Finite Group Databasebeta

← C71×#10026 #31830898

Subgroups — C71×#10026 #31830898

Order 18176.

564 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3245C211yesmaximal, characteristic, complemented
2C71×#3109C212yesmaximal, complemented
3C71×#1955C211yesmaximal, characteristic, complemented
4C71×#1530C212yesmaximal, complemented
5C71×#1290C211yesmaximal, characteristic, complemented
6#532×C71C2212yes—
7C2×C4×D4×C71C2211yescharacteristic
8#481×C71—22no—
9#476×C71—22no—
10#474×C71—22no—
11#466×C71—22no—
12#459×C71—22no—
13#459×C71—22no—
14#447×C71—22no—
15#442×C71—24no—
16#432×C71C2212yes—
17C2×C4⋊C8×C71C2211yescharacteristic
18C2×C22⋊C8×C71—22no—
19C2×#97×C71C2211yescharacteristic
20#329×C71—24no—
21C2×#41×C71—22no—
22C22×D4×C71D411yescharacteristic
23C23×C4×C71—21no—
24C2×D8×C71—24no—
25C2×D8×C71—22no—
26C22×C8×C71—21no—
27C22×C8×C71D412yes—
28C4⋊Q8×C71D412yes—
29#127×C71D412yes—
30#124×C71—22no—
31#121×C71—22no—
32#121×C71—22no—
33C4×D4×C71—22no—
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36C4×D4×C71—22no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C22⋊C4×C71—21no—
39C2×C42×C71C2311yescharacteristic
40#107×C71—22no—
41#106×C71—24no—
42C4⋊C8×C71—22no—
43C4⋊C8×C71—22no—
44#102×C71—24no—
45#102×C71—24no—
46#102×C71—24no—
47C22⋊C8×C71—22no—
48C22⋊C8×C71—44no—
49#97×C71—22no—
50#97×C71D414yes—
51C24×C71—21no—
52#41×C71—84no—
53C2×Q8×C71—22no—
54C2×D4×C71—22no—
55C2×D4×C71—22no—
56C2×D4×C71—22no—
57C2×D4×C71—22no—
58C2×D4×C71—42no—
59C2×D4×C71—42no—
60C22×C4×C71—21no—
61C22×C4×C71—42no—
62C22×C4×C71—42no—
63C22×C4×C71—41no—
64C22×C4×C71—42no—
65C22×C4×C71—41no—
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D411yescharacteristic
68C22×C4×C71#4111yescharacteristic
69D8×C71—48no—
70D8×C71—82no—
71C2×C8×C71—22no—
72C2×C8×C71—24no—
73C2×C8×C71—22no—
74C2×C8×C71—41no—
75C2×C8×C71—41no—
76C2×C8×C71—42no—
77C4⋊C4×C71—22no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—22no—
80C22⋊C4×C71—42no—
81C22⋊C4×C71—42no—
82C22⋊C4×C71—42no—
83C42×C71C2×D412yes—
84C42×C71C2×D412yes—
85C23×C71—22no—
86C23×C71—42no—
87C23×C71—41no—
88C23×C71—41no—
89C23×C71—42no—
90C23×C71—42no—
91C23×C71#12111yescharacteristic
92Q8×C71—82no—
93D4×C71—24no—
94D4×C71—42no—
95D4×C71—44no—
96D4×C71—42no—
97D4×C71—42no—
98D4×C71—82no—
99D4×C71—84no—
100C2×C4×C71—22no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—41no—
104C2×C4×C71—42no—
105C2×C4×C71—41no—
106C2×C4×C71—42no—
107C2×C4×C71—82no—
108C2×C4×C71—84no—
109C2×C4×C71—81no—
110C2×C4×C71—81no—
111C2×C4×C71#12012yes—
112C2×C4×C71#12011yescharacteristic
113C2×C4×C71#12112yes—
114C2×C4×C71#12711yescharacteristic
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—82no—
126C22×C71—82no—
127C22×C71—82no—
128C22×C71#39211yescharacteristic
129C22×C71#47212yes—
130C4×C71—21no—
131C4×C71—22no—
132C4×C71—22no—
133C4×C71—81no—
134C4×C71—82no—
135C4×C71#45311yescharacteristic
136C4×C71#45411yescharacteristic
137#10026C7111yesmaximal, characteristic, direct product factor, Sylow
138C2×C71—22no—
139C2×C71—82no—
