Subgroups — #31901715

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986404	C₂	1	1	yes	maximal, characteristic
2	#22589263	C₂	1	2	yes	maximal
3	C₂×#17985744	C₂	1	2	yes	maximal
4	#22586175	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	C₂²	1	2	yes	—
7	#17986404	—	2	4	no	—
8	C₂×#14556062	—	2	1	no	—
9	C₂×#14556062	—	2	1	no	—
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	C₂×#14556051	—	2	2	no	—
12	C₂×#14556051	—	2	2	no	—
13	#17985744	—	2	4	no	—
14	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
18	#336×C₇₁	—	2	1	no	—
19	C₂²×Dic₁₄₂	—	2	2	no	—
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	#14556128	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	—	2	2	no	—
24	#14556128	D₄	1	4	yes	—
25	C₂×C₄⋊Dic₇₁	—	2	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	#14556062	—	2	4	no	—
28	#14556062	—	2	4	no	—
29	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
31	#14556051	—	4	4	no	—
32	#14556051	—	4	4	no	—
33	C₂²×C₈×C₇₁	—	2	1	no	—
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	—	2	1	no	—
39	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₇₁	—	2	2	no	—
41	C₄⋊C₈×C₇₁	—	4	2	no	—
42	C₄×C₈×C₇₁	—	2	2	no	—
43	#95×C₇₁	—	2	1	no	—
44	C₂×Dic₁₄₂	—	2	4	no	—
45	C₂×Dic₁₄₂	—	4	2	no	—
46	C₂×Dic₁₄₂	—	4	2	no	—
47	C₂×Dic₁₄₂	—	4	2	no	—
48	C₂×Dic₁₄₂	—	4	2	no	—
49	C₂×Dic₁₄₂	—	4	2	no	—
50	C₂²×Dic₇₁	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	4	2	no	—
53	C₂×C₇₁⋊C₈	—	4	2	no	—
54	C₄⋊Dic₇₁	—	2	4	no	—
55	C₄⋊Dic₇₁	—	2	4	no	—
56	C₄⋊Dic₇₁	—	4	2	no	—
57	C₄⋊Dic₇₁	—	4	2	no	—
58	C₂²×C₄×C₇₁	—	2	1	no	—
59	C₂²×C₄×C₇₁	—	2	1	no	—
60	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
63	C₂×C₈×C₇₁	—	2	1	no	—
64	C₂×C₈×C₇₁	—	2	1	no	—
65	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	—	2	1	no	—
67	C₂×C₈×C₇₁	—	4	1	no	—
68	C₂×C₈×C₇₁	—	4	1	no	—
69	C₂×C₈×C₇₁	—	4	1	no	—
70	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
71	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
72	C₄⋊C₄×C₇₁	—	2	1	no	—
73	C₄⋊C₄×C₇₁	—	2	2	no	—
74	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
75	C₄²×C₇₁	—	4	1	no	—
76	C₄²×C₇₁	—	4	1	no	—
77	C₄²×C₇₁	C₂×D₄	1	2	yes	—
78	C₄²×C₇₁	#41	1	2	yes	—
79	C₂³×C₇₁	#121	1	1	yes	characteristic
80	C₂×C₄×C₇₁	—	2	1	no	—
81	C₂×C₄×C₇₁	—	2	1	no	—
82	C₂×C₄×C₇₁	—	2	2	no	—
83	C₂×C₄×C₇₁	—	4	1	no	—
84	C₂×C₄×C₇₁	—	4	1	no	—
85	C₂×C₄×C₇₁	—	4	1	no	—
86	C₂×C₄×C₇₁	—	4	1	no	—
87	C₂×C₄×C₇₁	—	4	1	no	—
88	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
89	C₂×C₄×C₇₁	#121	1	2	yes	—
90	C₂×C₄×C₇₁	#123	1	2	yes	—
91	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂×D₈	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	#135	1	1	yes	characteristic
96	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
98	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
99	C₂×Dic₇₁	—	4	4	no	—
100	C₂×Dic₇₁	—	8	2	no	—
101	C₂×Dic₇₁	—	8	2	no	—
102	Dic₁₄₂	—	4	4	no	—
103	Dic₁₄₂	—	4	4	no	—
104	Dic₁₄₂	—	8	2	no	—
105	Dic₁₄₂	—	8	2	no	—
106	C₈×C₇₁	—	2	2	no	—
107	C₈×C₇₁	—	2	2	no	—
108	C₈×C₇₁	—	8	1	no	—
109	C₇₁⋊C₈	—	8	2	no	—
110	C₂²×C₇₁	#394	1	1	yes	characteristic
111	C₂²×C₇₁	#461	1	1	yes	characteristic
112	C₂²×C₇₁	#462	1	1	yes	characteristic
113	C₂²×C₇₁	#466	1	1	yes	characteristic
114	C₂²×C₇₁	#469	1	1	yes	characteristic
115	C₂²×C₇₁	#473	1	1	yes	characteristic
116	C₂²×C₇₁	#473	1	1	yes	characteristic
117	C₄×C₇₁	—	2	2	no	—
118	C₄×C₇₁	—	4	2	no	—
119	C₄×C₇₁	—	8	1	no	—
120	C₄×C₇₁	#448	1	1	yes	characteristic
121	C₄×C₇₁	#451	1	1	yes	characteristic
122	C₄×C₇₁	#459	1	2	yes	—
123	C₄×C₇₁	#463	1	2	yes	—
124	C₄×C₇₁	#487	1	1	yes	characteristic
125	C₄×C₇₁	#491	1	1	yes	characteristic
126	Dic₇₁	—	8	4	no	—
127	#10545	—	71	1	no	maximal, Sylow
128	C₂×C₇₁	#1673	1	1	yes	characteristic
129	C₂×C₇₁	#1674	1	1	yes	characteristic
130	C₂×C₇₁	#1693	1	1	yes	characteristic
131	C₂×C₇₁	#1694	1	1	yes	characteristic
132	C₂×C₇₁	#2043	1	1	yes	characteristic
133	C₂×C₇₁	#2047	1	1	yes	characteristic
134	C₂×C₇₁	#2054	1	1	yes	characteristic
135	C₂×#500	—	71	1	no	—
136	#1981	—	71	2	no	—
