Subgroups — #34080436

Order 19328.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752762	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752509	C₂	1	2	yes	maximal, complemented
3	#23752507	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752502	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752446	C₂	1	2	yes	maximal, complemented
6	#23752443	C₂	1	2	yes	maximal, complemented
7	#23752424	C₂	1	2	yes	maximal, complemented
8	#23752412	C₂	1	1	yes	maximal, characteristic, complemented
9	#23752407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊(C₁₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14559004	—	2	1	no	—
13	C₂×#14558991	C₂²	1	2	yes	—
14	C₂×#14558989	C₂²	1	2	yes	—
15	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
16	#18012739	C₂²	1	1	yes	characteristic
17	#18012736	C₂²	1	1	yes	characteristic
18	#18012727	—	2	1	no	—
19	#18012713	—	2	1	no	—
20	#18012713	C₂²	1	2	yes	—
21	C₄×D₆₀₄	—	2	1	no	—
22	C₄×D₆₀₄	C₂²	1	2	yes	—
23	#18012672	C₂²	1	2	yes	—
24	#18012670	C₂²	1	2	yes	—
25	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	2	yes	—
28	#18012663	C₂²	1	2	yes	—
29	#18012660	C₂²	1	2	yes	—
30	#18012659	C₂²	1	2	yes	—
31	#18012659	C₂²	1	2	yes	—
32	#126×C₁₅₁	—	2	1	no	—
33	#125×C₁₅₁	C₂²	1	1	yes	characteristic
34	#124×C₁₅₁	C₂²	1	1	yes	characteristic
35	#123×C₁₅₁	—	2	1	no	—
36	C₂²⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
37	#121×C₁₅₁	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₁₅₁	—	2	1	no	—
40	C₄×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
41	#117×C₁₅₁	—	2	1	no	—
42	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
43	#14559004	—	4	4	no	—
44	C₂×Dic₃₀₂	—	2	1	no	—
45	C₂×D₆₀₄	—	2	1	no	—
46	C₂×D₆₀₄	—	2	1	no	—
47	C₂×D₆₀₄	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₁₅₁	—	2	2	no	—
49	C₂×C₄×D₁₅₁	—	2	2	no	—
50	#14558991	—	4	2	no	—
51	#14558989	—	4	2	no	—
52	C₂×C₁₅₁⋊C₈	—	4	2	no	—
53	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
54	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
55	C₄⋊Dic₁₅₁	—	2	1	no	—
56	C₄⋊Dic₁₅₁	C₂³	1	2	yes	—
57	(C₂×D₁₅₁)⋊C₄	—	2	2	no	—
58	(C₂×D₁₅₁)⋊C₄	—	2	2	no	—
59	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₅₁	—	2	1	no	—
61	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₅₁	—	2	1	no	—
63	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₅₁	—	2	1	no	—
67	C₄⋊C₄×C₁₅₁	—	2	1	no	—
68	C₄⋊C₄×C₁₅₁	—	2	1	no	—
69	C₄⋊C₄×C₁₅₁	—	2	1	no	—
70	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
73	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
74	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
75	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
76	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
77	C₄²×C₁₅₁	—	2	1	no	—
78	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
79	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
80	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
81	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
82	C₂²×D₁₅₁	—	2	2	no	—
83	Dic₃₀₂	—	4	2	no	—
84	D₆₀₄	—	2	2	no	—
85	D₆₀₄	—	2	2	no	—
86	D₆₀₄	—	4	2	no	—
87	D₆₀₄	—	4	2	no	—
88	C₂²⋊D₁₅₁	—	4	4	no	—
89	C₂×Dic₁₅₁	—	2	2	no	—
90	C₄×D₁₅₁	—	4	4	no	—
91	C₁₅₁⋊C₈	—	2	2	no	—
92	C₁₅₁⋊C₈	—	2	2	no	—
93	C₂³×C₁₅₁	—	2	1	no	—
94	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₁₅₁	—	2	1	no	—
96	Q₈×C₁₅₁	—	2	1	no	—
97	D₄×C₁₅₁	—	2	1	no	—
98	D₄×C₁₅₁	—	2	1	no	—
99	D₄×C₁₅₁	—	4	2	no	—
100	C₂×C₄×C₁₅₁	—	2	2	no	—
101	C₂×C₄×C₁₅₁	—	2	2	no	—
102	C₂×C₄×C₁₅₁	—	2	1	no	—
103	C₂×C₄×C₁₅₁	—	2	1	no	—
104	C₂×C₄×C₁₅₁	—	2	1	no	—
105	C₂×C₄×C₁₅₁	—	2	1	no	—
106	C₂×C₄×C₁₅₁	—	2	1	no	—
107	C₂×C₄×C₁₅₁	—	4	2	no	—
108	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
114	C₂²×C₁₅₁	—	2	1	no	—
115	C₂²×C₁₅₁	—	2	1	no	—
116	C₂²×C₁₅₁	—	4	1	no	—
117	C₂²×C₁₅₁	—	4	1	no	—
118	C₂²×C₁₅₁	—	4	1	no	—
119	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
120	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₅₁	—	4	2	no	—
122	C₂×D₁₅₁	—	4	2	no	—
123	C₂×D₁₅₁	—	4	2	no	—
124	C₄×C₁₅₁	—	2	1	no	—
125	C₄×C₁₅₁	—	2	1	no	—
126	C₄×C₁₅₁	—	2	1	no	—
127	C₄×C₁₅₁	—	2	1	no	—
128	C₄×C₁₅₁	—	4	1	no	—
129	C₄×C₁₅₁	—	4	1	no	—
130	C₄×C₁₅₁	—	4	1	no	—
