Subgroups — #30676004

Order 16768.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553645	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553628	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553609	C₂	1	2	yes	maximal, complemented
5	#22553524	C₂	1	2	yes	maximal, complemented
6	#22553462	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553445	C₂	1	1	yes	maximal, characteristic
8	C₄×#14523500	C₂	1	2	yes	maximal, complemented
9	C₄×#14523498	C₂	1	2	yes	maximal, complemented
10	#22553365	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅₂₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃₁	—	2	1	no	—
14	C₂×#14523500	C₂²	1	2	yes	—
15	C₂×#14523498	C₂²	1	2	yes	—
16	C₄×D₅₂₄	—	2	1	no	—
17	C₄×D₅₂₄	—	2	1	no	—
18	C₄×D₅₂₄	C₂²	1	2	yes	—
19	C₄×D₅₂₄	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
22	#16862179	C₂²	1	2	yes	complemented
23	#16862177	C₂²	1	2	yes	complemented
24	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	complemented
25	#16862170	C₂²	1	2	yes	complemented
26	#16862167	C₂²	1	2	yes	complemented
27	#16862166	C₂²	1	2	yes	—
28	#16862166	C₂²	1	2	yes	—
29	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
31	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	complemented
32	#126×C₁₃₁	—	2	1	no	—
33	#125×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
34	#124×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
35	#123×C₁₃₁	—	2	1	no	—
36	C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
37	#121×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₁₃₁	—	2	1	no	—
40	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
41	#117×C₁₃₁	—	2	1	no	—
42	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
43	C₂³×D₁₃₁	—	2	1	no	—
44	C₂×D₅₂₄	—	2	4	no	—
45	C₂×D₅₂₄	—	2	1	no	—
46	C₂×D₅₂₄	C₂³	1	2	yes	—
47	C₂²×Dic₁₃₁	—	2	1	no	—
48	C₂×C₄×D₁₃₁	—	2	2	no	—
49	C₂×C₄×D₁₃₁	—	2	2	no	—
50	C₂×C₄×D₁₃₁	—	4	4	no	—
51	#14523500	—	2	4	no	—
52	#14523498	—	2	4	no	—
53	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
54	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
55	C₄⋊Dic₁₃₁	—	2	1	no	—
56	C₄⋊Dic₁₃₁	C₂³	1	2	yes	—
57	(C₂×D₁₃₁)⋊C₄	—	2	2	no	—
58	(C₂×D₁₃₁)⋊C₄	—	2	2	no	—
59	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₃₁	—	2	1	no	—
61	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	—	2	1	no	—
63	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₁	—	2	1	no	—
67	C₄⋊C₄×C₁₃₁	—	2	1	no	—
68	C₄⋊C₄×C₁₃₁	—	2	1	no	—
69	C₄⋊C₄×C₁₃₁	—	2	1	no	—
70	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
74	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
76	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
77	C₄²×C₁₃₁	—	2	1	no	—
78	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
79	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
80	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
81	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
