Finite Group Databasebeta

← #31893237

Subgroups — #31893237

Order 18176.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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#22596671C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic, complemented
3C2×#17985732C212yesmaximal
4#22586219C212yesmaximal, complemented
5C2×#383×C71C211yesmaximal, characteristic, complemented
6C2×#14556111—21no—
7C2×#14556102C2211yescharacteristic
8C2×C22⋊(C71⋊C8)C2212yes—
9#17985901C412yescomplemented
10#17985897C412yescomplemented
11#17985890—22no—
12#17985878—21no—
13#17985859C2211yescharacteristic
14#17985754—22no—
15#17985732—24no—
16C22×C42×C71C2211yescharacteristic
17#383×C71—24no—
18C2×#95×C71—21no—
19C2×#95×C71—21no—
20C2×#95×C71C2212yes—
21C22×D284C2×C411yescharacteristic
22C22×C22⋊D71—21no—
23C22×C71⋊C8—22no—
24#14556111—44no—
25#14556102—22no—
26#14556102—22no—
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C4⋊Dic71C2×C412yes—
29C2×Dic71⋊C4—22no—
30C2×C22⋊Dic71—21no—
31C2×(C2×D71)⋊C4—22no—
32C2×(C2×D71)⋊C4—22no—
33C2×(C2×D71)⋊C4—21no—
34C22⋊(C71⋊C8)—22no—
35C22⋊(C71⋊C8)—22no—
36C71⋊#95—22no—
37C71⋊#95—21no—
38C23×C4×C71—22no—
39C23×C4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C23×C4×C71C2311yescharacteristic
42C2×C42×C71—22no—
43C2×C42×C71—22no—
44C2×C42×C71C2×C412yes—
45C2×C42×C71D412yes—
46#95×C71—24no—
47#95×C71—24no—
48#95×C71—22no—
49#95×C71D414yes—
50C23×D71—21no—
51C2×D284—22no—
52C2×D284—42no—
53C2×C22⋊D71—41no—
54C2×C22⋊D71—41no—
55C2×C22⋊D71—42no—
56C2×C22⋊D71—42no—
57C22×Dic71—21no—
58C22×Dic71—22no—
59C2×C71⋊C8—44no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—22no—
62C4⋊Dic71—42no—
63Dic71⋊C4—44no—
64C22⋊Dic71—42no—
65(C2×D71)⋊C4—44no—
66(C2×D71)⋊C4—44no—
67(C2×D71)⋊C4—42no—
68C24×C71C2×D411yescharacteristic
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71—24no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71—44no—
79C22×C4×C71—44no—
80C22×C4×C71—42no—
81C22×C4×C71—42no—
82C22×C4×C71C22⋊C412yes—
83C22×C4×C71C22⋊C412yes—
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D412yes—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71#4112yes—
88C42×C71—24no—
89C42×C71—24no—
90C23×C71—22no—
91C23×C71—22no—
92C23×C71C2×C22⋊C411yescharacteristic
93C23×C71#12011yescharacteristic
94C23×C71#12311yescharacteristic
95C23×C71#13512yes—
96C23×C71AGL(1,ℤ/8)12yes—
97C22×D71—41no—
98C22×D71—42no—
99C22×D71—41no—
100C22×D71—41no—
101C22×D71—42no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—22no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—42no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—44no—
117C2×C4×C71—44no—
118C2×C4×C71—44no—
119C2×C4×C71—84no—
120C2×C4×C71—84no—
121C2×C4×C71C4×D412yes—
122C2×C4×C71#12112yes—
123C22⋊D71—82no—
124C22⋊D71—82no—
125C2×Dic71—41no—
126C2×Dic71—42no—
127C2×Dic71—41no—
128C2×Dic71—42no—
129C2×Dic71—41no—
130C2×Dic71—42no—
131D284—82no—
132C71⋊C8—82no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—24no—
138C22×C71—24no—
139C22×C71#38611yescharacteristic
140C22×C71#41012yes—
141C22×C71#41611yescharacteristic
142C22×C71#41811yescharacteristic
143C22×C71#42112yes—
144C22×C71#44912yes—
145C22×C71#48212yes—
146C2×D71—81no—
147C2×D71—81no—
148C2×D71—81no—
149C2×D71—82no—
150C2×D71—82no—
151C4×C71—22no—
152C4×C71—22no—
153C4×C71—42no—
154C4×C71—42no—
155C4×C71—84no—
156Dic71—81no—
157Dic71—82no—
158#9157—711nomaximal, Sylow
159C2×C71—22no—
160C2×C71—22no—
