Subgroups — C₇₁×#7876 #31828748

Order 18176.

662 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₇₁	—	2	1	no	—
6	C₂²×M₁₆×C₇₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
8	#432×C₇₁	—	2	2	no	—
9	#432×C₇₁	C₄	1	4	yes	—
10	#423×C₇₁	—	2	2	no	—
11	#423×C₇₁	C₄	1	4	yes	—
12	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₇₁	—	2	1	no	—
16	C₄×M₁₆×C₇₁	—	2	2	no	—
17	C₄×M₁₆×C₇₁	—	2	2	no	—
18	C₂×#97×C₇₁	—	2	1	no	—
19	C₂×#97×C₇₁	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
21	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₁	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₇₁	—	2	4	no	—
25	C₂×M₁₆×C₇₁	—	2	4	no	—
26	C₂×M₁₆×C₇₁	—	2	8	no	—
27	C₂×M₁₆×C₇₁	—	4	4	no	—
28	C₂²×C₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	D₄	1	2	yes	—
36	C₂×C₄²×C₇₁	D₄	1	2	yes	—
37	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₁	—	2	4	no	—
40	C₄⋊C₈×C₇₁	—	2	2	no	—
41	C₄⋊C₈×C₇₁	—	2	2	no	—
42	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
43	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
44	C₂²⋊C₈×C₇₁	—	2	4	no	—
45	#97×C₇₁	—	2	4	no	—
46	#97×C₇₁	—	2	4	no	—
47	C₄×C₈×C₇₁	—	2	2	no	—
48	C₄×C₈×C₇₁	—	2	2	no	—
49	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	—	2	4	no	—
51	C₂²×C₄×C₇₁	—	2	2	no	—
52	C₂²×C₄×C₇₁	—	2	2	no	—
53	C₂²×C₄×C₇₁	—	4	2	no	—
54	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
55	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₇₁	—	2	16	no	—
67	M₁₆×C₇₁	—	4	8	no	—
68	C₂×C₈×C₇₁	—	2	4	no	—
69	C₂×C₈×C₇₁	—	2	4	no	—
70	C₂×C₈×C₇₁	—	2	4	no	—
71	C₂×C₈×C₇₁	—	2	4	no	—
72	C₂×C₈×C₇₁	—	4	2	no	—
73	C₂×C₈×C₇₁	—	4	4	no	—
74	C₂×C₈×C₇₁	—	4	2	no	—
75	C₂×C₈×C₇₁	M₁₆	1	8	yes	complemented
76	C₄²×C₇₁	C₄²	1	2	yes	—
77	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
81	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
82	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₇₁	—	2	2	no	—
87	C₂³×C₇₁	—	2	2	no	—
88	C₂³×C₇₁	—	2	2	no	—
89	C₂³×C₇₁	#95	1	2	yes	—
90	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	2	2	no	—
93	C₂×C₄×C₇₁	—	2	2	no	—
94	C₂×C₄×C₇₁	—	2	2	no	—
95	C₂×C₄×C₇₁	—	2	4	no	—
96	C₂×C₄×C₇₁	—	2	4	no	—
97	C₂×C₄×C₇₁	—	2	4	no	—
98	C₂×C₄×C₇₁	—	2	2	no	—
99	C₂×C₄×C₇₁	—	2	2	no	—
100	C₂×C₄×C₇₁	—	4	2	no	—
101	C₂×C₄×C₇₁	—	4	2	no	—
102	C₂×C₄×C₇₁	#95	1	2	yes	—
103	C₂×C₄×C₇₁	#95	1	2	yes	—
104	C₂×C₄×C₇₁	#95	1	2	yes	—
105	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₇₁	#137	1	2	yes	—
114	C₈×C₇₁	—	2	8	no	—
115	C₈×C₇₁	—	4	4	no	—
116	C₈×C₇₁	—	4	4	no	—
117	C₂²×C₇₁	—	2	2	no	—
118	C₂²×C₇₁	—	2	2	no	—
119	C₂²×C₇₁	—	2	2	no	—
120	C₂²×C₇₁	—	4	4	no	—
121	C₂²×C₇₁	C₂×#95	1	1	yes	characteristic
122	C₂²×C₇₁	#423	1	2	yes	—
123	C₂²×C₇₁	#428	1	2	yes	—
124	C₂²×C₇₁	#430	1	2	yes	—
125	C₄×C₇₁	—	2	2	no	—
126	C₄×C₇₁	—	2	2	no	—
127	C₄×C₇₁	—	2	2	no	—
128	C₄×C₇₁	—	2	1	no	—
129	C₄×C₇₁	—	2	1	no	—
130	C₄×C₇₁	C₄×M₁₆	1	2	yes	—
131	C₄×C₇₁	#407	1	2	yes	—
132	C₄×C₇₁	#417	1	2	yes	—
133	C₄×C₇₁	#418	1	1	yes	characteristic
134	C₄×C₇₁	#419	1	1	yes	characteristic
135	#7876	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₇₁	—	4	2	no	—
137	C₂×C₇₁	#1569	1	4	yes	—
138	C₂×C₇₁	#1728	1	1	yes	characteristic
139	C₂×C₇₁	#1736	1	1	yes	characteristic
140	C₂×C₇₁	#1738	1	1	yes	characteristic
141	C₂×#432	C₂×C₇₁	1	1	yes	characteristic
142	C₂×#423	C₂×C₇₁	1	1	yes	characteristic
143	C₂×C₄×M₁₆	C₂×C₇₁	1	1	yes	characteristic
144	#1277	C₂×C₇₁	1	4	yes	complemented
145	C₇₁	#7876	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×M₁₆	—	2	1	no	—
