Subgroups — #34078008

Order 19328.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23753009	C₂	1	1	yes	maximal, characteristic
2	#131×D₁₅₁	C₂	1	2	yes	maximal, complemented
3	#23752334	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752333	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752331	C₂	1	2	yes	maximal, complemented
6	#23752330	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752327	C₂	1	2	yes	maximal, complemented
8	#23752259	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	#18012771	C₄	1	2	yes	complemented
14	#18012768	C₄	1	2	yes	complemented
15	#18012766	C₄	1	2	yes	complemented
16	#18012763	C₄	1	2	yes	complemented
17	#18012761	C₂²	1	1	yes	characteristic
18	#18012759	C₂²	1	2	yes	—
19	#18012747	—	2	2	no	—
20	#18012747	C₂²	1	2	yes	complemented
21	#18012746	C₂²	1	2	yes	complemented
22	#18012745	—	2	2	no	—
23	#18012745	C₂²	1	2	yes	complemented
24	C₂×#14558987	C₂²	1	1	yes	characteristic
25	M₁₆×D₁₅₁	—	2	2	no	—
26	M₁₆×D₁₅₁	C₂²	1	2	yes	—
27	C₂×C₈×D₁₅₁	—	2	2	no	—
28	C₂×C₈×D₁₅₁	C₂²	1	1	yes	characteristic
29	#18012696	C₂²	1	1	yes	characteristic
30	#18012675	C₂²	1	1	yes	characteristic, complemented
31	#18012674	C₂²	1	2	yes	complemented
32	#18012667	C₂²	1	2	yes	complemented
33	#18012666	C₂²	1	2	yes	complemented
34	#18012665	C₂²	1	2	yes	complemented
35	#18012650	C₂²	1	1	yes	characteristic, complemented
36	#18012649	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic, complemented
38	#135×C₁₅₁	C₂²	1	1	yes	characteristic
39	#131×C₁₅₁	C₂²	1	2	yes	complemented
40	#108×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
41	#104×C₁₅₁	C₂²	1	2	yes	complemented
42	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
43	#14559005	C₂×C₄	1	2	yes	—
44	#14559004	—	2	2	no	—
45	#14559004	C₂³	1	2	yes	—
46	#14559003	—	2	2	no	—
47	#14559003	C₂×C₄	1	2	yes	—
48	Q₈×D₁₅₁	C₂×C₄	1	2	yes	—
49	D₄×D₁₅₁	—	2	2	no	—
50	D₄×D₁₅₁	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₅₁	—	2	2	no	—
52	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
53	Dic₆₀₄	C₂×C₄	1	1	yes	characteristic
54	#14558994	C₂×C₄	1	2	yes	—
55	#14558993	C₂×C₄	1	2	yes	—
56	#14558992	C₂×C₄	1	2	yes	—
57	#14558991	C₂×C₄	1	2	yes	—
58	D₁₂₀₈	C₂×C₄	1	1	yes	characteristic
59	#14558989	C₂×C₄	1	2	yes	—
60	#14558988	—	2	2	no	—
61	#14558988	C₂³	1	2	yes	—
62	#14558987	—	2	2	no	—
63	#14558987	—	2	2	no	—
64	#14558987	—	2	2	no	—
65	#14558987	C₂×C₄	1	2	yes	—
66	#14558987	C₂×C₄	1	2	yes	—
67	C₂×C₁₅₁⋊C₈	—	2	2	no	—
68	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×D₁₅₁	—	2	2	no	—
71	C₈×D₁₅₁	—	2	2	no	—
72	C₈×D₁₅₁	—	2	2	no	—
73	C₈×D₁₅₁	D₄	1	2	yes	—
74	C₈×D₁₅₁	D₄	1	2	yes	—
75	Dic₁₅₁⋊C₄	—	2	1	no	—
76	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
77	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
78	#41×C₁₅₁	C₂³	1	2	yes	—
79	Q₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
80	SD₁₆×C₁₅₁	C₂×C₄	1	2	yes	—
81	D₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
82	M₁₆×C₁₅₁	—	2	2	no	—
83	M₁₆×C₁₅₁	C₂³	1	2	yes	—
84	C₂×C₈×C₁₅₁	—	2	2	no	—
85	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
87	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
88	C₂²×D₁₅₁	—	2	2	no	—
89	C₂²×D₁₅₁	#41	1	1	yes	characteristic
