Subgroups — #34077526

Order 19328.

564 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14558987	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#117×D₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#23752170	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752169	C₂	1	2	yes	maximal, complemented
6	#23752168	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752167	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752166	C₂	1	2	yes	maximal, complemented
9	#23752164	C₂	1	2	yes	maximal, complemented
10	C₄×#14558987	C₂	1	2	yes	maximal, complemented
11	#405×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×#14558988	C₂²	1	1	yes	characteristic
14	C₂×#14558987	C₄	1	8	yes	complemented
15	C₂×#14558987	C₂²	1	2	yes	—
16	C₂×#14558987	C₂²	1	1	yes	characteristic
17	C₂×#14558987	C₂²	1	1	yes	characteristic
18	C₂×#14558987	C₂²	1	1	yes	characteristic
19	C₂×#14558987	C₂²	1	1	yes	characteristic
20	M₁₆×D₁₅₁	C₄	1	8	yes	complemented
21	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₅₁	C₂²	1	2	yes	—
23	#18012698	C₂²	1	1	yes	characteristic
24	#18012697	C₄	1	2	yes	—
25	#18012696	C₂²	1	2	yes	—
26	#18012695	C₄	1	2	yes	—
27	C₄⋊C₄×D₁₅₁	C₄	1	2	yes	—
28	C₂²⋊C₄×D₁₅₁	C₄	1	2	yes	—
29	C₄²×D₁₅₁	C₂²	1	2	yes	—
30	#18012650	C₂²	1	2	yes	complemented
31	#18012649	C₂²	1	2	yes	complemented
32	#18012649	C₂²	1	2	yes	complemented
33	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	complemented
34	C₈×Dic₁₅₁	C₂²	1	2	yes	complemented
35	C₈×Dic₁₅₁	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
38	#117×C₁₅₁	C₂²	1	1	yes	characteristic
39	#97×C₁₅₁	C₂²	1	2	yes	complemented
40	C₄×C₈×C₁₅₁	C₂²	1	2	yes	complemented
41	C₂³×D₁₅₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₅₁	C₂×C₄	1	4	yes	—
44	C₂×C₄×D₁₅₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₅₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₅₁	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₁₅₁	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
50	#14558988	C₂×C₄	1	4	yes	complemented
51	#14558987	C₂×C₄	1	8	yes	—
52	#14558987	C₂×C₄	1	4	yes	—
53	#14558987	C₂×C₄	1	4	yes	—
54	#14558987	C₂×C₄	1	4	yes	complemented
55	#14558987	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₅₁⋊C₈	C₂×C₄	1	4	yes	complemented
57	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
58	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
59	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
60	C₈×D₁₅₁	C₂×C₄	1	8	yes	complemented
61	C₄⋊Dic₁₅₁	C₂×C₄	1	2	yes	—
62	Dic₁₅₁⋊C₄	C₂×C₄	1	4	yes	—
63	C₂²⋊Dic₁₅₁	C₂×C₄	1	2	yes	—
64	(C₂×D₁₅₁)⋊C₄	C₂×C₄	1	4	yes	—
65	C₄×Dic₁₅₁	C₂×C₄	1	4	yes	—
66	C₄×Dic₁₅₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
68	M₁₆×C₁₅₁	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₁₅₁	C₂×C₄	1	4	yes	complemented
70	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
71	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₅₁	C₂×C₄	1	2	yes	—
74	C₂²⋊C₄×C₁₅₁	C₂×C₄	1	2	yes	—
75	C₄²×C₁₅₁	C₂³	1	2	yes	—
76	C₂²×D₁₅₁	—	4	2	no	—
77	C₂²×D₁₅₁	C₄²	1	4	yes	—
