Subgroups — C₇₁×#13290 #31834162

Order 18176.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₄×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1387	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
6	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
7	#500×C₇₁	—	2	1	no	—
8	#500×C₇₁	—	2	1	no	—
9	#500×C₇₁	C₄	1	2	yes	—
10	#500×C₇₁	C₄	1	2	yes	—
11	C₂×#108×C₇₁	—	2	1	no	—
12	C₂×#108×C₇₁	C₂²	1	2	yes	—
13	C₂×#107×C₇₁	—	2	1	no	—
14	C₂×#107×C₇₁	C₂²	1	2	yes	—
15	#423×C₇₁	—	2	1	no	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#346×C₇₁	—	2	1	no	—
18	C₄×C₁₆×C₇₁	—	2	1	no	—
19	C₂²×Q₈×C₇₁	—	4	1	no	—
20	C₂×M₁₆×C₇₁	—	2	2	no	—
21	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₇₁	—	2	2	no	—
24	C₄⋊Q₈×C₇₁	—	2	2	no	—
25	C₄⋊Q₈×C₇₁	—	4	1	no	—
26	C₄⋊Q₈×C₇₁	—	4	1	no	—
27	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
29	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
30	M₃₂×C₇₁	—	2	2	no	—
31	M₃₂×C₇₁	Q₈	1	4	yes	complemented
32	C₂×C₁₆×C₇₁	—	2	2	no	—
33	#108×C₇₁	—	2	4	no	—
34	#108×C₇₁	—	4	2	no	—
35	#107×C₇₁	—	2	2	no	—
36	#107×C₇₁	—	2	2	no	—
37	#107×C₇₁	—	2	2	no	—
38	#107×C₇₁	C₂×C₄	1	4	yes	—
39	C₄⋊C₈×C₇₁	—	4	1	no	—
40	C₄⋊C₈×C₇₁	—	4	1	no	—
41	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₇₁	—	4	2	no	—
46	C₂×Q₈×C₇₁	—	4	2	no	—
47	C₂×Q₈×C₇₁	—	8	4	no	—
48	C₂²×C₄×C₇₁	—	4	2	no	—
49	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	M₁₆×C₇₁	—	4	2	no	—
53	M₁₆×C₇₁	—	4	2	no	—
54	C₂×C₈×C₇₁	—	4	2	no	—
55	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
58	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₇₁	#41	1	2	yes	—
63	C₄⋊C₄×C₇₁	—	2	4	no	—
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₄⋊C₄×C₇₁	—	4	2	no	—
67	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₇₁	D₈	1	1	yes	characteristic
70	C₄²×C₇₁	D₈	1	1	yes	characteristic
71	C₁₆×C₇₁	—	2	4	no	—
72	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
73	Q₈×C₇₁	—	8	4	no	—
74	Q₈×C₇₁	—	8	4	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	4	4	no	—
78	C₂×C₄×C₇₁	—	8	4	no	—
79	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
89	C₈×C₇₁	—	8	2	no	—
90	C₈×C₇₁	#103	1	2	yes	—
91	C₈×C₇₁	#103	1	2	yes	—
92	C₈×C₇₁	C₄×Q₈	1	2	yes	—
93	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
94	C₂²×C₇₁	—	2	1	no	—
95	C₂²×C₇₁	—	2	1	no	—
96	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
97	C₂²×C₇₁	#487	1	1	yes	characteristic
98	C₂²×C₇₁	#498	1	1	yes	characteristic
99	C₄×C₇₁	—	2	1	no	—
100	C₄×C₇₁	—	2	1	no	—
101	C₄×C₇₁	—	8	4	no	—
102	C₄×C₇₁	#361	1	2	yes	—
103	C₄×C₇₁	#362	1	2	yes	—
104	C₄×C₇₁	#391	1	1	yes	characteristic
105	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
106	C₄×C₇₁	#486	1	1	yes	characteristic
107	C₄×C₇₁	#500	1	1	yes	characteristic
108	#13290	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₇₁	—	2	1	no	—
110	C₂×C₇₁	—	2	1	no	—
111	C₂×C₇₁	#1990	1	1	yes	characteristic
112	C₂×C₇₁	C₂×#361	1	1	yes	characteristic
113	C₂×C₇₁	C₂×#362	1	1	yes	characteristic
114	C₂×#500	C₂×C₇₁	1	1	yes	characteristic
115	C₄×M₃₂	C₂×C₇₁	1	1	yes	characteristic
116	#1848	C₂×C₇₁	1	1	yes	characteristic
117	#1387	C₂×C₇₁	1	4	yes	—
118	C₇₁	#13290	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂×M₃₂	C₂²×C₇₁	1	2	yes	—
121	#500	—	2	1	no	—
122	#500	—	2	1	no	—
