Subgroups — #34080298

Order 19328.

368 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752759	C₂	1	1	yes	maximal, characteristic
2	#125×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752620	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752619	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752575	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752575	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752532	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752529	C₂	1	1	yes	maximal, characteristic, complemented
9	#23752527	C₂	1	1	yes	maximal, characteristic, complemented
10	#23752524	C₂	1	1	yes	maximal, characteristic, complemented
11	#23752513	C₂	1	1	yes	maximal, characteristic, complemented
12	#23752512	C₂	1	1	yes	maximal, characteristic, complemented
13	#23752509	C₂	1	1	yes	maximal, characteristic, complemented
14	#23752268	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14559004	—	2	1	no	—
17	#18012738	C₂²	1	1	yes	characteristic
18	#18012735	C₂²	1	1	yes	characteristic
19	#18012731	C₂²	1	1	yes	characteristic
20	#18012730	C₂²	1	1	yes	characteristic
21	#18012730	C₂²	1	1	yes	characteristic
22	#18012729	—	2	1	no	—
23	#18012729	—	2	1	no	—
24	#18012725	—	2	1	no	—
25	#18012714	C₂²	1	1	yes	characteristic
26	#18012712	—	2	1	no	—
27	#18012710	C₂²	1	1	yes	characteristic
28	C₄×D₆₀₄	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₅₁	—	2	1	no	—
30	C₄⋊C₄×D₁₅₁	—	2	1	no	—
31	C₄⋊C₄×D₁₅₁	—	2	1	no	—
32	C₄⋊C₄×D₁₅₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₅₁	C₂²	1	1	yes	characteristic
34	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
35	#18012673	C₂²	1	1	yes	characteristic
36	#18012672	C₂²	1	1	yes	characteristic
37	#18012672	C₂²	1	1	yes	characteristic
38	#18012671	C₂²	1	1	yes	characteristic
39	#18012670	C₂²	1	1	yes	characteristic
40	#18012670	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#18012661	C₂²	1	1	yes	characteristic
43	#18012661	C₂²	1	1	yes	characteristic
44	#18012660	C₂²	1	1	yes	characteristic
45	#18012660	C₂²	1	1	yes	characteristic
46	#18012659	C₂²	1	1	yes	characteristic
47	#18012659	C₂²	1	1	yes	characteristic
48	(C₂×D₁₅₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₅₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#18012649	C₂²	1	1	yes	characteristic
51	#18012649	C₂²	1	1	yes	characteristic
52	#125×C₁₅₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
54	#107×C₁₅₁	C₂²	1	1	yes	characteristic
55	#106×C₁₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
57	#102×C₁₅₁	C₂²	1	1	yes	characteristic
58	#102×C₁₅₁	C₂²	1	1	yes	characteristic
59	#14559004	—	4	4	no	—
60	C₂×Dic₃₀₂	—	2	1	no	—
61	C₂×D₆₀₄	—	2	1	no	—
62	C₂×D₆₀₄	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₅₁	—	2	1	no	—
64	C₂×C₂²⋊D₁₅₁	—	2	1	no	—
65	C₂×C₂²⋊D₁₅₁	—	2	1	no	—
66	C₂×C₄×D₁₅₁	—	2	1	no	—
67	C₂×C₄×D₁₅₁	—	2	1	no	—
68	C₂×C₄×D₁₅₁	—	2	1	no	—
69	C₂×C₄×D₁₅₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₅₁	D₄	1	1	yes	characteristic
71	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
72	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₅₁	—	2	1	no	—
75	C₄⋊Dic₁₅₁	—	2	1	no	—
76	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
79	Dic₁₅₁⋊C₄	—	2	1	no	—
80	Dic₁₅₁⋊C₄	—	2	1	no	—
81	Dic₁₅₁⋊C₄	—	2	1	no	—
82	Dic₁₅₁⋊C₄	—	2	1	no	—
83	Dic₁₅₁⋊C₄	—	2	1	no	—
84	Dic₁₅₁⋊C₄	—	2	1	no	—
