173	#131	—	302	2	no	—
174	#131	—	302	2	no	—
175	#131	—	151	2	no	—
176	#131	—	151	2	no	—
177	#131	—	151	2	no	—
178	#131	C₂×D₁₅₁	1	2	yes	complemented
179	C₂×M₁₆	—	302	2	no	—
180	C₂×M₁₆	—	302	2	no	—
181	C₂×M₁₆	—	151	2	no	—
182	C₂×M₁₆	—	151	1	no	—
183	C₂²×C₈	—	151	1	no	—
184	#117	—	151	1	no	—
185	#108	—	151	2	no	—
186	#108	—	151	1	no	—
187	#108	C₂×D₁₅₁	1	1	yes	characteristic, complemented
188	#104	—	151	2	no	—
189	#104	—	151	2	no	—
190	#104	—	151	2	no	—
191	#104	C₂×D₁₅₁	1	2	yes	complemented
192	#97	—	151	1	no	—
193	#97	—	151	1	no	—
194	C₄×C₈	—	151	1	no	—
195	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic, complemented
196	#41	—	302	2	no	—
197	#41	—	302	2	no	—
198	#41	—	151	2	no	—
199	#41	—	151	2	no	—
200	#41	—	151	2	no	—
201	#41	C₂²×D₁₅₁	1	2	yes	—
202	C₂×Q₈	—	151	2	no	—
203	C₂×D₄	—	302	2	no	—
204	C₂×D₄	—	151	2	no	—
205	C₂²×C₄	—	302	2	no	—
206	C₂²×C₄	—	151	1	no	—
207	Q₁₆	—	151	2	no	—
208	Q₁₆	—	151	1	no	—
209	Q₁₆	C₄×D₁₅₁	1	1	yes	characteristic
210	SD₁₆	—	151	2	no	—
211	SD₁₆	—	151	2	no	—
212	SD₁₆	—	151	2	no	—
213	SD₁₆	C₄×D₁₅₁	1	2	yes	—
214	D₈	—	151	2	no	—
215	D₈	—	151	1	no	—
216	D₈	C₄×D₁₅₁	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	302	2	no	—
219	M₁₆	—	302	2	no	—
220	M₁₆	—	302	2	no	—
221	M₁₆	—	302	2	no	—
222	M₁₆	—	151	2	no	—
223	M₁₆	—	151	1	no	—
224	M₁₆	—	151	1	no	—
225	M₁₆	—	151	1	no	—
226	M₁₆	—	151	1	no	—
227	M₁₆	C₂²×D₁₅₁	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	302	2	no	—
230	C₂×C₈	—	302	2	no	—
231	C₂×C₈	—	302	2	no	—
232	C₂×C₈	—	302	2	no	—
233	C₂×C₈	—	151	1	no	—
234	C₂×C₈	—	151	1	no	—
235	C₂×C₈	—	151	1	no	—
236	C₂×C₈	—	151	1	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	—	302	1	no	—
242	C₂²⋊C₄	—	302	1	no	—
243	C₄²	—	151	1	no	—
244	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
245	C₂³	—	302	2	no	—
246	C₂³	—	151	1	no	—
247	Q₈	—	302	2	no	—
248	Q₈	—	151	2	no	—
249	Q₈	C₂×C₄×D₁₅₁	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	302	2	no	—
252	D₄	—	302	2	no	—
253	D₄	—	302	2	no	—
254	D₄	—	302	2	no	—
255	D₄	—	151	2	no	—
256	D₄	C₂×C₄×D₁₅₁	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	302	2	no	—
260	C₂×C₄	—	302	2	no	—
261	C₂×C₄	—	302	2	no	—
262	C₂×C₄	—	302	2	no	—
263	C₂×C₄	—	302	1	no	—
264	C₂×C₄	—	151	1	no	—
265	C₂×C₄	—	151	1	no	—
266	C₂×C₄	—	151	1	no	—
267	C₂×C₄	—	151	1	no	—
268	C₂×C₄	—	151	1	no	—
269	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	302	2	no	—
272	C₈	—	151	1	no	—
273	C₈	—	151	1	no	—
274	C₈	—	151	1	no	—
275	C₈	—	151	1	no	—
276	C₈	C₂×C₄×D₁₅₁	1	1	yes	characteristic
277	C₈	C₂×C₄×D₁₅₁	1	1	yes	characteristic
278	C₈	D₄×D₁₅₁	1	1	yes	characteristic
279	C₈	D₄×D₁₅₁	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	302	2	no	—
282	C₂²	—	604	1	no	—
283	C₂²	—	151	1	no	—
284	C₂²	—	151	1	no	—
285	C₂²	—	1208	2	no	—
286	C₂²	#41×D₁₅₁	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	302	2	no	—
290	C₄	—	604	1	no	—
291	C₄	—	151	1	no	—
292	C₄	—	151	1	no	—
293	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
294	C₄	C₂×D₄×D₁₅₁	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	302	1	no	—
298	C₂	—	302	1	no	—
299	C₂	—	604	2	no	—
300	C₂	C₄×D₄×D₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₅₁
commutator	C₄×C₁₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#443×C₁₅₁	C₂
socle	C₂×C₁₅₁	C₂×C₄×D₄
prime radical	#135×D₁₅₁	C₂
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	#443	D₁₅₁
hypocenter	C₁₅₁	#2957
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #23752977	C₂	semidirect
3. #23752332	C₂	semidirect
4. #23752331	C₂	semidirect
5. #23752330	C₂	semidirect
6. #23752329	C₂	semidirect
7. #23752328	C₂	semidirect
8. #108×D₁₅₁	C₂	semidirect
9. #23752227	C₂	semidirect
10. #23752166	C₂	semidirect
11. #443×C₁₅₁	C₂	semidirect
13. #18012762	C₄	semidirect
15. #18012760	C₄	semidirect
17. #18012758	C₄	semidirect
18. Q₁₆×D₁₅₁	C₄	semidirect
19. SD₁₆×D₁₅₁	C₄	semidirect
20. D₈×D₁₅₁	C₄	semidirect
22. #18012747	C₂²	semidirect
23. #18012746	C₂²	semidirect
25. #18012745	C₂²	semidirect
32. #18012675	C₂²	semidirect
33. #18012674	C₂²	semidirect
34. #18012667	C₂²	semidirect
35. #18012666	C₂²	semidirect
36. #18012665	C₂²	semidirect
37. #18012650	C₂²	semidirect
38. #18012649	C₂²	semidirect
39. C₈×Dic₁₅₁	C₂²	semidirect
41. #131×C₁₅₁	C₂²	semidirect
42. #108×C₁₅₁	C₂²	semidirect
43. #104×C₁₅₁	C₂²	semidirect
44. C₄×C₈×C₁₅₁	C₂²	semidirect
150. C₁₅₁	#2957	semidirect
159. #443	D₁₅₁	semidirect
178. #131	C₂×D₁₅₁	semidirect
187. #108	C₂×D₁₅₁	semidirect
191. #104	C₂×D₁₅₁	semidirect
195. C₄×C₈	C₂×D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#135×D₁₅₁	2
#23752977	2
#23752332	2
#23752331	2
#23752330	2
#23752329	2
#23752328	2
#108×D₁₅₁	2
#23752227	2
#23752166	2
#443×C₁₅₁	2
#2957	151