Subgroups — C₇₁×#12812 #31833684

Order 18176.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 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×#1845	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	—	2	1	no	—
12	C₂×#107×C₇₁	C₂²	1	2	yes	—
13	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#103×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#384×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×C₄⋊C₄×C₇₁	—	4	1	no	—
34	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
35	C₂×C₄²×C₇₁	—	4	1	no	—
36	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₇₁	—	2	2	no	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₂×C₁₆×C₇₁	—	2	2	no	—
40	#107×C₇₁	—	2	1	no	—
41	#107×C₇₁	—	2	2	no	—
42	#107×C₇₁	—	2	2	no	—
43	#107×C₇₁	—	4	2	no	—
44	#107×C₇₁	C₂×C₄	1	2	yes	—
45	#103×C₇₁	—	2	4	no	—
46	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₇₁	D₄	1	2	yes	—
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	complemented
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	C₄⋊C₄×C₇₁	—	8	2	no	—
79	C₄⋊C₄×C₇₁	—	8	2	no	—
80	C₄²×C₇₁	—	4	4	no	—
81	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₇₁	SD₁₆	1	2	yes	—
83	C₁₆×C₇₁	—	4	2	no	—
84	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₇₁	—	2	4	no	—
86	Q₈×C₇₁	—	4	2	no	—
87	Q₈×C₇₁	—	4	2	no	—
88	Q₈×C₇₁	—	4	2	no	—
89	C₂×C₄×C₇₁	—	2	1	no	—
90	C₂×C₄×C₇₁	—	2	1	no	—
91	C₂×C₄×C₇₁	—	4	2	no	—
92	C₂×C₄×C₇₁	—	4	2	no	—
93	C₂×C₄×C₇₁	—	4	2	no	—
94	C₂×C₄×C₇₁	—	4	2	no	—
95	C₂×C₄×C₇₁	—	8	1	no	—
96	C₂×C₄×C₇₁	—	8	1	no	—
97	C₂×C₄×C₇₁	—	8	2	no	—
98	C₂×C₄×C₇₁	—	8	2	no	—
99	C₂×C₄×C₇₁	—	8	2	no	—
100	C₂×C₄×C₇₁	#102	1	2	yes	—
101	C₂×C₄×C₇₁	#102	1	2	yes	—
102	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
103	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
104	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
110	C₈×C₇₁	—	2	2	no	—
111	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₇₁	#117	1	1	yes	characteristic
113	C₈×C₇₁	#121	1	1	yes	characteristic
114	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
116	C₂²×C₇₁	#459	1	1	yes	characteristic
117	C₂²×C₇₁	#474	1	1	yes	characteristic
118	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
120	C₂²×C₇₁	#509	1	1	yes	characteristic
121	C₂²×C₇₁	#510	1	1	yes	characteristic
122	C₄×C₇₁	—	2	2	no	—
123	C₄×C₇₁	—	4	4	no	—
124	C₄×C₇₁	—	8	2	no	—
125	C₄×C₇₁	—	16	2	no	—
126	C₄×C₇₁	#358	1	4	yes	complemented
127	C₄×C₇₁	#380	1	1	yes	characteristic
128	C₄×C₇₁	#417	1	1	yes	characteristic
129	C₄×C₇₁	#461	1	1	yes	characteristic
130	C₄×C₇₁	#476	1	1	yes	characteristic
131	#12812	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₇₁	#1802	1	1	yes	characteristic
133	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
134	C₂×C₇₁	#2139	1	1	yes	characteristic
135	C₂×C₇₁	#2207	1	1	yes	characteristic
136	C₂×C₇₁	#2209	1	1	yes	characteristic
137	C₂×C₇₁	#2225	1	1	yes	characteristic
138	C₂×C₇₁	#2227	1	1	yes	characteristic
139	C₂×C₄×Q₁₆	C₂×C₇₁	1	1	yes	characteristic
140	C₂×C₄⋊C₁₆	C₂×C₇₁	1	1	yes	characteristic
141	C₂×#358	C₂×C₇₁	1	2	yes	—
142	#1845	C₂×C₇₁	1	1	yes	characteristic
143	#1380	C₂×C₇₁	1	2	yes	—
144	C₇₁	#12812	1	1	yes	characteristic, direct product factor, Sylow
145	C₂²×Q₁₆	C₂²×C₇₁	1	1	yes	characteristic
146	C₂×C₄×Q₈	—	2	1	no	—
147	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
148	C₄×Q₁₆	—	2	2	no	—
149	C₄×Q₁₆	C₄×C₇₁	1	4	yes	complemented
