Finite Group Databasebeta

← #31907953

Subgroups — #31907953

Order 18176.

622 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#17985768C211yesmaximal, characteristic, complemented
2#22586670C211yesmaximal, characteristic, complemented
3#22586370C212yesmaximal, complemented
4#22586343C212yesmaximal, complemented
5C71×#2406C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17985963—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725—21no—
11#17985725C2212yes—
12C22×#41×C71C411yescharacteristic
13C2×#123×C71—21no—
14C22×C4⋊C4×C71C2211yescharacteristic
15#395×C71C411yescharacteristic
16#395×C71C2211yescharacteristic
17#394×C71C411yescharacteristic
18#394×C71C2211yescharacteristic
19#386×C71—21no—
20#385×C71—21no—
21#385×C71—21no—
22#381×C71C411yescharacteristic
23C2×#11765849—24no—
24C2×#11765849—22no—
25C2×#11765849—41no—
26C2×#11765849—42no—
27C2×#11765849—41no—
28C22×C71⋊C8—41no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—41no—
31C2×#41×C71—41no—
32C2×#41×C71—41no—
33C2×#41×C71—44no—
34C22×Q8×C71C2×C411yescharacteristic
35C22×D4×C71—21no—
36C22×D4×C71C2×C411yescharacteristic
37C23×C4×C71—21no—
38C23×C4×C71C2×C411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41#123×C71—44no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—41no—
45C2×C4⋊C4×C71—41no—
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C4⋊C4×C71C2311yescharacteristic
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71C2×C411yescharacteristic
59#95×C71C2×C411yescharacteristic
60#11765849—24no—
61#11765849—44no—
62#11765849—42no—
63#11765849—82no—
64C2×C71⋊C8—44no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—42no—
67C24×C71—21no—
68C24×C71C22⋊C411yescharacteristic
69#41×C71—44no—
70#41×C71—44no—
71#41×C71—44no—
72#41×C71—44no—
73C2×Q8×C71—42no—
74C2×Q8×C71—41no—
75C2×D4×C71—24no—
76C2×D4×C71—41no—
77C2×D4×C71—42no—
78C2×D4×C71—41no—
79C2×D4×C71—41no—
80C2×D4×C71—42no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—22no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71—42no—
94C22×C4×C71—42no—
95C22×C4×C71—42no—
96C22×C4×C71—42no—
97C22×C4×C71—42no—
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22⋊C411yescharacteristic
101C22×C4×C71C22×C411yescharacteristic
102C22×C4×C71C2×D411yescharacteristic
103C22×C4×C71C2×D411yescharacteristic
104C4⋊C4×C71—41no—
105C4⋊C4×C71—41no—
106C4⋊C4×C71—41no—
107C4⋊C4×C71—41no—
108C4⋊C4×C71—42no—
109C4⋊C4×C71—42no—
110C22⋊C4×C71—42no—
111C22⋊C4×C71—42no—
112C22⋊C4×C71—42no—
113C22⋊C4×C71—42no—
114C22⋊C4×C71—41no—
115C22⋊C4×C71—41no—
116C23×C71—22no—
117C23×C71—21no—
118C23×C71—21no—
119C23×C71—22no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71—41no—
123C23×C71—41no—
124C23×C71—41no—
125C23×C71—41no—
126C23×C71—41no—
127C23×C71#9911yescharacteristic
128C23×C71#9911yescharacteristic
129C23×C71C2×C22⋊C411yescharacteristic
130Q8×C71—42no—
131Q8×C71—42no—
132D4×C71—44no—
133D4×C71—42no—
134D4×C71—42no—
135D4×C71—44no—
136C2×C4×C71—22no—
137C2×C4×C71—22no—
138C2×C4×C71—22no—
139C2×C4×C71—22no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—22no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—41no—
150C2×C4×C71—42no—
151C2×C4×C71—48no—
152C2×C4×C71—42no—
153C2×C4×C71—42no—
154C2×C4×C71—41no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—41no—
160C2×C4×C71—41no—
161C2×C4×C71—41no—
162C2×C4×C71—41no—
163C2×C4×C71—41no—
164C2×C4×C71—41no—
165C2×C4×C71—81no—
166C2×C4×C71#10412yes—
167C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C71#13711yescharacteristic
169C71⋊C8—44no—
170C71⋊C8—82no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71—21no—
174C22×C71—21no—
175C22×C71—21no—
176C22×C71—21no—
177C22×C71—21no—
178C22×C71—42no—
179C22×C71—42no—
180C22×C71—42no—
181C22×C71—42no—
