Finite Group Databasebeta

← #31894858

Subgroups — #31894858

Order 18176.

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

normal subgroups only

This group has 534 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#22587012C212yesmaximal, complemented
4#22586243C212yesmaximal, complemented
5C71×#2547C211yesmaximal, characteristic
6#17986077—24no—
7#17986052C2212yes—
8#17986023C2212yes—
9C2×#14556045—21no—
10#17985963—22no—
11#17985768—21no—
12#17985745—24no—
13C2×C4⋊Q8×C71C2211yescharacteristic
14C2×#124×C71—21no—
15C2×C4×D4×C71C2211yescharacteristic
16#397×C71C414yes—
17#391×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×C71—21no—
30C22×D4×C71D411yescharacteristic
31C23×C4×C71D412yes—
32C4⋊Q8×C71—22no—
33C4⋊Q8×C71—22no—
34#124×C71—44no—
35C4×D4×C71—22no—
36C4×D4×C71—22no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C412yes—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71—24no—
42C2×C22⋊C4×C71C2×C412yes—
43C2×C42×C71—21no—
44C2×C42×C71C2311yescharacteristic
45#95×C71—24no—
46#95×C71C2×C412yes—
47#11765849—42no—
48#11765849—42no—
49#11765849—84no—
50C2×C71⋊C8—24no—
51C2×C71⋊C8—42no—
52C24×C71C22⋊C412yes—
53C2×Q8×C71—42no—
54C2×Q8×C71—42no—
55C2×Q8×C71—41no—
56C2×Q8×C71—41no—
57C2×D4×C71—22no—
58C2×D4×C71—22no—
59C2×D4×C71—21no—
60C2×D4×C71—21no—
61C22×C4×C71—22no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—44no—
66C22×C4×C71C22⋊C412yes—
67C22×C4×C71C22×C411yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C4⋊C4×C71—22no—
71C4⋊C4×C71—24no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—41no—
75C22⋊C4×C71—24no—
76C22⋊C4×C71—48no—
77C42×C71—42no—
78C42×C71C2×D412yes—
79C42×C71#4112yes—
80C23×C71—22no—
81C23×C71—22no—
82C23×C71—22no—
83C23×C71—42no—
84C23×C71—42no—
85C23×C71C2×C22⋊C411yescharacteristic
86Q8×C71—82no—
87Q8×C71—82no—
88D4×C71—42no—
89D4×C71—42no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—41no—
93C2×C4×C71—44no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—82no—
102C2×C4×C71C4×D412yes—
103C2×C4×C71#12011yescharacteristic
104C2×C4×C71#12112yes—
105C2×C4×C71#12311yescharacteristic
106C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
107C2×C4×C71AGL(1,ℤ/8)12yes—
108C2×C4×C71#13711yescharacteristic
109C71⋊C8—44no—
110C71⋊C8—82no—
111C22×C71—21no—
112C22×C71—21no—
113C22×C71—44no—
114C22×C71—42no—
115C22×C71—42no—
116C22×C71—42no—
117C22×C71—82no—
118C22×C71#38611yescharacteristic
119C22×C71#41711yescharacteristic
120C22×C71#41811yescharacteristic
121C4×C71—21no—
122C4×C71—22no—
123C4×C71—42no—
124C4×C71—82no—
125C4×C71—82no—
126C4×C71#45512yes—
127C4×C71#46812yes—
128C4×C71#46912yes—
129#9401—711nomaximal, Sylow
130C2×C71—21no—
131C2×C71—21no—
132C2×C71—82no—
133C2×C71#188511yescharacteristic
134C2×C71#189511yescharacteristic
135C2×C71#191511yescharacteristic
136#2975—711no—
137#2547D7111yescharacteristic
138#1800—711no—
139#1546—712no—
140#1303—712no—
141C71#940111yescharacteristic, complemented, Sylow
142C2×C4⋊Q8C2×D7111yescharacteristic
143C2×#124—21no—
144C2×C4×D4C2×D7111yescharacteristic
145#436—1424no—
146#431—712no—
147#423—712no—
148C2×#100—1421no—
149#407—1422no—
150#397Dic7114yes—
151#391—21no—
152#380—22no—
153C4×C4⋊C4C2×D7111yescharacteristic
154#343—1421no—
155#332—1424no—
156C22×Q8—21no—
157C22×D4C22⋊D7111yescharacteristic
158C23×C4C22⋊D7112yes—
159C2×M16—1422no—
160C2×M16—1424no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163#124—44no—
164C4×D4—22no—
165C4×D4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C2×Dic7112yes—
169C2×C4⋊C4C22⋊D7111yescharacteristic
170C2×C22⋊C4—24no—
171C2×C22⋊C4C2×Dic7112yes—
172C2×C42—21no—
173C2×C42C22×D7111yescharacteristic
174C4⋊C8—714no—
175C4⋊C8—1422no—
176C4⋊C8—1422no—
177#100—2842no—
178C22⋊C8—1424no—
179C22⋊C8—1422no—
180C4×C8—1422no—
181#95—24no—
182#95C2×Dic7112yes—
183C24C22⋊Dic7112yes—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—44no—
197C22×C4C22⋊Dic7112yes—
198C22×C4C22×Dic7111yescharacteristic
199C22×C4C2×C22⋊D7111yescharacteristic
200C22×C4C2×C22⋊D7111yescharacteristic
201M16—2842no—
202M16—2842no—
203M16—5684no—
204C2×C8—2842no—
205C2×C8—1424no—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C22⋊C4—24no—
212C22⋊C4—48no—
213C42—42no—
214C42D4×D7112yes—
215C42#1176586412yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×C22⋊Dic7111yescharacteristic
222Q8—82no—
223Q8—82no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4D4×Dic7112yes—
239C2×C4#1455609611yescharacteristic
240C2×C4#1455610012yes—
241C2×C4#1455610911yescharacteristic
242C2×C4#1455615311yescharacteristic
243C2×C4#1455615412yes—
244C2×C4#1455615811yescharacteristic
245C8—2844no—
246C8—5682no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22#1798587911yescharacteristic
255C22#1798600011yescharacteristic
256C22#1798600411yescharacteristic
257C4—21no—
258C4—22no—
259C4—42no—
260C4—82no—
261C4—82no—
262C4#1798617612yes—
263C4#1798624312yes—
264C4#1798625012yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#2258872911yescharacteristic
269C2#2258877911yescharacteristic
270C2#2258889311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2547C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2547D71
hypocenterC71#9401
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #22587012C2semidirect
4. #22586243C2semidirect
141. C71#9401semidirect

Maximal subgroups

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

subgroupindex
#225973812
#225883322
#225870122
#225862432
C71×#25472
#940171