Finite Group Databasebeta

← #31890887

Subgroups — #31890887

Order 18176.

514 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 135 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#22590745C211yesmaximal, characteristic
2#22588317C212yesmaximal
3#22586224C212yesmaximal, complemented
4#22586223C211yesmaximal, characteristic, complemented
5C71×#1296C211yesmaximal, characteristic, complemented
6C2×#14556086C412yes—
7C2×C42×D71C2211yescharacteristic
8C2×C4⋊(C71⋊C8)C2212yes—
9C2×(C2×D71)⋊C8—22no—
10#17985869C2211yescharacteristic
11#17985864C412yes—
12#17985856—22no—
13C4⋊C4×Dic71—22no—
14#17985753—22no—
15C2×C4⋊C8×C71C2212yes—
16#384×C71C2211yescharacteristic
17C22×C4×D71C2×C411yescharacteristic
18C22×C4×D71D412yes—
19C22×C71⋊C8—22no—
20#14556086—44no—
21C2×C4⋊Dic71—22no—
22C2×C4⋊Dic71C2×C412yes—
23C2×(C2×D71)⋊C4—22no—
24C2×C4×Dic71—22no—
25C2×C4×Dic71C2×C411yescharacteristic
26C2×C4×Dic71D412yes—
27C42×D71—21no—
28C42×D71—22no—
29C42×D71—21no—
30C4⋊(C71⋊C8)—22no—
31C4⋊(C71⋊C8)—22no—
32(C2×D71)⋊C8—44no—
33C71⋊#95—24no—
34C22×C8×C71—22no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71C2×C412yes—
37C2×C42×C71C2311yescharacteristic
38C4⋊C8×C71—22no—
39C4⋊C8×C71—22no—
40C23×D71C22⋊C411yescharacteristic
41C22×Dic71—22no—
42C22×Dic71C22⋊C412yes—
43C22×Dic71C22⋊C411yescharacteristic
44C2×C4×D71—22no—
45C2×C4×D71—21no—
46C2×C4×D71—21no—
47C2×C4×D71—22no—
48C2×C4×D71—22no—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—22no—
53C2×C4×D71—22no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—42no—
58C4⋊Dic71—42no—
59C4⋊Dic71—42no—
60(C2×D71)⋊C4—44no—
61C4×Dic71—21no—
62C4×Dic71—21no—
63C4×Dic71—22no—
64C4×Dic71—22no—
65C4×Dic71—44no—
66C22×C4×C71—22no—
67C22×C4×C71C22×C411yescharacteristic
68C22×C4×C71C2×D412yes—
69C2×C8×C71—42no—
70C2×C8×C71—42no—
71C2×C8×C71—42no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—42no—
74C4⋊C4×C71—42no—
75C42×C71#4111yescharacteristic
76C42×C71#4112yes—
77C42×C71#4111yescharacteristic
78C23×C71C2×C22⋊C411yescharacteristic
79C22×D71—21no—
80C22×D71—21no—
81C22×D71—22no—
82C22×D71—22no—
83C22×D71#10112yes—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—42no—
87C2×C4×C71—44no—
88C2×C4×C71#11711yescharacteristic
89C2×C4×C71#11711yescharacteristic
90C2×C4×C71#12312yes—
91C2×C4×C71#12312yes—
92C2×C4×C71#13512yes—
93C2×C4×C71#13512yes—
94C2×C4×C71AGL(1,ℤ/8)12yes—
95C2×C4×C71#13712yes—
96C2×Dic71—21no—
97C2×Dic71—21no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71—42no—
102C2×Dic71—44no—
103C2×Dic71#10012yes—
104C4×D71—24no—
105C4×D71—24no—
106C4×D71—42no—
107C4×D71—42no—
108C4×D71—42no—
109C4×D71—42no—
110C8×C71—82no—
111C71⋊C8—82no—
112C22×C71#38111yescharacteristic
113C22×C71C2×#10011yescharacteristic
114C22×C71C2×#10111yescharacteristic
115C22×C71#41612yes—
116C22×C71#41712yes—
117C2×D71—24no—
118C2×D71—22no—
119C2×D71—41no—
120C2×D71—41no—
121C2×D71—42no—
122C4×C71—21no—
123C4×C71—21no—
124C4×C71—82no—
125C4×C71#48212yes—
126C4×C71#48212yes—
127C4×C71#48512yes—
128C4×C71#48512yes—
129Dic71—22no—
130Dic71—42no—
131Dic71—82no—
132#8728—711nomaximal, Sylow
133C2×C71#154311yescharacteristic
134C2×C71#155212yes—
135C2×C71#156011yescharacteristic
136C2×C71#182111yescharacteristic
