Subgroups — #31890388

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588332	C₂	1	2	yes	maximal
2	#22587099	C₂	1	1	yes	maximal, characteristic, complemented
3	#22587096	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586215	C₂	1	2	yes	maximal, complemented
5	C₂×#436×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986023	C₂²	1	2	yes	—
7	#17986023	C₂²	1	2	yes	—
8	C₂×#14556045	—	2	2	no	—
9	#17985768	—	2	2	no	—
10	#17985752	—	2	1	no	—
11	#17985752	—	2	1	no	—
12	#17985750	—	2	1	no	—
13	#17985750	—	2	1	no	—
14	C₂²×M₁₆×C₇₁	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
16	#436×C₇₁	—	2	1	no	—
17	#436×C₇₁	—	2	1	no	—
18	#436×C₇₁	—	2	1	no	—
19	#436×C₇₁	—	2	1	no	—
20	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₇₁	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#11765849	—	2	2	no	—
24	C₂×#11765849	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
29	#14556045	—	4	4	no	—
30	C₂²×D₄×C₇₁	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₇₁	—	2	2	no	—
33	C₂×M₁₆×C₇₁	—	2	2	no	—
34	C₂×M₁₆×C₇₁	—	4	4	no	—
35	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₇₁	—	2	1	no	—
39	C₄×D₄×C₇₁	—	2	1	no	—
40	C₄×D₄×C₇₁	—	2	1	no	—
41	C₄×D₄×C₇₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₇₁	Q₈	1	2	yes	—
44	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₇₁	—	2	1	no	—
46	C₄⋊C₈×C₇₁	—	2	1	no	—
47	C₂²⋊C₈×C₇₁	—	2	2	no	—
48	C₂²⋊C₈×C₇₁	—	2	2	no	—
49	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
53	#11765849	—	4	2	no	—
54	#11765849	—	4	2	no	—
55	#11765849	—	4	2	no	—
56	#11765849	—	4	2	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₂×C₇₁⋊C₈	—	4	2	no	—
59	C₂⁴×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂×D₄×C₇₁	—	2	1	no	—
61	C₂×D₄×C₇₁	—	2	1	no	—
62	C₂×D₄×C₇₁	—	2	2	no	—
63	C₂×D₄×C₇₁	—	2	2	no	—
64	C₂²×C₄×C₇₁	—	2	2	no	—
65	C₂²×C₄×C₇₁	—	2	2	no	—
66	C₂²×C₄×C₇₁	—	4	2	no	—
67	C₂²×C₄×C₇₁	—	4	2	no	—
68	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
72	M₁₆×C₇₁	—	4	4	no	—
73	M₁₆×C₇₁	—	4	4	no	—
74	C₂×C₈×C₇₁	—	2	2	no	—
75	C₂×C₈×C₇₁	—	2	2	no	—
76	C₂×C₈×C₇₁	—	4	1	no	—
77	C₂×C₈×C₇₁	—	4	2	no	—
78	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₇₁	—	2	1	no	—
83	C₄⋊C₄×C₇₁	—	2	1	no	—
84	C₂²⋊C₄×C₇₁	—	2	2	no	—
85	C₂²⋊C₄×C₇₁	—	2	2	no	—
86	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₇₁	#41	1	1	yes	characteristic
89	C₄²×C₇₁	#41	1	1	yes	characteristic
90	C₂³×C₇₁	—	2	2	no	—
91	C₂³×C₇₁	—	2	4	no	—
92	C₂³×C₇₁	—	4	2	no	—
93	C₂³×C₇₁	—	4	2	no	—
94	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	D₄×C₇₁	—	4	2	no	—
96	D₄×C₇₁	—	4	1	no	—
97	D₄×C₇₁	—	4	1	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	2	no	—
101	C₂×C₄×C₇₁	—	4	1	no	—
102	C₂×C₄×C₇₁	—	4	4	no	—
103	C₂×C₄×C₇₁	—	4	2	no	—
104	C₂×C₄×C₇₁	—	4	2	no	—
105	C₂×C₄×C₇₁	—	4	2	no	—
106	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₇₁	#137	1	2	yes	—
114	C₈×C₇₁	—	2	1	no	—
115	C₈×C₇₁	—	2	1	no	—
116	C₈×C₇₁	—	2	1	no	—
117	C₈×C₇₁	—	4	2	no	—
118	C₈×C₇₁	#100	1	2	yes	complemented
119	C₇₁⋊C₈	—	8	2	no	—
120	C₇₁⋊C₈	—	8	2	no	—
121	C₂²×C₇₁	—	2	1	no	—
122	C₂²×C₇₁	—	2	1	no	—
123	C₂²×C₇₁	—	4	2	no	—
124	C₂²×C₇₁	—	4	2	no	—
125	C₂²×C₇₁	—	4	2	no	—
126	C₂²×C₇₁	—	8	4	no	—
127	C₂²×C₇₁	#380	1	1	yes	characteristic
128	C₂²×C₇₁	#428	1	2	yes	—
129	C₄×C₇₁	—	2	1	no	—
130	C₄×C₇₁	—	2	1	no	—
131	C₄×C₇₁	—	2	1	no	—
132	C₄×C₇₁	—	4	2	no	—
133	C₄×C₇₁	C₂×#100	1	1	yes	characteristic
134	C₄×C₇₁	#413	1	1	yes	characteristic
135	C₄×C₇₁	#468	1	2	yes	—
136	C₄×C₇₁	#469	1	1	yes	characteristic
137	C₄×C₇₁	#470	1	1	yes	characteristic
138	#8618	—	71	1	no	maximal, Sylow
