Finite Group Databasebeta

← #31890358

Subgroups — #31890358

Order 18176.

486 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×(C22⋊D71)⋊C8C211yesmaximal, characteristic
2#22588376C212yesmaximal, complemented
3#22586676C212yesmaximal
4#22586217C211yesmaximal, characteristic, complemented
5C71×#1293C211yesmaximal, characteristic, complemented
6C23×C71⋊C8C2211yescharacteristic
7C22×C8×D71C411yescharacteristic
8C2×C4×C22⋊D71C2211yescharacteristic
9(C22⋊D71)⋊C8—22no—
10(C22⋊D71)⋊C8—22no—
11C2×#14556063—22no—
12C2×Dic71⋊C8C411yescharacteristic
13#17985963C2212yes—
14C2×(C2×D71)⋊C8C411yescharacteristic
15C2×C8×Dic71C411yescharacteristic
16#17985763—22no—
17#17985724—22no—
18#17985722—22no—
19#407×C71C2212yes—
20C2×C22⋊C8×C71C2211yescharacteristic
21C22×C22⋊D71Q811yescharacteristic
22C22×C4×D71C2×C411yescharacteristic
23C2×#11765849—22no—
24C22×C71⋊C8—42no—
25C22×C71⋊C8C2×C411yescharacteristic
26C22×C71⋊C8C2×C411yescharacteristic
27C22×C71⋊C8D412yes—
28C22×C71⋊C8D412yes—
29C2×C8×D71—21no—
30C2×C8×D71—21no—
31C2×C8×D71—42no—
32C4×C22⋊D71—22no—
33C4×C22⋊D71—22no—
34C2×Dic71⋊C4D411yescharacteristic
35C2×C22⋊Dic71Q811yescharacteristic
36C2×(C2×D71)⋊C4D411yescharacteristic
37C2×C4×Dic71C2×C411yescharacteristic
38#14556063—44no—
39Dic71⋊C8—22no—
40C22⋊(C71⋊C8)—22no—
41C22⋊(C71⋊C8)—22no—
42(C2×D71)⋊C8—22no—
43C8×Dic71—21no—
44C8×Dic71—21no—
45C23×C4×C71C2311yescharacteristic
46C2×M16×C71—22no—
47C22×C8×C71C2×C411yescharacteristic
48C22×C8×C71C2×C411yescharacteristic
49C22⋊C8×C71—22no—
50C22⋊C8×C71—22no—
51C22⋊C8×C71—22no—
52C23×D71C4⋊C411yescharacteristic
53C2×C22⋊D71—22no—
54C2×C22⋊D71—22no—
55C2×C22⋊D71—22no—
56C22×Dic71C4⋊C411yescharacteristic
57C22×Dic71C4⋊C411yescharacteristic
58C22×Dic71C4⋊C411yescharacteristic
59C2×C4×D71—21no—
60C2×C4×D71—21no—
61C2×C4×D71—41no—
62C2×C4×D71—41no—
63#11765849—42no—
64#11765849—42no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—22no—
67C2×C71⋊C8—41no—
68C2×C71⋊C8—42no—
69C2×C71⋊C8—42no—
70C2×C71⋊C8C22⋊C411yescharacteristic
71C2×C71⋊C8C22⋊C412yes—
72C2×C71⋊C8C22⋊C411yescharacteristic
73C8×D71—42no—
74C8×D71—42no—
75Dic71⋊C4—22no—
76C22⋊Dic71—22no—
77(C2×D71)⋊C4—22no—
78C4×Dic71—21no—
79C4×Dic71—21no—
80C24×C71C2×Q811yescharacteristic
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D412yes—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4112yes—
87M16×C71—42no—
88M16×C71—42no—
89C2×C8×C71—21no—
90C2×C8×C71—21no—
91C2×C8×C71—41no—
92C2×C8×C71—41no—
93C2×C8×C71—42no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71C2×C4⋊C411yescharacteristic
99C23×C71C22⋊Q812yes—
100C22×D71—21no—
101C22×D71—21no—
102C22×D71—41no—
103C22×D71—41no—
104C22×D71#10812yescomplemented
105C2×C4×C71—21no—
106C2×C4×C71—22no—
107C2×C4×C71—21no—
108C2×C4×C71—22no—
109C2×C4×C71—41no—
110C2×C4×C71—42no—
111C2×C4×C71C2×C22⋊C411yescharacteristic
112C2×C4×C71#11711yescharacteristic
113C2×C4×C71#12112yes—
114C22⋊D71—28no—
115C2×Dic71—21no—
116C2×Dic71—21no—
117C2×Dic71—21no—
118C2×Dic71—21no—
119C2×Dic71—41no—
120C2×Dic71—41no—
121C2×Dic71#10812yescomplemented
122C4×D71—42no—
123C4×D71—41no—
124C4×D71—41no—
125C8×C71—41no—
126C8×C71—41no—
127C8×C71—82no—
128C71⋊C8—21no—
129C71⋊C8—22no—
130C71⋊C8—82no—
131C71⋊C8#9912yes—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—22no—
135C22×C71—22no—
136C22×C71—22no—
137C22×C71—42no—
138C22×C71—42no—
139C22×C71#38011yescharacteristic
140C22×C71C2×#10811yescharacteristic
141C22×C71C2×#10811yescharacteristic
142C2×D71—22no—
143C2×D71—42no—
144C2×D71—41no—
145C2×D71—41no—
146C2×D71—42no—
147C4×C71—22no—
148C4×C71—21no—
149C4×C71C2×#9911yescharacteristic
150C4×C71#41011yescharacteristic
151Dic71—22no—
