Finite Group Databasebeta

← #31908984

Subgroups — #31908984

Order 18176.

412 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#22590152C212yesmaximal, complemented
2#22586381C212yesmaximal, complemented
3#22586377C212yesmaximal, complemented
4C71×#3151C211yesmaximal, characteristic
5#17985820C2214yescomplemented
6#17985781—22no—
7#17985777—22no—
8C2×AGL(1,ℤ/8)×C71—22no—
9#532×C71C411yescharacteristic
10C2×#127×C71C411yescharacteristic
11#496×C71—22no—
12#493×C71C411yescharacteristic
13#493×C71C2211yescharacteristic
14#492×C71C411yescharacteristic
15#492×C71C2211yescharacteristic
16C4×M16×C71C2211yescharacteristic
17#14556068—24no—
18C2×#41×C71—21no—
19C22×D4×C71—21no—
20C22×D4×C71—41no—
21AGL(1,ℤ/8)×C71—48no—
22C2×SD16×C71—22no—
23C2×SD16×C71—22no—
24C2×D8×C71—22no—
25C2×D8×C71—22no—
26C2×M16×C71D412yes—
27C4⋊Q8×C71C2×C411yescharacteristic
28#127×C71—21no—
29#127×C71—41no—
30#127×C71C2×C411yescharacteristic
31#127×C71C2×C411yescharacteristic
32#127×C71C2×C411yescharacteristic
33#124×C71—21no—
34#121×C71—41no—
35C4×D4×C71—41no—
36C2×C42×C71C2×C411yescharacteristic
37#97×C71D412yes—
38C4×C8×C71C2×C411yescharacteristic
39C4×C8×C71C2311yescharacteristic
40C71⋊C16—44no—
41C24×C71—41no—
42#41×C71—44no—
43C2×Q8×C71—21no—
44C2×D4×C71—21no—
45C2×D4×C71—21no—
46C2×D4×C71—21no—
47C2×D4×C71—44no—
48C2×D4×C71—41no—
49C2×D4×C71—41no—
50C2×D4×C71—41no—
51C2×D4×C71—42no—
52C2×D4×C71—41no—
53C2×D4×C71—41no—
54C2×D4×C71—84no—
55C22×C4×C71—21no—
56C22×C4×C71—41no—
57C22×C4×C71C22⋊C411yescharacteristic
58SD16×C71—44no—
59SD16×C71—44no—
60D8×C71—44no—
61D8×C71—44no—
62M16×C71—24no—
63C2×C8×C71C22⋊C412yes—
64C2×C8×C71C2×D412yes—
65C4⋊C4×C71—41no—
66C22⋊C4×C71—42no—
67C42×C71—21no—
68C42×C71C22⋊C411yescharacteristic
69C42×C71C22×C411yescharacteristic
70C23×C71—41no—
71C23×C71—41no—
72C23×C71—41no—
73C23×C71—82no—
74C23×C71—84no—
75C23×C71#10011yescharacteristic
76Q8×C71—42no—
77D4×C71—42no—
78D4×C71—42no—
79D4×C71—42no—
80D4×C71—82no—
81D4×C71—82no—
82D4×C71—82no—
83D4×C71—82no—
84D4×C71—84no—
85D4×C71—82no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—42no—
90C2×C4×C71—41no—
91C2×C4×C71—82no—
92C2×C4×C71#10011yescharacteristic
93C2×C4×C71C2×C22⋊C411yescharacteristic
94C8×C71—22no—
95C8×C71AGL(1,ℤ/8)14yes—
96C22×C71—21no—
97C22×C71—41no—
98C22×C71—82no—
99C22×C71—82no—
100C22×C71—82no—
101C22×C71—81no—
102C22×C71—81no—
103C22×C71—81no—
104C22×C71—164no—
105C22×C71C2×#10011yescharacteristic
106C4×C71—22no—
107C4×C71—21no—
108C4×C71—41no—
109C4×C71—81no—
110C4×C71#41812yes—
111#11992—711nomaximal, Sylow
112C2×C71—21no—
113C2×C71—41no—
114C2×C71—161no—
115C2×C71—161no—
116C2×C71—161no—
117C2×C71#155511yescharacteristic
118#3151D7111yescharacteristic
119#2165—712no—
120#1359—712no—
121#1357—712no—
122C71#1199211yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—22no—
124#532Dic7111yescharacteristic
125C2×#127Dic7111yescharacteristic
126#496—22no—
127#493Dic7111yescharacteristic
128#493C2×D7111yescharacteristic
129#492Dic7111yescharacteristic
130#492C2×D7111yescharacteristic
131C4×M16C2×D7111yescharacteristic
132#364—714no—
133#349—1422no—
134#347—1422no—
135C2×#41—21no—
136C22×D4—21no—
137C22×D4—41no—
138AGL(1,ℤ/8)—48no—
139C2×SD16—22no—
140C2×SD16—22no—
141C2×D8—22no—
142C2×D8—22no—
143C2×M16C22⋊D7112yes—
144C4⋊Q8C2×Dic7111yescharacteristic
145#127—21no—
146#127—41no—
147#127C2×Dic7111yescharacteristic
148#127C2×Dic7111yescharacteristic
149#127C2×Dic7111yescharacteristic
150#124—21no—
151#121—41no—
152C4×D4—41no—
153C2×C42C2×Dic7111yescharacteristic
154M32—1424no—
155#97C22⋊D7112yes—
156C4×C8C2×Dic7111yescharacteristic
157C4×C8C22×D7111yescharacteristic
158C24—41no—
159#41—44no—
160C2×Q8—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—44no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—84no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4C22⋊Dic7111yescharacteristic
175SD16—44no—
176SD16—44no—
177D8—44no—
178D8—44no—
179M16—24no—
180C2×C8C22⋊Dic7112yes—
181C2×C8C2×C22⋊D7112yes—
182C4⋊C4—41no—
183C22⋊C4—42no—
184C42—21no—
185C42C22⋊Dic7111yescharacteristic
186C42C22×Dic7111yescharacteristic
187C16—2844no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—82no—
192C23—84no—
193C23#1455604511yescharacteristic
194Q8—42no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—82no—
199D4—82no—
200D4—82no—
201D4—82no—
202D4—84no—
203D4—82no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—82no—
210C2×C4#1455604511yescharacteristic
211C2×C4C2×C22⋊Dic7111yescharacteristic
212C8—22no—
213C8#1455615414yes—
214C22—21no—
215C22—41no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—164no—
223C22C2×#1455604511yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—81no—
228C4#1798600412yes—
229C2—21no—
230C2—41no—
231C2—161no—
232C2—161no—
233C2—161no—
234C2#2258705411yescharacteristic
subgroupG/H
centerC2#22587054
commutatorC42×C71C22×C4
FrattiniC4×C8C22×D71
Fitting, prime radicalC71×#3151C2
socleC2×C71#1555
cosocleC4×C8×C71C23
hypercenter#3151D71
hypocenterC71#11992
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #22590152C2semidirect
2. #22586381C2semidirect
3. #22586377C2semidirect
5. #17985820C22semidirect
122. C71#11992semidirect

Maximal subgroups

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

subgroupindex
#225901522
#225863812
#225863772
C71×#31512
#1199271