85	Dic₁₅₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₁₅₁	—	2	1	no	—
87	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
89	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
90	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
91	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₅₁	—	2	1	no	—
94	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₅₁	—	2	1	no	—
97	C₄⋊C₄×C₁₅₁	—	2	1	no	—
98	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
102	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
103	C₂²×D₁₅₁	—	2	1	no	—
104	C₂²×D₁₅₁	C₂×D₄	1	1	yes	characteristic
105	Dic₃₀₂	—	4	2	no	—
106	D₆₀₄	—	2	1	no	—
107	D₆₀₄	—	2	1	no	—
108	D₆₀₄	—	4	2	no	—
109	C₂²⋊D₁₅₁	—	4	2	no	—
110	C₂²⋊D₁₅₁	—	4	2	no	—
111	C₂²⋊D₁₅₁	—	4	2	no	—
112	C₂×Dic₁₅₁	—	2	1	no	—
113	C₂×Dic₁₅₁	—	2	1	no	—
114	C₂×Dic₁₅₁	—	2	1	no	—
115	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
118	C₄×D₁₅₁	—	2	2	no	—
119	C₄×D₁₅₁	—	2	1	no	—
120	C₄×D₁₅₁	—	2	1	no	—
121	C₄×D₁₅₁	—	4	1	no	—
122	C₄×D₁₅₁	—	4	2	no	—
123	C₄×D₁₅₁	—	4	2	no	—
124	C₄×D₁₅₁	—	4	2	no	—
125	C₁₅₁⋊C₈	—	2	1	no	—
126	C₁₅₁⋊C₈	—	2	1	no	—
127	C₂³×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	—	4	2	no	—
134	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₅₁	—	2	1	no	—
138	C₈×C₁₅₁	—	2	1	no	—
139	C₂²×C₁₅₁	—	4	1	no	—
140	C₂²×C₁₅₁	—	4	1	no	—
141	C₂²×C₁₅₁	—	4	1	no	—
142	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₅₁	—	2	1	no	—
144	C₂×D₁₅₁	—	2	1	no	—
145	C₂×D₁₅₁	—	4	1	no	—
146	C₂×D₁₅₁	—	4	1	no	—
147	C₂×D₁₅₁	—	4	1	no	—
148	C₂×D₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
149	C₄×C₁₅₁	—	2	1	no	—
150	C₄×C₁₅₁	—	2	1	no	—
151	C₄×C₁₅₁	—	4	1	no	—
152	C₄×C₁₅₁	—	4	1	no	—
153	C₄×C₁₅₁	—	4	1	no	—
154	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
155	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₁₅₁	—	2	1	no	—
157	Dic₁₅₁	—	2	1	no	—
158	Dic₁₅₁	—	2	1	no	—
159	Dic₁₅₁	—	4	1	no	—
160	Dic₁₅₁	—	4	1	no	—
161	Dic₁₅₁	—	4	1	no	—
162	C₂×C₁₅₁	—	8	1	no	—
163	C₂×C₁₅₁	#536	1	1	yes	characteristic
164	C₂×C₁₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₁₅₁	#575	1	1	yes	characteristic
166	D₁₅₁	—	2	2	no	—
167	D₁₅₁	—	8	1	no	—
168	C₁₅₁	#3234	1	1	yes	characteristic, complemented, Sylow
169	#3234	—	151	1	no	maximal, Sylow
170	#532	—	151	1	no	—
171	C₂×#125	—	151	1	no	—
172	#501	—	151	1	no	—
173	#501	—	151	1	no	—
174	#490	—	151	1	no	—
175	#490	—	151	1	no	—
176	#481	—	151	1	no	—
177	#481	—	151	1	no	—
178	#480	—	151	1	no	—
179	#480	—	151	1	no	—
180	#478	—	151	1	no	—
181	#478	—	151	1	no	—
182	#478	—	151	1	no	—
183	#478	D₁₅₁	1	1	yes	characteristic, complemented
184	#432	—	151	1	no	—
185	C₂×#41	—	302	1	no	—
186	C₄⋊Q₈	—	151	1	no	—
187	#127	—	151	1	no	—
188	#125	—	302	1	no	—
189	#125	—	302	1	no	—
190	#125	—	151	1	no	—
191	#125	—	151	1	no	—
192	#125	—	151	1	no	—
193	#125	C₂×D₁₅₁	1	1	yes	characteristic
194	#124	—	302	1	no	—
195	#121	—	302	1	no	—
196	#121	—	151	1	no	—
197	#121	—	151	1	no	—
198	C₄×D₄	—	302	1	no	—
199	C₄×D₄	—	151	1	no	—
200	C₄×D₄	C₂×D₁₅₁	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	302	1	no	—
202	C₂×C₄⋊C₄	—	302	1	no	—
203	C₂×C₄⋊C₄	—	151	1	no	—
204	C₂×C₄⋊C₄	—	151	1	no	—
205	C₂×C₄²	—	151	1	no	—
206	#107	—	151	1	no	—
207	#107	—	151	1	no	—
208	#107	—	151	1	no	—
209	#107	C₂×D₁₅₁	1	1	yes	characteristic
210	#106	—	151	1	no	—
211	#106	—	151	1	no	—
212	#106	—	151	1	no	—
213	#106	C₂×D₁₅₁	1	1	yes	characteristic