82	C₂²×D₁₃₁	—	2	4	no	—
83	C₂²×D₁₃₁	—	2	2	no	—
84	C₂²×D₁₃₁	—	4	2	no	—
85	C₂²×D₁₃₁	—	4	2	no	—
86	D₅₂₄	—	2	4	no	—
87	D₅₂₄	—	2	2	no	—
88	D₅₂₄	#41	1	4	yes	—
89	C₂×Dic₁₃₁	—	2	2	no	—
90	C₂×Dic₁₃₁	—	4	2	no	—
91	C₄×D₁₃₁	—	4	4	no	—
92	C₄×D₁₃₁	—	4	4	no	—
93	C₁₃₁⋊C₈	—	2	2	no	—
94	C₁₃₁⋊C₈	—	2	2	no	—
95	C₂³×C₁₃₁	—	2	1	no	—
96	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₁₃₁	—	2	2	no	—
98	D₄×C₁₃₁	—	2	2	no	—
99	D₄×C₁₃₁	—	4	2	no	—
100	C₂×C₄×C₁₃₁	—	2	2	no	—
101	C₂×C₄×C₁₃₁	—	2	2	no	—
102	C₂×C₄×C₁₃₁	—	2	2	no	—
103	C₂×C₄×C₁₃₁	—	2	1	no	—
104	C₂×C₄×C₁₃₁	—	2	1	no	—
105	C₂×C₄×C₁₃₁	—	2	1	no	—
106	C₂×C₄×C₁₃₁	—	4	2	no	—
107	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
113	C₂²×C₁₃₁	—	2	1	no	—
114	C₂²×C₁₃₁	—	2	1	no	—
115	C₂²×C₁₃₁	—	4	1	no	—
116	C₂²×C₁₃₁	—	4	1	no	—
117	C₂²×C₁₃₁	—	4	1	no	—
118	C₂²×C₁₃₁	#135	1	2	yes	—
119	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₃₁	—	2	4	no	—
121	C₂×D₁₃₁	—	4	8	no	—
122	C₂×D₁₃₁	—	4	2	no	—
123	C₂×D₁₃₁	—	4	2	no	—
124	C₄×C₁₃₁	—	2	1	no	—
125	C₄×C₁₃₁	—	2	1	no	—
126	C₄×C₁₃₁	—	2	1	no	—
127	C₄×C₁₃₁	—	2	1	no	—
128	C₄×C₁₃₁	—	4	1	no	—
129	C₄×C₁₃₁	—	4	1	no	—
130	C₄×C₁₃₁	—	4	1	no	—
131	C₄×C₁₃₁	AGL(1,ℤ/8)	1	2	yes	—
132	C₄×C₁₃₁	C₂×#41	1	2	yes	—
133	Dic₁₃₁	—	4	2	no	—
134	C₂×C₁₃₁	—	2	2	no	—
135	C₂×C₁₃₁	—	8	1	no	—
136	C₂×C₁₃₁	#525	1	1	yes	characteristic
137	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
138	C₂×C₁₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
139	D₁₃₁	—	4	4	no	—
140	C₁₃₁	#3105	1	1	yes	characteristic, complemented, Sylow
141	#3105	—	131	1	no	maximal, Sylow
142	#529	D₁₃₁	1	1	yes	characteristic, complemented
143	C₂×C₄×D₄	—	131	1	no	—
144	#484	—	131	1	no	—
145	#481	—	131	1	no	—
146	#478	—	131	2	no	—
147	#463	—	131	2	no	—
148	#450	—	131	1	no	—
149	#447	—	131	1	no	—
150	C₄×SD₁₆	—	131	2	no	—
151	C₄×D₈	—	131	2	no	—
152	#431	—	131	1	no	—
153	C₂²×D₄	—	131	1	no	—
154	C₂³×C₄	—	262	1	no	—
155	C₂×SD₁₆	—	131	2	no	—
156	C₂×D₈	—	131	2	no	—
157	#126	—	2	1	no	—
158	#125	C₂×D₁₃₁	1	1	yes	characteristic, complemented
159	#124	C₂×D₁₃₁	1	1	yes	characteristic, complemented
160	#123	—	2	1	no	—
161	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
162	#121	C₂×D₁₃₁	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	262	1	no	—
166	C₄×D₄	—	262	1	no	—
167	C₄×D₄	—	131	2	no	—
168	C₄×D₄	—	131	2	no	—
169	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic, complemented
170	#117	—	2	1	no	—
171	C₂×C₄⋊C₄	—	131	1	no	—
172	C₂×C₂²⋊C₄	—	262	1	no	—
173	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
174	#107	—	131	2	no	—
175	#106	—	131	2	no	—
176	C₄⋊C₈	—	131	2	no	—
177	#103	—	131	2	no	—
178	#102	—	131	2	no	—
179	#102	—	131	2	no	—
180	#102	—	131	2	no	—
181	C₂²⋊C₈	—	131	1	no	—
182	C₂²⋊C₈	—	131	1	no	—
183	C₄×C₈	—	131	2	no	—
184	C₂⁴	—	262	1	no	—
185	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	262	4	no	—
188	C₂×D₄	—	262	1	no	—