139	C₂×C₇₁	—	2	2	no	—
140	C₂×C₇₁	—	8	2	no	—
141	C₂×C₇₁	#1800	1	1	yes	characteristic
142	C₂×C₇₁	#1809	1	1	yes	characteristic
143	C₂×C₇₁	#1813	1	1	yes	characteristic
144	C₂×#436	D₇₁	1	1	yes	characteristic
145	#1800	—	71	2	no	—
146	#1567	—	71	1	no	—
147	#1566	—	71	1	no	—
148	#1292	—	71	2	no	—
149	C₇₁	#8618	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	Dic₇₁	1	2	yes	—
151	C₂×C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
152	#436	—	2	1	no	—
153	#436	—	2	1	no	—
154	#436	—	2	1	no	—
155	#436	—	2	1	no	—
156	#423	—	71	2	no	—
157	#423	—	71	2	no	—
158	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
159	C₂×#100	—	142	2	no	—
160	C₂×C₂²⋊C₈	Dic₇₁	1	2	yes	—
161	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
162	#343	—	142	2	no	—
163	#335	—	142	1	no	—
164	#335	—	142	1	no	—
165	#334	—	142	1	no	—
166	#334	—	142	1	no	—
167	C₂²×D₄	D₂₈₄	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₇₁	1	2	yes	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	4	4	no	—
172	C₂×M₁₆	—	142	2	no	—
173	C₂×M₁₆	—	142	2	no	—
174	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
175	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
176	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	Dic₁₄₂	1	2	yes	—
183	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	142	2	no	—
187	C₄⋊C₈	—	142	2	no	—
188	C₄⋊C₈	—	142	2	no	—
189	C₄⋊C₈	—	142	2	no	—
190	#100	—	284	4	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
197	C₂⁴	C₄⋊Dic₇₁	1	2	yes	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
210	M₁₆	—	4	4	no	—
211	M₁₆	—	4	4	no	—
212	M₁₆	—	284	2	no	—
213	M₁₆	—	284	2	no	—
214	M₁₆	—	284	2	no	—
215	M₁₆	—	284	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	284	2	no	—
221	C₂×C₈	—	284	2	no	—
222	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
232	C₄²	#11765865	1	1	yes	characteristic
233	C₄²	#11765865	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	D₄	—	4	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
251	C₂×C₄	#14556087	1	1	yes	characteristic
252	C₂×C₄	#14556102	1	1	yes	characteristic
253	C₂×C₄	#14556102	1	1	yes	characteristic
254	C₂×C₄	#14556107	1	1	yes	characteristic
255	C₂×C₄	#14556107	1	1	yes	characteristic
256	C₂×C₄	#14556156	1	2	yes	—
257	C₂×C₄	#14556161	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	4	2	no	—
262	C₈	—	568	2	no	—
263	C₈	—	568	2	no	—
264	C₈	#14556045	1	2	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	8	4	no	—
271	C₂²	#17985859	1	1	yes	characteristic
272	C₂²	#17986041	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂×#14556045	1	1	yes	characteristic
278	C₄	#17985983	1	1	yes	characteristic
279	C₄	#17986245	1	2	yes	—
280	C₄	#17986252	1	1	yes	characteristic
281	C₄	#17986257	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	8	2	no	—
284	C₂	#22588332	1	1	yes	characteristic
285	C₂	#22588373	1	1	yes	characteristic
286	C₂	#22588390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985859
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#436×C₇₁	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂²×D₄×C₇₁	D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#436	D₇₁
hypocenter	C₇₁	#8618
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. #22587099	C₂	semidirect
3. #22587096	C₂	semidirect
4. #22586215	C₂	semidirect
118. C₈×C₇₁	#100	semidirect
149. C₇₁	#8618	semidirect
264. C₈	#14556045	semidirect

Maximal subgroups

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

subgroup	index
#22588332	2
#22587099	2
#22587096	2
#22586215	2
C₂×#436×C₇₁	2
#8618	71