Subgroups — #34080897

Order 19328.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23753029	C₂	1	1	yes	maximal, characteristic, complemented
2	#23753019	C₂	1	2	yes	maximal, complemented
3	#23753017	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752596	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752487	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752482	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752476	C₂	1	2	yes	maximal, complemented
8	#23752391	C₂	1	2	yes	maximal, complemented
9	#23752386	C₂	1	2	yes	maximal, complemented
10	#23752330	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#18012786	C₂²	1	2	yes	complemented
13	#18012785	C₂²	1	2	yes	—
14	#18012772	—	2	2	no	—
15	#18012772	C₂²	1	2	yes	complemented
16	#18012771	—	2	2	no	—
17	#18012769	C₂²	1	2	yes	—
18	#18012767	—	2	2	no	—
19	#18012767	C₂²	1	2	yes	—
20	#18012764	C₂²	1	2	yes	complemented
21	#18012763	—	2	2	no	—
22	#18012761	—	2	1	no	—
23	#18012761	—	2	1	no	—
24	#18012761	C₂²	1	1	yes	characteristic
25	#18012760	—	2	2	no	—
26	C₂×Dic₆₀₄	—	2	1	no	—
27	C₂×Dic₆₀₄	C₂²	1	1	yes	characteristic, complemented
28	C₂×#14558993	C₂²	1	2	yes	complemented
29	SD₁₆×D₁₅₁	—	2	2	no	—
30	C₂×D₁₂₀₈	—	2	1	no	—
31	C₂×D₁₂₀₈	C₂²	1	1	yes	characteristic
32	#18012745	C₂²	1	2	yes	complemented
33	#18012727	C₂²	1	2	yes	complemented
34	#18012675	C₂²	1	1	yes	characteristic, complemented
35	#18012667	C₂²	1	2	yes	complemented
36	#18012657	C₂²	1	2	yes	complemented
37	#18012655	C₂²	1	2	yes	complemented
38	#135×C₁₅₁	C₂²	1	1	yes	characteristic
39	#131×C₁₅₁	C₂²	1	1	yes	characteristic
40	#131×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
41	#108×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
42	#104×C₁₅₁	C₂²	1	1	yes	characteristic
43	#104×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
45	#14559005	—	2	2	no	—
46	#14559004	—	2	2	no	—
47	#14559004	—	2	2	no	—
48	#14559004	C₂³	1	2	yes	—
49	#14559003	—	2	2	no	—
50	#14559003	—	4	2	no	—
51	C₂×Dic₃₀₂	—	2	2	no	—
52	C₂×Dic₃₀₂	C₂³	1	2	yes	—
53	Q₈×D₁₅₁	—	2	2	no	—
54	C₂×D₆₀₄	—	2	2	no	—
55	C₂×D₆₀₄	C₂³	1	2	yes	—
56	D₄×D₁₅₁	—	2	2	no	—
57	D₄×D₁₅₁	—	4	2	no	—
58	Dic₆₀₄	—	2	2	no	—
59	Dic₆₀₄	—	2	1	no	—
60	Dic₆₀₄	—	2	1	no	—
61	Dic₆₀₄	D₄	1	1	yes	characteristic
62	#14558994	—	2	2	no	—
63	#14558993	—	2	2	no	—
64	#14558993	—	2	2	no	—
65	#14558993	—	2	2	no	—
66	#14558993	—	2	2	no	—
67	#14558993	D₄	1	2	yes	—
68	#14558992	—	2	2	no	—