137C2×C71#182512yes—
138D71—42no—
139#2285—711no—
140#1797—712no—
141#1296—712no—
142#1296—711no—
143#1296D7111yescharacteristic, complemented
144C71#872811yescharacteristic, complemented, Sylow
145C2×#117—712no—
146C22×C42—711no—
147C2×C4⋊C8—712no—
148C2×C4⋊C8C2×D7112yes—
149C2×C22⋊C8—1422no—
150#384—711no—
151#384C2×D7111yescharacteristic
152#382—712no—
153#380—1422no—
154C4×C4⋊C4—1422no—
155#336—1422no—
156C23×C4—711no—
157C23×C4—712no—
158C22×C8—22no—
159C22×C8—1422no—
160#117—2844no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4—1422no—
164C2×C4⋊C4C4×D7112yes—
165C2×C22⋊C4—1422no—
166C2×C42—711no—
167C2×C42—712no—
168C2×C42—1422no—
169C2×C42—1421no—
170C2×C42—1422no—
171C2×C42—1421no—
172C2×C42C22×D7111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—1422no—
176C4⋊C8—1422no—
177C22⋊C8—2844no—
178#95—1424no—
179C24—711no—
180C22×C4—22no—
181C22×C4—711no—
182C22×C4—712no—
183C22×C4—1422no—
184C22×C4—1422no—
185C22×C4—1422no—
186C22×C4—1422no—
187C22×C4—1422no—
188C22×C4—1422no—
189C22×C4—1422no—
190C22×C4—1422no—
191C22×C4—1422no—
192C22×C4—1421no—
193C22×C4—1421no—
194C22×C4C2×C4×D7111yescharacteristic
195C22×C4D4×D7112yes—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—2842no—
200C2×C8—2842no—
201C2×C8—2842no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—2842no—
206C4⋊C4—2842no—
207C4⋊C4—2842no—
208C22⋊C4—2844no—
209C42—2844no—
210C42—1421no—
211C42—1421no—
212C42—1422no—
213C42—1422no—
214C42#1176586411yescharacteristic
215C42#1176586512yes—
216C42#1176586611yescharacteristic
217C23—712no—
218C23—1421no—
219C23—1421no—
220C23—1422no—
221C23—1422no—
222C23C22⋊C4×D7111yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—2842no—
228C2×C4—2842no—
229C2×C4—2842no—
230C2×C4—2842no—
231C2×C4—2842no—
232C2×C4—2844no—
233C2×C4—712no—
234C2×C4—1424no—
235C2×C4—1424no—
236C2×C4—1422no—
237C2×C4—1422no—
238C2×C4—1422no—
239C2×C4—1421no—
240C2×C4—1421no—
241C2×C4#1455608511yescharacteristic
242C2×C4#1455608611yescharacteristic
243C2×C4#1455611012yes—
244C2×C4#1455611312yes—
245C2×C4#1455614712yes—
246C2×C4#1455614912yes—
247C2×C4#1455615512yes—
248C2×C4#1455616012yes—
249C8—82no—
250C8—5682no—
251C22—2842no—
252C22—2841no—
253C22—2841no—
254C22—1422no—
255C22—1424no—
256C22#1798586111yescharacteristic
257C22#100×D7111yescharacteristic
258C22#101×D7111yescharacteristic
259C22#1798599212yes—
260C22#1798599912yes—
261C4—21no—
262C4—21no—
263C4—82no—
264C4—2842no—
265C4—5682no—
266C4—1422no—
267C4#1798632112yes—
268C4#1798632512yes—
269C4#1798633812yes—
270C4#1798634212yes—
271C2—2842no—
272C2#2258699911yescharacteristic
273C2#2258703712yes—
274C2#2258707411yescharacteristic
275C2#2258841911yescharacteristic
276C2#2258843312yes—
subgroupG/H
centerC23C22⋊C4×D71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1296C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1296D71
hypocenterC71#8728
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #22586224C2semidirect
4. #22586223C2semidirect
5. C71×#1296C2semidirect
143. #1296D71semidirect
144. C71#8728semidirect

Maximal subgroups

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

subgroupindex
#225907452
#225883172
#225862242
#225862232
C71×#12962
#872871