123	#500	C₄×C₇₁	1	2	yes	—
124	#500	C₄×C₇₁	1	2	yes	—
125	C₂×#108	—	2	1	no	—
126	C₂×#108	C₂²×C₇₁	1	2	yes	—
127	C₂×#107	—	2	1	no	—
128	C₂×#107	C₂²×C₇₁	1	2	yes	—
129	#423	—	2	1	no	—
130	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
131	#346	—	2	1	no	—
132	C₄×C₁₆	—	2	1	no	—
133	C₂²×Q₈	—	4	1	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	4	1	no	—
140	C₄⋊Q₈	—	4	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
144	M₃₂	—	2	2	no	—
145	M₃₂	Q₈×C₇₁	1	4	yes	complemented
146	C₂×C₁₆	—	2	2	no	—
147	#108	—	2	4	no	—
148	#108	—	4	2	no	—
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	#107	—	2	2	no	—
152	#107	C₂×C₄×C₇₁	1	4	yes	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄⋊C₈	—	4	1	no	—
155	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
156	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
157	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
158	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	8	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	SD₁₆×C₇₁	1	1	yes	characteristic
164	C₂²×C₄	SD₁₆×C₇₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
166	M₁₆	—	4	2	no	—
167	M₁₆	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
171	C₂×C₈	SD₁₆×C₇₁	1	2	yes	—
172	C₂×C₈	Q₁₆×C₇₁	1	2	yes	—
173	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
175	C₂×C₈	C₂×Q₈×C₇₁	1	2	yes	—
176	C₂×C₈	#41×C₇₁	1	2	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
183	C₄²	D₈×C₇₁	1	1	yes	characteristic
184	C₄²	D₈×C₇₁	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₂³	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	8	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
194	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
195	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
196	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
198	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
199	C₂×C₄	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
200	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
201	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
202	C₂×C₄	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
203	C₈	—	8	2	no	—
204	C₈	#103×C₇₁	1	2	yes	—
205	C₈	#103×C₇₁	1	2	yes	—
206	C₈	C₄×Q₈×C₇₁	1	2	yes	—
207	C₈	C₂²⋊Q₈×C₇₁	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×#102×C₇₁	1	1	yes	characteristic
211	C₂²	#487×C₇₁	1	1	yes	characteristic
212	C₂²	#498×C₇₁	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	4	no	—
216	C₄	#361×C₇₁	1	2	yes	—
217	C₄	#362×C₇₁	1	2	yes	—
218	C₄	#391×C₇₁	1	1	yes	characteristic
219	C₄	C₂×#103×C₇₁	1	1	yes	characteristic
220	C₄	#486×C₇₁	1	1	yes	characteristic
221	C₄	#500×C₇₁	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	C₇₁×#1990	1	1	yes	characteristic
225	C₂	C₂×#361×C₇₁	1	1	yes	characteristic
226	C₂	C₂×#362×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
prime radical	C₂³×C₇₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
31. M₃₂×C₇₁	Q₈	semidirect
108. #13290	C₇₁	direct (N × H)
118. C₇₁	#13290	direct (N × H)
145. M₃₂	Q₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₇₁	2
C₄×M₃₂×C₇₁	2
C₇₁×#1848	2
C₇₁×#1387	2
#13290	71

Subgroups — C71×#13290 #31834162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13290 #31834162