Finite Group Databasebeta

← #31894848

Subgroups — #31894848

Order 18176.

514 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 123 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#22597381C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22587021C212yesmaximal, complemented
4#22586243C212yesmaximal, complemented
5C71×#2563C211yesmaximal, characteristic
6#17986077—24no—
7#17986052C2212yes—
8#17986023C2212yes—
9C2×#14556045—21no—
10#17985963—22no—
11#17985768—21no—
12#17985748—24no—
13C2×C4⋊Q8×C71C2211yescharacteristic
14C2×C4×D4×C71C2211yescharacteristic
15#393×C71C414yes—
16#390×C71—21no—
17#382×C71—21no—
18#380×C71—22no—
19C4×C4⋊C4×C71C2211yescharacteristic
20C2×#11765849—24no—
21C2×#11765849—22no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)D414yes—
25#14556045—42no—
26C22⋊(C71⋊C8)—24no—
27C22⋊(C71⋊C8)—22no—
28C4×C71⋊C8—22no—
29C22×Q8×C71C2×C412yes—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71D412yes—
32C4⋊Q8×C71—22no—
33C4⋊Q8×C71—22no—
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C22⋊C4×C71—24no—
41C2×C22⋊C4×C71C2×C412yes—
42C2×C42×C71—21no—
43C2×C42×C71C2311yescharacteristic
44#95×C71—24no—
45#95×C71C2×C412yes—
46#11765849—42no—
47#11765849—42no—
48#11765849—84no—
49C2×C71⋊C8—24no—
50C2×C71⋊C8—42no—
51C24×C71C22⋊C412yes—
52C2×Q8×C71—24no—
53C2×Q8×C71—82no—
54C2×D4×C71—22no—
55C2×D4×C71—21no—
56C2×D4×C71—21no—
57C2×D4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—44no—
63C22×C4×C71C22⋊C412yes—
64C22×C4×C71C22×C411yescharacteristic
65C22×C4×C71C2×D411yescharacteristic
66C22×C4×C71C2×D411yescharacteristic
67C4⋊C4×C71—22no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—41no—
72C22⋊C4×C71—24no—
73C22⋊C4×C71—84no—
74C42×C71—41no—
75C42×C71—41no—
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
84Q8×C71—84no—
85D4×C71—42no—
86D4×C71—42no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—41no—
90C2×C4×C71—44no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—44no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—82no—
98C2×C4×C71C4×D412yes—
99C2×C4×C71#12011yescharacteristic
100C2×C4×C71#12112yes—
101C2×C4×C71#12311yescharacteristic
102C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C71#13711yescharacteristic
104C2×C4×C71#13712yes—
105C71⋊C8—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—82no—
121C4×C71—82no—
122C4×C71#45512yes—
123C4×C71#46812yes—
124C4×C71#47012yes—
125#9399—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71—82no—
129C2×C71#188511yescharacteristic
130C2×C71#189411yescharacteristic
131C2×C71#191411yescharacteristic
132#2975—711no—
133#2563D7111yescharacteristic
134#1800—711no—
135#1548—712no—
136#1303—712no—
137C71#939911yescharacteristic, complemented, Sylow
138C2×C4⋊Q8C2×D7111yescharacteristic
139C2×C4×D4C2×D7111yescharacteristic
140#436—1424no—
141#431—712no—
142#423—712no—
143C2×#100—1421no—
144#407—1422no—
145#393Dic7114yes—
146#390—21no—
147#382—21no—
148#380—22no—
149C4×C4⋊C4C2×D7111yescharacteristic
150#343—1421no—
151#333—1424no—
152C22×Q8C2×Dic7112yes—
153C22×D4C22⋊D7111yescharacteristic
154C23×C4C22⋊D7112yes—
155C2×M16—1422no—
156C2×M16—1424no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22⋊D7111yescharacteristic
165C2×C22⋊C4—24no—
166C2×C22⋊C4C2×Dic7112yes—
167C2×C42—21no—
168C2×C42C22×D7111yescharacteristic
169C4⋊C8—714no—
170C4⋊C8—1422no—
171C4⋊C8—1422no—
172#100—2842no—
173C22⋊C8—1424no—
174C22⋊C8—1422no—
175C4×C8—1422no—
176#95—24no—
177#95C2×Dic7112yes—
178C24C22⋊Dic7112yes—
179C2×Q8—24no—
180C2×Q8—82no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—44no—
190C22×C4C22⋊Dic7112yes—
191C22×C4C22×Dic7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×C22⋊D7111yescharacteristic
194M16—2842no—
195M16—2842no—
196M16—5684no—
197C2×C8—2842no—
198C2×C8—1424no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—41no—
204C22⋊C4—24no—
205C22⋊C4—84no—
206C42—41no—
207C42—41no—
208C42D4×D7112yes—
209C42#1176586412yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic7111yescharacteristic
216Q8—84no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4D4×Dic7112yes—
231C2×C4#1455609611yescharacteristic
232C2×C4#1455610012yes—
233C2×C4#1455610911yescharacteristic
234C2×C4#1455615311yescharacteristic
235C2×C4#1455615811yescharacteristic
236C2×C4#1455615912yes—
237C8—2844no—
238C8—5682no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1798587911yescharacteristic
247C22#1798600011yescharacteristic
248C22#1798600811yescharacteristic
249C4—21no—
250C4—22no—
251C4—42no—
252C4—82no—
253C4—82no—
254C4#1798617612yes—
255C4#1798624712yes—
256C4#1798625512yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2258872911yescharacteristic
261C2#2258877411yescharacteristic
262C2#2258888811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2563C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2563D71
hypocenterC71#9399
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #22587021C2semidirect
4. #22586243C2semidirect
137. C71#9399semidirect

Maximal subgroups

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

subgroupindex
#225973812
#225883322
#225870212
#225862432
C71×#25632
#939971