Finite Group Databasebeta

← #31885432

Subgroups — #31885432

Order 18176.

516 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 143 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#22588188C212yesmaximal
3#22586166C211yesmaximal, characteristic
4#22586165C212yesmaximal
5C71×#1272C211yesmaximal, characteristic
6C2×#14556126C411yescharacteristic
7C2×Q8×Dic71C411yescharacteristic
8C2×#14556093C411yescharacteristic
9C2×Dic71⋊C8—22no—
10C2×C8×Dic71—22no—
11#17985905C2211yescharacteristic
12#17985896—22no—
13#17985868C411yescharacteristic
14#17985850—22no—
15C42×Dic71C2211yescharacteristic
16#17985754C2212yes—
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—
36Dic71⋊C8—28no—
37C8×Dic71—44no—
38C71⋊#95—22no—
39C22×Q8×C71C2×C411yescharacteristic
40C22×C8×C71—22no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C42×C71C2311yescharacteristic
43#95×C71—22no—
44C2×Dic142—21no—
45C2×Dic142—41no—
46C2×Dic142—41no—
47C2×Dic142C2×C812yes—
48C22×Dic71—21no—
49C22×Dic71—21no—
50C22×Dic71C22⋊C411yescharacteristic
51C22×Dic71C22⋊C412yes—
52C22×Dic71C22⋊C411yescharacteristic
53C2×C71⋊C8—24no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C4⋊Dic71—42no—
57C4⋊Dic71—41no—
58Dic71⋊C4—28no—
59Dic71⋊C4—42no—
60Dic71⋊C4—44no—
61C4×Dic71—22no—
62C4×Dic71—24no—
63C4×Dic71—42no—
64C4×Dic71—42no—
65C4×Dic71C2×C812yes—
66C4×Dic71M1612yes—
67C4×Dic71SD1614yes—
68C4×Dic71Q1614yes—
69C2×Q8×C71—21no—
70C2×Q8×C71—41no—
71C2×Q8×C71—41no—
72C2×Q8×C71C2×C812yes—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D412yes—
77C2×C8×C71—24no—
78C2×C8×C71—42no—
79C2×C8×C71—42no—
80C4⋊C4×C71—41no—
81C42×C71—41no—
82C23×C71C2×C22⋊C411yescharacteristic
83Q8×C71—42no—
84Q8×C71—42no—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—41no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—42no—
94C2×C4×C71C22×C811yescharacteristic
95C2×C4×C71C2×M1611yescharacteristic
96C2×C4×C71C2×SD1612yes—
97C2×C4×C71C2×Q1612yes—
98C2×Dic71—22no—
99C2×Dic71—21no—
100C2×Dic71—22no—
101C2×Dic71—41no—
102C2×Dic71—41no—
103C2×Dic71—41no—
104C2×Dic71—42no—
105C2×Dic71C22⋊C814yes—
106C2×Dic71#10012yes—
107C2×Dic71#10314yes—
108C2×Dic71#10314yes—
109C2×Dic71#10414yes—
110Dic142—42no—
111Dic142—42no—
112C8×C71—44no—
113C71⋊C8—44no—
114C22×C71C2×C22⋊C811yescharacteristic
115C22×C71#41011yescharacteristic
116C22×C71C2×#10011yescharacteristic
117C22×C71C2×#10312yes—
118C22×C71C2×#10412yes—
119C4×C71—21no—
120C4×C71—21no—
121C4×C71—22no—
122C4×C71—22no—
123C4×C71—42no—
124C4×C71—81no—
125Dic71—42no—
126Dic71—41no—
127Dic71—81no—
128Dic71Q8⋊C818yes—
129Dic71#33314yes—
130#7571—711nomaximal, Sylow
131C2×C71#146011yescharacteristic
132C2×C71C2×Q8⋊C812yes—
133C2×C71#150312yes—
134C2×C71#152511yescharacteristic
135C2×C71C2×#33311yescharacteristic
