Finite Group Databasebeta

← #31916109

Subgroups — #31916109

Order 18176.

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

normal subgroups only

This group has 488 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#22592657C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22586457C214yesmaximal
4C4×#99×C71C211yesmaximal, characteristic
5C22×C4⋊Dic71C411yescharacteristic, complemented
6#17986023—21no—
7C2×#14556045C2212yes—
8#17985938—21no—
9#17985935—21no—
10#17985897—21no—
11#17985879C2212yes—
12#17985871C412yescomplemented
13#17985859—21no—
14#17985859C411yescharacteristic, complemented
15#17985768—21no—
16C2×C4×D4×C71C2211yescharacteristic
17C2×#99×C71C2212yes—
18C4×C22⋊C4×C71—21no—
19#342×C71—21no—
20C23×Dic71C2×C412yes—
21C2×#11765849—22no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71—41no—
27C2×Dic71⋊C4—22no—
28C2×C22⋊Dic71—22no—
29C2×C22⋊Dic71C2×C412yes—
30C4⋊(C71⋊C8)—41no—
31C4⋊(C71⋊C8)—41no—
32#14556045—24no—
33C71⋊#95—22no—
34C71⋊#95—21no—
35C71⋊#95—22no—
36C22×D4×C71C2311yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C4×D4×C71—41no—
40C4×D4×C71—41no—
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71C2×C411yescharacteristic
44C2×C22⋊C4×C71D411yescharacteristic
45C2×C42×C71—41no—
46C2×C42×C71D411yescharacteristic
47#99×C71—22no—
48#99×C71Q814yes—
49#95×C71—41no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52C22×Dic71—42no—
53C22×Dic71—82no—
54#11765849—42no—
55#11765849—42no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—42no—
58C4⋊Dic71—42no—
59C4⋊Dic71—41no—
60C4⋊Dic71—41no—
61Dic71⋊C4—82no—
62C22⋊Dic71—42no—
63C22⋊Dic71—82no—
64C24×C71C22×C411yescharacteristic
65C24×C71C2×D411yescharacteristic
66C2×D4×C71—41no—
67C2×D4×C71—41no—
68C2×D4×C71C2×Q812yes—
69C2×D4×C71#4112yes—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—42no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—41no—
83C22⋊C4×C71—24no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—42no—
87C42×C71—21no—
88C42×C71—21no—
89C42×C71—82no—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—41no—
95C23×C71—41no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71C2×C22⋊C411yescharacteristic
99C23×C71C4×Q812yes—
100C23×C71C22⋊Q812yes—
101D4×C71—42no—
102D4×C71—42no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—22no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—44no—
116C2×C4×C71—81no—
117C2×C4×C71—82no—
118C2×C4×C71—82no—
119C2×C4×C71#9911yescharacteristic
120C2×C4×C71#9911yescharacteristic
121C2×C4×C71#12411yescharacteristic
122C2×C4×C71C4⋊Q811yescharacteristic
123C2×Dic71—42no—
124C2×Dic71—42no—
125C2×Dic71—42no—
126C2×Dic71—42no—
127C2×Dic71—42no—
128C2×Dic71—42no—
129C2×Dic71—84no—
130C71⋊C8—82no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71—42no—
136C22×C71—42no—
137C22×C71—42no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71—41no—
142C22×C71—81no—
143C22×C71#39111yescharacteristic
144C22×C71C2×#9911yescharacteristic
145C22×C71#41011yescharacteristic
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—21no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—41no—
152C4×C71—84no—
153C4×C71#35212yes—
154Dic71—82no—
155Dic71—82no—
156#13295—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—21no—
159C2×C71—42no—
160C2×C71—42no—
161C2×C71#199611yescharacteristic
162C2×C71C2×#35211yescharacteristic
163C2×C71#212211yescharacteristic
164#2487—711no—
165#1800—711no—
166C4×#99D7111yescharacteristic
167#1390—714no—
168C71#1329511yescharacteristic, complemented, Sylow
169C2×C4×D4C2×D7111yescharacteristic
170C22×C4⋊C4—711no—
171#423—1421no—
172C2×#100—712no—
173C2×#99C2×D7112yes—
174#400—1421no—
175#399—1421no—
176#389—1421no—
177#386—712no—
178#385—712no—
179#380—711no—
180#380—1421no—
181C4×C22⋊C4—21no—
182#343—1421no—
183#342—21no—
184C22×D4C22×D7111yescharacteristic
185C23×C4—712no—
186C23×C4C4×D7111yescharacteristic
187C23×C4C22⋊D7111yescharacteristic
188C2×M16—1422no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—2841no—
192C2×C4⋊C4—2841no—
193C2×C4⋊C4—1421no—
194C2×C4⋊C4—1421no—
195C2×C4⋊C4—1421no—
196C2×C4⋊C4—1422no—
197C2×C4⋊C4D28411yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—712no—
200C2×C22⋊C4—1422no—
201C2×C22⋊C4C4×D7111yescharacteristic
202C2×C22⋊C4C22⋊D7111yescharacteristic
203C2×C42—41no—
204C2×C42D28411yescharacteristic
205C4⋊C8—2841no—
206C4⋊C8—2841no—
207#100—1424no—
208#99—22no—
209#99Dic14214yes—
210#95—41no—
211#95—1421no—
212#95—1422no—
213#95—1422no—
214C24C2×C4×D7111yescharacteristic
215C24C2×C22⋊D7111yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4C2×Dic14212yes—
219C2×D4#1176586512yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—2842no—
228C22×C4—5682no—
229C22×C4—1422no—
230C22×C4—1422no—
231C22×C4(C2×D71)⋊C411yescharacteristic
232C22×C4(C2×D71)⋊C411yescharacteristic
233C22×C4(C2×D71)⋊C411yescharacteristic
234C22×C4(C2×D71)⋊C411yescharacteristic
235C22×C4C2×D28411yescharacteristic
236M16—2842no—
237M16—2842no—
238C2×C8—2842no—
239C4⋊C4—41no—
240C4⋊C4—2842no—
241C4⋊C4—2842no—
242C4⋊C4—2841no—
243C4⋊C4—2841no—
244C4⋊C4—5682no—
245C22⋊C4—24no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—2842no—
250C22⋊C4—5682no—
251C42—21no—
252C42—21no—
253C42—82no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×(C2×D71)⋊C411yescharacteristic
263C23C4×Dic14212yes—
264C23#1455610712yes—
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—2842no—
284C2×C4—2842no—
285C2×C4—2842no—
286C2×C4—2842no—
287C2×C4—2842no—
288C2×C4—2842no—
289C2×C4—5684no—
290C2×C4#1455604211yescharacteristic
291C2×C4#1455604211yescharacteristic
292C2×C4#1455611611yescharacteristic
293C2×C4#1455612811yescharacteristic
294C8—5682no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1798590511yescharacteristic
308C22C2×#1455604211yescharacteristic
309C22#1798597211yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—84no—
317C4—5682no—
318C4—5682no—
319C4#1798578812yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#2258932511yescharacteristic
325C2C2×#1798578811yescharacteristic
326C2#2258998911yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC24×C71C22×C4
FrattiniC22×D4C22×D71
FittingC4×#99×C71C2
socleC22×C71C2×#99
prime radical, cosocleC22×D4×C71C23
hypercenterC4×#99D71
hypocenterC71#13295
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Dic71C4semidirect
12. #17985871C4semidirect
14. #17985859C4semidirect
168. C71#13295semidirect

Maximal subgroups

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

subgroupindex
#225926572
#225883322
#225864572
C4×#99×C712
#1329571