Finite Group Databasebeta

← #31916110

Subgroups — #31916110

Order 18176.

488 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 111 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#22588332C211yesmaximal, characteristic
2C4×#14556043C211yesmaximal, characteristic
3#22586458C214yesmaximal
4C71×#2487C211yesmaximal, characteristic
5#17986023—21no—
6C2×#14556045C2212yes—
7C2×#14556043C2212yes—
8C4×C22⋊Dic71—21no—
9#17985768—21no—
10#17985766—21no—
11C2×C4×D4×C71C2211yescharacteristic
12C22×C4⋊C4×C71C411yescharacteristic, complemented
13#400×C71—21no—
14#399×C71—21no—
15#389×C71—21no—
16#386×C71C2212yes—
17#385×C71C412yescomplemented
18#380×C71—21no—
19#380×C71C411yescharacteristic, complemented
20C2×#11765849—22no—
21C2×C22⋊Dic71—22no—
22C2×C4×Dic71—41no—
23C4⋊(C71⋊C8)—41no—
24C4⋊(C71⋊C8)—41no—
25#14556045—24no—
26#14556043—22no—
27#14556043Q814yes—
28C71⋊#95—41no—
29C22×D4×C71C2311yescharacteristic
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71C2×C412yes—
32C23×C4×C71D411yescharacteristic
33C4×D4×C71—41no—
34C4×D4×C71—41no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—41no—
40C2×C4⋊C4×C71—41no—
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71C2×C412yes—
44C2×C22⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47#95×C71—22no—
48#95×C71—21no—
49#95×C71—22no—
50C22×Dic71—42no—
51#11765849—42no—
52#11765849—42no—
53C2×C71⋊C8—42no—
54C22⋊Dic71—24no—
55C22⋊Dic71—42no—
56C4×Dic71—82no—
57C24×C71C22×C411yescharacteristic
58C24×C71C2×D411yescharacteristic
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61C2×D4×C71C2×Q812yes—
62C2×D4×C71#4112yes—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71—82no—
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—42no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71—82no—
84C22⋊C4×C71—41no—
85C22⋊C4×C71—41no—
86C22⋊C4×C71—42no—
87C22⋊C4×C71—82no—
88C42×C71—21no—
89C42×C71—21no—
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—42no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—81no—
122C2×C4×C71—84no—
123C2×C4×C71#9911yescharacteristic
124C2×C4×C71#9911yescharacteristic
125C2×C4×C71#12411yescharacteristic
126C2×C4×C71C4⋊Q811yescharacteristic
127C2×Dic71—44no—
128C2×Dic71—82no—
129C2×Dic71—82no—
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—82no—
153C4×C71—82no—
154C4×C71#35212yes—
155Dic71—84no—
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#2487D7111yescharacteristic
165#1800—711no—
166C4×#99—711no—
167#1390—714no—
168C71#1329511yescharacteristic, complemented, Sylow
169C2×C4×D4C2×D7111yescharacteristic
170C22×C4⋊C4Dic7111yescharacteristic, complemented
171#423—1421no—
172C2×#100—712no—
173C2×#99—712no—
174#400—21no—
175#399—21no—
176#389—21no—
177#386C2×D7112yes—
178#385Dic7112yescomplemented
179#380—21no—
180#380Dic7111yescharacteristic, complemented
181C4×C22⋊C4—1421no—
182#343—1421no—
183#342—1421no—
184C22×D4C22×D7111yescharacteristic
185C23×C4C2×Dic7111yescharacteristic
186C23×C4C2×Dic7112yes—
187C23×C4C22⋊D7111yescharacteristic
188C2×M16—1422no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C22⋊D7111yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1422no—
200C2×C22⋊C4C2×Dic7112yes—
201C2×C22⋊C4C2×Dic7111yescharacteristic
202C2×C22⋊C4C22⋊D7111yescharacteristic
203C2×C42—2841no—
204C2×C42C22⋊D7111yescharacteristic
205C4⋊C8—2841no—
206C4⋊C8—2841no—
207#100—1424no—
208#99—714no—
209#99—1422no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—2841no—
214C24C22×Dic7111yescharacteristic
215C24C2×C22⋊D7111yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4Q8×D7112yes—
219C2×D4#1176586612yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—2842no—
231C22×C4C22⋊Dic7111yescharacteristic
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4C22⋊Dic7111yescharacteristic
234C22×C4C22⋊Dic7111yescharacteristic
235C22×C4C2×C22⋊D7111yescharacteristic
236M16—2842no—
237M16—2842no—
238C2×C8—2842no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—82no—
249C22⋊C4—2842no—
250C22⋊C4—1424no—
251C42—21no—
252C42—21no—
253C42—5682no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊Dic7111yescharacteristic
263C23Q8×Dic7112yes—
264C23#1455610412yes—
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—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—84no—
287C2×C4—2844no—
288C2×C4—5682no—
289C2×C4—5682no—
290C2×C4#1455604311yescharacteristic
291C2×C4#1455604311yescharacteristic
292C2×C4#1455611711yescharacteristic
293C2×C4#1455612611yescharacteristic
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#1798590611yescharacteristic
308C22C2×#1455604311yescharacteristic
309C22#1798597311yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—82no—
318C4—5684no—
319C4#1798578912yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#2258932611yescharacteristic
325C2C2×#1798578911yescharacteristic
326C2#2258999011yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC22×D4C22×D71
FittingC71×#2487C2
socleC22×C71C2×#99
prime radical, cosocleC22×D4×C71C23
hypercenter#2487D71
hypocenterC71#13295
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
12. C22×C4⋊C4×C71C4semidirect
17. #385×C71C4semidirect
19. #380×C71C4semidirect
168. C71#13295semidirect
170. C22×C4⋊C4Dic71semidirect
178. #385Dic71semidirect
180. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
#225883322
C4×#145560432
#225864582
C71×#24872
#1329571