69	#14558991	—	2	2	no	—
70	D₁₂₀₈	—	2	2	no	—
71	D₁₂₀₈	—	2	1	no	—
72	D₁₂₀₈	—	2	1	no	—
73	D₁₂₀₈	D₄	1	1	yes	characteristic
74	#14558989	—	2	2	no	—
75	#14558988	C₂³	1	2	yes	—
76	#14558987	—	2	2	no	—
77	C₈×D₁₅₁	—	2	2	no	—
78	(C₂×D₁₅₁)⋊C₄	—	4	2	no	—
79	#41×C₁₅₁	C₂³	1	1	yes	characteristic
80	#41×C₁₅₁	C₂³	1	1	yes	characteristic
81	Q₁₆×C₁₅₁	D₄	1	1	yes	characteristic
82	SD₁₆×C₁₅₁	D₄	1	1	yes	characteristic
83	SD₁₆×C₁₅₁	D₄	1	1	yes	characteristic
84	D₈×C₁₅₁	D₄	1	1	yes	characteristic
85	M₁₆×C₁₅₁	—	2	1	no	—
86	M₁₆×C₁₅₁	—	2	1	no	—
87	M₁₆×C₁₅₁	C₂³	1	1	yes	characteristic
88	M₁₆×C₁₅₁	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₁₅₁	—	2	1	no	—
90	C₂×C₈×C₁₅₁	—	2	1	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	—	4	2	no	—
96	Dic₃₀₂	—	2	2	no	—
97	Dic₃₀₂	—	2	2	no	—
98	Dic₃₀₂	—	4	2	no	—
99	Dic₃₀₂	C₂×D₄	1	2	yes	—
100	D₆₀₄	—	2	2	no	—
101	D₆₀₄	—	2	2	no	—
102	D₆₀₄	—	4	2	no	—
103	D₆₀₄	C₂×D₄	1	2	yes	—
104	C₂²⋊D₁₅₁	—	2	2	no	—
105	C₂²⋊D₁₅₁	—	4	2	no	—
106	C₂²⋊D₁₅₁	—	4	2	no	—
107	C₂×Dic₁₅₁	—	2	2	no	—
108	C₂×Dic₁₅₁	—	4	2	no	—
109	C₄×D₁₅₁	—	2	2	no	—
110	C₄×D₁₅₁	—	4	2	no	—
111	C₄×D₁₅₁	—	4	2	no	—
112	C₁₅₁⋊C₈	—	2	2	no	—
113	Q₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
115	D₄×C₁₅₁	—	2	1	no	—
116	D₄×C₁₅₁	—	2	1	no	—
117	D₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
118	D₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₅₁	—	2	1	no	—
120	C₂×C₄×C₁₅₁	—	2	1	no	—
121	C₂×C₄×C₁₅₁	—	2	1	no	—
122	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
123	C₈×C₁₅₁	—	2	1	no	—
124	C₈×C₁₅₁	—	2	1	no	—
125	C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
126	C₈×C₁₅₁	C₂×D₄	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	1	no	—
130	C₂²×C₁₅₁	—	2	1	no	—
131	C₂²×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₁₅₁	—	2	2	no	—
133	C₂×D₁₅₁	—	4	2	no	—
134	C₂×D₁₅₁	—	8	2	no	—
135	C₂×D₁₅₁	—	16	2	no	—
136	C₄×C₁₅₁	—	2	1	no	—
137	C₄×C₁₅₁	—	2	1	no	—
138	C₄×C₁₅₁	—	4	1	no	—
139	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
141	Dic₁₅₁	—	2	2	no	—
142	Dic₁₅₁	—	4	2	no	—
143	C₂×C₁₅₁	—	2	1	no	—
144	C₂×C₁₅₁	—	4	1	no	—
145	C₂×C₁₅₁	—	4	1	no	—
146	C₂×C₁₅₁	D₄²	1	1	yes	characteristic
147	D₁₅₁	—	4	2	no	—
148	D₁₅₁	—	8	2	no	—
149	C₁₅₁	#3290	1	1	yes	characteristic, complemented, Sylow
150	#3290	—	151	1	no	maximal, Sylow
151	#578	—	151	1	no	—
152	#577	—	151	2	no	—
153	#576	—	151	1	no	—
154	#495	—	151	1	no	—
155	#473	—	151	1	no	—
156	#472	—	151	1	no	—
157	#471	—	151	2	no	—