136C4×C4⋊Q8—711no—
137C4×C4⋊C8—712no—
138#1272—711no—
139#1272—712no—
140#1272D7111yescharacteristic
141C71#757111yescharacteristic, complemented, Sylow
142C2×C4⋊Q8—711no—
143C2×C4×Q8—711no—
144C2×C4×Q8—711no—
145C2×C4⋊C8—1422no—
146C2×C4×C8—1422no—
147#391—711no—
148#391C2×D7111yescharacteristic
149#389—1422no—
150#384—711no—
151C4×C4⋊C4—1422no—
152C43—711no—
153#336—712no—
154#336C2×D7112yes—
155C22×Q8—711no—
156C22×Q8C4×D7111yescharacteristic
157C22×C8—22no—
158C22×C8—1422no—
159C4⋊Q8—714no—
160C4⋊Q8—1422no—
161C4×Q8—2842no—
162C4×Q8—2842no—
163C2×C4⋊C4—711no—
164C2×C4⋊C4—1421no—
165C2×C4⋊C4—1421no—
166C2×C4⋊C4—1422no—
167C2×C4⋊C4—1422no—
168C2×C4⋊C4C4×D7111yescharacteristic
169C2×C42—711no—
170C2×C42—712no—
171C2×C42—711no—
172C2×C42—712no—
173C2×C42—1421no—
174C2×C42—1421no—
175C2×C42C22×D7111yescharacteristic
176C4⋊C8—1428no—
177C4×C8—2844no—
178#95—22no—
179#95—1422no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—2841no—
184C2×Q8—2841no—
185C2×Q8—712no—
186C2×Q8—1421no—
187C2×Q8C8×D7112yes—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—711no—
191C22×C4—712no—
192C22×C4—711no—
193C22×C4—1421no—
194C22×C4—1421no—
195C22×C4C2×C4×D7111yescharacteristic
196C22×C4D4×D7112yes—
197C2×C8—24no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—2842no—
201C2×C8—2842no—
202C2×C8—1424no—
203C4⋊C4—41no—
204C4⋊C4—2841no—
205C4⋊C4—2844no—
206C4⋊C4—2842no—
207C4⋊C4—2842no—
208C4⋊C4—1428no—
209C42—41no—
210C42—2842no—
211C42—2842no—
212C42—712no—
213C42—712no—
214C42—714no—
215C42—714no—
216C42—1424no—
217C42—1422no—
218C23C22⋊C4×D7111yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—2842no—
222Q8—2842no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—2841no—
233C2×C4—2841no—
234C2×C4—2841no—
235C2×C4—2842no—
236C2×C4—714no—
237C2×C4—712no—
238C2×C4—714no—
239C2×C4—714no—
240C2×C4—714no—
241C2×C4—1422no—
242C2×C4—1421no—
243C2×C4—1422no—
244C2×C4C2×C8×D7111yescharacteristic
245C2×C4M16×D7111yescharacteristic
246C2×C4SD16×D7112yes—
247C2×C4Q16×D7112yes—
248C8—44no—
249C8—2844no—
250C22C22⋊C8×D7111yescharacteristic
251C22#1798597111yescharacteristic
252C22#100×D7111yescharacteristic
253C22#103×D7112yes—
254C22#104×D7112yes—
255C4—21no—
256C4—21no—
257C4—22no—
258C4—22no—
259C4—42no—
260C4—81no—
261C4—2842no—
262C4—2841no—
263C4—5681no—
264C4—718no—
265C4—714no—
266C2#2258665911yescharacteristic
267C2Q8⋊C8×D7112yes—
268C2#2258683712yes—
269C2#2258693011yescharacteristic
270C2#333×D7111yescharacteristic
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#7571
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
141. C71#7571semidirect

Maximal subgroups

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

subgroupindex
#225909092
#225881882
#225861662
#225861652
C71×#12722
#757171