Subgroups — #31890552

Order 18176.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986129	C₂	1	1	yes	maximal, characteristic
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	#22588346	C₂	1	2	yes	maximal
4	#22586224	C₂	1	2	yes	maximal
5	C₂×#334×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
7	#17986129	—	2	2	no	—
8	#17986129	C₄	1	4	yes	complemented
9	C₂×#14556062	—	2	1	no	—
10	C₂×#14556062	—	2	1	no	—
11	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₁⋊C₈	—	2	1	no	—
13	C₂×#14556053	—	2	2	no	—
14	C₂×#14556038	—	2	1	no	—
15	#17985869	C₂²	1	2	yes	—
16	#17985762	—	2	2	no	—
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	#334×C₇₁	—	2	2	no	—
20	C₂²×Dic₁₄₂	D₄	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	—	2	1	no	—
22	C₄×Dic₁₄₂	—	2	1	no	—
23	C₄×Dic₁₄₂	—	2	1	no	—
24	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₄₂	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	—	2	2	no	—
28	C₂×C₄⋊Dic₇₁	—	2	2	no	—
29	C₂×C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
30	C₂×Dic₇₁⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₇₁	—	2	1	no	—
32	#14556062	—	4	2	no	—
33	#14556062	—	4	2	no	—
34	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
35	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₇₁⋊C₈	—	2	4	no	—
37	#14556053	—	4	4	no	—
38	#14556038	—	2	4	no	—
39	C₂²×C₈×C₇₁	—	2	2	no	—
40	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₁	—	2	2	no	—
44	C₄⋊C₈×C₇₁	—	2	2	no	—
45	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₁	D₄	1	2	yes	—
47	C₂×Dic₁₄₂	—	2	2	no	—
48	C₂×Dic₁₄₂	—	2	2	no	—
49	C₂×Dic₁₄₂	—	2	1	no	—
50	C₂×Dic₁₄₂	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₇₁	—	2	1	no	—
52	C₂²×Dic₇₁	—	2	1	no	—
53	C₂²×Dic₇₁	—	4	2	no	—
54	C₂×C₇₁⋊C₈	—	2	2	no	—
55	C₂×C₇₁⋊C₈	—	4	1	no	—
56	C₂×C₇₁⋊C₈	—	4	1	no	—
57	C₄⋊Dic₇₁	—	2	1	no	—
58	C₄⋊Dic₇₁	—	4	2	no	—
59	C₄⋊Dic₇₁	—	4	2	no	—
60	C₄⋊Dic₇₁	—	4	2	no	—
61	C₄⋊Dic₇₁	—	4	2	no	—
62	C₄⋊Dic₇₁	—	8	2	no	—
63	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
64	Dic₇₁⋊C₄	—	2	2	no	—
65	Dic₇₁⋊C₄	—	2	2	no	—
66	C₄×Dic₇₁	—	2	2	no	—
67	C₄×Dic₇₁	—	2	2	no	—
68	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
71	C₂×C₈×C₇₁	—	2	1	no	—
72	C₂×C₈×C₇₁	—	2	1	no	—
73	C₂×C₈×C₇₁	—	4	2	no	—
74	C₂×C₈×C₇₁	—	4	2	no	—
75	C₂×C₈×C₇₁	—	4	2	no	—
76	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
79	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
80	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₇₁	#41	1	1	yes	characteristic
84	C₂³×C₇₁	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₁	—	2	1	no	—
86	C₂×C₄×C₇₁	—	2	1	no	—
87	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
98	C₂×C₄×C₇₁	#137	1	2	yes	—
99	C₂×Dic₇₁	—	2	2	no	—
100	C₂×Dic₇₁	—	2	2	no	—
101	C₂×Dic₇₁	—	4	1	no	—
102	C₂×Dic₇₁	—	4	1	no	—
103	C₂×Dic₇₁	—	4	1	no	—
104	C₂×Dic₇₁	—	4	1	no	—
105	C₂×Dic₇₁	—	8	2	no	—