140C2×C71—162no—
141C2×C71#200611yescharacteristic
142C2×C71#201011yescharacteristic
143C2×C71#201811yescharacteristic
144#3245C2×C7111yescharacteristic, complemented
145#3109C2×C7112yescomplemented
146#1955C2×C7111yescharacteristic, complemented
147#1530C2×C7112yescomplemented
148#1290C2×C7111yescharacteristic, complemented
149C71#1002611yescharacteristic, direct product factor, Sylow
150#532C22×C7112yes—
151C2×C4×D4C22×C7111yescharacteristic
152#481—22no—
153#476—22no—
154#474—22no—
155#466—22no—
156#459—22no—
157#459—22no—
158#447—22no—
159#442—24no—
160#432C22×C7112yes—
161C2×C4⋊C8C22×C7111yescharacteristic
162C2×C22⋊C8—22no—
163C2×#97C22×C7111yescharacteristic
164#329—24no—
165C2×#41—22no—
166C22×D4D4×C7111yescharacteristic
167C23×C4—21no—
168C2×D8—24no—
169C2×D8—22no—
170C22×C8—21no—
171C22×C8D4×C7112yes—
172C4⋊Q8D4×C7112yes—
173#127D4×C7112yes—
174#124—22no—
175#121—22no—
176#121—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C2×C4⋊C4D4×C7111yescharacteristic
182C2×C22⋊C4—21no—
183C2×C42C23×C7111yescharacteristic
184#107—22no—
185#106—24no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188#102—24no—
189#102—24no—
190#102—24no—
191C22⋊C8—22no—
192C22⋊C8—44no—
193#97—22no—
194#97D4×C7114yes—
195C24—21no—
196#41—84no—
197C2×Q8—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4C2×D4×C7111yescharacteristic
211C22×C4C2×D4×C7111yescharacteristic
212C22×C4#41×C7111yescharacteristic
213D8—48no—
214D8—82no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C42C2×D4×C7112yes—
228C42C2×D4×C7112yes—
229C23—22no—
230C23—42no—
231C23—41no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23#121×C7111yescharacteristic
236Q8—82no—
237D4—24no—
238D4—42no—
239D4—44no—
240D4—42no—
241D4—42no—
242D4—82no—
243D4—84no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4—84no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4#120×C7112yes—
256C2×C4#120×C7111yescharacteristic
257C2×C4#121×C7112yes—
258C2×C4#127×C7111yescharacteristic
259C8—42no—
260C8—44no—
261C8—42no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—42no—
266C22—82no—
267C22—81no—
268C22—81no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22#392×C7111yescharacteristic
273C22#472×C7112yes—
274C4—22no—
275C4—21no—
276C4—22no—
277C4—82no—
278C4—81no—
279C4#453×C7111yescharacteristic
280C4#454×C7111yescharacteristic
281C2—22no—
282C2—82no—
283C2—162no—
284C2C71×#200611yescharacteristic
285C2C71×#201011yescharacteristic
286C2C71×#201811yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutator, Frattini, cosocleC2×C42C23×C71
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×#3245C2semidirect
2. C71×#3109C2semidirect
3. C71×#1955C2semidirect
4. C71×#1530C2semidirect
5. C71×#1290C2semidirect
137. #10026C71direct (N × H)
144. #3245C2×C71semidirect
145. #3109C2×C71semidirect
146. #1955C2×C71semidirect
147. #1530C2×C71semidirect
148. #1290C2×C71semidirect
149. C71#10026direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32452
C71×#31092
C71×#19552
C71×#15302
C71×#12902
#1002671