90	Dic₃₀₂	—	2	2	no	—
91	Dic₃₀₂	C₂²×C₄	1	2	yes	—
92	D₆₀₄	—	2	2	no	—
93	D₆₀₄	C₂²×C₄	1	2	yes	—
94	C₂²⋊D₁₅₁	—	2	2	no	—
95	C₂²⋊D₁₅₁	—	2	2	no	—
96	C₂²⋊D₁₅₁	—	2	2	no	—
97	C₂×Dic₁₅₁	—	2	2	no	—
98	C₂×Dic₁₅₁	—	2	1	no	—
99	C₂×Dic₁₅₁	#41	1	1	yes	characteristic
100	C₄×D₁₅₁	—	2	2	no	—
101	C₄×D₁₅₁	—	2	2	no	—
102	C₄×D₁₅₁	—	2	2	no	—
103	C₄×D₁₅₁	C₂²×C₄	1	2	yes	—
104	C₄×D₁₅₁	C₂×D₄	1	2	yes	—
105	C₁₅₁⋊C₈	—	2	2	no	—
106	C₁₅₁⋊C₈	C₂²×C₄	1	2	yes	—
107	C₁₅₁⋊C₈	C₂×D₄	1	2	yes	—
108	Q₈×C₁₅₁	C₂²×C₄	1	2	yes	—
109	D₄×C₁₅₁	—	2	2	no	—
110	D₄×C₁₅₁	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₁₅₁	—	2	2	no	—
112	C₂×C₄×C₁₅₁	—	2	1	no	—
113	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
114	C₈×C₁₅₁	—	2	2	no	—
115	C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
116	C₈×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
118	C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₁₅₁	—	2	2	no	—
120	C₂²×C₁₅₁	C₂×#41	1	1	yes	characteristic
121	C₂×D₁₅₁	—	2	2	no	—
122	C₂×D₁₅₁	—	4	1	no	—
123	C₂×D₁₅₁	—	4	4	no	—
124	C₂×D₁₅₁	C₄×D₄	1	2	yes	—
125	C₄×C₁₅₁	—	2	2	no	—
126	C₄×C₁₅₁	—	4	1	no	—
127	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
129	Dic₁₅₁	—	2	2	no	—
130	Dic₁₅₁	—	4	1	no	—
131	Dic₁₅₁	C₄×D₄	1	2	yes	—
132	C₂×C₁₅₁	—	2	1	no	—
133	C₂×C₁₅₁	—	4	2	no	—
134	C₂×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
135	D₁₅₁	—	2	2	no	—
136	D₁₅₁	—	4	2	no	—
137	C₁₅₁	#2955	1	1	yes	characteristic, complemented, Sylow
138	#2955	—	151	1	no	maximal, Sylow
139	#575	—	151	1	no	—
140	C₂×#131	—	151	2	no	—
141	#444	—	151	1	no	—
142	#444	—	151	1	no	—
143	#444	—	151	2	no	—
144	#443	—	151	2	no	—
145	#443	—	151	1	no	—
146	#443	D₁₅₁	1	1	yes	characteristic, complemented
147	#430	—	151	1	no	—
148	#421	—	151	2	no	—
149	#405	—	151	1	no	—
150	C₂×#41	—	151	2	no	—
151	#137	—	151	2	no	—
152	#137	—	151	2	no	—
153	AGL(1,ℤ/8)	—	151	2	no	—
154	AGL(1,ℤ/8)	—	151	2	no	—
155	#135	—	151	2	no	—
156	#135	—	151	1	no	—
157	#135	C₂×D₁₅₁	1	1	yes	characteristic
158	#131	—	302	2	no	—
159	#131	—	302	2	no	—
160	#131	—	151	2	no	—
161	#131	—	151	2	no	—
162	#131	—	151	2	no	—
163	#131	C₂×D₁₅₁	1	2	yes	complemented
164	C₂×M₁₆	—	302	2	no	—
165	C₂×M₁₆	—	151	2	no	—
166	C₂×M₁₆	—	151	1	no	—
167	C₂²×C₈	—	302	2	no	—
168	C₂²×C₈	—	151	1	no	—
169	#117	—	151	1	no	—
170	#108	—	151	2	no	—
171	#108	—	151	1	no	—
172	#108	C₂×D₁₅₁	1	1	yes	characteristic, complemented
173	#104	—	151	2	no	—
174	#104	—	151	2	no	—
175	#104	—	151	2	no	—
176	#104	C₂×D₁₅₁	1	2	yes	complemented
177	#97	—	151	1	no	—
178	#97	—	151	1	no	—
179	C₄×C₈	—	151	1	no	—
180	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic, complemented
181	#41	—	302	2	no	—
182	#41	—	302	2	no	—
183	#41	—	151	2	no	—
184	#41	—	151	2	no	—
185	#41	—	151	2	no	—
186	#41	C₂²×D₁₅₁	1	2	yes	—
187	C₂×Q₈	—	151	2	no	—
188	C₂×D₄	—	302	2	no	—
189	C₂×D₄	—	151	2	no	—
190	C₂²×C₄	—	302	2	no	—
191	C₂²×C₄	—	151	1	no	—
192	Q₁₆	—	151	2	no	—
193	Q₁₆	—	151	1	no	—
194	Q₁₆	C₄×D₁₅₁	1	1	yes	characteristic
195	SD₁₆	—	151	2	no	—
196	SD₁₆	—	151	2	no	—
197	SD₁₆	—	151	2	no	—
198	SD₁₆	C₄×D₁₅₁	1	2	yes	—
199	D₈	—	151	2	no	—
200	D₈	—	151	1	no	—
201	D₈	C₄×D₁₅₁	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	302	2	no	—
204	M₁₆	—	302	2	no	—
205	M₁₆	—	302	2	no	—
