Subgroups — C₇₁×#12814 #31833686

Order 18176.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#358×C₇₁	C₂	1	2	yes	maximal
4	C₇₁×#1846	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1380	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₇₁	—	2	1	no	—
8	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₇₁	—	2	2	no	—
10	C₄×Q₁₆×C₇₁	C₄	1	4	yes	complemented
11	C₂×#107×C₇₁	C₂²	1	2	yes	—
12	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#106×C₇₁	—	2	1	no	—
14	C₂×#103×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#382×C₇₁	—	2	1	no	—
17	C₄⋊C₁₆×C₇₁	—	2	2	no	—
18	#358×C₇₁	—	2	4	no	—
19	C₂²×Q₈×C₇₁	—	2	1	no	—
20	C₂×Q₁₆×C₇₁	—	2	1	no	—
21	C₂×Q₁₆×C₇₁	—	2	1	no	—
22	C₂×Q₁₆×C₇₁	—	2	1	no	—
23	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
24	C₂×Q₁₆×C₇₁	D₄	1	2	yes	—
25	C₂×Q₁₆×C₇₁	Q₈	1	2	yes	—
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₇₁	—	2	4	no	—
29	C₄×Q₈×C₇₁	—	4	2	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
33	C₂×C₄²×C₇₁	—	4	1	no	—
34	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	—	2	2	no	—
37	C₂×C₁₆×C₇₁	—	2	2	no	—
38	#107×C₇₁	—	2	1	no	—
39	#107×C₇₁	—	2	2	no	—
40	#107×C₇₁	—	2	2	no	—
41	#107×C₇₁	C₂×C₄	1	2	yes	—
42	#106×C₇₁	—	4	2	no	—
43	#103×C₇₁	—	2	4	no	—
44	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₁	D₄	1	2	yes	—
46	#95×C₇₁	—	4	1	no	—
47	#95×C₇₁	—	4	1	no	—
48	C₂×Q₈×C₇₁	—	2	2	no	—
49	C₂×Q₈×C₇₁	—	2	2	no	—
50	C₂×Q₈×C₇₁	—	2	2	no	—
51	C₂×Q₈×C₇₁	—	4	1	no	—
52	C₂×Q₈×C₇₁	—	4	1	no	—
53	C₂×Q₈×C₇₁	—	4	1	no	—
54	C₂²×C₄×C₇₁	—	4	1	no	—
55	C₂²×C₄×C₇₁	—	4	1	no	—
56	C₂²×C₄×C₇₁	—	4	2	no	—
57	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₇₁	—	2	2	no	—
61	Q₁₆×C₇₁	—	2	2	no	—
62	Q₁₆×C₇₁	—	2	2	no	—
63	Q₁₆×C₇₁	C₄⋊C₄	1	4	yes	—
64	C₂×C₈×C₇₁	—	2	1	no	—
65	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
72	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
73	C₄⋊C₄×C₇₁	—	2	2	no	—
74	C₄⋊C₄×C₇₁	—	4	4	no	—
75	C₄⋊C₄×C₇₁	—	4	1	no	—
76	C₄⋊C₄×C₇₁	—	4	2	no	—
77	C₄⋊C₄×C₇₁	—	4	2	no	—
78	C₄²×C₇₁	—	4	4	no	—
79	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₇₁	SD₁₆	1	2	yes	—
81	C₁₆×C₇₁	—	4	2	no	—
82	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₇₁	—	2	4	no	—
84	Q₈×C₇₁	—	4	2	no	—
85	Q₈×C₇₁	—	4	2	no	—
86	Q₈×C₇₁	—	4	2	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	—	2	2	no	—
89	C₂×C₄×C₇₁	—	4	2	no	—
90	C₂×C₄×C₇₁	—	4	2	no	—
91	C₂×C₄×C₇₁	—	4	2	no	—
92	C₂×C₄×C₇₁	—	4	2	no	—
93	C₂×C₄×C₇₁	—	8	2	no	—
94	C₂×C₄×C₇₁	—	8	2	no	—
95	C₂×C₄×C₇₁	—	8	2	no	—
96	C₂×C₄×C₇₁	—	8	2	no	—
97	C₂×C₄×C₇₁	#102	1	2	yes	—
98	C₂×C₄×C₇₁	#102	1	2	yes	—
99	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
105	C₈×C₇₁	—	2	2	no	—
106	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₇₁	#117	1	1	yes	characteristic
108	C₈×C₇₁	#121	1	1	yes	characteristic
109	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
111	C₂²×C₇₁	#459	1	1	yes	characteristic
112	C₂²×C₇₁	#474	1	1	yes	characteristic
113	C₂²×C₇₁	#508	1	2	yes	—
114	C₂²×C₇₁	#510	1	2	yes	—
115	C₄×C₇₁	—	2	2	no	—
116	C₄×C₇₁	—	2	2	no	—
117	C₄×C₇₁	—	4	4	no	—
118	C₄×C₇₁	—	8	2	no	—
119	C₄×C₇₁	—	16	2	no	—
120	C₄×C₇₁	#380	1	1	yes	characteristic
121	C₄×C₇₁	#417	1	1	yes	characteristic
122	C₄×C₇₁	#461	1	1	yes	characteristic
123	C₄×C₇₁	#476	1	1	yes	characteristic
124	#12814	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₇₁	#1802	1	1	yes	characteristic
126	C₂×C₇₁	#2138	1	1	yes	characteristic
127	C₂×C₇₁	#2141	1	1	yes	characteristic
128	C₂×C₇₁	#2211	1	2	yes	—
129	C₂×C₇₁	#2229	1	2	yes	—
130	C₂×C₄×Q₁₆	C₂×C₇₁	1	1	yes	characteristic
