Finite Group Databasebeta

← #31908579

Subgroups — #31908579

Order 18176.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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#22592657C211yesmaximal, characteristic
2#22586661C211yesmaximal, characteristic
3#22586363C214yesmaximal, complemented
4C71×#1799C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic, complemented
6#17986052—21no—
7#17985938C412yescomplemented
8#17985935C2212yes—
9#17985897—21no—
10#17985879—21no—
11#17985871—21no—
12#17985859—21no—
13#17985859C411yescharacteristic, complemented
14#17985723—21no—
15#17985723C2212yes—
16C2×C4×D4×C71C2211yescharacteristic
17C2×#99×C71—21no—
18#380×C71—21no—
19#342×C71C2212yes—
20C23×Dic71—21no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—41no—
24C2×C4⋊Dic71—41no—
25C2×C4⋊Dic71C2×C412yes—
26C2×Dic71⋊C4—22no—
27C2×C22⋊Dic71—21no—
28C2×C22⋊Dic71—22no—
29C4⋊(C71⋊C8)—41no—
30C22⋊(C71⋊C8)—22no—
31C4×C71⋊C8—41no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95C2×C412yes—
35C22×D4×C71D411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C4×D4×C71—41no—
39C4×D4×C71—41no—
40C2×C4⋊C4×C71—41no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71C2311yescharacteristic
45C2×C42×C71D411yescharacteristic
46#99×C71—42no—
47#95×C71—41no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50C22×Dic71—42no—
51C22×Dic71—82no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—42no—
54C4⋊Dic71—41no—
55C4⋊Dic71—41no—
56C4⋊Dic71—42no—
57Dic71⋊C4—82no—
58C22⋊Dic71—42no—
59C22⋊Dic71—82no—
60C24×C71C22⋊C411yescharacteristic
61C24×C71C2×D411yescharacteristic
62C2×D4×C71—41no—
63C2×D4×C71—41no—
64C2×D4×C71SD1612yes—
65C2×D4×C71Q1612yes—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C4⋊C4×C71—41no—
79C4⋊C4×C71—82no—
80C22⋊C4×C71—41no—
81C22⋊C4×C71—41no—
82C22⋊C4×C71—42no—
83C22⋊C4×C71—42no—
84C42×C71—21no—
85C42×C71—21no—
86C23×C71—21no—
87C23×C71—21no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—41no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71—41no—
94C23×C71#10312yes—
95C23×C71C2×C22⋊C411yescharacteristic
96C23×C71C2×SD1611yescharacteristic
97C23×C71C2×Q1611yescharacteristic
98D4×C71—42no—
99D4×C71—42no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—22no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—41no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—81no—
113C2×C4×C71—82no—
114C2×C4×C71—82no—
115C2×C4×C71—82no—
116C2×C4×C71#9911yescharacteristic
117C2×C4×C71#9911yescharacteristic
118C2×C4×C71#10312yes—
119C2×Dic71—42no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124C2×Dic71—42no—
125C2×Dic71—84no—
126C71⋊C8—82no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—22no—
131C22×C71—21no—
132C22×C71—42no—
133C22×C71—41no—
134C22×C71—41no—
135C22×C71—42no—
136C22×C71—41no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—81no—
141C22×C71C2×#9911yescharacteristic
142C22×C71C2×#10311yescharacteristic
143C22×C71#42111yescharacteristic
144C4×C71—21no—
145C4×C71—22no—
146C4×C71—41no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—162no—
150C4×C71#35212yes—
151Dic71—82no—
152Dic71—82no—
153#11899—711nomaximal, Sylow
154C2×C71—21no—
155C2×C71—21no—
156C2×C71—41no—
157C2×C71—42no—
158C2×C71—41no—
159C2×C71#152011yescharacteristic
160C2×C71C2×#35211yescharacteristic
161C2×C71#212811yescharacteristic
162#2487—711no—
163#1799D7111yescharacteristic
164#1460—711no—
165#1352—714no—
166C71#1189911yescharacteristic, complemented, Sylow
167C2×C4×D4C2×D7111yescharacteristic
168C22×C4⋊C4—711no—
169#431—1421no—
170C2×#99—21no—
171#400—712no—
172#399—712no—
173#389—1421no—
174#386—1421no—
175#385—1421no—
176#380—21no—
177#380—711no—
178#380—1421no—
179#342C2×D7112yes—
180#323—712no—
181#323—1421no—
182C22×D4C22⋊D7111yescharacteristic
183C23×C4—1421no—
184C23×C4C4×D7111yescharacteristic
185C23×C4C22⋊D7111yescharacteristic
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—2841no—
190C2×C4⋊C4—2841no—
191C2×C4⋊C4—712no—
192C2×C4⋊C4—1421no—
193C2×C4⋊C4—1421no—
194C2×C4⋊C4—1422no—
195C2×C4⋊C4C4×D7111yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—1421no—
198C2×C22⋊C4—1422no—
199C2×C22⋊C4D28411yescharacteristic
200C2×C22⋊C4C22×D7111yescharacteristic
201C2×C42D28411yescharacteristic
202C4⋊C8—2841no—
203#99—42no—
204C22⋊C8—1422no—
205C4×C8—2841no—
206#95—41no—
207#95—712no—
208#95—1422no—
209#95—1422no—
210C24(C2×D71)⋊C411yescharacteristic
211C24C2×C22⋊D7111yescharacteristic
212C2×D4—41no—
213C2×D4—41no—
214C2×D4#1176585312yes—
215C2×D4#1176585512yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—2842no—
224C22×C4—5682no—
225C22×C4—1422no—
226C22×C4—1422no—
227C22×C4(C2×D71)⋊C411yescharacteristic
228C22×C4(C2×D71)⋊C411yescharacteristic
229C22×C4(C2×D71)⋊C411yescharacteristic
230C22×C4C2×C4×D7111yescharacteristic
231C22×C4C2×D28411yescharacteristic
232C2×C8—2842no—
233C4⋊C4—41no—
234C4⋊C4—82no—
235C4⋊C4—2842no—
236C4⋊C4—2842no—
237C4⋊C4—2841no—
238C4⋊C4—2841no—
239C4⋊C4—5682no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—2842no—
245C22⋊C4—5682no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#1455605112yes—
257C23C2×(C2×D71)⋊C411yescharacteristic
258C23C2×#1176585311yescharacteristic
259C23C2×#1176585511yescharacteristic
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—2842no—
279C2×C4—2842no—
280C2×C4—2842no—
281C2×C4—2842no—
282C2×C4—2842no—
283C2×C4—2842no—
284C2×C4—5684no—
285C2×C4#1455604211yescharacteristic
286C2×C4#1455604211yescharacteristic
287C2×C4#1455605112yes—
288C8—5682no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1455604211yescharacteristic
304C22C2×#1455605111yescharacteristic
305C22#1798601711yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—162no—
312C4—5682no—
313C4—5682no—
314C4#1798578812yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#2258691111yescharacteristic
321C2C2×#1798578811yescharacteristic
322C2#2259001111yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#1799C2
socleC22×C71C2×#99
prime radicalC22×D4×C71D4
cosocleC2×C22⋊C4×C71C23
hypercenter#1799D71
hypocenterC71#11899
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #22586363C2semidirect
5. C22×C4⋊Dic71C4semidirect
7. #17985938C4semidirect
13. #17985859C4semidirect
166. C71#11899semidirect

Maximal subgroups

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

subgroupindex
#225926572
#225866612
#225863632
C71×#17992
#1189971