206	M₁₆	—	302	2	no	—
207	M₁₆	—	151	2	no	—
208	M₁₆	—	151	2	no	—
209	M₁₆	—	151	2	no	—
210	M₁₆	C₂²×D₁₅₁	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	302	2	no	—
213	C₂×C₈	—	302	2	no	—
214	C₂×C₈	—	302	2	no	—
215	C₂×C₈	—	302	2	no	—
216	C₂×C₈	—	151	1	no	—
217	C₂×C₈	—	151	1	no	—
218	C₂×C₈	—	151	2	no	—
219	C₂×C₈	—	151	2	no	—
220	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
222	C₄⋊C₄	—	302	1	no	—
223	C₂²⋊C₄	—	302	1	no	—
224	C₄²	—	151	1	no	—
225	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
226	C₂³	—	302	2	no	—
227	C₂³	—	151	1	no	—
228	Q₈	—	302	2	no	—
229	Q₈	—	151	2	no	—
230	Q₈	C₂×C₄×D₁₅₁	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	302	2	no	—
233	D₄	—	302	2	no	—
234	D₄	—	302	2	no	—
235	D₄	—	302	2	no	—
236	D₄	—	151	2	no	—
237	D₄	C₂×C₄×D₁₅₁	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	302	2	no	—
241	C₂×C₄	—	302	2	no	—
242	C₂×C₄	—	302	2	no	—
243	C₂×C₄	—	302	2	no	—
244	C₂×C₄	—	302	1	no	—
245	C₂×C₄	—	151	1	no	—
246	C₂×C₄	—	151	2	no	—
247	C₂×C₄	—	151	2	no	—
248	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	302	2	no	—
251	C₈	—	151	2	no	—
252	C₈	—	151	2	no	—
253	C₈	C₂×C₄×D₁₅₁	1	1	yes	characteristic
254	C₈	C₂×C₄×D₁₅₁	1	1	yes	characteristic
255	C₈	D₄×D₁₅₁	1	1	yes	characteristic
256	C₈	D₄×D₁₅₁	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	302	2	no	—
259	C₂²	—	604	1	no	—
260	C₂²	—	604	4	no	—
261	C₂²	—	151	2	no	—
262	C₂²	#41×D₁₅₁	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	302	2	no	—
266	C₄	—	604	1	no	—
267	C₄	—	151	2	no	—
268	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
269	C₄	C₂×D₄×D₁₅₁	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	302	2	no	—
273	C₂	—	604	2	no	—
274	C₂	C₄×D₄×D₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₁₅₁
commutator	C₄×C₁₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#443×C₁₅₁	C₂
socle	C₂×C₁₅₁	C₂×C₄×D₄
prime radical	#23753009	C₂
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	#443	D₁₅₁
hypocenter	C₁₅₁	#2955
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₁₅₁	C₂	semidirect
3. #23752334	C₂	semidirect
4. #23752333	C₂	semidirect
5. #23752331	C₂	semidirect
6. #23752330	C₂	semidirect
7. #23752327	C₂	semidirect
8. #23752259	C₂	semidirect
9. #23752227	C₂	semidirect
10. #23752166	C₂	semidirect
11. #443×C₁₅₁	C₂	semidirect
13. #18012771	C₄	semidirect
14. #18012768	C₄	semidirect
15. #18012766	C₄	semidirect
16. #18012763	C₄	semidirect
20. #18012747	C₂²	semidirect
21. #18012746	C₂²	semidirect
23. #18012745	C₂²	semidirect
30. #18012675	C₂²	semidirect
31. #18012674	C₂²	semidirect
32. #18012667	C₂²	semidirect
33. #18012666	C₂²	semidirect
34. #18012665	C₂²	semidirect
35. #18012650	C₂²	semidirect
36. #18012649	C₂²	semidirect
37. C₈×Dic₁₅₁	C₂²	semidirect
39. #131×C₁₅₁	C₂²	semidirect
40. #108×C₁₅₁	C₂²	semidirect
41. #104×C₁₅₁	C₂²	semidirect
42. C₄×C₈×C₁₅₁	C₂²	semidirect
137. C₁₅₁	#2955	semidirect
146. #443	D₁₅₁	semidirect
163. #131	C₂×D₁₅₁	semidirect
172. #108	C₂×D₁₅₁	semidirect
176. #104	C₂×D₁₅₁	semidirect
180. 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
#23753009	2
#131×D₁₅₁	2
#23752334	2
#23752333	2
#23752331	2
#23752330	2
#23752327	2
#23752259	2
#23752227	2
#23752166	2
#443×C₁₅₁	2
#2955	151