Finite Group Databasebeta

← #31907776

Subgroups — #31907776

Order 18176.

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

normal subgroups only

This group has 648 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
1C4×#14556043C211yesmaximal, characteristic, complemented
2#22586661C211yesmaximal, characteristic, complemented
3#22586361C212yesmaximal, complemented
4#22586337C212yesmaximal, complemented
5C71×#2388C211yesmaximal, characteristic
6#17986052—21no—
7C2×#14556043—21no—
8C4×C22⋊Dic71—21no—
9#17985766C2212yes—
10#17985723—21no—
11#17985723C2212yes—
12C22×#41×C71C411yescharacteristic, complemented
13C2×C4×D4×C71—21no—
14C2×C4×D4×C71C2211yescharacteristic
15#395×C71C411yescharacteristic, complemented
16#394×C71—21no—
17#394×C71C411yescharacteristic, complemented
18#393×C71C2211yescharacteristic
19#392×C71—21no—
20#392×C71C2211yescharacteristic
21C4×C22⋊C4×C71—21no—
22C4×C22⋊C4×C71C411yescharacteristic, complemented
23C2×C22⋊Dic71—22no—
24C2×C4×Dic71—41no—
25C4⋊(C71⋊C8)—41no—
26#14556043—24no—
27C22⋊(C71⋊C8)—22no—
28C4×C71⋊C8—41no—
29C71⋊#95—41no—
30C2×#41×C71—44no—
31C2×#41×C71—41no—
32C2×#41×C71—41no—
33C22×Q8×C71C2×C411yescharacteristic
34C22×D4×C71—21no—
35C22×D4×C71—21no—
36C22×D4×C71C2×C411yescharacteristic
37C22×D4×C71D411yescharacteristic
38C23×C4×C71—21no—
39C23×C4×C71C2×C411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C4×D4×C71—41no—
42C4×D4×C71—41no—
43C4×D4×C71—44no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71D411yescharacteristic
52C2×C22⋊C4×C71C2311yescharacteristic
53C2×C42×C71—21no—
54C2×C42×C71D411yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71C2×C411yescharacteristic
58#95×C71C2×C411yescharacteristic
59C22×Dic71—42no—
60C2×C71⋊C8—42no—
61C22⋊Dic71—24no—
62C22⋊Dic71—42no—
63C4×Dic71—82no—
64C24×C71—21no—
65C24×C71C22⋊C411yescharacteristic
66C24×C71C2×D411yescharacteristic
67#41×C71—44no—
68#41×C71—44no—
69#41×C71—44no—
70#41×C71—44no—
71C2×Q8×C71—42no—
72C2×Q8×C71—41no—
73C2×D4×C71—22no—
74C2×D4×C71—22no—
75C2×D4×C71—41no—
76C2×D4×C71—41no—
77C2×D4×C71—42no—
78C2×D4×C71—42no—
79C2×D4×C71—42no—
80C2×D4×C71—42no—
81C2×D4×C71—42no—
82C2×D4×C71—42no—
83C2×D4×C71—42no—
84C22×C4×C71—24no—
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—42no—
93C22×C4×C71—42no—
94C22×C4×C71—42no—
95C22×C4×C71—41no—
96C22×C4×C71—41no—
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22×C411yescharacteristic
101C22×C4×C71C2×D411yescharacteristic
102C4⋊C4×C71—41no—
103C4⋊C4×C71—42no—
104C22⋊C4×C71—41no—
105C22⋊C4×C71—41no—
106C22⋊C4×C71—42no—
107C22⋊C4×C71—42no—
108C22⋊C4×C71—42no—
109C22⋊C4×C71—42no—
110C22⋊C4×C71—42no—
111C42×C71—21no—
112C42×C71—21no—
113C42×C71—42no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—22no—
117C23×C71—21no—
118C23×C71—21no—
119C23×C71—21no—
120C23×C71—41no—
121C23×C71—41no—
122C23×C71—41no—
123C23×C71—41no—
124C23×C71—41no—
125C23×C71—41no—
126C23×C71—41no—
127C23×C71—41no—
128C23×C71—41no—
129C23×C71—41no—
130C23×C71C2×C22⋊C411yescharacteristic
131C23×C71#13512yes—
132Q8×C71—42no—
133Q8×C71—42no—
134D4×C71—44no—
135D4×C71—42no—
136D4×C71—48no—
137D4×C71—48no—
138D4×C71—42no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—22no—
142C2×C4×C71—22no—
143C2×C4×C71—22no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—44no—
148C2×C4×C71—42no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—44no—
154C2×C4×C71—42no—
