Finite Group Databasebeta

← C71×#7490 #31828362

Subgroups — C71×#7490 #31828362

Order 18176.

274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2126C211yesmaximal, characteristic
2C71×#1884C211yesmaximal, characteristic
3C71×#1543C211yesmaximal, characteristic
4#568×C71C411yescharacteristic, complemented
5#532×C71C2211yescharacteristic
6#431×C71—21no—
7#421×C71—21no—
8#410×C71—21no—
9#410×C71C411yescharacteristic, complemented
10#354×C71C412yescomplemented
11#353×C71—21no—
12#339×C71—21no—
13#333×C71C412yescomplemented
14#331×C71—21no—
15C2×#41×C71C2×C411yescharacteristic
16C2×#41×C71D411yescharacteristic
17#131×C71—42no—
18C2×M16×C71—21no—
19C2×M16×C71—41no—
20C4⋊Q8×C71C2×C411yescharacteristic
21#127×C71D411yescharacteristic
22#124×C71C2×C411yescharacteristic
23#124×C71Q811yescharacteristic
24#121×C71—41no—
25C4×D4×C71—41no—
26#117×C71—21no—
27#117×C71—41no—
28C2×C42×C71D411yescharacteristic
29C4⋊C8×C71—22no—
30#104×C71—42no—
31#99×C71—22no—
32#99×C71—41no—
33#99×C71—41no—
34C22⋊C8×C71—41no—
35C4×C8×C71—41no—
36#41×C71—41no—
37#41×C71—41no—
38#41×C71—41no—
39#41×C71—41no—
40C2×Q8×C71C4211yescharacteristic
41C2×Q8×C71C4⋊C411yescharacteristic
42C2×D4×C71—21no—
43C2×D4×C71—21no—
44C2×D4×C71—41no—
45C2×D4×C71C22⋊C411yescharacteristic
46C2×D4×C71C4⋊C411yescharacteristic
47C22×C4×C71—21no—
48C22×C4×C71—21no—
49C22×C4×C71—21no—
50C22×C4×C71C22⋊C411yescharacteristic
51M16×C71—22no—
52M16×C71—41no—
53M16×C71—42no—
54C2×C8×C71—41no—
55C2×C8×C71—42no—
56C2×C8×C71—42no—
57C4⋊C4×C71—41no—
58C4⋊C4×C71—41no—
59C4⋊C4×C71—81no—
60C22⋊C4×C71—41no—
61C22⋊C4×C71—41no—
62C22⋊C4×C71—42no—
63C22⋊C4×C71—41no—
64C42×C71—21no—
65C42×C71—41no—
66C42×C71—81no—
67C42×C71C22⋊C411yescharacteristic
68C42×C71C4⋊C411yescharacteristic
69C23×C71—21no—
70C23×C71—21no—
71C23×C71#9911yescharacteristic
72Q8×C71—41no—
73Q8×C71—41no—
74D4×C71—41no—
75D4×C71—41no—
76D4×C71—42no—
77D4×C71—82no—
78D4×C71—81no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—42no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—81no—
91C2×C4×C71—82no—
92C2×C4×C71#9511yescharacteristic
93C2×C4×C71#9911yescharacteristic
94C8×C71—42no—
95C8×C71—82no—
96C22×C71—21no—
97C22×C71—41no—
98C22×C71—41no—
99C22×C71—41no—
100C22×C71—81no—
101C22×C71—81no—
102C22×C71#34211yescharacteristic
103C4×C71—22no—
104C4×C71—21no—
105C4×C71—41no—
106C4×C71—41no—
107C4×C71—41no—
108C4×C71—161no—
109C4×C71—162no—
110C4×C71AΣL(1,𝔽16)11yescharacteristic
111C4×C71#35211yescharacteristic
112#7490C7111yesmaximal, characteristic, direct product factor, Sylow
113C2×C71—21no—
114C2×C71—41no—
115C2×C71—81no—
116C2×C71—81no—
117C2×C71#139011yescharacteristic
118#2126C2×C7111yescharacteristic
119#1884C2×C7111yescharacteristic
120#1543C2×C7111yescharacteristic
121C71#749011yescharacteristic, direct product factor, Sylow
122#568C4×C7111yescharacteristic, complemented
123#532C22×C7111yescharacteristic
124#431—21no—
125#421—21no—
126#410—21no—
127#410C4×C7111yescharacteristic, complemented
128#354C4×C7112yescomplemented
129#353—21no—
130#339—21no—
131#333C4×C7112yescomplemented
132#331—21no—
133C2×#41C2×C4×C7111yescharacteristic
134C2×#41D4×C7111yescharacteristic
135#131—42no—
136C2×M16—21no—
137C2×M16—41no—
138C4⋊Q8C2×C4×C7111yescharacteristic
139#127D4×C7111yescharacteristic
140#124C2×C4×C7111yescharacteristic
141#124Q8×C7111yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—21no—
145#117—41no—
146C2×C42D4×C7111yescharacteristic
147C4⋊C8—22no—
148#104—42no—
149#99—22no—
150#99—41no—
151#99—41no—
152C22⋊C8—41no—
153C4×C8—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8C42×C7111yescharacteristic
159C2×Q8C4⋊C4×C7111yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C22⋊C4×C7111yescharacteristic
164C2×D4C4⋊C4×C7111yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C22⋊C4×C7111yescharacteristic
169M16—22no—
170M16—41no—
171M16—42no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—42no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—81no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—41no—
182C42—21no—
183C42—41no—
184C42—81no—
185C42C22⋊C4×C7111yescharacteristic
186C42C4⋊C4×C7111yescharacteristic
187C23—21no—
188C23—21no—
189C23#99×C7111yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—82no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4#95×C7111yescharacteristic
211C2×C4#99×C7111yescharacteristic
212C8—42no—
213C8—82no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22#342×C7111yescharacteristic
221C4—22no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—161no—
227C4—162no—
228C4AΣL(1,𝔽16)×C7111yescharacteristic
229C4#352×C7111yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2C71×#139011yescharacteristic
subgroupG/H
centerC4×C71AΣL(1,𝔽16)
commutatorC2×Q8C42×C71
Frattini, cosocle#532C22×C71
socleC2×C71#1390
prime radical#532×C71C22
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C71 ◁ C2×C4×C71 ◁ C22×C4×C71 ◁ #532×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. #568×C71C4semidirect
9. #410×C71C4semidirect
10. #354×C71C4semidirect
13. #333×C71C4semidirect
112. #7490C71direct (N × H)
121. C71#7490direct (N × H)
122. #568C4×C71semidirect
127. #410C4×C71semidirect
128. #354C4×C71semidirect
131. #333C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#21262
C71×#18842
C71×#15432
#749071