147	C₂²×M₁₆	C₄×C₇₁	1	2	yes	—
148	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
149	#432	—	2	2	no	—
150	#432	C₄×C₇₁	1	4	yes	—
151	#423	—	2	2	no	—
152	#423	C₄×C₇₁	1	4	yes	—
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₄×M₁₆	—	2	2	no	—
158	C₄×M₁₆	—	2	2	no	—
159	C₂×#97	—	2	1	no	—
160	C₂×#97	C₄×C₇₁	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
162	C₂³×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
163	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
164	C₂³×C₄	Q₈×C₇₁	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	8	no	—
168	C₂×M₁₆	—	4	4	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	C₂×C₄×C₇₁	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
174	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
175	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
176	C₂×C₄²	D₄×C₇₁	1	2	yes	—
177	C₂×C₄²	D₄×C₇₁	1	2	yes	—
178	C₂×C₄²	Q₈×C₇₁	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂×C₄×C₇₁	1	4	yes	—
184	C₄⋊C₈	C₂×C₄×C₇₁	1	4	yes	—
185	C₂²⋊C₈	—	2	4	no	—
186	#97	—	2	4	no	—
187	#97	—	2	4	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₂⁴	C₄⋊C₄×C₇₁	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄²×C₇₁	1	2	yes	—
196	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
199	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
201	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊C₄×C₇₁	1	2	yes	—
203	C₂²×C₄	C₄⋊C₄×C₇₁	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
207	M₁₆	—	2	16	no	—
208	M₁₆	—	4	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	M₁₆×C₇₁	1	8	yes	complemented
217	C₄²	C₄²×C₇₁	1	2	yes	—
218	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
219	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
220	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
221	C₄²	C₄⋊C₄×C₇₁	1	2	yes	—
222	C₄²	C₄⋊C₄×C₇₁	1	2	yes	—
223	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
226	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	#95×C₇₁	1	2	yes	—
231	C₂³	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	#95×C₇₁	1	2	yes	—
244	C₂×C₄	#95×C₇₁	1	2	yes	—
245	C₂×C₄	#95×C₇₁	1	2	yes	—
246	C₂×C₄	C₂×C₄²×C₇₁	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
252	C₂×C₄	C₂×M₁₆×C₇₁	1	4	yes	—
253	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
254	C₂×C₄	#137×C₇₁	1	2	yes	—
255	C₈	—	2	8	no	—
256	C₈	—	4	4	no	—
257	C₈	—	4	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×#95×C₇₁	1	1	yes	characteristic
263	C₂²	#423×C₇₁	1	2	yes	—
264	C₂²	#428×C₇₁	1	2	yes	—
265	C₂²	#430×C₇₁	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	C₄×M₁₆×C₇₁	1	2	yes	—
272	C₄	#407×C₇₁	1	2	yes	—
273	C₄	#417×C₇₁	1	2	yes	—
274	C₄	#418×C₇₁	1	1	yes	characteristic
275	C₄	#419×C₇₁	1	1	yes	characteristic
276	C₂	—	4	2	no	—
277	C₂	C₇₁×#1569	1	4	yes	—
278	C₂	C₇₁×#1728	1	1	yes	characteristic
279	C₂	C₇₁×#1736	1	1	yes	characteristic
280	C₂	C₇₁×#1738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂³×C₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₁	C₄⋊C₄
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
4. C₇₁×#1277	C₂	semidirect
75. C₂×C₈×C₇₁	M₁₆	semidirect
135. #7876	C₇₁	direct (N × H)
144. #1277	C₂×C₇₁	semidirect
145. C₇₁	#7876	direct (N × H)
216. C₂×C₈	M₁₆×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₇₁	2
C₂×#423×C₇₁	2
C₂×C₄×M₁₆×C₇₁	2
C₇₁×#1277	2
#7876	71

Subgroups — C71×#7876 #31828748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7876 #31828748