158	#455	—	151	2	no	—
159	#454	—	151	2	no	—
160	#443	—	151	1	no	—
161	#443	D₁₅₁	1	1	yes	characteristic, complemented
162	2₋¹⁺⁴	—	151	2	no	—
163	H(5,𝔽₂)	—	151	2	no	—
164	#137	—	302	2	no	—
165	#137	—	302	2	no	—
166	#137	—	151	2	no	—
167	#137	—	151	2	no	—
168	AGL(1,ℤ/8)	—	302	2	no	—
169	AGL(1,ℤ/8)	—	302	2	no	—
170	AGL(1,ℤ/8)	—	151	2	no	—
171	AGL(1,ℤ/8)	—	151	2	no	—
172	#135	—	302	1	no	—
173	#135	—	302	1	no	—
174	#135	—	302	2	no	—
175	#135	—	151	1	no	—
176	#135	C₂×D₁₅₁	1	1	yes	characteristic
177	C₂×Q₁₆	—	302	1	no	—
178	C₂×Q₁₆	—	151	1	no	—
179	C₂×SD₁₆	—	302	2	no	—
180	C₂×SD₁₆	—	151	2	no	—
181	C₂×D₈	—	302	1	no	—
182	C₂×D₈	—	151	1	no	—
183	#131	—	151	2	no	—
184	#131	C₂×D₁₅₁	1	1	yes	characteristic
185	#131	C₂×D₁₅₁	1	1	yes	characteristic, complemented
186	#124	—	151	2	no	—
187	#108	—	151	1	no	—
188	#108	C₂×D₁₅₁	1	1	yes	characteristic, complemented
189	#104	—	151	2	no	—
190	#104	C₂×D₁₅₁	1	1	yes	characteristic
191	#104	C₂×D₁₅₁	1	1	yes	characteristic, complemented
192	#101	—	151	2	no	—
193	#100	—	151	2	no	—
194	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic, complemented
195	#41	—	302	2	no	—
196	#41	—	302	2	no	—
197	#41	—	302	2	no	—
198	#41	—	302	2	no	—
199	#41	—	604	2	no	—
200	#41	—	151	2	no	—
201	#41	C₂²×D₁₅₁	1	1	yes	characteristic
202	#41	C₂²×D₁₅₁	1	1	yes	characteristic
203	C₂×Q₈	—	302	2	no	—
204	C₂×Q₈	—	302	2	no	—
205	C₂×Q₈	—	151	2	no	—
206	C₂×D₄	—	302	2	no	—
207	C₂×D₄	—	302	2	no	—
208	C₂×D₄	—	604	2	no	—
209	C₂×D₄	—	151	2	no	—
210	Q₁₆	—	302	2	no	—
211	Q₁₆	—	302	2	no	—
212	Q₁₆	—	302	1	no	—
213	Q₁₆	—	302	1	no	—
214	Q₁₆	—	151	1	no	—
215	Q₁₆	D₆₀₄	1	1	yes	characteristic
216	SD₁₆	—	302	2	no	—
217	SD₁₆	—	302	2	no	—
218	SD₁₆	—	302	2	no	—
219	SD₁₆	—	302	2	no	—
220	SD₁₆	—	302	2	no	—
221	SD₁₆	—	302	2	no	—
222	SD₁₆	—	151	2	no	—
223	SD₁₆	D₆₀₄	1	1	yes	characteristic
224	SD₁₆	D₆₀₄	1	1	yes	characteristic
225	D₈	—	302	2	no	—
226	D₈	—	302	2	no	—
227	D₈	—	302	1	no	—
228	D₈	—	302	1	no	—
229	D₈	—	151	1	no	—
230	D₈	D₆₀₄	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	302	2	no	—
234	M₁₆	—	151	2	no	—
235	M₁₆	C₂²×D₁₅₁	1	1	yes	characteristic
236	M₁₆	C₂²×D₁₅₁	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	302	2	no	—
240	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
242	C₂²⋊C₄	—	604	2	no	—
243	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
244	C₂³	—	302	2	no	—
245	C₂³	—	604	2	no	—
246	Q₈	—	302	2	no	—
247	Q₈	—	302	2	no	—
248	Q₈	—	604	2	no	—
