Finite Group Databasebeta

← #31893416

Subgroups — #31893416

Order 18176.

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

normal subgroups only

This group has 532 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#22597369C211yesmaximal, characteristic
2C4×#14556043C211yesmaximal, characteristic
3#22586860C212yesmaximal, complemented
4#22586192C212yesmaximal, complemented
5C71×#2638C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8C2×#14556043—21no—
9#17985963C2212yes—
10C2×C22⋊(C71⋊C8)—21no—
11C4×C22⋊Dic71C2212yes—
12#17985766—21no—
13#17985736—24no—
14C2×C22⋊Q8×C71C2211yescharacteristic
15C2×C4×D4×C71C2211yescharacteristic
16#398×C71C411yescharacteristic, complemented
17#397×C71—21no—
18#397×C71C411yescharacteristic, complemented
19#395×C71C411yescharacteristic, complemented
20#394×C71—21no—
21#393×C71C411yescharacteristic, complemented
22#387×C71—21no—
23#385×C71—21no—
24#381×C71C2211yescharacteristic
25C2×#11765849—22no—
26C22×C71⋊C8—22no—
27C2×C22⋊Dic71—22no—
28C2×C4×Dic71—22no—
29C4⋊(C71⋊C8)—22no—
30#14556043—24no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)D414yes—
34#14556039—22no—
35C71⋊#95—22no—
36C22×Q8×C71C2×C411yescharacteristic
37C22×D4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C22⋊Q8×C71—22no—
41C22⋊Q8×C71—22no—
42C4×D4×C71—22no—
43C4×D4×C71—22no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71C2×C411yescharacteristic
54C2×C22⋊C4×C71D411yescharacteristic
55C2×C42×C71D411yescharacteristic
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71C2×C411yescharacteristic
62#95×C71C2×C411yescharacteristic
63C22×Dic71—42no—
64#11765849—82no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—82no—
67C22⋊Dic71—24no—
68C22⋊Dic71—42no—
69C4×Dic71—82no—
70C24×C71C22⋊C411yescharacteristic
71C24×C71C2×D411yescharacteristic
72C2×Q8×C71—22no—
73C2×Q8×C71—81no—
74C2×D4×C71—22no—
75C2×D4×C71—22no—
76C2×D4×C71—41no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—81no—
84C22×C4×C71—81no—
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D412yes—
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71#4112yes—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—22no—
94C4⋊C4×C71—42no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—81no—
98C22⋊C4×C71—22no—
99C22⋊C4×C71—22no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—81no—
103C22⋊C4×C71—81no—
104C42×C71—21no—
105C42×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71#10412yes—
115C23×C71C2×C22⋊C411yescharacteristic
116C23×C71#12112yes—
117Q8×C71—82no—
118D4×C71—44no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—42no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—81no—
144C2×C4×C71C4×D412yes—
145C2×C4×C71#12011yescharacteristic
146C2×C4×C71#12311yescharacteristic
147C2×Dic71—44no—
148C2×Dic71—82no—
149C2×Dic71—82no—
150C71⋊C8—44no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—22no—
156C22×C71—22no—
157C22×C71—42no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—81no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71#38611yescharacteristic
164C22×C71C2×#10411yescharacteristic
165C22×C71#42111yescharacteristic
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—22no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—81no—
173C4×C71—81no—
174C4×C71—81no—
175C4×C71—81no—
176Dic71—84no—
177#9183—711nomaximal, Sylow
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—42no—
181C2×C71—42no—
182C2×C71#187111yescharacteristic
183C2×C71#190911yescharacteristic
184C2×C71#191311yescharacteristic
185#2973—711no—
186#2638D7111yescharacteristic
187C4×#99—711no—
188#1508—712no—
189#1284—712no—
190C71#918311yescharacteristic, complemented, Sylow
191C2×C22⋊Q8C2×D7111yescharacteristic
192C2×C4×D4C2×D7111yescharacteristic
193#435—1424no—
194#432—1421no—
195C2×#99—1421no—
196#407—712no—
197C2×C22⋊C8—1421no—
198#398Dic7111yescharacteristic, complemented
199#397—21no—
200#397Dic7111yescharacteristic, complemented
201#395Dic7111yescharacteristic, complemented
202#394—21no—
203#393Dic7111yescharacteristic, complemented
204#387—21no—
205#385—21no—
206#381C2×D7111yescharacteristic
207C4×C22⋊C4—712no—
208#342—1421no—
209#328—1424no—
210C22×Q8C2×Dic7111yescharacteristic
211C22×D4C22⋊D7111yescharacteristic
212C23×C4C22⋊D7111yescharacteristic
213C23×C4C22×D7111yescharacteristic
214C2×M16—1422no—
215C22×C8—1422no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C4×D4—22no—
219C4×D4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C2×Dic7111yescharacteristic
224C2×C4⋊C4C2×Dic7111yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—1422no—
230C2×C22⋊C4C2×Dic7111yescharacteristic
231C2×C22⋊C4C22⋊D7111yescharacteristic
232C2×C42—1422no—
233C2×C42C22⋊D7111yescharacteristic
234C4⋊C8—1422no—
235#99—1424no—
236C22⋊C8—714no—
237C22⋊C8—1422no—
238C22⋊C8—1422no—
239#97—1422no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—1422no—
246#95C2×Dic7111yescharacteristic
247#95C2×Dic7111yescharacteristic
248C24C22⋊Dic7111yescharacteristic
249C24C2×C22⋊D7111yescharacteristic
250C2×Q8—22no—
251C2×Q8—81no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—2842no—
264C22×C4C22⋊Dic7111yescharacteristic
265C22×C4C22⋊Dic7111yescharacteristic
266C22×C4C22⋊Dic7111yescharacteristic
267C22×C4C22×Dic7111yescharacteristic
268C22×C4D4×D7112yes—
269C22×C4C2×C22⋊D7111yescharacteristic
270C22×C4#1176586412yes—
271M16—5682no—
272C2×C8—5682no—
273C2×C8—1424no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—2842no—
287C22⋊C4—1424no—
288C42—21no—
289C42—21no—
290C42—5682no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#1455605512yes—
300C23C2×C22⋊Dic7111yescharacteristic
301C23#1455610012yes—
302Q8—82no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—2844no—
330C2×C4—5682no—
331C2×C4—5682no—
332C2×C4D4×Dic7112yes—
333C2×C4#1455609611yescharacteristic
334C2×C4#1455610911yescharacteristic
335C8—2844no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1798587911yescharacteristic
349C22C2×#1455605511yescharacteristic
350C22#1798601611yescharacteristic
351C4—21no—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—5684no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—42no—
366C2#2258864311yescharacteristic
367C2#2258886311yescharacteristic
368C2#2258888311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2638C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2638D71
hypocenterC71#9183
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #22586860C2semidirect
4. #22586192C2semidirect
16. #398×C71C4semidirect
18. #397×C71C4semidirect
19. #395×C71C4semidirect
21. #393×C71C4semidirect
190. C71#9183semidirect
198. #398Dic71semidirect
200. #397Dic71semidirect
201. #395Dic71semidirect
203. #393Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973692
C4×#145560432
#225868602
#225861922
C71×#26382
#918371