Finite Group Databasebeta

← C71×#10258 #31831130

Subgroups — C71×#10258 #31831130

Order 18176.

526 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3249C211yesmaximal, characteristic, complemented
2C71×#3077C212yesmaximal
3C71×#1954C211yesmaximal, characteristic
4C71×#1493C212yesmaximal, complemented
5C71×#1292C211yesmaximal, characteristic, complemented
6#533×C71C2212yes—
7C2×C4×D4×C71C2211yescharacteristic
8#478×C71—24no—
9#477×C71—22no—
10#476×C71—22no—
11#474×C71—22no—
12#467×C71—22no—
13#462×C71—22no—
14#428×C71—22no—
15#423×C71C2212yes—
16C2×C4⋊C8×C71C2211yescharacteristic
17#418×C71—22no—
18C2×C22⋊C8×C71—22no—
19C2×C4×C8×C71C2211yescharacteristic
20Q8⋊C8×C71—24no—
21C22×D4×C71D411yescharacteristic
22C23×C4×C71—21no—
23C2×Q16×C71—22no—
24C2×M16×C71—22no—
25C22×C8×C71—21no—
26C22×C8×C71D412yes—
27C4⋊Q8×C71D412yes—
28C4⋊Q8×C71D412yes—
29#125×C71—22no—
30C22⋊Q8×C71—22no—
31C22⋊Q8×C71—22no—
32C4×Q8×C71—22no—
33C4×Q8×C71—22no—
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36#117×C71—22no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—21no—
39C2×C42×C71C2311yescharacteristic
40#107×C71—24no—
41#107×C71—22no—
42#106×C71—24no—
43C4⋊C8×C71—22no—
44C4⋊C8×C71—22no—
45C4⋊C8×C71D414yes—
46#103×C71—24no—
47#102×C71—24no—
48#102×C71—24no—
49C22⋊C8×C71—44no—
50C4×C8×C71—22no—
51C24×C71—21no—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×D4×C71—22no—
55C2×D4×C71—21no—
56C2×D4×C71—22no—
57C2×D4×C71Q1612yescomplemented
58C22×C4×C71—21no—
59C22×C4×C71—42no—
60C22×C4×C71—42no—
61C22×C4×C71—41no—
62C22×C4×C71—41no—
63C22×C4×C71C2×D411yescharacteristic
64C22×C4×C71C2×D411yescharacteristic
65C22×C4×C71#4111yescharacteristic
66Q16×C71—82no—
67M16×C71—42no—
68M16×C71—42no—
69C2×C8×C71—22no—
70C2×C8×C71—24no—
71C2×C8×C71—22no—
72C2×C8×C71—41no—
73C2×C8×C71—41no—
74C2×C8×C71—42no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—22no—
77C4⋊C4×C71—22no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—22no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—42no—
82C22⋊C4×C71—42no—
83C22⋊C4×C71—42no—
84C42×C71—42no—
85C42×C71C2×D412yes—
86C42×C71C2×D412yes—
87C23×C71—22no—
88C23×C71—42no—
89C23×C71—42no—
90C23×C71—41no—
91C23×C71—41no—
92C23×C71#12111yescharacteristic
93Q8×C71—44no—
94Q8×C71—42no—
95Q8×C71—42no—
96D4×C71—24no—
97D4×C71—42no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—41no—
105C2×C4×C71—82no—
106C2×C4×C71—81no—
107C2×C4×C71—81no—
108C2×C4×C71#12012yes—
109C2×C4×C71#12011yescharacteristic
110C2×C4×C71#12112yes—
111C2×C4×C71#12711yescharacteristic
112C2×C4×C71C2×Q1611yescharacteristic
113C2×C4×C71#13511yescharacteristic
114C2×C4×C71#13712yes—
115C8×C71—42no—
116C8×C71—44no—
117C8×C71—42no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—44no—
121C22×C71—42no—
122C22×C71—82no—
123C22×C71—81no—
124C22×C71—81no—
125C22×C71#39211yescharacteristic
126C22×C71#46711yescharacteristic
127C22×C71#47011yescharacteristic
128C4×C71—21no—
129C4×C71—22no—
130C4×C71—81no—
131C4×C71—82no—
132C4×C71—82no—
133C4×C71#45112yes—
134C4×C71#45212yes—
135C4×C71#45311yescharacteristic
136C4×C71#45411yescharacteristic