137	#1939	D₇₁	1	1	yes	characteristic
138	C₂×#332	—	71	2	no	—
139	#1277	—	71	1	no	—
140	C₇₁	#10545	1	1	yes	characteristic, complemented, Sylow
141	C₂×C₄⋊Q₈	—	71	2	no	—
142	#500	—	142	4	no	—
143	C₂×#107	—	142	1	no	—
144	C₂×#107	—	142	1	no	—
145	C₂×C₄⋊C₈	—	2	1	no	—
146	C₂×C₄⋊C₈	—	71	2	no	—
147	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
148	C₂×#103	—	142	2	no	—
149	C₂×#103	—	142	2	no	—
150	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
151	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
152	#336	—	2	1	no	—
153	#332	—	142	4	no	—
154	C₂²×Q₈	—	142	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	142	2	no	—
157	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
159	C₄⋊Q₈	—	71	4	no	—
160	C₄⋊Q₈	—	142	2	no	—
161	C₄⋊Q₈	—	142	2	no	—
162	C₄⋊Q₈	—	142	2	no	—
163	C₂×C₄⋊C₄	—	142	2	no	—
164	C₂×C₄⋊C₄	—	142	2	no	—
165	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
166	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
169	#107	—	142	4	no	—
170	#107	—	142	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	4	2	no	—
173	C₄⋊C₈	—	142	2	no	—
174	C₄⋊C₈	—	142	2	no	—
175	#103	—	284	4	no	—
176	#103	—	284	4	no	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	2	1	no	—
179	C₂×Q₈	—	284	2	no	—
180	C₂×Q₈	—	284	2	no	—
181	C₂×Q₈	—	284	2	no	—
182	C₂×Q₈	—	284	2	no	—
183	C₂×Q₈	—	284	2	no	—
184	C₂×Q₈	—	142	4	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	284	2	no	—
188	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
190	C₂²×C₄	#11765865	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	284	2	no	—
199	C₂×C₈	—	284	2	no	—
200	C₂×C₈	—	284	2	no	—
201	C₂×C₈	#11765853	1	2	yes	—
202	C₂×C₈	#11765855	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	284	2	no	—
206	C₄⋊C₄	—	284	2	no	—
207	C₄⋊C₄	—	142	4	no	—
208	C₄⋊C₄	—	142	4	no	—
209	C₄⋊C₄	D₅₆₈	1	2	yes	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	D₄×D₇₁	1	2	yes	—
213	C₄²	#11765866	1	2	yes	—
214	C₂³	#14556102	1	1	yes	characteristic
215	Q₈	—	284	4	no	—
216	Q₈	—	284	4	no	—
217	Q₈	—	568	2	no	—
218	Q₈	—	568	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	284	4	no	—
228	C₂×C₄	—	568	2	no	—
229	C₂×C₄	—	568	2	no	—
230	C₂×C₄	#14556096	1	1	yes	characteristic
231	C₂×C₄	#14556103	1	2	yes	—
232	C₂×C₄	#14556113	1	2	yes	—
233	C₂×C₄	#14556117	1	1	yes	characteristic
234	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
236	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
237	C₂×C₄	#14556150	1	1	yes	characteristic
238	C₂×C₄	#14556156	1	2	yes	—
239	C₂×C₄	#14556158	1	1	yes	characteristic
240	C₂×C₄	#14556158	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	8	1	no	—
244	C₈	—	568	2	no	—
245	C₂²	#17985916	1	1	yes	characteristic
246	C₂²	#17986204	1	1	yes	characteristic
247	C₂²	#17986209	1	1	yes	characteristic
248	C₂²	#17986235	1	1	yes	characteristic
249	C₂²	#17986252	1	1	yes	characteristic
250	C₂²	#17986274	1	1	yes	characteristic
251	C₂²	#17986274	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	8	1	no	—
255	C₄	—	568	4	no	—
256	C₄	#17986137	1	1	yes	characteristic
257	C₄	#17986154	1	1	yes	characteristic
258	C₄	#17986195	1	2	yes	—
259	C₄	#17986215	1	2	yes	—
260	C₄	#17986351	1	1	yes	characteristic
261	C₄	#17986369	1	1	yes	characteristic
262	C₂	#22587729	1	1	yes	characteristic
263	C₂	#22587734	1	1	yes	characteristic
264	C₂	#22587853	1	1	yes	characteristic
265	C₂	#22587858	1	1	yes	characteristic
266	C₂	#22589586	1	1	yes	characteristic
267	C₂	#22589608	1	1	yes	characteristic
268	C₂	#22589651	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556102
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1939	C₂
socle, prime radical	C₂³×C₇₁	#121
hypercenter	#1939	D₇₁
hypocenter	C₇₁	#10545
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939

Complements

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

N	H ≅ G/N	type
140. C₇₁	#10545	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986404	2
#22589263	2
C₂×#17985744	2
#22586175	2
C₇₁×#1939	2
#10545	71