Subgroups — C₇₁×#8587 #31829459

Order 18176.

516 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1842	C₂	1	2	yes	maximal
3	C₂×#333×C₇₁	C₂	1	2	yes	maximal
4	C₇₁×#1296	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₇₁	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₇₁	—	2	2	no	—
8	C₂×C₄⋊C₈×C₇₁	—	2	2	no	—
9	C₂×C₄⋊C₈×C₇₁	C₂²	1	4	yes	—
10	#391×C₇₁	C₄	1	2	yes	—
11	#389×C₇₁	—	2	2	no	—
12	#384×C₇₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₁	—	2	2	no	—
14	C₄³×C₇₁	C₂²	1	1	yes	characteristic
15	#333×C₇₁	—	2	4	no	—
16	C₂²×Q₈×C₇₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₇₁	—	2	4	no	—
18	C₄⋊Q₈×C₇₁	—	2	2	no	—
19	C₄⋊Q₈×C₇₁	D₄	1	2	yes	—
20	C₄⋊Q₈×C₇₁	Q₈	1	2	yes	—
21	C₄×Q₈×C₇₁	—	4	4	no	—
22	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₁	—	2	2	no	—
27	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	D₄	1	2	yes	—
30	C₂×C₄²×C₇₁	D₄	1	2	yes	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₁	—	2	4	no	—
33	C₄⋊C₈×C₇₁	—	2	4	no	—
34	C₄⋊C₈×C₇₁	—	4	4	no	—
35	C₄⋊C₈×C₇₁	—	4	4	no	—
36	#95×C₇₁	—	2	4	no	—
37	C₂×Q₈×C₇₁	—	2	2	no	—
38	C₂×Q₈×C₇₁	—	4	2	no	—
39	C₂×Q₈×C₇₁	—	4	2	no	—
40	C₂×Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₇₁	—	2	2	no	—
42	C₂²×C₄×C₇₁	—	2	2	no	—
43	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₇₁	—	4	4	no	—
49	C₂×C₈×C₇₁	—	4	4	no	—
50	C₂×C₈×C₇₁	—	4	4	no	—
51	C₄⋊C₄×C₇₁	—	2	4	no	—
52	C₄⋊C₄×C₇₁	—	2	4	no	—
53	C₄⋊C₄×C₇₁	—	4	2	no	—
54	C₄⋊C₄×C₇₁	—	4	4	no	—
55	C₄⋊C₄×C₇₁	—	4	2	no	—
56	C₄⋊C₄×C₇₁	—	4	2	no	—
57	C₄²×C₇₁	—	2	2	no	—
58	C₄²×C₇₁	—	2	2	no	—
59	C₄²×C₇₁	—	2	2	no	—
60	C₄²×C₇₁	—	4	1	no	—
61	C₄²×C₇₁	—	4	4	no	—
62	C₄²×C₇₁	SD₁₆	1	4	yes	—
63	C₄²×C₇₁	Q₁₆	1	4	yes	—
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₇₁	#41	1	2	yes	—
67	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	Q₈×C₇₁	—	4	4	no	—
69	Q₈×C₇₁	—	4	4	no	—
70	C₂×C₄×C₇₁	—	2	1	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	2	1	no	—
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	4	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	4	2	no	—
78	C₂×C₄×C₇₁	—	4	2	no	—
79	C₂×C₄×C₇₁	—	4	2	no	—
80	C₂×C₄×C₇₁	—	4	4	no	—
81	C₂×C₄×C₇₁	#100	1	2	yes	—
82	C₂×C₄×C₇₁	#103	1	4	yes	—
83	C₂×C₄×C₇₁	#103	1	4	yes	—
84	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#121	1	2	yes	—
87	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₇₁	#137	1	2	yes	—
92	C₈×C₇₁	—	8	4	no	—
93	C₂²×C₇₁	#380	1	1	yes	characteristic
94	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
95	C₂²×C₇₁	#413	1	1	yes	characteristic
96	C₂²×C₇₁	C₂×#103	1	2	yes	—
97	C₂²×C₇₁	#417	1	2	yes	—
98	C₄×C₇₁	—	2	1	no	—
99	C₄×C₇₁	—	2	1	no	—
100	C₄×C₇₁	—	2	4	no	—
101	C₄×C₇₁	—	4	4	no	—
102	C₄×C₇₁	—	4	1	no	—
103	C₄×C₇₁	—	8	2	no	—
104	C₄×C₇₁	#333	1	4	yes	complemented
105	C₄×C₇₁	#460	1	2	yes	—
106	C₄×C₇₁	#462	1	2	yes	—
107	C₄×C₇₁	#475	1	2	yes	—
108	C₄×C₇₁	#477	1	2	yes	—
109	#8587	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₇₁	C₂×#333	1	1	yes	characteristic
111	C₂×C₇₁	#1550	1	2	yes	—
112	C₂×C₇₁	#1560	1	1	yes	characteristic
113	C₂×C₇₁	#1800	1	1	yes	characteristic
114	C₂×C₇₁	#1803	1	2	yes	—
