Subgroups — C₇₁×#10864 #31831736

Order 18176.

806 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	C₂²×#108×C₇₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#341×C₇₁	C₂	1	2	yes	maximal
4	C₇₁×#1728	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₇₁	C₂²	1	4	yes	—
6	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×#108×C₇₁	—	2	4	no	—
10	C₂×#108×C₇₁	—	2	4	no	—
11	C₂×#108×C₇₁	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₇₁	—	2	4	no	—
13	C₂×C₄×C₈×C₇₁	C₄	1	4	yes	—
14	#341×C₇₁	—	2	4	no	—
15	#336×C₇₁	—	2	4	no	—
16	C₂³×C₄×C₇₁	Q₈	1	2	yes	—
17	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₇₁	—	2	4	no	—
19	C₂×M₁₆×C₇₁	—	2	4	no	—
20	C₂×M₁₆×C₇₁	—	2	4	no	—
21	C₂×M₁₆×C₇₁	—	2	4	no	—
22	C₂×M₁₆×C₇₁	—	2	4	no	—
23	C₂×M₁₆×C₇₁	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₇₁	—	2	2	no	—
25	C₂²×C₈×C₇₁	—	2	2	no	—
26	C₂²×C₈×C₇₁	—	2	4	no	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₁	D₄	1	2	yes	—
31	C₂²×C₈×C₇₁	D₄	1	2	yes	—
32	C₂²×C₈×C₇₁	D₄	1	4	yes	—
33	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
34	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
35	C₂×C₄²×C₇₁	—	2	2	no	—
36	C₂×C₄²×C₇₁	—	2	2	no	—
37	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₁	D₄	1	2	yes	—
39	#108×C₇₁	—	2	16	no	—
40	C₄×C₈×C₇₁	—	2	8	no	—
41	C₂⁴×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	—	2	4	no	—
43	C₂²×C₄×C₇₁	—	2	8	no	—
44	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	#41	1	2	yes	—
54	C₂²×C₄×C₇₁	#41	1	2	yes	—
55	M₁₆×C₇₁	—	2	8	no	—
56	M₁₆×C₇₁	—	2	8	no	—
57	M₁₆×C₇₁	—	4	4	no	—
58	M₁₆×C₇₁	—	4	4	no	—
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	—	2	2	no	—
61	C₂×C₈×C₇₁	—	2	4	no	—
62	C₂×C₈×C₇₁	—	2	4	no	—
63	C₂×C₈×C₇₁	—	2	8	no	—
64	C₂×C₈×C₇₁	—	4	4	no	—
65	C₂×C₈×C₇₁	—	4	4	no	—
66	C₂×C₈×C₇₁	C₄²	1	4	yes	—
67	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
68	C₂×C₈×C₇₁	C₂²⋊C₄	1	8	yes	—
69	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₇₁	C₄⋊C₄	1	8	yes	—
72	C₄²×C₇₁	—	2	2	no	—
73	C₄²×C₇₁	—	2	2	no	—
74	C₄²×C₇₁	—	2	4	no	—
75	C₂³×C₇₁	—	2	4	no	—
76	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₇₁	C₄×Q₈	1	2	yes	—
78	C₂³×C₇₁	C₂²⋊Q₈	1	2	yes	—
79	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₇₁	—	2	16	no	—
82	C₂×C₄×C₇₁	—	2	4	no	—
83	C₂×C₄×C₇₁	—	2	4	no	—
84	C₂×C₄×C₇₁	#108	1	8	yes	complemented
85	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₇₁	#121	1	2	yes	—
97	C₂×C₄×C₇₁	#121	1	2	yes	—
98	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₇₁	#124	1	2	yes	—
100	C₂×C₄×C₇₁	#125	1	2	yes	—
101	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
103	C₈×C₇₁	—	2	4	no	—
104	C₈×C₇₁	—	4	8	no	—
105	C₈×C₇₁	#95	1	8	yes	—
106	C₂²×C₇₁	—	2	4	no	—
107	C₂²×C₇₁	—	2	2	no	—
108	C₂²×C₇₁	—	2	2	no	—
109	C₂²×C₇₁	—	2	4	no	—
110	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₇₁	#380	1	1	yes	characteristic
112	C₂²×C₇₁	#384	1	1	yes	characteristic
113	C₂²×C₇₁	#389	1	2	yes	—
114	C₂²×C₇₁	#391	1	2	yes	—
115	C₂²×C₇₁	C₂×#108	1	4	yes	—
116	C₄×C₇₁	—	2	8	no	—
117	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
118	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₇₁	#380	1	1	yes	characteristic
120	C₄×C₇₁	#382	1	1	yes	characteristic
121	C₄×C₇₁	#389	1	2	yes	—
122	C₄×C₇₁	#390	1	2	yes	—
123	#10864	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₇₁	—	2	2	no	—
125	C₂×C₇₁	—	2	2	no	—
126	C₂×C₇₁	C₄⋊#95	1	1	yes	characteristic
127	C₂×C₇₁	C₄×#108	1	2	yes	—
128	C₂×C₇₁	#1810	1	2	yes	—
129	C₂×C₇₁	#1848	1	2	yes	—
130	C₂²×#108	C₂×C₇₁	1	2	yes	—
131	C₂²×C₄×C₈	C₂×C₇₁	1	1	yes	characteristic
132	C₂×#341	C₂×C₇₁	1	2	yes	—
133	#1728	C₂×C₇₁	1	2	yes	—
134	C₇₁	#10864	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²×M₁₆	C₂²×C₇₁	1	4	yes	—
136	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
137	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
