Finite Group Databasebeta

← C71×#8494 #31829366

Subgroups — C71×#8494 #31829366

Order 18176.

732 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3548C211yesmaximal, characteristic
2C71×#1995C211yesmaximal, characteristic, complemented
3C71×#1557C212yesmaximal, complemented
4C71×#1537C211yesmaximal, characteristic, complemented
5C71×#1505C212yesmaximal, complemented
6#562×C71—22no—
7#554×C71—22no—
8#550×C71—22no—
9#547×C71C414yescomplemented
10#540×C71—22no—
11#537×C71—24no—
12#533×C71C2211yescharacteristic
13#532×C71C2212yes—
14#432×C71C2212yes—
15#423×C71C2211yescharacteristic
16#376×C71C2211yescharacteristic
17#339×C71—22no—
18#332×C71—22no—
19#331×C71—22no—
20#330×C71—22no—
21#327×C71—24no—
22C2×#41×C71—24no—
23C2×#41×C71—22no—
24C2×M16×C71—21no—
25C4⋊Q8×C71C2×C412yes—
26C4⋊Q8×C71D412yes—
27#127×C71—22no—
28#127×C71D412yes—
29#125×C71—22no—
30#125×C71D412yes—
31#124×C71—22no—
32#123×C71—28no—
33C22⋊Q8×C71—28no—
34C22⋊Q8×C71—22no—
35C22⋊Q8×C71—24no—
36#121×C71—28no—
37#121×C71—28no—
38#121×C71—24no—
39#121×C71—22no—
40#121×C71D414yes—
41C4×Q8×C71—22no—
42C4×D4×C71—24no—
43C4×D4×C71—22no—
44C4×D4×C71C2×C414yes—
45#117×C71D412yes—
46C2×C4⋊C4×C71—24no—
47C2×C42×C71—21no—
48C2×C42×C71C2311yescharacteristic
49C4⋊C8×C71—22no—
50C4⋊C8×C71—22no—
51C22⋊C8×C71—22no—
52#97×C71—22no—
53#95×C71—22no—
54#41×C71—84no—
55#41×C71—88no—
56C2×Q8×C71—22no—
57C2×Q8×C71—22no—
58C2×D4×C71—28no—
59C2×D4×C71—24no—
60C2×D4×C71—22no—
61C2×D4×C71—22no—
62C2×D4×C71C22⋊C414yes—
63C22×C4×C71—28no—
64C22×C4×C71—24no—
65C22×C4×C71—41no—
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D412yes—
68M16×C71—42no—
69C2×C8×C71—42no—
70C2×C8×C71—41no—
71C4⋊C4×C71—28no—
72C4⋊C4×C71—22no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—24no—
75C4⋊C4×C71C22⋊C414yes—
76C4⋊C4×C71C22⋊C414yes—
77C22⋊C4×C71—28no—
78C22⋊C4×C71—22no—
79C22⋊C4×C71—24no—
80C22⋊C4×C71—24no—
81C42×C71—22no—
82C42×C71—22no—
83C42×C71—41no—
84C42×C71C22×C411yescharacteristic
85C42×C71C2×D411yescharacteristic
86C42×C71C2×D412yes—
87C23×C71—24no—
88C23×C71#12011yescharacteristic
89Q8×C71—44no—
90Q8×C71—82no—
91D4×C71—44no—
92D4×C71—42no—
93D4×C71—42no—
94D4×C71—82no—
95D4×C71—84no—
96D4×C71—88no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—24no—
100C2×C4×C71—24no—
101C2×C4×C71—42no—
102C2×C4×C71—48no—
103C2×C4×C71—42no—
104C2×C4×C71—88no—
105C2×C4×C71—82no—
106C2×C4×C71C2×C22⋊C411yescharacteristic
107C2×C4×C71C2×C22⋊C412yes—
108C2×C4×C71#12012yes—
109C2×C4×C71#12011yescharacteristic
110C8×C71—82no—
111C8×C71—81no—
112C22×C71—21no—
113C22×C71—22no—
114C22×C71—44no—
115C22×C71—44no—
116C22×C71—44no—
117C22×C71#37911yescharacteristic
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—44no—
122C4×C71—44no—
123C4×C71—82no—
124C4×C71#45411yescharacteristic
125C4×C71#45611yescharacteristic
126#8494C7111yesmaximal, characteristic, direct product factor, Sylow
127C2×C71—41no—
128C2×C71—84no—
129C2×C71#179311yescharacteristic
130C2×C71#179512yes—
