Finite Group Databasebeta

← #31885578

Subgroups — #31885578

Order 18176.

440 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 111 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#22590909C211yesmaximal, characteristic
2#22588482C212yesmaximal
3#22586166C211yesmaximal, characteristic
4#22586165C212yesmaximal
5C71×#1272C211yesmaximal, characteristic
6C2×#14556126C411yescharacteristic
7C2×Q8×Dic71C411yescharacteristic
8C2×#14556093C411yescharacteristic
9#17985905C2211yescharacteristic
10#17985896—22no—
11#17985868C411yescharacteristic
12#17985850—22no—
13C42×Dic71C2211yescharacteristic
14#17985754C2212yes—
15#17985753—22no—
16#17985753—22no—
17#391×C71C2211yescharacteristic
18#336×C71C2212yes—
19C22×Dic142C2×C411yescharacteristic
20C22×C71⋊C8—22no—
21#14556126—22no—
22#14556126C814yescomplemented
23Q8×Dic71—42no—
24#14556093—42no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4—21no—
30C2×C4×Dic71—21no—
31C2×C4×Dic71—21no—
32C2×C4×Dic71C2×C411yescharacteristic
33C2×C4×Dic71C2×C411yescharacteristic
34C2×C4×Dic71D412yes—
35C2×C4×Dic71D412yes—
36C71⋊#95—22no—
37C22×Q8×C71C2×C411yescharacteristic
38C22×C8×C71—22no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C42×C71C2311yescharacteristic
41#95×C71—22no—
42C2×Dic142—21no—
43C2×Dic142—41no—
44C2×Dic142—41no—
45C2×Dic142C2×C812yes—
46C22×Dic71—21no—
47C22×Dic71—21no—
48C22×Dic71C22⋊C411yescharacteristic
49C22×Dic71C22⋊C412yes—
50C22×Dic71C22⋊C411yescharacteristic
51C2×C71⋊C8—42no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—42no—
54C4⋊Dic71—42no—
55C4⋊Dic71—41no—
56Dic71⋊C4—28no—
57Dic71⋊C4—44no—
58Dic71⋊C4—42no—
59C4×Dic71—22no—
60C4×Dic71—22no—
61C4×Dic71—22no—
62C4×Dic71—42no—
63C4×Dic71—42no—
64C4×Dic71—42no—
65C4×Dic71C2×C812yes—
66C4×Dic71M1612yes—
67C2×Q8×C71—21no—
68C2×Q8×C71—41no—
69C2×Q8×C71—41no—
70C2×Q8×C71C2×C812yes—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D412yes—
75C2×C8×C71—42no—
76C2×C8×C71—42no—
77C2×C8×C71—42no—
78C4⋊C4×C71—41no—
79C42×C71—41no—
80C23×C71C2×C22⋊C411yescharacteristic
81Q8×C71—42no—
82Q8×C71—42no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—41no—
91C2×C4×C71—42no—
92C2×C4×C71C22×C811yescharacteristic
93C2×C4×C71C2×M1611yescharacteristic
94C2×C4×C71#13712yes—
95C2×C4×C71#13712yes—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—41no—
103C2×Dic71—41no—
104C2×Dic71—41no—
105C2×Dic71—42no—
106C2×Dic71C22⋊C814yes—
107C2×Dic71#10012yes—
108Dic142—42no—
109Dic142—42no—
110C8×C71—82no—
111C71⋊C8—82no—
112C22×C71C2×C22⋊C811yescharacteristic
113C22×C71#41011yescharacteristic
114C22×C71C2×#10011yescharacteristic
115C22×C71#41912yes—
116C22×C71#42112yes—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—42no—
122C4×C71—81no—
123Dic71—22no—
124Dic71—24no—
125Dic71—41no—
126Dic71—42no—
127Dic71—81no—
128#7605—711nomaximal, Sylow
129C2×C71#146011yescharacteristic
130C2×C71#147912yes—
131C2×C71#151512yes—