106	C₂×Dic₇₁	—	8	2	no	—
107	C₂×Dic₇₁	—	8	2	no	—
108	Dic₁₄₂	—	2	4	no	—
109	Dic₁₄₂	—	4	2	no	—
110	C₈×C₇₁	—	2	2	no	—
111	C₈×C₇₁	—	8	2	no	—
112	C₈×C₇₁	#103	1	4	yes	complemented
113	C₇₁⋊C₈	—	4	2	no	—
114	C₂²×C₇₁	#380	1	1	yes	characteristic
115	C₂²×C₇₁	#460	1	1	yes	characteristic
116	C₂²×C₇₁	#462	1	1	yes	characteristic
117	C₂²×C₇₁	#469	1	1	yes	characteristic
118	C₂²×C₇₁	#470	1	1	yes	characteristic
119	C₂²×C₇₁	#472	1	1	yes	characteristic
120	C₂²×C₇₁	#473	1	1	yes	characteristic
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
124	C₄×C₇₁	#417	1	1	yes	characteristic
125	C₄×C₇₁	#428	1	2	yes	—
126	C₄×C₇₁	#468	1	2	yes	—
127	C₄×C₇₁	#475	1	1	yes	characteristic
128	C₄×C₇₁	#477	1	1	yes	characteristic
129	Dic₇₁	—	4	2	no	—
130	Dic₇₁	—	4	2	no	—
131	Dic₇₁	—	16	2	no	—
132	#8653	—	71	1	no	maximal, Sylow
133	C₂×C₇₁	#1688	1	1	yes	characteristic
134	C₂×C₇₁	#1690	1	1	yes	characteristic
135	C₂×C₇₁	#1694	1	1	yes	characteristic
136	C₂×C₇₁	#1696	1	1	yes	characteristic
137	C₂×C₇₁	#1803	1	1	yes	characteristic
138	C₂×C₇₁	#1809	1	1	yes	characteristic
139	C₂×C₇₁	#1816	1	1	yes	characteristic
140	C₂×#446	—	71	1	no	—
141	#1845	—	71	1	no	—
142	#1803	—	71	2	no	—
143	C₂×#334	D₇₁	1	1	yes	characteristic
144	#1296	—	71	2	no	—
145	C₇₁	#8653	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄×Q₈	—	71	1	no	—
147	#446	—	71	4	no	—
148	#446	—	142	2	no	—
149	C₂×#107	—	142	1	no	—
150	C₂×#107	—	142	1	no	—
151	C₂×C₄⋊C₈	—	71	1	no	—
152	C₂×C₄⋊C₈	—	142	1	no	—
153	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
154	C₂×#103	—	142	2	no	—
155	C₂×#97	—	142	1	no	—
156	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
157	#384	—	71	2	no	—
158	#340	—	142	2	no	—
159	#334	—	2	2	no	—
160	C₂²×Q₈	—	71	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	142	1	no	—
163	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
164	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
165	C₄×Q₈	—	71	2	no	—
166	C₄×Q₈	—	71	2	no	—
167	C₄×Q₈	—	142	1	no	—
168	C₄×Q₈	—	142	1	no	—
169	C₂×C₄⋊C₄	—	71	1	no	—
170	C₂×C₄⋊C₄	—	142	2	no	—
171	C₂×C₄⋊C₄	—	142	2	no	—
172	C₂×C₄⋊C₄	—	142	2	no	—
173	C₂×C₄⋊C₄	—	142	1	no	—
174	C₂×C₄²	—	142	1	no	—
175	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
176	#107	—	284	2	no	—
177	#107	—	284	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	71	2	no	—
181	C₄⋊C₈	—	142	4	no	—
182	C₄⋊C₈	—	142	1	no	—
183	#103	—	284	4	no	—
184	#97	—	142	4	no	—
185	C₄×C₈	C₄×D₇₁	1	2	yes	—
186	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
187	C₂×Q₈	—	71	2	no	—
188	C₂×Q₈	—	142	2	no	—
189	C₂×Q₈	—	142	2	no	—
190	C₂×Q₈	—	142	1	no	—
191	C₂²×C₄	—	284	2	no	—
192	C₂²×C₄	—	142	1	no	—
193	C₂²×C₄	—	142	1	no	—
194	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
196	C₂²×C₄	#11765866	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	284	1	no	—
203	C₂×C₈	—	284	1	no	—
204	C₂×C₈	—	142	2	no	—