131#3548C2×C7111yescharacteristic
132#1995C2×C7111yescharacteristic, complemented
133#1557C2×C7112yescomplemented
134#1537C2×C7111yescharacteristic, complemented
135#1505C2×C7112yescomplemented
136C71#849411yescharacteristic, direct product factor, Sylow
137#562—22no—
138#554—22no—
139#550—22no—
140#547C4×C7114yescomplemented
141#540—22no—
142#537—24no—
143#533C22×C7111yescharacteristic
144#532C22×C7112yes—
145#432C22×C7112yes—
146#423C22×C7111yescharacteristic
147#376C22×C7111yescharacteristic
148#339—22no—
149#332—22no—
150#331—22no—
151#330—22no—
152#327—24no—
153C2×#41—24no—
154C2×#41—22no—
155C2×M16—21no—
156C4⋊Q8C2×C4×C7112yes—
157C4⋊Q8D4×C7112yes—
158#127—22no—
159#127D4×C7112yes—
160#125—22no—
161#125D4×C7112yes—
162#124—22no—
163#123—28no—
164C22⋊Q8—28no—
165C22⋊Q8—22no—
166C22⋊Q8—24no—
167#121—28no—
168#121—28no—
169#121—24no—
170#121—22no—
171#121D4×C7114yes—
172C4×Q8—22no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4C2×C4×C7114yes—
176#117D4×C7112yes—
177C2×C4⋊C4—24no—
178C2×C42—21no—
179C2×C42C23×C7111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C22⋊C8—22no—
183#97—22no—
184#95—22no—
185#41—84no—
186#41—88no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4C22⋊C4×C7114yes—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—41no—
197C22×C4C2×D4×C7111yescharacteristic
198C22×C4C2×D4×C7112yes—
199M16—42no—
200C2×C8—42no—
201C2×C8—41no—
202C4⋊C4—28no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4C22⋊C4×C7114yes—
207C4⋊C4C22⋊C4×C7114yes—
208C22⋊C4—28no—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C42—22no—
213C42—22no—
214C42—41no—
215C42C22×C4×C7111yescharacteristic
216C42C2×D4×C7111yescharacteristic
217C42C2×D4×C7112yes—
218C23—24no—
219C23#120×C7111yescharacteristic
220Q8—44no—
221Q8—82no—
222D4—44no—
223D4—42no—
224D4—42no—
225D4—82no—
226D4—84no—
227D4—88no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—42no—
235C2×C4—88no—
236C2×C4—82no—
237C2×C4C2×C22⋊C4×C7111yescharacteristic
238C2×C4C2×C22⋊C4×C7112yes—
239C2×C4#120×C7112yes—
240C2×C4#120×C7111yescharacteristic
241C8—82no—
242C8—81no—
243C22—21no—
244C22—22no—
245C22—44no—
246C22—44no—
247C22—44no—
248C22#379×C7111yescharacteristic
249C4—21no—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4#454×C7111yescharacteristic
256C4#456×C7111yescharacteristic
257C2—41no—
258C2—84no—
259C2C71×#179311yescharacteristic
260C2C71×#179512yes—
subgroupG/H
center, socleC22×C71#379
commutatorC42C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC71×#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#1995C2semidirect
3. C71×#1557C2semidirect
4. C71×#1537C2semidirect
5. C71×#1505C2semidirect
9. #547×C71C4semidirect
126. #8494C71direct (N × H)
132. #1995C2×C71semidirect
133. #1557C2×C71semidirect
134. #1537C2×C71semidirect
135. #1505C2×C71semidirect
136. C71#8494direct (N × H)
140. #547C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#35482
C71×#19952
C71×#15572
C71×#15372
C71×#15052
#849471