131	C₄×C₁₅₁	#135	1	2	yes	—
132	C₄×C₁₅₁	H(5,𝔽₂)	1	1	yes	characteristic
133	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
134	Dic₁₅₁	—	4	2	no	—
135	C₂×C₁₅₁	—	2	2	no	—
136	C₂×C₁₅₁	—	8	1	no	—
137	C₂×C₁₅₁	#537	1	1	yes	characteristic
138	C₂×C₁₅₁	C₂×#135	1	1	yes	characteristic
139	C₂×C₁₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
140	D₁₅₁	—	8	2	no	—
141	C₁₅₁	#3247	1	1	yes	characteristic, complemented, Sylow
142	#3247	—	151	1	no	maximal, Sylow
143	#532	—	151	1	no	—
144	#529	D₁₅₁	1	1	yes	characteristic, complemented
145	#478	—	151	2	no	—
146	#477	—	151	1	no	—
147	#476	—	151	1	no	—
148	#466	—	151	2	no	—
149	#465	—	151	2	no	—
150	#463	—	151	2	no	—
151	#460	—	151	1	no	—
152	#459	—	151	1	no	—
153	C₂×C₄⋊C₈	—	151	1	no	—
154	C₂×#41	—	302	1	no	—
155	C₂×SD₁₆	—	151	2	no	—
156	C₂×D₈	—	151	2	no	—
157	C₂²×C₈	—	151	2	no	—
158	C₄⋊Q₈	—	151	1	no	—
159	#127	—	151	1	no	—
160	#126	—	2	1	no	—
161	#125	C₂×D₁₅₁	1	1	yes	characteristic
162	#124	—	302	1	no	—
163	#124	C₂×D₁₅₁	1	1	yes	characteristic
164	#123	—	2	1	no	—
165	C₂²⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
166	#121	—	302	1	no	—
167	#121	—	151	2	no	—
168	#121	C₂×D₁₅₁	1	1	yes	characteristic
169	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	—	302	1	no	—
172	C₄×D₄	—	151	2	no	—
173	C₄×D₄	C₂×D₁₅₁	1	1	yes	characteristic
174	#117	—	2	1	no	—
175	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
176	#107	—	151	2	no	—
177	#106	—	151	2	no	—
178	C₄⋊C₈	—	151	1	no	—
179	C₄⋊C₈	—	151	1	no	—
180	C₄⋊C₈	—	151	2	no	—
181	#103	—	151	2	no	—
182	#102	—	151	2	no	—
183	#102	—	151	2	no	—
184	#102	—	151	2	no	—
185	#41	—	604	4	no	—
186	C₂×Q₈	—	302	1	no	—
187	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	302	2	no	—
190	C₂×D₄	—	302	1	no	—
191	C₂×D₄	—	302	1	no	—
192	C₂×D₄	—	151	2	no	—
193	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	302	2	no	—
196	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
199	SD₁₆	—	604	2	no	—
200	D₈	—	604	2	no	—
201	C₂×C₈	—	604	2	no	—
202	C₂×C₈	—	151	2	no	—
203	C₂×C₈	—	151	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	302	1	no	—
209	C₄⋊C₄	—	151	2	no	—
210	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
211	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	302	2	no	—
218	C₂²⋊C₄	—	302	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
221	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
222	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
223	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	302	2	no	—
226	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	604	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	1	no	—
232	D₄	—	4	2	no	—
233	D₄	—	302	2	no	—
234	D₄	—	302	2	no	—
235	D₄	—	604	2	no	—
236	D₄	—	604	4	no	—
237	D₄	—	604	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	302	2	no	—
247	C₂×C₄	—	604	4	no	—
248	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
254	C₈	—	302	2	no	—
255	C₈	—	302	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	604	2	no	—
262	C₂²	—	604	2	no	—
263	C₂²	—	604	2	no	—
264	C₂²	#18012764	1	2	yes	—
265	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	604	2	no	—
274	C₄	#18012759	1	2	yes	—
275	C₄	#18012785	1	1	yes	characteristic
276	C₄	#18012786	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	1208	2	no	—
280	C₂	#23752790	1	1	yes	characteristic
281	C₂	C₂×#18012759	1	1	yes	characteristic
282	C₂	C₂×#18012764	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#529×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
hypercenter	#529	D₁₅₁
hypocenter	C₁₅₁	#3247
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #23752762	C₂	semidirect
2. #23752509	C₂	semidirect
3. #23752507	C₂	semidirect
4. #23752502	C₂	semidirect
5. #23752446	C₂	semidirect
6. #23752443	C₂	semidirect
7. #23752424	C₂	semidirect
8. #23752412	C₂	semidirect
9. #23752407	C₂	semidirect
10. C₂×C₄⋊(C₁₅₁⋊C₈)	C₂	semidirect
11. #529×C₁₅₁	C₂	semidirect
141. C₁₅₁	#3247	semidirect
144. #529	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752762	2
#23752509	2
#23752507	2
#23752502	2
#23752446	2
#23752443	2
#23752424	2
#23752412	2
#23752407	2
C₂×C₄⋊(C₁₅₁⋊C₈)	2
#529×C₁₅₁	2
#3247	151