Subgroups — C₇₁×#13158 #31834030

Order 18176.

316 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×#361×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#361×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1978	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	#430×C₇₁	—	2	1	no	—
12	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
15	#418×C₇₁	—	2	1	no	—
16	C₂×#100×C₇₁	—	2	1	no	—
17	#405×C₇₁	C₂²	1	1	yes	characteristic
18	#361×C₇₁	—	2	2	no	—
19	#361×C₇₁	C₄	1	4	yes	complemented
20	#347×C₇₁	—	2	2	no	—
21	#344×C₇₁	—	2	1	no	—
22	C₂²×D₄×C₇₁	—	2	1	no	—
23	C₂×D₈×C₇₁	—	2	1	no	—
24	C₂×D₈×C₇₁	—	2	1	no	—
25	C₂×D₈×C₇₁	—	2	1	no	—
26	C₂×D₈×C₇₁	—	2	1	no	—
27	C₂×D₈×C₇₁	—	2	1	no	—
28	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₇₁	—	2	1	no	—
30	C₂×M₁₆×C₇₁	—	2	1	no	—
31	C₂×M₁₆×C₇₁	—	2	1	no	—
32	C₂×M₁₆×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	#117×C₇₁	D₄	1	1	yes	characteristic
35	M₃₂×C₇₁	—	2	1	no	—
36	M₃₂×C₇₁	—	2	1	no	—
37	M₃₂×C₇₁	C₂×C₄	1	2	yes	—
38	M₃₂×C₇₁	D₄	1	2	yes	—
39	C₂×C₁₆×C₇₁	—	2	1	no	—
40	C₂×C₁₆×C₇₁	—	2	1	no	—
41	#108×C₇₁	—	2	1	no	—
42	#108×C₇₁	—	2	1	no	—
43	#108×C₇₁	—	2	1	no	—
44	#108×C₇₁	—	2	1	no	—
45	#108×C₇₁	—	2	1	no	—
46	#108×C₇₁	—	4	2	no	—
47	#108×C₇₁	C₂×C₄	1	2	yes	—
48	#102×C₇₁	—	4	2	no	—
49	#100×C₇₁	—	4	2	no	—
50	#97×C₇₁	—	2	1	no	—
51	C₄×C₈×C₇₁	—	2	1	no	—
52	C₂⁴×C₇₁	—	4	1	no	—
53	C₂×D₄×C₇₁	—	2	2	no	—
54	C₂×D₄×C₇₁	—	4	1	no	—
55	C₂×D₄×C₇₁	—	4	1	no	—
56	C₂×D₄×C₇₁	—	4	1	no	—
57	C₂×D₄×C₇₁	—	4	1	no	—
58	C₂×D₄×C₇₁	—	4	1	no	—
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	D₈×C₇₁	—	2	2	no	—
61	D₈×C₇₁	—	2	2	no	—
62	D₈×C₇₁	—	4	1	no	—
63	D₈×C₇₁	—	4	1	no	—
64	M₁₆×C₇₁	—	2	2	no	—
65	M₁₆×C₇₁	—	2	1	no	—
66	M₁₆×C₇₁	—	2	1	no	—
67	M₁₆×C₇₁	—	4	1	no	—
68	M₁₆×C₇₁	—	4	1	no	—
69	M₁₆×C₇₁	—	4	1	no	—
70	M₁₆×C₇₁	—	4	1	no	—
71	M₁₆×C₇₁	—	4	1	no	—
72	C₂×C₈×C₇₁	—	2	1	no	—
73	C₂×C₈×C₇₁	—	4	1	no	—
74	C₂×C₈×C₇₁	—	4	1	no	—
75	C₂×C₈×C₇₁	—	4	1	no	—
76	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
79	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
80	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
81	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₇₁	—	2	1	no	—
83	C₂²⋊C₄×C₇₁	—	2	1	no	—
84	C₄²×C₇₁	—	2	1	no	—
85	C₁₆×C₇₁	—	2	2	no	—
86	C₁₆×C₇₁	—	2	2	no	—
87	C₂³×C₇₁	—	4	2	no	—
88	C₂³×C₇₁	—	8	1	no	—
89	C₂³×C₇₁	—	8	1	no	—
90	C₂³×C₇₁	—	16	2	no	—
91	C₂³×C₇₁	#137	1	1	yes	characteristic
92	D₄×C₇₁	—	4	2	no	—
93	D₄×C₇₁	—	4	2	no	—
94	D₄×C₇₁	—	8	1	no	—
95	D₄×C₇₁	—	8	1	no	—
96	C₂×C₄×C₇₁	—	4	1	no	—
97	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
99	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
100	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₈×C₇₁	—	4	2	no	—
104	C₈×C₇₁	—	4	1	no	—
105	C₈×C₇₁	—	8	1	no	—
106	C₈×C₇₁	—	8	1	no	—
107	C₈×C₇₁	#117	1	1	yes	characteristic
108	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
109	C₈×C₇₁	#123	1	1	yes	characteristic
110	C₈×C₇₁	#126	1	1	yes	characteristic
111	C₂²×C₇₁	—	4	1	no	—
112	C₂²×C₇₁	—	8	2	no	—
113	C₂²×C₇₁	—	16	2	no	—
114	C₂²×C₇₁	—	16	2	no	—
115	C₂²×C₇₁	—	16	2	no	—
116	C₂²×C₇₁	#440	1	1	yes	characteristic
117	C₂²×C₇₁	#468	1	1	yes	characteristic
118	C₂²×C₇₁	#489	1	1	yes	characteristic
119	C₄×C₇₁	—	8	1	no	—
120	C₄×C₇₁	#388	1	1	yes	characteristic
121	C₄×C₇₁	#442	1	1	yes	characteristic
122	C₄×C₇₁	#469	1	1	yes	characteristic
123	C₄×C₇₁	#490	1	1	yes	characteristic
124	#13158	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₇₁	—	2	1	no	—
126	C₂×C₇₁	—	2	1	no	—
127	C₂×C₇₁	—	2	1	no	—
128	C₂×C₇₁	—	16	2	no	—
129	C₂×C₇₁	#1936	1	1	yes	characteristic
130	C₂×#361	C₂×C₇₁	1	1	yes	characteristic
131	C₂×#361	C₂×C₇₁	1	1	yes	characteristic
132	#2110	C₂×C₇₁	1	1	yes	characteristic, complemented
133	#1978	C₂×C₇₁	1	1	yes	characteristic
134	#1953	C₂×C₇₁	1	1	yes	characteristic, complemented
135	#1388	C₂×C₇₁	1	1	yes	characteristic, complemented