137#10258C7111yesmaximal, characteristic, direct product factor, Sylow
138C2×C71—21no—
139C2×C71—21no—
140C2×C71—82no—
141C2×C71#200111yescharacteristic
142C2×C71#200611yescharacteristic
143C2×C71#201711yescharacteristic
144#3249C2×C7111yescharacteristic, complemented
145#3077C2×C7112yes—
146#1954C2×C7111yescharacteristic
147#1493C2×C7112yescomplemented
148#1292C2×C7111yescharacteristic, complemented
149C71#1025811yescharacteristic, direct product factor, Sylow
150#533C22×C7112yes—
151C2×C4×D4C22×C7111yescharacteristic
152#478—24no—
153#477—22no—
154#476—22no—
155#474—22no—
156#467—22no—
157#462—22no—
158#428—22no—
159#423C22×C7112yes—
160C2×C4⋊C8C22×C7111yescharacteristic
161#418—22no—
162C2×C22⋊C8—22no—
163C2×C4×C8C22×C7111yescharacteristic
164Q8⋊C8—24no—
165C22×D4D4×C7111yescharacteristic
166C23×C4—21no—
167C2×Q16—22no—
168C2×M16—22no—
169C22×C8—21no—
170C22×C8D4×C7112yes—
171C4⋊Q8D4×C7112yes—
172C4⋊Q8D4×C7112yes—
173#125—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—22no—
180#117—22no—
181C2×C4⋊C4D4×C7111yescharacteristic
182C2×C22⋊C4—21no—
183C2×C42C23×C7111yescharacteristic
184#107—24no—
185#107—22no—
186#106—24no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8D4×C7114yes—
190#103—24no—
191#102—24no—
192#102—24no—
193C22⋊C8—44no—
194C4×C8—22no—
195C24—21no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4Q16×C7112yescomplemented
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C2×D4×C7111yescharacteristic
208C22×C4C2×D4×C7111yescharacteristic
209C22×C4#41×C7111yescharacteristic
210Q16—82no—
211M16—42no—
212M16—42no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—42no—
229C42C2×D4×C7112yes—
230C42C2×D4×C7112yes—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23#121×C7111yescharacteristic
237Q8—44no—
238Q8—42no—
239Q8—42no—
240D4—24no—
241D4—42no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4#120×C7112yes—
253C2×C4#120×C7111yescharacteristic
254C2×C4#121×C7112yes—
255C2×C4#127×C7111yescharacteristic
256C2×C4C2×Q16×C7111yescharacteristic
257C2×C4#135×C7111yescharacteristic
258C2×C4#137×C7112yes—
259C8—42no—
260C8—44no—
261C8—42no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—42no—
266C22—82no—
267C22—81no—
268C22—81no—
269C22#392×C7111yescharacteristic
270C22#467×C7111yescharacteristic
271C22#470×C7111yescharacteristic
272C4—22no—
273C4—21no—
274C4—82no—
275C4—81no—
276C4—82no—
277C4#451×C7112yes—
278C4#452×C7112yes—
279C4#453×C7111yescharacteristic
280C4#454×C7111yescharacteristic
281C2—21no—
282C2—21no—
283C2—82no—
284C2C71×#200111yescharacteristic
285C2C71×#200611yescharacteristic
286C2C71×#201711yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutator, Frattini, cosocleC2×C42C23×C71
prime radicalC22×D4×C71D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#3249C2semidirect
4. C71×#1493C2semidirect
5. C71×#1292C2semidirect
57. C2×D4×C71Q16semidirect
137. #10258C71direct (N × H)
144. #3249C2×C71semidirect
147. #1493C2×C71semidirect
148. #1292C2×C71semidirect
149. C71#10258direct (N × H)
201. C2×D4Q16×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#32492
C71×#30772
C71×#19542
C71×#14932
C71×#12922
#1025871