205	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
206	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
207	C₂×C₈	#11765854	1	2	yes	—
208	C₂×C₈	Dic₂₈₄	1	2	yes	—
209	C₄⋊C₄	—	284	2	no	—
210	C₄⋊C₄	—	284	2	no	—
211	C₄⋊C₄	—	284	2	no	—
212	C₄⋊C₄	—	284	2	no	—
213	C₄⋊C₄	—	568	2	no	—
214	C₄⋊C₄	—	71	2	no	—
215	C₄⋊C₄	—	142	2	no	—
216	C₄⋊C₄	—	142	1	no	—
217	C₄⋊C₄	—	142	2	no	—
218	C₄²	—	142	2	no	—
219	C₄²	—	142	2	no	—
220	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
222	C₄²	Q₈×D₇₁	1	1	yes	characteristic
223	C₄²	#11765864	1	1	yes	characteristic
224	C₂³	#14556103	1	1	yes	characteristic
225	Q₈	—	284	2	no	—
226	Q₈	—	142	4	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	284	1	no	—
230	C₂×C₄	—	284	1	no	—
231	C₂×C₄	—	284	1	no	—
232	C₂×C₄	—	284	1	no	—
233	C₂×C₄	—	568	2	no	—
234	C₂×C₄	—	568	2	no	—
235	C₂×C₄	—	568	2	no	—
236	C₂×C₄	—	142	2	no	—
237	C₂×C₄	—	142	2	no	—
238	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
240	C₂×C₄	#14556086	1	1	yes	characteristic
241	C₂×C₄	#14556099	1	1	yes	characteristic
242	C₂×C₄	#14556104	1	1	yes	characteristic
243	C₂×C₄	#14556108	1	1	yes	characteristic
244	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
245	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
246	C₂×C₄	#14556155	1	2	yes	—
247	C₂×C₄	#14556156	1	1	yes	characteristic
248	C₂×C₄	#14556157	1	2	yes	—
249	C₂×C₄	#14556161	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	2	no	—
252	C₈	—	284	2	no	—
253	C₈	#14556053	1	4	yes	complemented
254	C₂²	#17985856	1	1	yes	characteristic
255	C₂²	#17986200	1	1	yes	characteristic
256	C₂²	#17986210	1	1	yes	characteristic
257	C₂²	#17986253	1	1	yes	characteristic
258	C₂²	#17986258	1	1	yes	characteristic
259	C₂²	#17986270	1	1	yes	characteristic
260	C₂²	#17986275	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	284	2	no	—
264	C₄	—	284	2	no	—
265	C₄	—	1136	2	no	—
266	C₄	C₂×#14556053	1	1	yes	characteristic
267	C₄	#17986001	1	1	yes	characteristic
268	C₄	#17986038	1	2	yes	—
269	C₄	#17986242	1	2	yes	—
270	C₄	#17986285	1	1	yes	characteristic
271	C₄	#17986295	1	1	yes	characteristic
272	C₂	#22587825	1	1	yes	characteristic
273	C₂	#22587835	1	1	yes	characteristic
274	C₂	#22587859	1	1	yes	characteristic
275	C₂	#22587869	1	1	yes	characteristic
276	C₂	#22588346	1	1	yes	characteristic
277	C₂	#22588370	1	1	yes	characteristic
278	C₂	#22588401	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556103
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#334×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#334	D₇₁
hypocenter	C₇₁	#8653
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #17986129	C₄	semidirect
112. C₈×C₇₁	#103	semidirect
145. C₇₁	#8653	semidirect
253. C₈	#14556053	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986129	2
#22588500	2
#22588346	2
#22586224	2
C₂×#334×C₇₁	2
#8653	71