152Dic71—41no—
153Dic71—41no—
154#8611—711nomaximal, Sylow
155C2×C71—21no—
156C2×C71—21no—
157C2×C71—42no—
158C2×C71#179911yescharacteristic
159C2×C71#181011yescharacteristic
160C2×C71#181311yescharacteristic
161D71—42no—
162C2×C8×D4—711no—
163#1810—712no—
164#1465—712no—
165#1293—711no—
166#1293D7111yescharacteristic, complemented
167C71#861111yescharacteristic, complemented, Sylow
168C23×C8—711no—
169C23×C8—711no—
170C2×C4×D4—711no—
171C8×D4—1422no—
172C8×D4—1422no—
173C2×#108—1422no—
174C2×C4⋊C8—711no—
175#407—712no—
176#407C2×D7112yes—
177C2×C22⋊C8—711no—
178C2×C22⋊C8C2×D7111yescharacteristic
179C2×C4×C8—711no—
180#341—1422no—
181#324—1422no—
182#323—1422no—
183C22×D4—711no—
184C23×C4—711no—
185C23×C4C22×D7111yescharacteristic
186C2×M16—22no—
187C2×M16—1422no—
188C22×C8—2842no—
189C22×C8—2842no—
190C22×C8—711no—
191C22×C8—712no—
192C22×C8—711no—
193C22×C8—712no—
194C22×C8—1421no—
195C22×C8—1421no—
196C22×C8C4×D7111yescharacteristic
197C22×C8C4×D7111yescharacteristic
198C4×D4—1422no—
199C4×D4—1422no—
200C2×C4⋊C4—711no—
201C2×C22⋊C4—711no—
202C2×C22⋊C4—711no—
203C2×C42—711no—
204#108—2844no—
205C4⋊C8—1422no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—1422no—
210C22⋊C8—1422no—
211C22⋊C8—1422no—
212C4×C8—1421no—
213C4×C8—1421no—
214C24—711no—
215C24Q8×D7111yescharacteristic
216C2×D4—1422no—
217C2×D4—1422no—
218C2×D4—1422no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—2841no—
222C22×C4—2841no—
223C22×C4—711no—
224C22×C4—711no—
225C22×C4—711no—
226C22×C4—1421no—
227C22×C4—1421no—
228C22×C4C2×C4×D7111yescharacteristic
229C22×C4D4×D7112yes—
230C22×C4D4×D7111yescharacteristic
231C22×C4#1176586512yes—
232M16—42no—
233M16—42no—
234M16—2842no—
235M16—2842no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—2842no—
242C2×C8—2842no—
243C2×C8—2842no—
244C2×C8—2842no—
245C2×C8—2841no—
246C2×C8—711no—
247C2×C8—711no—
248C2×C8—712no—
249C2×C8—1424no—
250C2×C8—1422no—
251C4⋊C4—1422no—
252C22⋊C4—1422no—
253C22⋊C4—1422no—
254C42—1421no—
255C42—1421no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—2841no—
261C23—2841no—
262C23—712no—
263C23—1421no—
264C23—1421no—
265C23C4⋊C4×D7111yescharacteristic
266C23#1455610612yes—
267D4—1428no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—2842no—
275C2×C4—2841no—
276C2×C4—2841no—
277C2×C4—2841no—
278C2×C4—2841no—
279C2×C4—712no—
280C2×C4—1421no—
281C2×C4—1421no—
282C2×C4—1421no—
283C2×C4—1421no—
284C2×C4C22⋊C4×D7111yescharacteristic
285C2×C4#1455608511yescharacteristic
286C2×C4#1455610112yes—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—5682no—
291C8—712no—
292C8—1421no—
293C8—1422no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—2842no—
302C22—2842no—
303C22—2841no—
304C22—2841no—
305C22—1422no—
306C22#1798585811yescharacteristic
307C22#108×D7111yescharacteristic
308C22#108×D7111yescharacteristic
309C4—22no—
310C4—21no—
311C4—2841no—
312C4—2841no—
313C4—1422no—
314C4#99×D7111yescharacteristic
315C4#1798597111yescharacteristic
316C2—21no—
317C2—21no—
318C2—42no—
319C2—2842no—
320C2#2258832711yescharacteristic
321C2#2258837711yescharacteristic
322C2#2258838911yescharacteristic
subgroupG/H
centerC22#17985858
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#1293C2
socleC22×C71#380
prime radicalC22×C22⋊D71Q8
cosocleC23×C4×C71C23
hypercenter#1293D71
hypocenterC71#8611
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #22588376C2semidirect
4. #22586217C2semidirect
5. C71×#1293C2semidirect
104. C22×D71#108semidirect
121. C2×Dic71#108semidirect
166. #1293D71semidirect
167. C71#8611semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D71)⋊C82
#225883762
#225866762
#225862172
C71×#12932
#861171