182C22×C71—42no—
183C22×C71—81no—
184C22×C71—81no—
185C22×C71—81no—
186C22×C71—81no—
187C22×C71AΣL(1,𝔽16)12yes—
188C22×C71C2×#9911yescharacteristic
189C22×C71#41711yescharacteristic
190C22×C71C2×#10411yescharacteristic
191C4×C71—21no—
192C4×C71—22no—
193C4×C71—21no—
194C4×C71—41no—
195C4×C71—41no—
196C4×C71—42no—
197C4×C71—41no—
198C4×C71—41no—
199C4×C71—81no—
200C4×C71—81no—
201#11762—711nomaximal, Sylow
202C2×C71—21no—
203C2×C71—21no—
204C2×C71—22no—
205C2×C71—41no—
206C2×C71—82no—
207C2×C71#150411yescharacteristic
208C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
209C2×C71#213311yescharacteristic
210#2406D7111yescharacteristic
211C2×#343—711no—
212#1463—711no—
213#1354—712no—
214#1345—712no—
215C71#1176211yescharacteristic, complemented, Sylow
216C22×#41Dic7111yescharacteristic
217C22×M16—1421no—
218C2×#123—21no—
219C22×C4⋊C4C2×D7111yescharacteristic
220#407—1421no—
221#395Dic7111yescharacteristic
222#395C2×D7111yescharacteristic
223#394Dic7111yescharacteristic
224#394C2×D7111yescharacteristic
225#386—21no—
226#385—21no—
227#385—21no—
228#381Dic7111yescharacteristic
229#343—712no—
230#343—1421no—
231#324—712no—
232#324—1421no—
233C2×#41—41no—
234C2×#41—41no—
235C2×#41—44no—
236C22×Q8C2×Dic7111yescharacteristic
237C22×D4—21no—
238C22×D4C2×Dic7111yescharacteristic
239C23×C4—21no—
240C23×C4C2×Dic7111yescharacteristic
241C23×C4C22⋊D7111yescharacteristic
242C23×C4C22⋊D7111yescharacteristic
243C2×M16—2841no—
244C2×M16—2842no—
245C2×M16—2841no—
246C2×M16—1424no—
247C2×M16—1422no—
248C22×C8—2841no—
249#123—44no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4C2×Dic7111yescharacteristic
255C2×C4⋊C4C22⋊D7111yescharacteristic
256C2×C4⋊C4C22⋊D7111yescharacteristic
257C2×C4⋊C4C22×D7111yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C22⋊C8—2841no—
264C22⋊C8—1422no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95C2×Dic7111yescharacteristic
269#95C2×Dic7111yescharacteristic
270C24—21no—
271C24C22⋊Dic7111yescharacteristic
272#41—44no—
273#41—44no—
274#41—44no—
275#41—44no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×D4—24no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C22⋊Dic7111yescharacteristic
302C22×C4C22⋊Dic7111yescharacteristic
303C22×C4C22⋊Dic7111yescharacteristic
304C22×C4C22×Dic7111yescharacteristic
305C22×C4C2×C22⋊D7111yescharacteristic
306C22×C4C2×C22⋊D7111yescharacteristic
307M16—2844no—
308M16—2842no—
309M16—5682no—
310M16—1424no—
311C2×C8—2844no—
312C2×C8—2842no—
313C2×C8—2842no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#1455604311yescharacteristic
338C23#1455604311yescharacteristic
339C23C2×C22⋊Dic7111yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—44no—
343D4—42no—
344D4—42no—
345D4—44no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4#1455605512yes—
377C2×C4#1455615311yescharacteristic
378C2×C4#1455615811yescharacteristic
379C8—2844no—
380C8—5682no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#1798578612yes—
398C22C2×#1455604311yescharacteristic
399C22#1798600011yescharacteristic
400C22C2×#1455605511yescharacteristic
401C4—21no—
402C4—22no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—82no—
416C2#2258684411yescharacteristic
417C2C2×#1798578611yescharacteristic
418C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2406C2
socleC22×C71C2×#99
prime radicalC22×#41×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2406D71
hypocenterC71#11762
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. C2×#17985768C2semidirect
2. #22586670C2semidirect
3. #22586370C2semidirect
4. #22586343C2semidirect
215. C71#11762semidirect

Maximal subgroups

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

subgroupindex
C2×#179857682
#225866702
#225863702
#225863432
C71×#24062
#1176271