161C2×C71#151412yes—
162C2×C71#152112yes—
163C2×C71#187511yescharacteristic
164C2×C71#189711yescharacteristic
165C2×C71#189911yescharacteristic
166D71—161no—
167#2854—711no—
168C2×#383D7111yescharacteristic, complemented
169#1817—711no—
170C2×#327—712no—
171#1294—712no—
172C71#915711yescharacteristic, complemented, Sylow
173C2×#123—1421no—
174C2×#121—711no—
175C22×C42C2×D7111yescharacteristic
176C2×C22⋊C8—712no—
177#390—712no—
178#389—712no—
179#388—1422no—
180#386—1421no—
181#383—24no—
182#380—711no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95C2×D7112yes—
186#336—1422no—
187#327—1424no—
188C22×D4—711no—
189C22×D4—1421no—
190C23×C4—22no—
191C23×C4C22⋊D7111yescharacteristic
192C23×C4C22⋊D7111yescharacteristic
193C23×C4C22×D7111yescharacteristic
194C22×C8—1422no—
195#123—2844no—
196#121—1422no—
197#121—1422no—
198C2×C4⋊C4—712no—
199C2×C4⋊C4—711no—
200C2×C4⋊C4—1422no—
201C2×C22⋊C4—1421no—
202C2×C22⋊C4—1422no—
203C2×C22⋊C4—1422no—
204C2×C22⋊C4—1421no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D7112yes—
208C2×C42D28412yes—
209C22⋊C8—1422no—
210C22⋊C8—1422no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—1422no—
215#95—1421no—
216#95C22⋊D7114yes—
217C24—1421no—
218C24C2×C22⋊D7111yescharacteristic
219C2×D4—2842no—
220C2×D4—2841no—
221C2×D4—2841no—
222C2×D4—2842no—
223C2×D4—2842no—
224C2×D4—1422no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—1422no—
239C22×C4—1421no—
240C22×C4(C2×D71)⋊C412yes—
241C22×C4(C2×D71)⋊C412yes—
242C22×C4C2×C4×D7111yescharacteristic
243C22×C4C2×C22⋊D7112yes—
244C22×C4C2×D28411yescharacteristic
245C22×C4#1176586512yes—
246C2×C8—2844no—
247C2×C8—2842no—
248C4⋊C4—2842no—
249C4⋊C4—2844no—
250C4⋊C4—1422no—
251C22⋊C4—2844no—
252C22⋊C4—2844no—
253C22⋊C4—2842no—
254C22⋊C4—2842no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—2841no—
260C23—2842no—
261C23—2841no—
262C23—2841no—
263C23—2842no—
264C23C2×(C2×D71)⋊C411yescharacteristic
265C23#1455609711yescharacteristic
266C23#1455611111yescharacteristic
267C23#1455614812yes—
268C23#1455615312yes—
269D4—5682no—
270D4—5682no—
271D4—5682no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—2841no—
292C2×C4—2842no—
293C2×C4—2841no—
294C2×C4—2842no—
295C2×C4—2841no—
296C2×C4—2842no—
297C2×C4C4×C22⋊D7112yes—
298C2×C4#1455610212yes—
299C8—5682no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—5682no—
307C22—5682no—
308C22—5681no—
309C22—5681no—
310C22—5681no—
311C22#1798587811yescharacteristic
312C22#1798597212yes—
313C22#1798599311yescharacteristic
314C22#1798600311yescharacteristic
315C22#1798601712yes—
316C22#1798614412yes—
317C22#1798632312yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—84no—
323C4—5682no—
324C4—5681no—
325C2—22no—
326C2—22no—
327C2—11361no—
328C2#2258688612yes—
329C2#2258691512yes—
330C2#2258866211yescharacteristic
331C2#2258878811yescharacteristic
332C2#2258879811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#383×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#14556102C22
cosocleC23×C4×C71C23
hypercenterC2×#383D71
hypocenterC71#9157
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
2. #22588408C2semidirect
4. #22586219C2semidirect
5. C2×#383×C71C2semidirect
9. #17985901C4semidirect
10. #17985897C4semidirect
168. C2×#383D71semidirect
172. C71#9157semidirect

Maximal subgroups

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

subgroupindex
#225966712
#225884082
C2×#179857322
#225862192
C2×#383×C712
#915771