Finite Group Databasebeta

← #31894847

Subgroups — #31894847

Order 18176.

514 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 119 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#22593513C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22587020C212yesmaximal, complemented
4#22586242C212yesmaximal, complemented
5C71×#2975C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986023C2212yes—
8C2×#14556045—21no—
9#17985911C414yes—
10#17985902—21no—
11#17985864—21no—
12#17985859—22no—
13C4×C4⋊Dic71C2211yescharacteristic
14#17985768—21no—
15#17985747—24no—
16C2×C4×D4×C71C2211yescharacteristic
17#436×C71—24no—
18#431×C71C2212yes—
19#407×C71—22no—
20C22×Dic142C2×C412yes—
21C2×#11765849—22no—
22#14556128—22no—
23#14556128—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C22⋊Dic71—24no—
28C2×C4×Dic71—21no—
29C4⋊(C71⋊C8)—22no—
30C4⋊(C71⋊C8)—22no—
31#14556045—42no—
32C71⋊#95—24no—
33C71⋊#95C2×C412yes—
34C22×D4×C71D411yescharacteristic
35C23×C4×C71D412yes—
36C2×M16×C71—24no—
37C4×D4×C71—22no—
38C4×D4×C71—22no—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C22⋊C4×C71C2×C412yes—
41C2×C42×C71C2311yescharacteristic
42C4⋊C8×C71D414yes—
43C22⋊C8×C71—24no—
44C22⋊C8×C71—22no—
45C4×C8×C71—22no—
46C2×Dic142—24no—
47C2×Dic142—82no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50#11765849—42no—
51#11765849—42no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—42no—
54C4⋊Dic71—41no—
55C4⋊Dic71—42no—
56C4⋊Dic71—41no—
57C22⋊Dic71—84no—
58C4×Dic71—41no—
59C4×Dic71—41no—
60C24×C71C22⋊C412yes—
61C2×D4×C71—22no—
62C2×D4×C71—21no—
63C2×D4×C71—22no—
64C2×D4×C71—21no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—44no—
68C22×C4×C71C22⋊C412yes—
69C22×C4×C71C22×C411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72M16×C71—84no—
73C2×C8×C71—24no—
74C4⋊C4×C71—22no—
75C22⋊C4×C71—24no—
76C42×C71C2×D412yes—
77C42×C71#4112yes—
78C23×C71—22no—
79C23×C71—22no—
80C23×C71—22no—
81C23×C71—42no—
82C23×C71—42no—
83C23×C71C2×C22⋊C411yescharacteristic
84D4×C71—42no—
85D4×C71—42no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—41no—
89C2×C4×C71—44no—
90C2×C4×C71—42no—
91C2×C4×C71—82no—
92C2×C4×C71C4×D412yes—
93C2×C4×C71#12011yescharacteristic
94C2×C4×C71#12112yes—
95C2×C4×C71#12311yescharacteristic
96C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
97C2×C4×C71#13712yes—
98C2×C4×C71#13711yescharacteristic
99C2×Dic71—42no—
100C2×Dic71—44no—
101C2×Dic71—42no—
102C2×Dic71—42no—
103C2×Dic71—42no—
104Dic142—84no—
105C8×C71—44no—
106C71⋊C8—82no—
107C22×C71—21no—
108C22×C71—21no—
109C22×C71—44no—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71—42no—
113C22×C71—82no—
114C22×C71#38611yescharacteristic
115C22×C71#41711yescharacteristic
116C22×C71#41911yescharacteristic
117C4×C71—21no—
118C4×C71—22no—
119C4×C71—42no—
120C4×C71#45512yes—
121C4×C71#46812yes—
122C4×C71#47012yes—
123Dic71—82no—
124Dic71—82no—
125#9399—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71—82no—
129C2×C71#188511yescharacteristic
130C2×C71#189411yescharacteristic
131C2×C71#191411yescharacteristic
132#2975D7111yescharacteristic
133#2563—711no—
134#1800—711no—
135#1548—712no—
136#1303—712no—
137C71#939911yescharacteristic, complemented, Sylow
138C2×C4⋊Q8—711no—
139C2×C4×D4C2×D7111yescharacteristic
140#436—24no—
141#431C2×D7112yes—
142#423—712no—
143C2×#100—1421no—
144#407—22no—
145#393—714no—
146#390—1421no—
147#382—1421no—
148#380—1422no—
149C4×C4⋊C4—711no—
150#343—1421no—
151#333—1424no—
152C22×Q8—712no—
153C22×D4D28411yescharacteristic
154C23×C4C22⋊D7112yes—
155C2×M16—24no—
156C2×M16—1422no—
157C4⋊Q8—1422no—
158C4⋊Q8—1422no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—1421no—
162C2×C4⋊C4—1421no—
163C2×C4⋊C4—1421no—
164C2×C4⋊C4D28411yescharacteristic
165C2×C22⋊C4—1424no—
166C2×C22⋊C4C4×D7112yes—
167C2×C42—1421no—
168C2×C42C22×D7111yescharacteristic
169C4⋊C8—1422no—
170C4⋊C8—1422no—
171C4⋊C8C22⋊D7114yes—
172#100—2842no—
173C22⋊C8—24no—
174C22⋊C8—22no—
175C4×C8—22no—
176#95—712no—
177#95—1424no—
178C24(C2×D71)⋊C412yes—
179C2×Q8—5682no—
180C2×Q8—1424no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—1422no—
189C22×C4—1422no—
190C22×C4(C2×D71)⋊C412yes—
191C22×C4C2×C4×D7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×D28411yescharacteristic
194M16—84no—
195M16—2842no—
196M16—2842no—
197C2×C8—24no—
198C2×C8—2842no—
199C4⋊C4—22no—
200C4⋊C4—2842no—
201C4⋊C4—2841no—
202C4⋊C4—2842no—
203C4⋊C4—2841no—
204C22⋊C4—24no—
205C22⋊C4—5684no—
206C42—2841no—
207C42—2841no—
208C42C2×C22⋊D7112yes—
209C42#1176586512yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×(C2×D71)⋊C411yescharacteristic
216Q8—5684no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—2842no—
226C2×C4—2844no—
227C2×C4—2842no—
228C2×C4—2842no—
229C2×C4—2842no—
230C2×C4C4×C22⋊D7112yes—
231C2×C4#1455609711yescharacteristic
232C2×C4#1455610212yes—
233C2×C4#1455611111yescharacteristic
234C2×C4#1455615611yescharacteristic
235C2×C4#1455616112yes—
236C2×C4#1455616111yescharacteristic
237C8—44no—
238C8—5682no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1798587811yescharacteristic
247C22#1798600111yescharacteristic
248C22#1798600911yescharacteristic
249C4—21no—
250C4—22no—
251C4—42no—
252C4—5682no—
253C4—5682no—
254C4#1798617712yes—
255C4#1798624512yes—
256C4#1798625712yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2258872811yescharacteristic
261C2#2258877311yescharacteristic
262C2#2258888711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2975C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2975D71
hypocenterC71#9399
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #22587020C2semidirect
4. #22586242C2semidirect
137. C71#9399semidirect

Maximal subgroups

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

subgroupindex
#225935132
#225883322
#225870202
#225862422
C71×#29752
#939971