Finite Group Databasebeta

← #31895152

Subgroups — #31895152

Order 18176.

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

normal subgroups only

This group has 554 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#22597369C211yesmaximal, characteristic
2#22587057C212yesmaximal, complemented
3#22586681C211yesmaximal, characteristic
4#22586196C212yesmaximal, complemented
5C71×#2542C211yesmaximal, characteristic
6#17986071—24no—
7#17986056C2212yes—
8#17986056C2212yes—
9#17985963—21no—
10C2×C22⋊(C71⋊C8)—21no—
11#17985745—24no—
12#17985723—21no—
13#17985723—21no—
14C2×C4⋊Q8×C71C2211yescharacteristic
15C2×C22⋊Q8×C71—21no—
16C2×C22⋊Q8×C71—21no—
17C2×C4×D4×C71C2211yescharacteristic
18#399×C71—21no—
19#399×C71—21no—
20#390×C71C412yes—
21#384×C71C412yes—
22C4×C4⋊C4×C71C2211yescharacteristic
23C2×#11765849—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)D414yes—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31#14556039—22no—
32#14556039—22no—
33C22×Q8×C71—21no—
34C22×D4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C4⋊Q8×C71—22no—
38C4⋊Q8×C71—22no—
39C22⋊Q8×C71—44no—
40C22⋊Q8×C71—44no—
41C4×D4×C71—22no—
42C4×D4×C71—22no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71—22no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71C2×C412yes—
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71—22no—
50C2×C22⋊C4×C71—22no—
51C2×C22⋊C4×C71D411yescharacteristic
52C2×C22⋊C4×C71D411yescharacteristic
53C2×C42×C71C2×C412yes—
54C2×C42×C71C2311yescharacteristic
55#95×C71—21no—
56#11765849—82no—
57C2×C71⋊C8—24no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—82no—
60C24×C71C22⋊C411yescharacteristic
61C24×C71C22⋊C411yescharacteristic
62C2×Q8×C71—42no—
63C2×Q8×C71—42no—
64C2×Q8×C71—41no—
65C2×Q8×C71—41no—
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71—41no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—24no—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—24no—
85C4⋊C4×C71—44no—
86C4⋊C4×C71—44no—
87C4⋊C4×C71—42no—
88C22⋊C4×C71—22no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—44no—
91C22⋊C4×C71—44no—
92C42×C71—42no—
93C42×C71C2×D412yes—
94C42×C71#4112yes—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71#10112yes—
105C23×C71C2×C22⋊C411yescharacteristic
106Q8×C71—82no—
107Q8×C71—82no—
108D4×C71—44no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—22no—
112C2×C4×C71—24no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—41no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—81no—
126C2×C4×C71—81no—
127C2×C4×C71C4×D412yes—
128C2×C4×C71#12011yescharacteristic
129C2×C4×C71#12112yes—
130C2×C4×C71#12311yescharacteristic
131C71⋊C8—44no—
132C71⋊C8—82no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—22no—
136C22×C71—42no—
137C22×C71—42no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71—81no—
144C22×C71—81no—
145C22×C71#38611yescharacteristic
146C22×C71C2×#10111yescharacteristic
147C22×C71#41311yescharacteristic
148C4×C71—21no—
149C4×C71—22no—
150C4×C71—22no—
151C4×C71—44no—
152C4×C71—41no—
153C4×C71—41no—
154C4×C71—82no—
155C4×C71#45312yes—
156#9455—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—21no—
159C2×C71—42no—
160C2×C71—81no—
161C2×C71#188711yescharacteristic
162C2×C71#190111yescharacteristic
163C2×C71#190911yescharacteristic
164#2973—711no—
165#2542D7111yescharacteristic
166#1556—712no—
167#1466—711no—
168#1285—712no—
169C71#945511yescharacteristic, complemented, Sylow
170C2×C4⋊Q8C2×D7111yescharacteristic
171C2×C22⋊Q8—21no—
172C2×C22⋊Q8—21no—
173C2×C4×D4C2×D7111yescharacteristic
174#435—1424no—
175#432—712no—
176#432—712no—
177#407—1421no—
178C2×C22⋊C8—1421no—
179#399—21no—
180#399—21no—
181#390Dic7112yes—
182#384Dic7112yes—
183C4×C4⋊C4C2×D7111yescharacteristic
184#332—1424no—
185#323—1421no—
186#323—1421no—
187C22×Q8—21no—
188C22×D4C2×Dic7111yescharacteristic
189C23×C4C22⋊D7111yescharacteristic
190C23×C4C22⋊D7111yescharacteristic
191C2×M16—1422no—
192C22×C8—1422no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C2×Dic7112yes—
204C2×C4⋊C4C2×Dic7111yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4C22⋊D7111yescharacteristic
208C2×C22⋊C4C22⋊D7111yescharacteristic
209C2×C42C2×Dic7112yes—
210C2×C42C22×D7111yescharacteristic
211C4⋊C8—714no—
212C4⋊C8—1422no—
213C22⋊C8—1422no—
214C22⋊C8—1422no—
215C22⋊C8—1422no—
216C22⋊C8—1422no—
217#97—1422no—
218#97—1422no—
219#95—21no—
220C24C22⋊Dic7111yescharacteristic
221C24C22⋊Dic7111yescharacteristic
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4C22⋊Dic7111yescharacteristic
238C22×C4C22⋊Dic7111yescharacteristic
239C22×C4C22×Dic7111yescharacteristic
240C22×C4C2×C22⋊D7111yescharacteristic
241C22×C4C2×C22⋊D7111yescharacteristic
242M16—5682no—
243C2×C8—2842no—
244C2×C8—5682no—
245C2×C8—1424no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C42—42no—
257C42D4×D7112yes—
258C42#1176586412yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#1455604712yes—
269C23C2×C22⋊Dic7111yescharacteristic
270Q8—82no—
271Q8—82no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4D4×Dic7112yes—
292C2×C4#1455609611yescharacteristic
293C2×C4#1455610012yes—
294C2×C4#1455610911yescharacteristic
295C8—2844no—
296C8—5682no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1798587911yescharacteristic
310C22C2×#1455604711yescharacteristic
311C22#1798598311yescharacteristic
312C4—21no—
313C4—22no—
314C4—22no—
315C4—44no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4#1798616612yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#2258873911yescharacteristic
325C2#2258880911yescharacteristic
326C2#2258886311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2542C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#2542D71
hypocenterC71#9455
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. #22587057C2semidirect
4. #22586196C2semidirect
169. C71#9455semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225870572
#225866812
#225861962
C71×#25422
#945571