249	Q₈	—	151	2	no	—
250	Q₈	C₂×D₆₀₄	1	1	yes	characteristic
251	Q₈	C₂×D₆₀₄	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	302	2	no	—
255	D₄	—	302	2	no	—
256	D₄	—	302	2	no	—
257	D₄	—	604	2	no	—
258	D₄	—	604	2	no	—
259	D₄	—	604	2	no	—
260	D₄	—	151	2	no	—
261	D₄	C₂×D₆₀₄	1	1	yes	characteristic
262	D₄	C₂×D₆₀₄	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	302	2	no	—
267	C₂×C₄	—	302	2	no	—
268	C₂×C₄	—	604	2	no	—
269	C₂×C₄	—	604	2	no	—
270	C₂×C₄	—	604	2	no	—
271	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	302	2	no	—
275	C₈	D₄×D₁₅₁	1	1	yes	characteristic
276	C₈	D₄×D₁₅₁	1	1	yes	characteristic
277	C₈	C₂×D₆₀₄	1	1	yes	characteristic
278	C₈	C₂×D₆₀₄	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	302	2	no	—
282	C₂²	—	604	2	no	—
283	C₂²	—	2416	2	no	—
284	C₂²	—	1208	2	no	—
285	C₂²	#18012785	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	302	2	no	—
290	C₄	—	604	2	no	—
291	C₄	C₂×D₄×D₁₅₁	1	1	yes	characteristic
292	C₄	C₂²×D₆₀₄	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	604	2	no	—
297	C₂	—	1208	2	no	—
298	C₂	D₄×D₆₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₀₄
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#443×C₁₅₁	C₂
socle	C₂×C₁₅₁	D₄²
hypercenter	#443	D₁₅₁
hypocenter	C₁₅₁	#3290
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #23753029	C₂	semidirect
2. #23753019	C₂	semidirect
3. #23753017	C₂	semidirect
4. #23752596	C₂	semidirect
5. #23752487	C₂	semidirect
6. #23752482	C₂	semidirect
7. #23752476	C₂	semidirect
8. #23752391	C₂	semidirect
9. #23752386	C₂	semidirect
10. #23752330	C₂	semidirect
11. #443×C₁₅₁	C₂	semidirect
12. #18012786	C₂²	semidirect
15. #18012772	C₂²	semidirect
20. #18012764	C₂²	semidirect
27. C₂×Dic₆₀₄	C₂²	semidirect
28. C₂×#14558993	C₂²	semidirect
32. #18012745	C₂²	semidirect
33. #18012727	C₂²	semidirect
34. #18012675	C₂²	semidirect
35. #18012667	C₂²	semidirect
36. #18012657	C₂²	semidirect
37. #18012655	C₂²	semidirect
40. #131×C₁₅₁	C₂²	semidirect
41. #108×C₁₅₁	C₂²	semidirect
43. #104×C₁₅₁	C₂²	semidirect
44. C₄×C₈×C₁₅₁	C₂²	semidirect
149. C₁₅₁	#3290	semidirect
161. #443	D₁₅₁	semidirect
185. #131	C₂×D₁₅₁	semidirect
188. #108	C₂×D₁₅₁	semidirect
191. #104	C₂×D₁₅₁	semidirect
194. 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
#23753029	2
#23753019	2
#23753017	2
#23752596	2
#23752487	2
#23752482	2
#23752476	2
#23752391	2
#23752386	2
#23752330	2
#443×C₁₅₁	2
#3290	151