Finite Group Databasebeta

← #31884487

Subgroups — #31884487

Order 18176.

276 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 67 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#22588423C211yesmaximal, characteristic
2#22588106C211yesmaximal, characteristic
3C71×#1464C211yesmaximal, characteristic
4#17985963—21no—
5C2×C22⋊(C71⋊C8)—21no—
6C2×#14556039C411yescharacteristic
7C2×C4×C71⋊C8C411yescharacteristic
8#17985768—21no—
9#17985768C412yes—
10#17985754—21no—
11C2×#117×C71C2211yescharacteristic
12#432×C71—21no—
13#324×C71—21no—
14#323×C71C412yes—
15C2×#11765849—22no—
16C22×C71⋊C8—21no—
17C22×C71⋊C8—21no—
18C22⋊(C71⋊C8)—24no—
19C22⋊(C71⋊C8)—41no—
20C22⋊(C71⋊C8)—41no—
21#14556039—41no—
22C4×C71⋊C8—41no—
23C23×C4×C71D411yescharacteristic
24#117×C71—41no—
25#117×C71—41no—
26C2×C4⋊C4×C71C2×C411yescharacteristic
27C2×C4⋊C4×C71D411yescharacteristic
28C2×C22⋊C4×C71C2×C411yescharacteristic
29C2×C22⋊C4×C71Q811yescharacteristic
30C2×C42×C71C2×C411yescharacteristic
31C2×C42×C71D411yescharacteristic
32C4⋊C8×C71—41no—
33C22⋊C8×C71—22no—
34#97×C71—41no—
35#11765849—42no—
36#11765849—42no—
37C2×C71⋊C8—22no—
38C2×C71⋊C8—22no—
39C2×C71⋊C8—41no—
40C2×C71⋊C8—41no—
41C2×C71⋊C8—41no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—42no—
44C24×C71C4⋊C411yescharacteristic
45C22×C4×C71—21no—
46C22×C4×C71—21no—
47C22×C4×C71—22no—
48C22×C4×C71—41no—
49C22×C4×C71C4211yescharacteristic
50C22×C4×C71C22⋊C411yescharacteristic
51C22×C4×C71C22⋊C411yescharacteristic
52C22×C4×C71C22⋊C411yescharacteristic
53C22×C4×C71C4⋊C411yescharacteristic
54C22×C4×C71C4⋊C411yescharacteristic
55C2×C8×C71—42no—
56C4⋊C4×C71—41no—
57C4⋊C4×C71—41no—
58C22⋊C4×C71—41no—
59C22⋊C4×C71—41no—
60C42×C71—21no—
61C42×C71—21no—
62C42×C71—21no—
63C42×C71—21no—
64C23×C71—21no—
65C23×C71—21no—
66C23×C71—21no—
67C23×C71—41no—
68C23×C71—41no—
69C23×C71#9511yescharacteristic
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—41no—
81C2×C4×C71—41no—
82C2×C4×C71—41no—
83C2×C4×C71—41no—
84C2×C4×C71#9911yescharacteristic
85C2×C4×C71#9911yescharacteristic
86C2×C4×C71#10412yes—
87C8×C71—82no—
88C71⋊C8—42no—
89C71⋊C8—42no—
90C71⋊C8—82no—
91C22×C71—21no—
92C22×C71—21no—
93C22×C71—42no—
94C22×C71—41no—
95C22×C71—41no—
96C22×C71—41no—
97C22×C71—81no—
98C22×C71H(3,ℤ/4)11yescharacteristic
99C22×C71#33911yescharacteristic
100C22×C71#34211yescharacteristic
101C4×C71—21no—
102C4×C71—21no—
103C4×C71—22no—
104C4×C71—41no—
105C4×C71—41no—
106C4×C71—41no—
107C4×C71—41no—
108#7216—711nomaximal, Sylow
109C2×C71—21no—
110C2×C71—21no—
111C2×C71—81no—
112C2×C71#128411yescharacteristic
113C2×C71#139211yescharacteristic
114C2×C71#139511yescharacteristic
115#1822—711no—
116#1750—711no—
117#1464D7111yescharacteristic
118C71#721611yescharacteristic, complemented, Sylow
119C2×#117C2×D7111yescharacteristic
120#432—21no—
121#407—1421no—
122C2×C22⋊C8—1421no—
123C2×#97—711no—
124C2×C4×C8—711no—
125#343—712no—
126#343—1421no—
127#336—1421no—
128#324—21no—
129#323Dic7112yes—
130C23×C4D28411yescharacteristic
131C2×M16—1422no—
132C22×C8—1421no—
133C22×C8—1421no—
134#117—41no—
135#117—41no—
136C2×C4⋊C4C4×D7111yescharacteristic
137C2×C4⋊C4C22⋊D7111yescharacteristic
138C2×C22⋊C4Dic14211yescharacteristic
139C2×C22⋊C4C2×Dic7111yescharacteristic
140C2×C42C4×D7111yescharacteristic
141C2×C42C22⋊D7111yescharacteristic
142C4⋊C8—41no—
143C22⋊C8—22no—
144C22⋊C8—2841no—
145C22⋊C8—2841no—
146C22⋊C8—1424no—
147#97—41no—
148#97—2841no—
149C4×C8—2841no—
150C24C4⋊Dic7111yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4C4×Dic7111yescharacteristic
156C22×C4(C2×D71)⋊C411yescharacteristic
157C22×C4(C2×D71)⋊C411yescharacteristic
158C22×C4C22⋊Dic7111yescharacteristic
159C22×C4Dic71⋊C411yescharacteristic
160C22×C4Dic71⋊C411yescharacteristic
161M16—2842no—
162M16—2842no—
163C2×C8—42no—
164C2×C8—2841no—
165C2×C8—2841no—
166C2×C8—2841no—
167C2×C8—2841no—
168C2×C8—2842no—
169C2×C8—1422no—
170C2×C8—1422no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23C71⋊#9511yescharacteristic
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4#1455604211yescharacteristic
200C2×C4#1455604211yescharacteristic
201C2×C4#1455605412yes—
202C8—82no—
203C8—2842no—
204C8—2842no—
205C8—5682no—
206C22—21no—
207C22—21no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—81no—
213C22#1798575611yescharacteristic
214C22#1798576011yescharacteristic
215C22#1798576511yescharacteristic
216C4—21no—
217C4—21no—
218C4—22no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—41no—
223C2—21no—
224C2—21no—
225C2—81no—
226C2#2258619411yescharacteristic
227C2#2258646211yescharacteristic
228C2#2258646811yescharacteristic
subgroupG/H
centerC22#17985765
commutatorC22×C4×C71C42
FrattiniC2×#117C2×D71
FittingC71×#1464C2
socleC22×C71#342
prime radicalC24×C71C4⋊C4
cosocleC2×#117×C71C22
hypercenter#1464D71
hypocenterC71#7216
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
118. C71#7216semidirect

Maximal subgroups

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

subgroupindex
#225884232
#225881062
C71×#14642
#721671