Finite Group Databasebeta

← #31890907

Subgroups — #31890907

Order 18176.

404 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 99 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#22590837C211yesmaximal, characteristic
2#22586652C212yesmaximal
3#22586173C211yesmaximal, characteristic, complemented
4#22586168C212yesmaximal, complemented
5C71×#1276C211yesmaximal, characteristic, complemented
6C2×#14556085C2211yescharacteristic
7#17986052C2212yes—
8#17985890—22no—
9#17985883—22no—
10C4×Dic71⋊C4C412yes—
11C4×(C2×D71)⋊C4C412yes—
12#17985845C2211yescharacteristic
13#17985724—22no—
14#17985722—22no—
15#431×C71C2212yes—
16C43×C71C2211yescharacteristic
17C22×C4×D71C2×C411yescharacteristic
18#14556085—21no—
19#14556085—22no—
20#14556085—21no—
21C2×Dic71⋊C4—22no—
22C2×Dic71⋊C4D412yes—
23C2×(C2×D71)⋊C4—22no—
24C2×(C2×D71)⋊C4D412yes—
25C2×C4×Dic71C2×C412yes—
26C2×C4×Dic71C2×C411yescharacteristic
27C4⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C4×C71⋊C8—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C71⋊#95—22no—
33C2×C42×C71—22no—
34C2×C42×C71C2×C412yes—
35C2×C42×C71C2311yescharacteristic
36C4⋊C8×C71—22no—
37C22⋊C8×C71—22no—
38C4×C8×C71—22no—
39C23×D71C22⋊C411yescharacteristic
40C22×Dic71—22no—
41C22×Dic71C22⋊C411yescharacteristic
42C22×Dic71C22⋊C412yes—
43C2×C4×D71—21no—
44C2×C4×D71—21no—
45C2×C4×D71—22no—
46C2×C4×D71—22no—
47C2×C71⋊C8—42no—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—44no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—44no—
54C4×Dic71—21no—
55C4×Dic71—21no—
56C4×Dic71—22no—
57C4×Dic71—22no—
58C22×C4×C71—22no—
59C22×C4×C71C22×C411yescharacteristic
60C22×C4×C71C2×D412yes—
61C2×C8×C71—42no—
62C42×C71—22no—
63C42×C71—22no—
64C42×C71—44no—
65C42×C71#4111yescharacteristic
66C42×C71#4112yes—
67C42×C71#4111yescharacteristic
68C23×C71C2×C22⋊C411yescharacteristic
69C22×D71—21no—
70C22×D71—22no—
71C22×D71—42no—
72C2×C4×C71—24no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71#11711yescharacteristic
77C2×C4×C71#11711yescharacteristic
78C2×C4×C71#12312yes—
79C2×C4×C71#12312yes—
80C2×C4×C71#13112yes—
81C2×C4×C71#13112yes—
82C2×Dic71—22no—
83C2×Dic71—21no—
84C2×Dic71—22no—
85C2×Dic71—42no—
86C2×Dic71—42no—
87C2×Dic71—42no—
88C2×Dic71—42no—
89C4×D71—41no—
90C4×D71—42no—
91C4×D71—41no—
92C8×C71—82no—
93C71⋊C8—82no—
94C22×C71—22no—
95C22×C71#38111yescharacteristic
96C22×C71#40812yes—
97C2×D71—41no—
98C2×D71—42no—
99C2×D71—82no—
100C4×C71—22no—
101C4×C71—22no—
102C4×C71—44no—
103C4×C71#43311yescharacteristic
104C4×C71#43312yes—
105C4×C71#43311yescharacteristic
106Dic71—41no—
107Dic71—42no—
108Dic71—82no—
109#8733—711nomaximal, Sylow
110C2×C71—22no—
111C2×C71#181811yescharacteristic
112C2×C71#182712yes—
113D71—81no—
114#2298—711no—
115#1458—712no—
116#1276—711no—
117#1276D7111yescharacteristic, complemented
118#1274—712no—
119C71#873311yescharacteristic, complemented, Sylow
120C2×#117—711no—
121#431—712no—
122#431C2×D7112yes—
123#388—1422no—
124#387—1422no—
125C4×C4⋊C4—712no—
126C4×C22⋊C4—712no—
127#376—711no—
128C43C2×D7111yescharacteristic
129#324—1422no—
130#323—1422no—
131C23×C4—711no—
132#117—1421no—
133#117—1422no—
134#117—1421no—
135C2×C4⋊C4—712no—
136C2×C4⋊C4—1422no—
137C2×C22⋊C4—712no—
138C2×C22⋊C4—1422no—
139C2×C42—22no—
140C2×C42—711no—
141C2×C42—712no—
142C2×C42C4×D7112yes—
143C2×C42C22×D7111yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—1422no—
146C22⋊C8—22no—
147C22⋊C8—1422no—
148C4×C8—22no—
149C4×C8—1422no—
150#95—1422no—
151#95—1422no—
152#95—1422no—
153C24—711no—
154C22×C4—22no—
155C22×C4—712no—
156C22×C4—711no—
157C22×C4—1422no—
158C22×C4—1421no—
159C22×C4—1421no—
160C22×C4—1422no—
161C22×C4—1422no—
162C22×C4C2×C4×D7111yescharacteristic
163C22×C4D4×D7112yes—
164C2×C8—42no—
165C2×C8—2842no—
166C4⋊C4—2844no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C22⋊C4—2844no—
170C22⋊C4—1422no—
171C22⋊C4—1422no—
172C42—22no—
173C42—22no—
174C42—44no—
175C42—1422no—
176C42—1422no—
177C42—1421no—
178C42—1421no—
179C42#1176586411yescharacteristic
180C42#1176586512yes—
181C42#1176586611yescharacteristic
182C23—2842no—
183C23—1421no—
184C23—1422no—
185C23C22⋊C4×D7111yescharacteristic
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—2841no—
191C2×C4—2842no—
192C2×C4—2841no—
193C2×C4—2842no—
194C2×C4—2842no—
195C2×C4—2842no—
196C2×C4—2842no—
197C2×C4—1422no—
198C2×C4—1421no—
199C2×C4—1422no—
200C2×C4#1455608511yescharacteristic
201C2×C4#1455608611yescharacteristic
202C2×C4#1455611012yes—
203C2×C4#1455611312yes—
204C2×C4#1455613412yes—
205C2×C4#1455613612yes—
206C8—82no—
207C8—5682no—
208C22—22no—
209C22—2842no—
210C22—2841no—
211C22—5682no—
212C22#1798586111yescharacteristic
213C22#1798596412yes—
214C4—22no—
215C4—22no—
216C4—44no—
217C4—2842no—
218C4—2841no—
219C4—5682no—
220C4#1798605811yescharacteristic
221C4#1798605912yes—
222C4#1798606111yescharacteristic
223C2—22no—
224C2—5681no—
225C2#2258841011yescharacteristic
226C2#2258844312yes—
subgroupG/H
centerC2×C4#14556086
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1276C2
socleC22×C71#381
prime radicalC23×D71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1276D71
hypocenterC71#8733
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
3. #22586173C2semidirect
4. #22586168C2semidirect
5. C71×#1276C2semidirect
117. #1276D71semidirect
119. C71#8733semidirect

Maximal subgroups

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

subgroupindex
#225908372
#225866522
#225861732
#225861682
C71×#12762
#873371