189	C₂×D₄	—	131	2	no	—
190	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	262	1	no	—
193	C₂²×C₄	—	262	2	no	—
194	C₂²×C₄	—	262	2	no	—
195	C₂²×C₄	—	524	4	no	—
196	C₂²×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
199	SD₁₆	—	262	4	no	—
200	D₈	—	262	4	no	—
201	C₂×C₈	—	131	2	no	—
202	C₂×C₈	—	131	2	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	262	1	no	—
208	C₄⋊C₄	—	131	2	no	—
209	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
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	—	262	2	no	—
217	C₂²⋊C₄	—	262	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
220	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
221	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
222	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
223	C₂³	—	2	1	no	—
224	C₂³	—	262	4	no	—
225	C₂³	—	262	2	no	—
226	C₂³	—	524	2	no	—
227	C₂³	—	524	2	no	—
228	C₂³	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	262	4	no	—
233	D₄	—	262	2	no	—
234	D₄	—	131	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	262	2	no	—
243	C₂×C₄	—	524	4	no	—
244	C₂×C₄	—	524	2	no	—
245	C₂×C₄	—	524	4	no	—
246	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
252	C₈	—	262	2	no	—
253	C₈	—	262	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	262	4	no	—
260	C₂²	—	524	8	no	—
261	C₂²	—	524	2	no	—
262	C₂²	—	524	2	no	—
263	C₂²	#16862266	1	2	yes	—
264	C₂²	C₂²×C₂²⋊D₁₃₁	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	524	2	no	—
273	C₄	#16862271	1	2	yes	—
274	C₄	#41×D₁₃₁	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	—	8	1	no	—
277	C₂	—	524	4	no	—
278	C₂	#22553819	1	1	yes	characteristic
279	C₂	C₂×#16862266	1	1	yes	characteristic
280	C₂	C₂×#16862271	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#529×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	#22553445	C₂
hypercenter	#529	D₁₃₁
hypocenter	C₁₃₁	#3105
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₅₂₄	C₂	semidirect
2. #22553645	C₂	semidirect
3. #22553628	C₂	semidirect
4. #22553609	C₂	semidirect
5. #22553524	C₂	semidirect
6. #22553462	C₂	semidirect
8. C₄×#14523500	C₂	semidirect
9. C₄×#14523498	C₂	semidirect
10. #22553365	C₂	semidirect
11. #529×C₁₃₁	C₂	semidirect
22. #16862179	C₂²	semidirect
23. #16862177	C₂²	semidirect
24. C₄⋊(C₁₃₁⋊C₈)	C₂²	semidirect
25. #16862170	C₂²	semidirect
26. #16862167	C₂²	semidirect
31. C₄×C₁₃₁⋊C₈	C₂²	semidirect
33. #125×C₁₃₁	C₂²	semidirect
34. #124×C₁₃₁	C₂²	semidirect
38. C₄×Q₈×C₁₃₁	C₂²	semidirect
40. C₄×D₄×C₁₃₁	C₂²	semidirect
140. C₁₃₁	#3105	semidirect
142. #529	D₁₃₁	semidirect
158. #125	C₂×D₁₃₁	semidirect
159. #124	C₂×D₁₃₁	semidirect
163. C₄×Q₈	C₂×D₁₃₁	semidirect
169. C₄×D₄	C₂×D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₂₄	2
#22553645	2
#22553628	2
#22553609	2
#22553524	2
#22553462	2
#22553445	2
C₄×#14523500	2
C₄×#14523498	2
#22553365	2
#529×C₁₃₁	2
#3105	131