150	C₂×#107	—	2	1	no	—
151	C₂×#107	C₂²×C₇₁	1	2	yes	—
152	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
153	C₂×#103	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
155	#384	—	2	1	no	—
156	C₄⋊C₁₆	—	2	2	no	—
157	#358	—	2	4	no	—
158	C₂²×Q₈	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	C₂×C₄×C₇₁	1	2	yes	—
163	C₂×Q₁₆	D₄×C₇₁	1	2	yes	—
164	C₂×Q₁₆	Q₈×C₇₁	1	2	yes	—
165	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
167	C₄×Q₈	—	2	4	no	—
168	C₄×Q₈	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₄⋊C₄	—	4	1	no	—
174	C₂×C₄²	—	4	1	no	—
175	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	#107	—	2	1	no	—
180	#107	—	2	2	no	—
181	#107	—	2	2	no	—
182	#107	—	4	2	no	—
183	#107	C₂×C₄×C₇₁	1	2	yes	—
184	#103	—	2	4	no	—
185	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
186	C₄×C₈	D₄×C₇₁	1	2	yes	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
197	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	2	2	no	—
201	Q₁₆	—	2	2	no	—
202	Q₁₆	C₄⋊C₄×C₇₁	1	4	yes	complemented
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
209	C₂×C₈	C₂×Q₈×C₇₁	1	1	yes	characteristic
210	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
211	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	8	2	no	—
218	C₄⋊C₄	—	8	2	no	—
219	C₄²	—	4	4	no	—
220	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
221	C₄²	SD₁₆×C₇₁	1	2	yes	—
222	C₁₆	—	4	2	no	—
223	C₂³	C₂×D₈×C₇₁	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	4	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	#102×C₇₁	1	2	yes	—
240	C₂×C₄	#102×C₇₁	1	2	yes	—
241	C₂×C₄	SD₃₂×C₇₁	1	2	yes	—
242	C₂×C₄	Q₃₂×C₇₁	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
244	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
246	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
247	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
248	C₂×C₄	#137×C₇₁	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
251	C₈	#117×C₇₁	1	1	yes	characteristic
252	C₈	#121×C₇₁	1	1	yes	characteristic
253	C₈	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
254	C₂²	C₂×#102×C₇₁	1	1	yes	characteristic
255	C₂²	#459×C₇₁	1	1	yes	characteristic
256	C₂²	#474×C₇₁	1	1	yes	characteristic
257	C₂²	C₂×SD₃₂×C₇₁	1	1	yes	characteristic
258	C₂²	C₂×Q₃₂×C₇₁	1	1	yes	characteristic
259	C₂²	#509×C₇₁	1	1	yes	characteristic
260	C₂²	#510×C₇₁	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₄	—	16	2	no	—
265	C₄	#358×C₇₁	1	4	yes	complemented
266	C₄	#380×C₇₁	1	1	yes	characteristic
267	C₄	#417×C₇₁	1	1	yes	characteristic
268	C₄	#461×C₇₁	1	1	yes	characteristic
269	C₄	#476×C₇₁	1	1	yes	characteristic
270	C₂	C₇₁×#1802	1	1	yes	characteristic
271	C₂	C₂×#358×C₇₁	1	1	yes	characteristic
272	C₂	C₇₁×#2139	1	1	yes	characteristic
273	C₂	C₇₁×#2207	1	1	yes	characteristic
274	C₂	C₇₁×#2209	1	1	yes	characteristic
275	C₂	C₇₁×#2225	1	1	yes	characteristic
276	C₂	C₇₁×#2227	1	1	yes	characteristic

	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
63. Q₁₆×C₇₁	C₄⋊C₄	semidirect
126. C₄×C₇₁	#358	semidirect
131. #12812	C₇₁	direct (N × H)
144. C₇₁	#12812	direct (N × H)
149. C₄×Q₁₆	C₄×C₇₁	semidirect
202. Q₁₆	C₄⋊C₄×C₇₁	semidirect
265. C₄	#358×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×#1845	2
C₇₁×#1380	2
#12812	71

Subgroups — C71×#12812 #31833684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#12812 #31833684