Subgroups — C₇₁×#13252 #31834124

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#357×C₇₁	C₂	1	4	yes	maximal, complemented
4	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₇₁	—	2	1	no	—
6	C₂²×D₈×C₇₁	C₂²	1	2	yes	—
7	C₂×#127×C₇₁	—	2	1	no	—
8	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
9	#493×C₇₁	—	2	2	no	—
10	#493×C₇₁	C₄	1	4	yes	complemented
11	C₂×#107×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
14	C₄⋊C₁₆×C₇₁	—	2	2	no	—
15	#357×C₇₁	—	2	8	no	—
16	#340×C₇₁	—	2	1	no	—
17	C₂²×D₄×C₇₁	—	2	2	no	—
18	C₂²×D₄×C₇₁	—	4	1	no	—
19	C₂²×D₄×C₇₁	—	4	1	no	—
20	C₂×D₈×C₇₁	—	2	4	no	—
21	C₂×D₈×C₇₁	—	2	4	no	—
22	C₂×D₈×C₇₁	—	2	2	no	—
23	C₂×D₈×C₇₁	—	2	4	no	—
24	C₂×D₈×C₇₁	—	4	4	no	—
25	C₂×D₈×C₇₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	#127×C₇₁	—	2	4	no	—
29	#127×C₇₁	—	4	2	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₂×C₄²×C₇₁	—	4	1	no	—
32	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₁	—	2	4	no	—
34	C₂×C₁₆×C₇₁	—	2	2	no	—
35	#107×C₇₁	—	2	2	no	—
36	#107×C₇₁	D₄	1	4	yes	complemented
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₁	D₄	1	2	yes	—
39	#95×C₇₁	—	4	1	no	—
40	C₂⁴×C₇₁	—	4	2	no	—
41	C₂×D₄×C₇₁	—	2	4	no	—
42	C₂×D₄×C₇₁	—	4	4	no	—
43	C₂×D₄×C₇₁	—	4	4	no	—
44	C₂×D₄×C₇₁	—	4	4	no	—
45	C₂×D₄×C₇₁	—	4	2	no	—
46	C₂×D₄×C₇₁	—	4	4	no	—
47	C₂×D₄×C₇₁	—	8	4	no	—
48	C₂×D₄×C₇₁	—	8	2	no	—
49	C₂×D₄×C₇₁	—	8	2	no	—
50	C₂²×C₄×C₇₁	—	4	2	no	—
51	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	D₈×C₇₁	—	2	8	no	—
55	D₈×C₇₁	—	4	8	no	—
56	D₈×C₇₁	—	4	2	no	—
57	D₈×C₇₁	—	4	2	no	—
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	2	yes	—
64	C₄⋊C₄×C₇₁	—	2	4	no	—
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₄²×C₇₁	—	8	2	no	—
67	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₇₁	SD₁₆	1	2	yes	—
69	C₁₆×C₇₁	—	4	2	no	—
70	C₂³×C₇₁	—	4	4	no	—
71	C₂³×C₇₁	—	8	4	no	—
72	C₂³×C₇₁	—	8	2	no	—
73	C₂³×C₇₁	—	8	2	no	—
74	C₂³×C₇₁	—	8	4	no	—
75	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
76	D₄×C₇₁	—	4	4	no	—
77	D₄×C₇₁	—	4	4	no	—
78	D₄×C₇₁	—	8	8	no	—
79	D₄×C₇₁	—	8	8	no	—
80	D₄×C₇₁	—	8	4	no	—
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	—	4	4	no	—
83	C₂×C₄×C₇₁	—	8	2	no	—
84	C₂×C₄×C₇₁	—	8	2	no	—
85	C₂×C₄×C₇₁	#102	1	2	yes	—
86	C₂×C₄×C₇₁	#102	1	2	yes	—
87	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
88	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#121	1	2	yes	—
91	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
93	C₈×C₇₁	—	2	2	no	—
94	C₈×C₇₁	C₄×D₄	1	2	yes	—
95	C₈×C₇₁	#124	1	1	yes	characteristic
96	C₈×C₇₁	#127	1	1	yes	characteristic
97	C₂²×C₇₁	—	8	4	no	—
98	C₂²×C₇₁	—	8	4	no	—
99	C₂²×C₇₁	—	8	4	no	—
100	C₂²×C₇₁	—	16	4	no	—
101	C₂²×C₇₁	—	16	4	no	—
102	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
103	C₂²×C₇₁	#459	1	2	yes	—
104	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
105	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
106	C₂²×C₇₁	#509	1	2	yes	—
107	C₄×C₇₁	—	2	2	no	—
108	C₄×C₇₁	—	8	4	no	—
109	C₄×C₇₁	#357	1	4	yes	complemented
110	C₄×C₇₁	#390	1	1	yes	characteristic
111	C₄×C₇₁	#418	1	1	yes	characteristic
112	C₄×C₇₁	#460	1	2	yes	—
113	#13252	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₇₁	—	16	4	no	—
115	C₂×C₇₁	#1982	1	1	yes	characteristic
116	C₂×C₇₁	C₂×#357	1	1	yes	characteristic
117	C₂×C₇₁	#2140	1	1	yes	characteristic
118	C₂×C₇₁	#2206	1	2	yes	—
119	C₂×C₇₁	#2207	1	2	yes	—
120	C₂×#493	C₂×C₇₁	1	1	yes	characteristic
121	C₂×C₄⋊C₁₆	C₂×C₇₁	1	1	yes	characteristic, complemented
122	C₂×#357	C₂×C₇₁	1	4	yes	complemented
123	C₄×#107	C₂×C₇₁	1	1	yes	characteristic, complemented
124	C₇₁	#13252	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×D₈	—	2	1	no	—
126	C₂²×D₈	C₂²×C₇₁	1	2	yes	—
127	C₂×#127	—	2	1	no	—
