Finite Group Databasebeta

← C71×#8770 #31829642

Subgroups — C71×#8770 #31829642

Order 18176.

426 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2357C211yesmaximal, characteristic
2C71×#1800C211yesmaximal, characteristic
3C71×#1460C211yesmaximal, characteristic
4C71×#1303C212yesmaximal, complemented
5C71×#1286C212yesmaximal, complemented
6C2×C4×D4×C71C2211yescharacteristic
7#431×C71C2212yes—
8#423×C71C2212yes—
9C2×#100×C71—21no—
10#390×C71—21no—
11#388×C71—22no—
12#382×C71C2211yescharacteristic
13#381×C71C412yes—
14C4×C4⋊C4×C71—21no—
15C4×C4⋊C4×C71C2211yescharacteristic
16C4×C22⋊C4×C71C412yes—
17#343×C71—21no—
18#323×C71—22no—
19C22×D4×C71D411yescharacteristic
20C23×C4×C71C2×C412yes—
21C2×M16×C71—22no—
22C4×D4×C71—22no—
23C4×D4×C71—22no—
24C2×C4⋊C4×C71—21no—
25C2×C4⋊C4×C71—21no—
26C2×C4⋊C4×C71D411yescharacteristic
27C2×C22⋊C4×C71—22no—
28C2×C22⋊C4×C71—22no—
29C2×C22⋊C4×C71D412yes—
30C2×C42×C71—21no—
31C2×C42×C71—22no—
32C2×C42×C71C2311yescharacteristic
33C4⋊C8×C71—22no—
34C4⋊C8×C71—22no—
35C4⋊C8×C71—22no—
36#100×C71—42no—
37C22⋊C8×C71—22no—
38C4×C8×C71—22no—
39#95×C71—22no—
40#95×C71—21no—
41#95×C71C2×C411yescharacteristic
42#95×C71C2×C412yes—
43#95×C71C2×C411yescharacteristic
44C24×C71C22⋊C412yes—
45C2×D4×C71—22no—
46C2×D4×C71—22no—
47C2×D4×C71—21no—
48C2×D4×C71—21no—
49C22×C4×C71—21no—
50C22×C4×C71—22no—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—21no—
54C22×C4×C71—82no—
55C22×C4×C71C22⋊C412yes—
56C22×C4×C71C22×C411yescharacteristic
57C22×C4×C71C2×D411yescharacteristic
58C22×C4×C71C2×D411yescharacteristic
59M16×C71—42no—
60M16×C71—42no—
61C2×C8×C71—42no—
62C2×C8×C71—42no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—42no—
65C4⋊C4×C71—42no—
66C22⋊C4×C71—24no—
67C22⋊C4×C71—44no—
68C22⋊C4×C71—82no—
69C42×C71—41no—
70C42×C71—41no—
71C42×C71—82no—
72C42×C71#4112yes—
73C42×C71#4112yes—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—22no—
77C23×C71—42no—
78C23×C71—42no—
79C23×C71C2×C22⋊C411yescharacteristic
80D4×C71—42no—
81D4×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—42no—
94C2×C4×C71—41no—
95C2×C4×C71—82no—
96C2×C4×C71—82no—
97C2×C4×C71#11711yescharacteristic
98C2×C4×C71#11711yescharacteristic
99C2×C4×C71#12312yes—
100C2×C4×C71#12312yes—
101C2×C4×C71#13512yes—
102C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C71#13711yescharacteristic
104C8×C71—82no—
105C8×C71—82no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71—42no—
111C22×C71—82no—
112C22×C71—82no—
113C22×C71#38111yescharacteristic
114C22×C71#41611yescharacteristic
115C22×C71#41711yescharacteristic
116C4×C71—22no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—42no—
120C4×C71—42no—
121C4×C71—81no—
122C4×C71—82no—
123C4×C71#48212yes—
124C4×C71#48512yes—
125#8770C7111yesmaximal, characteristic, direct product factor, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71—82no—
129C2×C71#182511yescharacteristic
130C2×C71#182711yescharacteristic
131C2×C71#182911yescharacteristic
132#2357C2×C7111yescharacteristic
133#1800C2×C7111yescharacteristic
134#1460C2×C7111yescharacteristic
135#1303C2×C7112yescomplemented
136#1286C2×C7112yescomplemented
137C71#877011yescharacteristic, direct product factor, Sylow
138C2×C4×D4C22×C7111yescharacteristic
139#431C22×C7112yes—
140#423C22×C7112yes—
141C2×#100—21no—
142#390—21no—
143#388—22no—
144#382C22×C7111yescharacteristic
145#381C4×C7112yes—
146C4×C4⋊C4—21no—
147C4×C4⋊C4C22×C7111yescharacteristic
148C4×C22⋊C4C4×C7112yes—
149#343—21no—
150#323—22no—
151C22×D4D4×C7111yescharacteristic
152C23×C4C2×C4×C7112yes—
153C2×M16—22no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4D4×C7111yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4D4×C7112yes—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C23×C7111yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168#100—42no—
169C22⋊C8—22no—
170C4×C8—22no—
171#95—22no—
172#95—21no—
173#95C2×C4×C7111yescharacteristic
174#95C2×C4×C7112yes—
175#95C2×C4×C7111yescharacteristic
176C24C22⋊C4×C7112yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—82no—
187C22×C4C22⋊C4×C7112yes—
188C22×C4C22×C4×C7111yescharacteristic
189C22×C4C2×D4×C7111yescharacteristic
190C22×C4C2×D4×C7111yescharacteristic
191M16—42no—
192M16—42no—
193C2×C8—42no—
194C2×C8—42no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—24no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C42—41no—
202C42—41no—
203C42—82no—
204C42#41×C7112yes—
205C42#41×C7112yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×C22⋊C4×C7111yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4#117×C7111yescharacteristic
230C2×C4#117×C7111yescharacteristic
231C2×C4#123×C7112yes—
232C2×C4#123×C7112yes—
233C2×C4#135×C7112yes—
234C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
235C2×C4#137×C7111yescharacteristic
236C8—82no—
237C8—82no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#381×C7111yescharacteristic
246C22#416×C7111yescharacteristic
247C22#417×C7111yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—82no—
255C4#482×C7112yes—
256C4#485×C7112yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2C71×#182511yescharacteristic
261C2C71×#182711yescharacteristic
262C2C71×#182911yescharacteristic
subgroupG/H
center, socleC22×C71#381
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC22×D4×C71D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1303C2semidirect
5. C71×#1286C2semidirect
125. #8770C71direct (N × H)
135. #1303C2×C71semidirect
136. #1286C2×C71semidirect
137. C71#8770direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#23572
C71×#18002
C71×#14602
C71×#13032
C71×#12862
#877071