78	C₂²×D₁₅₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×D₁₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₅₁	C₄²	1	4	yes	—
81	C₂×Dic₁₅₁	C₂²×C₄	1	4	yes	—
82	C₂×Dic₁₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₅₁	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₁₅₁	—	4	4	no	—
85	C₄×D₁₅₁	C₄²	1	4	yes	—
86	C₄×D₁₅₁	C₄²	1	4	yes	—
87	C₄×D₁₅₁	C₂²×C₄	1	2	yes	—
88	C₄×D₁₅₁	C₂²×C₄	1	2	yes	—
89	C₄×D₁₅₁	C₂²×C₄	1	2	yes	—
90	C₄×D₁₅₁	C₂²×C₄	1	2	yes	—
91	C₁₅₁⋊C₈	C₂²×C₄	1	4	yes	—
92	C₁₅₁⋊C₈	C₂²×C₄	1	2	yes	—
93	C₁₅₁⋊C₈	C₂²×C₄	1	2	yes	—
94	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₅₁	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₅₁	C₂²×C₄	1	4	yes	—
101	C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
102	C₈×C₁₅₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₁₅₁	—	2	2	no	—
104	C₂²×C₁₅₁	C₂×C₄²	1	2	yes	—
105	C₂²×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₁₅₁	—	2	2	no	—
107	C₂×D₁₅₁	—	2	2	no	—
108	C₂×D₁₅₁	—	4	4	no	—
109	C₂×D₁₅₁	C₂×C₄²	1	2	yes	—
110	C₂×D₁₅₁	C₂×C₄²	1	2	yes	—
111	C₄×C₁₅₁	—	2	4	no	—
112	C₄×C₁₅₁	C₂×C₄²	1	2	yes	—
113	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
115	Dic₁₅₁	—	2	4	no	—
116	Dic₁₅₁	C₂×C₄²	1	2	yes	—
117	Dic₁₅₁	C₂×C₄²	1	2	yes	—
118	C₂×C₁₅₁	—	2	2	no	—
119	C₂×C₁₅₁	C₂²×C₄²	1	1	yes	characteristic
120	C₂×C₁₅₁	#568	1	2	yes	—
121	D₁₅₁	—	2	2	no	—
122	D₁₅₁	—	2	2	no	—
123	C₁₅₁	#2873	1	1	yes	characteristic, complemented, Sylow
124	#2873	—	151	1	no	maximal, Sylow
125	C₂²×M₁₆	—	151	1	no	—
126	C₂²×M₁₆	—	151	1	no	—
127	C₂×#117	—	151	1	no	—
128	#405	—	151	1	no	—
129	#405	—	151	1	no	—
130	#405	—	151	1	no	—
131	#405	—	151	2	no	—
132	#405	—	151	2	no	—
133	#405	D₁₅₁	1	1	yes	characteristic, complemented
134	C₄×M₁₆	—	151	2	no	—
135	C₄×M₁₆	—	151	2	no	—
136	C₂³×C₄	—	151	1	no	—
137	C₂×M₁₆	—	151	1	no	—
138	C₂×M₁₆	—	151	8	no	—
139	C₂×M₁₆	—	151	8	no	—
140	C₂×M₁₆	—	151	1	no	—
141	C₂×M₁₆	—	151	2	no	—
142	C₂×M₁₆	—	151	1	no	—
143	C₂×M₁₆	—	151	1	no	—
144	C₂×M₁₆	—	151	1	no	—
145	C₂×M₁₆	C₂×D₁₅₁	1	1	yes	characteristic
146	C₂²×C₈	—	151	1	no	—
147	C₂²×C₈	—	151	2	no	—
148	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
149	#117	—	151	1	no	—
150	#117	—	151	2	no	—
151	#117	—	151	2	no	—
152	#117	—	151	2	no	—
153	#117	C₂×D₁₅₁	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	151	2	no	—
155	C₂×C₂²⋊C₄	—	151	2	no	—
156	C₂×C₄²	—	151	2	no	—
157	#97	—	151	2	no	—
158	#97	—	151	2	no	—
159	#97	—	151	2	no	—
160	#97	C₂×D₁₅₁	1	2	yes	complemented
161	C₄×C₈	—	151	2	no	—
162	C₄×C₈	—	151	2	no	—
163	C₄×C₈	—	151	2	no	—
164	C₄×C₈	C₂×D₁₅₁	1	2	yes	complemented
165	C₂⁴	—	151	1	no	—
166	C₂²×C₄	—	151	4	no	—
167	C₂²×C₄	—	151	1	no	—
168	C₂²×C₄	—	151	1	no	—
169	C₂²×C₄	—	151	1	no	—
170	C₂²×C₄	—	151	4	no	—
171	C₂²×C₄	—	151	4	no	—
172	C₂²×C₄	—	151	1	no	—
173	C₂²×C₄	—	151	1	no	—
174	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
175	M₁₆	—	151	4	no	—
176	M₁₆	—	151	8	no	—
177	M₁₆	—	151	4	no	—
178	M₁₆	—	151	4	no	—
179	M₁₆	—	151	4	no	—
180	M₁₆	—	151	4	no	—
181	M₁₆	C₄×D₁₅₁	1	4	yes	complemented
182	C₂×C₈	—	151	4	no	—
183	C₂×C₈	—	151	2	no	—
184	C₂×C₈	—	151	1	no	—
185	C₂×C₈	—	151	1	no	—
