Finite Group Databasebeta

← C71×#8735 #31829607

Subgroups — C71×#8735 #31829607

Order 18176.

514 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#2274C211yesmaximal, characteristic
2C2×#324×C71C212yesmaximal
3C71×#1282C214yesmaximal
4C2×#117×C71C411yescharacteristic
5C2×#117×C71C411yescharacteristic
6C22×C4⋊C4×C71C2211yescharacteristic
7C2×C22⋊C8×C71C2214yes—
8C4×C4⋊C4×C71—22no—
9C4×C4⋊C4×C71—22no—
10C4×C22⋊C4×C71C412yes—
11C4×C22⋊C4×C71C2212yes—
12#324×C71—22no—
13#324×C71—22no—
14C23×C4×C71C2×C411yescharacteristic
15C23×C4×C71C2×C411yescharacteristic
16C23×C4×C71C2311yescharacteristic
17C22×C8×C71—24no—
18#117×C71—22no—
19#117×C71—24no—
20#117×C71C814yescomplemented
21C2×C4⋊C4×C71—21no—
22C2×C4⋊C4×C71—21no—
23C2×C4⋊C4×C71—24no—
24C2×C4⋊C4×C71D412yes—
25C2×C4⋊C4×C71D412yes—
26C2×C22⋊C4×C71C2×C412yes—
27C2×C22⋊C4×C71C2×C412yes—
28C2×C42×C71—21no—
29C2×C42×C71—21no—
30C2×C42×C71—22no—
31C2×C42×C71—22no—
32C22⋊C8×C71—24no—
33C22⋊C8×C71—24no—
34#95×C71—22no—
35#95×C71—22no—
36C24×C71C22×C411yescharacteristic
37C22×C4×C71—22no—
38C22×C4×C71—22no—
39C22×C4×C71—22no—
40C22×C4×C71—22no—
41C22×C4×C71—22no—
42C22×C4×C71—81no—
43C22×C4×C71—81no—
44C22×C4×C71C22⋊C411yescharacteristic
45C22×C4×C71C22⋊C411yescharacteristic
46C22×C4×C71C22⋊C412yes—
47C22×C4×C71C2×C812yes—
48C22×C4×C71M1612yes—
49C22×C4×C71C2×D412yes—
50C22×C4×C71#4114yes—
51C2×C8×C71—44no—
52C2×C8×C71—84no—
53C4⋊C4×C71—24no—
54C4⋊C4×C71—24no—
55C4⋊C4×C71—24no—
56C4⋊C4×C71—24no—
57C22⋊C4×C71—22no—
58C22⋊C4×C71—22no—
59C22⋊C4×C71C2×C814yes—
60C22⋊C4×C71M1614yes—
61C42×C71—24no—
62C42×C71—24no—
63C42×C71—82no—
64C42×C71—82no—
65C23×C71—41no—
66C23×C71—41no—
67C23×C71C2×C22⋊C411yescharacteristic
68C23×C71#11712yes—
69C23×C71C22×C811yescharacteristic
70C23×C71C2×M1611yescharacteristic
71C23×C71C2×M1612yes—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—24no—
77C2×C4×C71—24no—
78C2×C4×C71—22no—
79C2×C4×C71—24no—
80C2×C4×C71—24no—
81C2×C4×C71—42no—
82C2×C4×C71—44no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71—82no—
89C2×C4×C71C22⋊C812yes—
90C2×C4×C71C22⋊C812yes—
91C2×C4×C71#12314yes—
92C8×C71—84no—
93C22×C71—21no—
94C22×C71—22no—
95C22×C71—21no—
96C22×C71—22no—
97C22×C71—41no—
98C22×C71—41no—
99C22×C71—42no—
100C22×C71#38111yescharacteristic
101C22×C71C2×C22⋊C811yescharacteristic
102C22×C71#40711yescharacteristic
103C22×C71#43112yes—
104C22×C71#43212yes—
105C4×C71—22no—
106C4×C71—22no—
107C4×C71—24no—
108C4×C71—44no—
109C4×C71—44no—
110C4×C71—42no—
111#8735C7111yesmaximal, characteristic, direct product factor, Sylow
112C2×C71—21no—
113C2×C71—21no—
114C2×C71—22no—
115C2×C71—22no—
116C2×C71#181711yescharacteristic
117C2×C71#182811yescharacteristic
118C2×C71#183011yescharacteristic
119#2274C2×C7111yescharacteristic
120C2×#324C2×C7112yes—
121#1282C2×C7114yes—
122C71#873511yescharacteristic, direct product factor, Sylow
123C2×#117C4×C7111yescharacteristic
124C2×#117C4×C7111yescharacteristic
125C22×C4⋊C4C22×C7111yescharacteristic
126C2×C22⋊C8C22×C7114yes—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C22⋊C4C4×C7112yes—
130C4×C22⋊C4C22×C7112yes—
131#324—22no—
132#324—22no—
133C23×C4C2×C4×C7111yescharacteristic
134C23×C4C2×C4×C7111yescharacteristic
135C23×C4C23×C7111yescharacteristic
136C22×C8—24no—
137#117—22no—
138#117—24no—
139#117C8×C7114yescomplemented
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4D4×C7112yes—
144C2×C4⋊C4D4×C7112yes—
145C2×C22⋊C4C2×C4×C7112yes—
146C2×C22⋊C4C2×C4×C7112yes—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C22⋊C8—24no—
152C22⋊C8—24no—
153#95—22no—
154#95—22no—
155C24C22×C4×C7111yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4C22⋊C4×C7111yescharacteristic
164C22×C4C22⋊C4×C7111yescharacteristic
165C22×C4C22⋊C4×C7112yes—
166C22×C4C2×C8×C7112yes—
167C22×C4M16×C7112yes—
168C22×C4C2×D4×C7112yes—
169C22×C4#41×C7114yes—
170C2×C8—44no—
171C2×C8—84no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4C2×C8×C7114yes—
179C22⋊C4M16×C7114yes—
180C42—24no—
181C42—24no—
182C42—82no—
183C42—82no—
184C23—41no—
185C23—41no—
186C23C2×C22⋊C4×C7111yescharacteristic
187C23#117×C7112yes—
188C23C22×C8×C7111yescharacteristic
189C23C2×M16×C7111yescharacteristic
190C23C2×M16×C7112yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4C22⋊C8×C7112yes—
209C2×C4C22⋊C8×C7112yes—
210C2×C4#123×C7114yes—
211C8—84no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—22no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22#381×C7111yescharacteristic
220C22C2×C22⋊C8×C7111yescharacteristic
221C22#407×C7111yescharacteristic
222C22#431×C7112yes—
223C22#432×C7112yes—
224C4—22no—
225C4—22no—
226C4—24no—
227C4—44no—
228C4—44no—
229C4—42no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2C71×#181711yescharacteristic
235C2C71×#182811yescharacteristic
236C2C71×#183011yescharacteristic
subgroupG/H
center, socleC22×C71#381
commutatorC23C22×C8×C71
Frattini, cosocleC23×C4C23×C71
prime radicalC24×C71C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
20. #117×C71C8semidirect
111. #8735C71direct (N × H)
122. C71#8735direct (N × H)
139. #117C8×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#22742
C2×#324×C712
C71×#12822
#873571