132C2×C71#153611yescharacteristic
133C2×C71#155811yescharacteristic
134C4×C4⋊Q8—711no—
135#1839—712no—
136#1272—711no—
137#1272—712no—
138#1272D7111yescharacteristic
139C71#760511yescharacteristic, complemented, Sylow
140C2×C4⋊Q8—711no—
141C2×C4×Q8—711no—
142C2×C4×Q8—711no—
143#391—711no—
144#391C2×D7111yescharacteristic
145#389—1422no—
146#384—711no—
147C4×C4⋊C4—1422no—
148C43—711no—
149#336—712no—
150#336—1422no—
151#336—1422no—
152#336C2×D7112yes—
153C22×Q8—711no—
154C22×Q8C4×D7111yescharacteristic
155C22×C8—22no—
156C22×C8—1422no—
157C4⋊Q8—714no—
158C4⋊Q8—1422no—
159C4×Q8—2842no—
160C4×Q8—2842no—
161C2×C4⋊C4—711no—
162C2×C4⋊C4—1421no—
163C2×C4⋊C4—1421no—
164C2×C4⋊C4—1422no—
165C2×C4⋊C4—1422no—
166C2×C4⋊C4C4×D7111yescharacteristic
167C2×C42—712no—
168C2×C42—711no—
169C2×C42—711no—
170C2×C42—712no—
171C2×C42—1421no—
172C2×C42—1421no—
173C2×C42C22×D7111yescharacteristic
174#95—22no—
175#95—1422no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—2841no—
180C2×Q8—2841no—
181C2×Q8—712no—
182C2×Q8—1421no—
183C2×Q8C8×D7112yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—711no—
187C22×C4—712no—
188C22×C4—711no—
189C22×C4—1421no—
190C22×C4—1421no—
191C22×C4C2×C4×D7111yescharacteristic
192C22×C4D4×D7112yes—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—2842no—
197C2×C8—2842no—
198C2×C8—2842no—
199C4⋊C4—41no—
200C4⋊C4—2841no—
201C4⋊C4—2844no—
202C4⋊C4—2842no—
203C4⋊C4—2842no—
204C4⋊C4—1428no—
205C42—41no—
206C42—2842no—
207C42—2842no—
208C42—2842no—
209C42—712no—
210C42—712no—
211C42—1422no—
212C42—1422no—
213C42—1422no—
214C23C22⋊C4×D7111yescharacteristic
215Q8—42no—
216Q8—42no—
217Q8—2842no—
218Q8—2842no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—2841no—
229C2×C4—2841no—
230C2×C4—2841no—
231C2×C4—2842no—
232C2×C4—714no—
233C2×C4—712no—
234C2×C4—1422no—
235C2×C4—1421no—
236C2×C4—1422no—
237C2×C4—1422no—
238C2×C4—1422no—
239C2×C4—1422no—
240C2×C4C2×C8×D7111yescharacteristic
241C2×C4M16×D7111yescharacteristic
242C2×C4#1455615712yes—
243C2×C4#1455616012yes—
244C8—82no—
245C8—5682no—
246C22C22⋊C8×D7111yescharacteristic
247C22#1798597111yescharacteristic
248C22#100×D7111yescharacteristic
249C22#1798600612yes—
250C22#1798601412yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—42no—
256C4—81no—
257C4—2842no—
258C4—2841no—
259C4—5681no—
260C4—1424no—
261C4—1422no—
262C2#2258665911yescharacteristic
263C2#2258673312yes—
264C2#2258688912yes—
265C2#2258697311yescharacteristic
266C2#2258706411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC71×#1272C2
socle, prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1272D71
hypocenterC71#7605
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
22. #14556126C8semidirect
139. C71#7605semidirect

Maximal subgroups

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

subgroupindex
#225909092
#225884822
#225861662
#225861652
C71×#12722
#760571