128	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
129	#493	—	2	2	no	—
130	#493	C₄×C₇₁	1	4	yes	complemented
131	C₂×#107	C₂²×C₇₁	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₁₆	—	2	2	no	—
135	#357	—	2	8	no	—
136	#340	—	2	1	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	4	1	no	—
139	C₂²×D₄	—	4	1	no	—
140	C₂×D₈	—	2	4	no	—
141	C₂×D₈	—	2	4	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	4	no	—
144	C₂×D₈	—	4	4	no	—
145	C₂×D₈	C₂×C₄×C₇₁	1	4	yes	—
146	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
148	#127	—	2	4	no	—
149	#127	—	4	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄²	—	4	1	no	—
152	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
153	C₂×C₁₆	—	2	4	no	—
154	C₂×C₁₆	—	2	2	no	—
155	#107	—	2	2	no	—
156	#107	D₄×C₇₁	1	4	yes	complemented
157	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
158	C₄×C₈	D₄×C₇₁	1	2	yes	—
159	#95	—	4	1	no	—
160	C₂⁴	—	4	2	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	4	no	—
167	C₂×D₄	—	8	4	no	—
168	C₂×D₄	—	8	2	no	—
169	C₂×D₄	—	8	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
172	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
174	D₈	—	2	8	no	—
175	D₈	—	4	8	no	—
176	D₈	—	4	2	no	—
177	D₈	—	4	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
180	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
182	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
183	C₂×C₈	#41×C₇₁	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄²	—	8	2	no	—
187	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
188	C₄²	SD₁₆×C₇₁	1	2	yes	—
189	C₁₆	—	4	2	no	—
190	C₂³	—	4	4	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	2	no	—
193	C₂³	—	8	2	no	—
194	C₂³	—	8	4	no	—
195	C₂³	C₂×D₈×C₇₁	1	1	yes	characteristic
196	D₄	—	4	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	8	8	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	#102×C₇₁	1	2	yes	—
206	C₂×C₄	#102×C₇₁	1	2	yes	—
207	C₂×C₄	D₁₆×C₇₁	1	2	yes	—
208	C₂×C₄	SD₃₂×C₇₁	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
210	C₂×C₄	#121×C₇₁	1	2	yes	—
211	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
212	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄×D₄×C₇₁	1	2	yes	—
215	C₈	#124×C₇₁	1	1	yes	characteristic
216	C₈	#127×C₇₁	1	1	yes	characteristic
217	C₂²	—	8	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	8	4	no	—
220	C₂²	—	16	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	C₂×#102×C₇₁	1	1	yes	characteristic
223	C₂²	#459×C₇₁	1	2	yes	—
224	C₂²	C₂×D₁₆×C₇₁	1	1	yes	characteristic
225	C₂²	C₂×SD₃₂×C₇₁	1	1	yes	characteristic
226	C₂²	#509×C₇₁	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	8	4	no	—
229	C₄	#357×C₇₁	1	4	yes	complemented
230	C₄	#390×C₇₁	1	1	yes	characteristic
231	C₄	#418×C₇₁	1	1	yes	characteristic
232	C₄	#460×C₇₁	1	2	yes	—
233	C₂	—	16	4	no	—
234	C₂	C₇₁×#1982	1	1	yes	characteristic
235	C₂	C₂×#357×C₇₁	1	1	yes	characteristic
236	C₂	C₇₁×#2140	1	1	yes	characteristic
237	C₂	C₇₁×#2206	1	2	yes	—
238	C₂	C₇₁×#2207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₂²×C₈	C₂³×C₇₁
prime radical	C₂×#493×C₇₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
2. C₂×C₄⋊C₁₆×C₇₁	C₂	semidirect
3. C₂×#357×C₇₁	C₂	semidirect
4. C₄×#107×C₇₁	C₂	semidirect
10. #493×C₇₁	C₄	semidirect
36. #107×C₇₁	D₄	semidirect
109. C₄×C₇₁	#357	semidirect
113. #13252	C₇₁	direct (N × H)
121. C₂×C₄⋊C₁₆	C₂×C₇₁	semidirect
122. C₂×#357	C₂×C₇₁	semidirect
123. C₄×#107	C₂×C₇₁	semidirect
124. C₇₁	#13252	direct (N × H)
130. #493	C₄×C₇₁	semidirect
156. #107	D₄×C₇₁	semidirect
229. C₄	#357×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₇₁	2
C₂×C₄⋊C₁₆×C₇₁	2
C₂×#357×C₇₁	2
C₄×#107×C₇₁	2
#13252	71

Subgroups — C71×#13252 #31834124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13252 #31834124