186	C₂×C₈	—	151	8	no	—
187	C₂×C₈	C₄×D₁₅₁	1	4	yes	complemented
188	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₁₅₁	1	2	yes	—
190	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
191	C₄⋊C₄	—	151	4	no	—
192	C₄⋊C₄	—	151	2	no	—
193	C₄⋊C₄	C₄×D₁₅₁	1	2	yes	—
194	C₂²⋊C₄	—	151	2	no	—
195	C₂²⋊C₄	—	151	4	no	—
196	C₂²⋊C₄	C₄×D₁₅₁	1	2	yes	—
197	C₄²	—	151	4	no	—
198	C₄²	—	151	2	no	—
199	C₄²	C₂²×D₁₅₁	1	2	yes	—
200	C₂³	—	604	2	no	—
201	C₂³	—	151	1	no	—
202	C₂³	—	151	1	no	—
203	C₂³	—	151	4	no	—
204	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
205	C₂×C₄	—	604	4	no	—
206	C₂×C₄	—	151	4	no	—
207	C₂×C₄	—	151	4	no	—
208	C₂×C₄	—	151	1	no	—
209	C₂×C₄	—	151	1	no	—
210	C₂×C₄	—	151	4	no	—
211	C₂×C₄	—	151	4	no	—
212	C₂×C₄	—	151	2	no	—
213	C₂×C₄	—	151	2	no	—
214	C₂×C₄	—	151	2	no	—
215	C₂×C₄	—	151	2	no	—
216	C₂×C₄	C₂×C₄×D₁₅₁	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
220	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
221	C₈	—	151	2	no	—
222	C₈	—	151	2	no	—
223	C₈	—	151	4	no	—
224	C₈	C₂×C₄×D₁₅₁	1	4	yes	—
225	C₈	C₂×C₄×D₁₅₁	1	2	yes	—
226	C₈	C₂×C₄×D₁₅₁	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	302	2	no	—
229	C₂²	—	302	2	no	—
230	C₂²	—	604	4	no	—
231	C₂²	—	151	2	no	—
232	C₂²	—	151	2	no	—
233	C₂²	C₄²×D₁₅₁	1	2	yes	—
234	C₂²	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	302	4	no	—
237	C₄	—	151	2	no	—
238	C₄	—	151	2	no	—
239	C₄	C₄²×D₁₅₁	1	2	yes	—
240	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
241	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	302	2	no	—
244	C₂	—	302	2	no	—
245	C₂	C₂×C₄²×D₁₅₁	1	1	yes	characteristic
246	C₂	#23752977	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₁
commutator	C₂×C₁₅₁	C₂²×C₄²
Fitting	#405×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂³×C₄
prime radical	C₂³×D₁₅₁	C₂×C₄
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	#405	D₁₅₁
hypocenter	C₁₅₁	#2873
LCS	G ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #23752170	C₂	semidirect
5. #23752169	C₂	semidirect
6. #23752168	C₂	semidirect
7. #23752167	C₂	semidirect
8. #23752166	C₂	semidirect
9. #23752164	C₂	semidirect
10. C₄×#14558987	C₂	semidirect
11. #405×C₁₅₁	C₂	semidirect
14. C₂×#14558987	C₄	semidirect
20. M₁₆×D₁₅₁	C₄	semidirect
30. #18012650	C₂²	semidirect
31. #18012649	C₂²	semidirect
32. #18012649	C₂²	semidirect
33. C₄×C₁₅₁⋊C₈	C₂²	semidirect
34. C₈×Dic₁₅₁	C₂²	semidirect
35. C₈×Dic₁₅₁	C₂²	semidirect
39. #97×C₁₅₁	C₂²	semidirect
40. C₄×C₈×C₁₅₁	C₂²	semidirect
50. #14558988	C₂×C₄	semidirect
54. #14558987	C₂×C₄	semidirect
55. #14558987	C₂×C₄	semidirect
56. C₂×C₁₅₁⋊C₈	C₂×C₄	semidirect
60. C₈×D₁₅₁	C₂×C₄	semidirect
68. M₁₆×C₁₅₁	C₂×C₄	semidirect
69. C₂×C₈×C₁₅₁	C₂×C₄	semidirect
123. C₁₅₁	#2873	semidirect
133. #405	D₁₅₁	semidirect
160. #97	C₂×D₁₅₁	semidirect
164. C₄×C₈	C₂×D₁₅₁	semidirect
181. M₁₆	C₄×D₁₅₁	semidirect
187. C₂×C₈	C₄×D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14558987	2
C₂×M₁₆×D₁₅₁	2
#117×D₁₅₁	2
#23752170	2
#23752169	2
#23752168	2
#23752167	2
#23752166	2
#23752164	2
C₄×#14558987	2
#405×C₁₅₁	2
#2873	151