136	#1388	C₂×C₇₁	1	1	yes	characteristic, complemented
137	C₇₁	#13158	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×D₈	C₂²×C₇₁	1	1	yes	characteristic
139	C₂×M₃₂	C₂²×C₇₁	1	1	yes	characteristic
140	C₂×M₃₂	C₂²×C₇₁	1	1	yes	characteristic
141	#430	—	2	1	no	—
142	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic
143	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic
144	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic
145	#418	—	2	1	no	—
146	C₂×#100	—	2	1	no	—
147	#405	C₂²×C₇₁	1	1	yes	characteristic
148	#361	—	2	2	no	—
149	#361	C₄×C₇₁	1	4	yes	complemented
150	#347	—	2	2	no	—
151	#344	—	2	1	no	—
152	C₂²×D₄	—	2	1	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	—	2	1	no	—
158	C₂×D₈	C₂×C₄×C₇₁	1	2	yes	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	D₄×C₇₁	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
164	#117	D₄×C₇₁	1	1	yes	characteristic
165	M₃₂	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	C₂×C₄×C₇₁	1	2	yes	—
168	M₃₂	D₄×C₇₁	1	2	yes	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	2	1	no	—
171	#108	—	2	1	no	—
172	#108	—	2	1	no	—
173	#108	—	2	1	no	—
174	#108	—	2	1	no	—
175	#108	—	2	1	no	—
176	#108	—	4	2	no	—
177	#108	C₂×C₄×C₇₁	1	2	yes	—
178	#102	—	4	2	no	—
179	#100	—	4	2	no	—
180	#97	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₂⁴	—	4	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
190	D₈	—	2	2	no	—
191	D₈	—	2	2	no	—
192	D₈	—	4	1	no	—
193	D₈	—	4	1	no	—
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	4	1	no	—
198	M₁₆	—	4	1	no	—
199	M₁₆	—	4	1	no	—
200	M₁₆	—	4	1	no	—
201	M₁₆	—	4	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
208	C₂×C₈	#41×C₇₁	1	1	yes	characteristic
209	C₂×C₈	#41×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	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₁₆	—	2	2	no	—
216	C₁₆	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	16	2	no	—
221	C₂³	#137×C₇₁	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	1	no	—
225	D₄	—	8	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
228	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
229	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
230	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
231	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
232	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
233	C₈	—	4	2	no	—
234	C₈	—	4	1	no	—
235	C₈	—	8	1	no	—
236	C₈	—	8	1	no	—
237	C₈	#117×C₇₁	1	1	yes	characteristic
238	C₈	C₄×D₄×C₇₁	1	1	yes	characteristic
239	C₈	#123×C₇₁	1	1	yes	characteristic
240	C₈	#126×C₇₁	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	16	2	no	—
244	C₂²	—	16	2	no	—
245	C₂²	—	16	2	no	—
246	C₂²	#440×C₇₁	1	1	yes	characteristic
247	C₂²	#468×C₇₁	1	1	yes	characteristic
248	C₂²	#489×C₇₁	1	1	yes	characteristic
249	C₄	—	8	1	no	—
250	C₄	#388×C₇₁	1	1	yes	characteristic
251	C₄	#442×C₇₁	1	1	yes	characteristic
252	C₄	#469×C₇₁	1	1	yes	characteristic
253	C₄	#490×C₇₁	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	C₇₁×#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇₁	#1936
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
prime radical	C₂²×D₈×C₇₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ #405×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₁×#2110	C₂	semidirect
5. C₇₁×#1953	C₂	semidirect
6. C₇₁×#1388	C₂	semidirect
7. C₇₁×#1388	C₂	semidirect
19. #361×C₇₁	C₄	semidirect
124. #13158	C₇₁	direct (N × H)
132. #2110	C₂×C₇₁	semidirect
134. #1953	C₂×C₇₁	semidirect
135. #1388	C₂×C₇₁	semidirect
136. #1388	C₂×C₇₁	semidirect
137. C₇₁	#13158	direct (N × H)
149. #361	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×#361×C₇₁	2
C₇₁×#2110	2
C₇₁×#1978	2
C₇₁×#1953	2
C₇₁×#1388	2
#13158	71

Subgroups — C71×#13158 #31834030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13158 #31834030