214	C₄⋊C₈	—	151	1	no	—
215	C₄⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
216	#102	—	151	1	no	—
217	#102	—	151	1	no	—
218	#102	—	151	1	no	—
219	#102	—	151	1	no	—
220	#102	—	151	1	no	—
221	#102	—	151	1	no	—
222	#102	C₂×D₁₅₁	1	1	yes	characteristic
223	#102	C₂×D₁₅₁	1	1	yes	characteristic
224	C₂²⋊C₈	—	151	1	no	—
225	C₂²⋊C₈	—	151	1	no	—
226	#97	—	151	1	no	—
227	#97	—	151	1	no	—
228	#41	—	604	4	no	—
229	C₂×Q₈	—	302	1	no	—
230	C₂×D₄	—	302	1	no	—
231	C₂×D₄	—	302	1	no	—
232	C₂×D₄	—	302	1	no	—
233	C₂×D₄	—	302	1	no	—
234	C₂×D₄	—	151	1	no	—
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	C₂²×C₄	—	151	1	no	—
241	C₂²×C₄	—	151	1	no	—
242	C₂²×C₄	—	151	1	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	—	2	1	no	—
248	C₄⋊C₄	—	2	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	—	302	1	no	—
255	C₄⋊C₄	—	302	1	no	—
256	C₄⋊C₄	—	302	1	no	—
257	C₄⋊C₄	—	302	1	no	—
258	C₄⋊C₄	—	151	1	no	—
259	C₄⋊C₄	—	151	1	no	—
260	C₄⋊C₄	—	151	1	no	—
261	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	302	1	no	—
266	C₂²⋊C₄	—	302	1	no	—
267	C₂²⋊C₄	—	302	1	no	—
268	C₄²	—	151	1	no	—
269	C₄²	—	151	1	no	—
270	C₄²	—	151	1	no	—
271	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	302	1	no	—
274	C₂³	—	151	1	no	—
275	Q₈	—	604	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	2	1	no	—
278	D₄	—	302	1	no	—
279	D₄	—	302	1	no	—
280	D₄	—	604	2	no	—
281	D₄	—	604	2	no	—
282	D₄	—	604	2	no	—
283	D₄	—	604	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	302	2	no	—
289	C₂×C₄	—	302	1	no	—
290	C₂×C₄	—	302	1	no	—
291	C₂×C₄	—	302	1	no	—
292	C₂×C₄	—	302	1	no	—
293	C₂×C₄	—	302	1	no	—
294	C₂×C₄	—	604	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	D₄×D₁₅₁	1	1	yes	characteristic
302	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
303	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	302	1	no	—
307	C₈	—	302	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	302	1	no	—
312	C₂²	—	302	1	no	—
313	C₂²	—	604	1	no	—
314	C₂²	—	604	1	no	—
315	C₂²	—	604	1	no	—
316	C₂²	—	151	2	no	—
317	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	302	1	no	—
324	C₄	—	302	1	no	—
325	C₄	—	302	1	no	—
326	C₄	—	604	1	no	—
327	C₄	—	604	1	no	—
328	C₄	—	604	1	no	—
329	C₄	#18012786	1	1	yes	characteristic
330	C₄	#18012787	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	—	302	2	no	—
333	C₂	—	1208	1	no	—
334	C₂	#23752785	1	1	yes	characteristic
335	C₂	AGL(1,ℤ/8)×D₁₅₁	1	1	yes	characteristic
336	C₂	#23753009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#478×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	#23752759	C₂
hypercenter	#478	D₁₅₁
hypocenter	C₁₅₁	#3234
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #125×D₁₅₁	C₂	semidirect
3. #23752620	C₂	semidirect
4. #23752619	C₂	semidirect
5. #23752575	C₂	semidirect
6. #23752575	C₂	semidirect
7. #23752532	C₂	semidirect
8. #23752529	C₂	semidirect
9. #23752527	C₂	semidirect
10. #23752524	C₂	semidirect
11. #23752513	C₂	semidirect
12. #23752512	C₂	semidirect
13. #23752509	C₂	semidirect
14. #23752268	C₂	semidirect
15. #478×C₁₅₁	C₂	semidirect
168. C₁₅₁	#3234	semidirect
183. #478	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752759	2
#125×D₁₅₁	2
#23752620	2
#23752619	2
#23752575	2
#23752532	2
#23752529	2
#23752527	2
#23752524	2
#23752513	2
#23752512	2
#23752509	2
#23752268	2
#478×C₁₅₁	2
#3234	151