115	C₄×C₄⋊Q₈	C₂×C₇₁	1	1	yes	characteristic
116	#1842	C₂×C₇₁	1	2	yes	—
117	C₂×#333	C₂×C₇₁	1	2	yes	—
118	#1296	C₂×C₇₁	1	2	yes	—
119	C₇₁	#8587	1	1	yes	characteristic, direct product factor, Sylow
120	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₄×C₇₁	1	2	yes	—
122	C₂×C₄⋊C₈	—	2	2	no	—
123	C₂×C₄⋊C₈	—	2	2	no	—
124	C₂×C₄⋊C₈	C₂²×C₇₁	1	4	yes	—
125	#391	C₄×C₇₁	1	2	yes	—
126	#389	—	2	2	no	—
127	#384	C₂²×C₇₁	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	2	2	no	—
129	C₄³	C₂²×C₇₁	1	1	yes	characteristic
130	#333	—	2	4	no	—
131	C₂²×Q₈	C₂×C₄×C₇₁	1	2	yes	—
132	C₂²×C₈	—	2	4	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	D₄×C₇₁	1	2	yes	—
135	C₄⋊Q₈	Q₈×C₇₁	1	2	yes	—
136	C₄×Q₈	—	4	4	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
143	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
144	C₂×C₄²	D₄×C₇₁	1	2	yes	—
145	C₂×C₄²	D₄×C₇₁	1	2	yes	—
146	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	2	4	no	—
149	C₄⋊C₈	—	4	4	no	—
150	C₄⋊C₈	—	4	4	no	—
151	#95	—	2	4	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	C₄⋊C₄×C₇₁	1	4	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
159	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
160	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
163	C₂×C₈	—	4	4	no	—
164	C₂×C₈	—	4	4	no	—
165	C₂×C₈	—	4	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	SD₁₆×C₇₁	1	4	yes	—
178	C₄²	Q₁₆×C₇₁	1	4	yes	—
179	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
180	C₄²	C₂×Q₈×C₇₁	1	1	yes	characteristic
181	C₄²	#41×C₇₁	1	2	yes	—
182	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	#100×C₇₁	1	2	yes	—
197	C₂×C₄	#103×C₇₁	1	4	yes	—
198	C₂×C₄	#103×C₇₁	1	4	yes	—
199	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
200	C₂×C₄	#117×C₇₁	1	1	yes	characteristic
201	C₂×C₄	#121×C₇₁	1	2	yes	—
202	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
203	C₂×C₄	C₂×SD₁₆×C₇₁	1	2	yes	—
204	C₂×C₄	C₂×Q₁₆×C₇₁	1	2	yes	—
205	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
206	C₂×C₄	#137×C₇₁	1	2	yes	—
207	C₈	—	8	4	no	—
208	C₂²	#380×C₇₁	1	1	yes	characteristic
209	C₂²	C₂×#100×C₇₁	1	1	yes	characteristic
210	C₂²	#413×C₇₁	1	1	yes	characteristic
211	C₂²	C₂×#103×C₇₁	1	2	yes	—
212	C₂²	#417×C₇₁	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	4	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	1	no	—
218	C₄	—	8	2	no	—
219	C₄	#333×C₇₁	1	4	yes	complemented
220	C₄	#460×C₇₁	1	2	yes	—
221	C₄	#462×C₇₁	1	2	yes	—
222	C₄	#475×C₇₁	1	2	yes	—
223	C₄	#477×C₇₁	1	2	yes	—
224	C₂	C₂×#333×C₇₁	1	1	yes	characteristic
225	C₂	C₇₁×#1550	1	2	yes	—
226	C₂	C₇₁×#1560	1	1	yes	characteristic
227	C₂	C₇₁×#1800	1	1	yes	characteristic
228	C₂	C₇₁×#1803	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₄³×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
104. C₄×C₇₁	#333	semidirect
109. #8587	C₇₁	direct (N × H)
119. C₇₁	#8587	direct (N × H)
219. C₄	#333×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₇₁	2
C₇₁×#1842	2
C₂×#333×C₇₁	2
C₇₁×#1296	2
#8587	71

Subgroups — C71×#8587 #31829459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#8587 #31829459