Finite Group Databasebeta

← #31895202

Subgroups — #31895202

Order 18176.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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#22590943C211yesmaximal, characteristic, complemented
2#22588420C211yesmaximal, characteristic, complemented
3#22586869C212yesmaximal
4#22586193C212yesmaximal, complemented
5C71×#2504C211yesmaximal, characteristic
6#17985963C2212yes—
7#17985885—24no—
8#17985862C2212yes—
9#17985859—21no—
10C4×C22⋊Dic71—21no—
11#17985768—22no—
12#17985733—24no—
13C2×#121×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#397×C71—22no—
16#390×C71C412yescomplemented
17#386×C71—21no—
18#386×C71—21no—
19#382×C71C412yescomplemented
20C4×C22⋊C4×C71C2211yescharacteristic
21C2×#11765849—22no—
22C2×C4⋊Dic71—22no—
23C2×C22⋊Dic71—22no—
24C2×C4×Dic71—22no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C71⋊#95—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C71⋊#95D414yes—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C23×C4×C71C2311yescharacteristic
36#121×C71—22no—
37#121×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C412yes—
42C2×C4⋊C4×C71D412yes—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71—22no—
47C2×C42×C71C2×C412yes—
48#95×C71—22no—
49#95×C71—22no—
50#95×C71—21no—
51C22×Dic71—24no—
52#11765849—44no—
53C2×C71⋊C8—42no—
54C4⋊Dic71—82no—
55C22⋊Dic71—44no—
56C4×Dic71—82no—
57C24×C71—21no—
58C24×C71C2×D411yescharacteristic
59C2×D4×C71—22no—
60C2×D4×C71—42no—
61C2×D4×C71—44no—
62C2×D4×C71—41no—
63C2×D4×C71—41no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—42no—
71C22×C4×C71—81no—
72C22×C4×C71—81no—
73C22×C4×C71C22⋊C412yes—
74C22×C4×C71C22⋊C412yes—
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D412yes—
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71#4112yes—
79C4⋊C4×C71—24no—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—22no—
82C22⋊C4×C71—42no—
83C22⋊C4×C71—42no—
84C22⋊C4×C71—82no—
85C22⋊C4×C71—82no—
86C42×C71—42no—
87C23×C71—22no—
88C23×C71—22no—
89C23×C71—41no—
90C23×C71—42no—
91C23×C71—42no—
92C23×C71—41no—
93C23×C71—42no—
94C23×C71C2×C22⋊C411yescharacteristic
95C23×C71#12011yescharacteristic
96C23×C71#12311yescharacteristic
97D4×C71—82no—
98D4×C71—82no—
99D4×C71—82no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—42no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—81no—
116C2×C4×C71—81no—
117C2×C4×C71C4×D412yes—
118C2×C4×C71#12112yes—
119C2×Dic71—44no—
120C2×Dic71—44no—
121C2×Dic71—44no—
122C71⋊C8—82no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—44no—
131C22×C71—84no—
132C22×C71—81no—
133C22×C71—81no—
134C22×C71—81no—
135C22×C71#38611yescharacteristic
136C22×C71#41012yes—
137C22×C71#45412yes—
138C4×C71—22no—
139C4×C71—22no—
140C4×C71—42no—
141C4×C71—42no—
142C4×C71—82no—
143C4×C71—81no—
144Dic71—84no—
145#9465—711nomaximal, Sylow
146C2×C71—22no—
147C2×C71—22no—
148C2×C71—41no—
149C2×C71—161no—
150C2×C71#188811yescharacteristic
151C2×C71#189911yescharacteristic
152C2×C71#191111yescharacteristic
153#2504D7111yescharacteristic
154#2313—711no—
155#1821—711no—
156#1510—712no—
157#1284—712no—
158C71#946511yescharacteristic, complemented, Sylow
159C2×#121C2×D7111yescharacteristic
160C22×C4⋊C4C2×D7111yescharacteristic
161#407—712no—
162#397—22no—
163#390Dic7112yescomplemented
164#387—1424no—
165#386—21no—
166#386—21no—
167#382Dic7112yescomplemented
168#381—712no—
169#380—1421no—
170C4×C22⋊C4—1421no—
171C4×C22⋊C4C2×D7111yescharacteristic
172#343—1422no—
173#327—1424no—
174C22×D4—21no—
175C22×D4C2×Dic7111yescharacteristic
176C23×C4C22⋊D7111yescharacteristic
177C23×C4C22⋊D7111yescharacteristic
178C23×C4C22×D7111yescharacteristic
179C2×M16—1422no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—1422no—
185C2×C4⋊C4C2×Dic7111yescharacteristic
186C2×C4⋊C4C2×Dic7112yes—
187C2×C4⋊C4C22⋊D7112yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1422no—
193C2×C42—1422no—
194C2×C42C2×Dic7112yes—
195C22⋊C8—1422no—
196C22⋊C8—1422no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—714no—
201#95—1422no—
202#95—1422no—
203#95—1422no—
204C24—21no—
205C24C2×C22⋊D7111yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—1424no—
221C22×C4C22⋊Dic7112yes—
222C22×C4C22⋊Dic7112yes—
223C22×C4C22×Dic7111yescharacteristic
224C22×C4D4×D7112yes—
225C22×C4C2×C22⋊D7111yescharacteristic
226C22×C4#1176586412yes—
227M16—2844no—
228C2×C8—2842no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—5682no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—2844no—
238C42—42no—
239C42—5682no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23C2×C22⋊Dic7111yescharacteristic
248C23#1455609611yescharacteristic
249C23#1455610911yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—2844no—
271C2×C4—2844no—
272C2×C4—2844no—
273C2×C4D4×Dic7112yes—
274C2×C4#1455610012yes—
275C8—5682no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#1798587911yescharacteristic
289C22#1798597312yes—
290C22#1798617112yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—81no—
297C4—5684no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#2258874411yescharacteristic
303C2#2258879911yescharacteristic
304C2#2258887311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2504C2
socleC22×C71#386
prime radicalC2×#121×C71C22
cosocleC23×C4×C71C23
hypercenter#2504D71
hypocenterC71#9465
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. #22590943C2semidirect
2. #22588420C2semidirect
4. #22586193C2semidirect
16. #390×C71C4semidirect
19. #382×C71C4semidirect
158. C71#9465semidirect
163. #390Dic71semidirect
167. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225884202
#225868692
#225861932
C71×#25042
#946571