138	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
139	C₂×#108	—	2	4	no	—
140	C₂×#108	—	2	4	no	—
141	C₂×#108	C₄×C₇₁	1	8	yes	complemented
142	C₂×C₄×C₈	—	2	4	no	—
143	C₂×C₄×C₈	C₄×C₇₁	1	4	yes	—
144	#341	—	2	4	no	—
145	#336	—	2	4	no	—
146	C₂³×C₄	Q₈×C₇₁	1	2	yes	—
147	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	C₂×C₄×C₇₁	1	8	yes	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂×C₄×C₇₁	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₇₁	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₇₁	1	4	yes	—
160	C₂²×C₈	D₄×C₇₁	1	2	yes	—
161	C₂²×C₈	D₄×C₇₁	1	4	yes	—
162	C₂²×C₈	D₄×C₇₁	1	2	yes	—
163	C₂²×C₈	Q₈×C₇₁	1	2	yes	—
164	C₂²×C₈	Q₈×C₇₁	1	2	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
168	C₂×C₄²	D₄×C₇₁	1	2	yes	—
169	#108	—	2	16	no	—
170	C₄×C₈	—	2	8	no	—
171	C₂⁴	C₂×Q₈×C₇₁	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	C₄⋊C₄×C₇₁	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
178	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
181	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
182	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
183	C₂²×C₄	#41×C₇₁	1	2	yes	—
184	C₂²×C₄	#41×C₇₁	1	2	yes	—
185	M₁₆	—	2	8	no	—
186	M₁₆	—	2	8	no	—
187	M₁₆	—	4	4	no	—
188	M₁₆	—	4	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	C₄²×C₇₁	1	4	yes	—
197	C₂×C₈	C₂²⋊C₄×C₇₁	1	4	yes	—
198	C₂×C₈	C₂²⋊C₄×C₇₁	1	8	yes	—
199	C₂×C₈	C₄⋊C₄×C₇₁	1	4	yes	—
200	C₂×C₈	C₄⋊C₄×C₇₁	1	4	yes	—
201	C₂×C₈	C₄⋊C₄×C₇₁	1	8	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	4	no	—
205	C₂³	—	2	4	no	—
206	C₂³	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
207	C₂³	C₄×Q₈×C₇₁	1	2	yes	—
208	C₂³	C₂²⋊Q₈×C₇₁	1	2	yes	—
209	C₂³	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
210	C₂³	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	#108×C₇₁	1	8	yes	complemented
215	C₂×C₄	C₂×C₄²×C₇₁	1	1	yes	characteristic
216	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
221	C₂×C₄	#117×C₇₁	1	1	yes	characteristic
222	C₂×C₄	#117×C₇₁	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
224	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
225	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
226	C₂×C₄	#121×C₇₁	1	2	yes	—
227	C₂×C₄	#121×C₇₁	1	2	yes	—
228	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
229	C₂×C₄	#124×C₇₁	1	2	yes	—
230	C₂×C₄	#125×C₇₁	1	2	yes	—
231	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
232	C₂×C₄	C₄⋊Q₈×C₇₁	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	4	8	no	—
235	C₈	#95×C₇₁	1	8	yes	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	C₄×C₄⋊C₄×C₇₁	1	1	yes	characteristic
241	C₂²	#380×C₇₁	1	1	yes	characteristic
242	C₂²	#384×C₇₁	1	1	yes	characteristic
243	C₂²	#389×C₇₁	1	2	yes	—
244	C₂²	#391×C₇₁	1	2	yes	—
245	C₂²	C₂×#108×C₇₁	1	4	yes	—
246	C₄	—	2	8	no	—
247	C₄	C₂×#95×C₇₁	1	1	yes	characteristic
248	C₄	C₄×C₄⋊C₄×C₇₁	1	1	yes	characteristic
249	C₄	#380×C₇₁	1	1	yes	characteristic
250	C₄	#382×C₇₁	1	1	yes	characteristic
251	C₄	#389×C₇₁	1	2	yes	—
252	C₄	#390×C₇₁	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	C₇₁×C₄⋊#95	1	1	yes	characteristic
256	C₂	C₄×#108×C₇₁	1	2	yes	—
257	C₂	C₇₁×#1810	1	2	yes	—
258	C₂	C₇₁×#1848	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇₁
Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
socle	C₂³×C₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₁ ◁ C₂²×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×#108×C₇₁	C₄	semidirect
84. C₂×C₄×C₇₁	#108	semidirect
123. #10864	C₇₁	direct (N × H)
134. C₇₁	#10864	direct (N × H)
141. C₂×#108	C₄×C₇₁	semidirect
214. C₂×C₄	#108×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#108×C₇₁	2
C₂²×C₄×C₈×C₇₁	2
C₂×#341×C₇₁	2
C₇₁×#1728	2
#10864	71

Subgroups — C71×#10864 #31831736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#10864 #31831736