131	C₂×C₄⋊C₁₆	C₂×C₇₁	1	1	yes	characteristic
132	C₂×#358	C₂×C₇₁	1	2	yes	—
133	#1846	C₂×C₇₁	1	1	yes	characteristic
134	#1380	C₂×C₇₁	1	2	yes	—
135	C₇₁	#12814	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×Q₁₆	C₂²×C₇₁	1	1	yes	characteristic
137	C₂×C₄×Q₈	—	2	1	no	—
138	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
139	C₄×Q₁₆	—	2	2	no	—
140	C₄×Q₁₆	C₄×C₇₁	1	4	yes	complemented
141	C₂×#107	C₂²×C₇₁	1	2	yes	—
142	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
143	C₂×#106	—	2	1	no	—
144	C₂×#103	—	2	1	no	—
145	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
146	#382	—	2	1	no	—
147	C₄⋊C₁₆	—	2	2	no	—
148	#358	—	2	4	no	—
149	C₂²×Q₈	—	2	1	no	—
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	—	2	1	no	—
152	C₂×Q₁₆	—	2	1	no	—
153	C₂×Q₁₆	C₂×C₄×C₇₁	1	2	yes	—
154	C₂×Q₁₆	D₄×C₇₁	1	2	yes	—
155	C₂×Q₁₆	Q₈×C₇₁	1	2	yes	—
156	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
158	C₄×Q₈	—	2	4	no	—
159	C₄×Q₈	—	4	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	4	1	no	—
163	C₂×C₄²	—	4	1	no	—
164	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#107	—	2	1	no	—
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#107	C₂×C₄×C₇₁	1	2	yes	—
172	#106	—	4	2	no	—
173	#103	—	2	4	no	—
174	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
175	C₄×C₈	D₄×C₇₁	1	2	yes	—
176	#95	—	4	1	no	—
177	#95	—	4	1	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
188	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
190	Q₁₆	—	2	2	no	—
191	Q₁₆	—	2	2	no	—
192	Q₁₆	—	2	2	no	—
193	Q₁₆	C₄⋊C₄×C₇₁	1	4	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
197	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
200	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
201	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
202	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	4	4	no	—
209	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
210	C₄²	SD₁₆×C₇₁	1	2	yes	—
211	C₁₆	—	4	2	no	—
212	C₂³	C₂×D₈×C₇₁	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	#102×C₇₁	1	2	yes	—
228	C₂×C₄	#102×C₇₁	1	2	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
230	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
231	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
232	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
233	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
234	C₂×C₄	#137×C₇₁	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
237	C₈	#117×C₇₁	1	1	yes	characteristic
238	C₈	#121×C₇₁	1	1	yes	characteristic
239	C₈	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
240	C₂²	C₂×#102×C₇₁	1	1	yes	characteristic
241	C₂²	#459×C₇₁	1	1	yes	characteristic
242	C₂²	#474×C₇₁	1	1	yes	characteristic
243	C₂²	#508×C₇₁	1	2	yes	—
244	C₂²	#510×C₇₁	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	16	2	no	—
250	C₄	#380×C₇₁	1	1	yes	characteristic
251	C₄	#417×C₇₁	1	1	yes	characteristic
252	C₄	#461×C₇₁	1	1	yes	characteristic
253	C₄	#476×C₇₁	1	1	yes	characteristic
254	C₂	C₇₁×#1802	1	1	yes	characteristic
255	C₂	C₇₁×#2138	1	1	yes	characteristic
256	C₂	C₇₁×#2141	1	1	yes	characteristic
257	C₂	C₇₁×#2211	1	2	yes	—
258	C₂	C₇₁×#2229	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×Q₁₆×C₇₁	C₄	semidirect
124. #12814	C₇₁	direct (N × H)
135. C₇₁	#12814	direct (N × H)
140. C₄×Q₁₆	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₇₁	2
C₂×C₄⋊C₁₆×C₇₁	2
C₂×#358×C₇₁	2
C₇₁×#1846	2
C₇₁×#1380	2
#12814	71

Subgroups — C71×#12814 #31833686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12814 #31833686