155C2×C4×C71—42no—
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—81no—
163C2×C4×C71#9911yescharacteristic
164C2×C4×C71#9911yescharacteristic
165C2×C4×C71#10412yes—
166C2×Dic71—44no—
167C2×Dic71—82no—
168C2×Dic71—82no—
169C71⋊C8—82no—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—22no—
173C22×C71—42no—
174C22×C71—42no—
175C22×C71—42no—
176C22×C71—41no—
177C22×C71—41no—
178C22×C71—41no—
179C22×C71—42no—
180C22×C71—41no—
181C22×C71—41no—
182C22×C71—81no—
183C22×C71—81no—
184C22×C71—81no—
185C22×C71—81no—
186C22×C71—81no—
187C22×C71—81no—
188C22×C71—81no—
189C22×C71C2×#9911yescharacteristic
190C22×C71#41611yescharacteristic
191C22×C71C2×#10411yescharacteristic
192C4×C71—21no—
193C4×C71—22no—
194C4×C71—42no—
195C4×C71—42no—
196C4×C71—41no—
197C4×C71—41no—
198C4×C71—41no—
199C4×C71—81no—
200C4×C71AΣL(1,𝔽16)12yes—
201Dic71—84no—
202#11725—711nomaximal, Sylow
203C2×C71—21no—
204C2×C71—21no—
205C2×C71—42no—
206C2×C71—81no—
207C2×C71—82no—
208C2×C71#150111yescharacteristic
209C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
210C2×C71#212611yescharacteristic
211#2388D7111yescharacteristic
212C4×#99—711no—
213#1460—711no—
214#1351—712no—
215#1343—712no—
216C71#1172511yescharacteristic, complemented, Sylow
217C22×#41Dic7111yescharacteristic, complemented
218C2×C4×D4—21no—
219C2×C4×D4C2×D7111yescharacteristic
220#431—1421no—
221C2×#99—1421no—
222#395Dic7111yescharacteristic, complemented
223#394—21no—
224#394Dic7111yescharacteristic, complemented
225#393C2×D7111yescharacteristic
226#392—21no—
227#392C2×D7111yescharacteristic
228C4×C22⋊C4—21no—
229C4×C22⋊C4—1421no—
230C4×C22⋊C4Dic7111yescharacteristic, complemented
231#342—712no—
232#323—712no—
233#323—1421no—
234C2×#41—44no—
235C2×#41—41no—
236C2×#41—41no—
237C22×Q8C2×Dic7111yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×Dic7111yescharacteristic
241C22×D4C22⋊D7111yescharacteristic
242C23×C4—21no—
243C23×C4C2×Dic7111yescharacteristic
244C23×C4C22⋊D7111yescharacteristic
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—44no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C2×Dic7111yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—1422no—
256C2×C22⋊C4C22⋊D7111yescharacteristic
257C2×C22⋊C4C22×D7111yescharacteristic
258C2×C42—21no—
259C2×C42—2841no—
260C2×C42C22⋊D7111yescharacteristic
261C4⋊C8—2841no—
262#99—1424no—
263C22⋊C8—1422no—
264C4×C8—2841no—
265#95—21no—
266#95—21no—
267#95—2841no—
268#95C2×Dic7111yescharacteristic
269#95C2×Dic7111yescharacteristic
270C24—21no—
271C24C22⋊Dic7111yescharacteristic
272C24C2×C22⋊D7111yescharacteristic
273#41—44no—
274#41—44no—
275#41—44no—
276#41—44no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—2842no—
304C22×C4C22⋊Dic7111yescharacteristic
305C22×C4C22⋊Dic7111yescharacteristic
306C22×C4C22⋊Dic7111yescharacteristic
307C22×C4C22×Dic7111yescharacteristic
308C22×C4C2×C22⋊D7111yescharacteristic
309C2×C8—2842no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—2842no—
320C22⋊C4—1424no—
321C42—21no—
322C42—21no—
323C42—42no—
324C42—5682no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×C22⋊Dic7111yescharacteristic
342C23#1455614812yes—
343Q8—42no—
344Q8—42no—
345D4—44no—
346D4—42no—
347D4—48no—
348D4—42no—
349D4—48no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—2844no—
375C2×C4—5682no—
376C2×C4—5682no—
377C2×C4#1455604311yescharacteristic
378C2×C4#1455604311yescharacteristic
379C2×C4#1455605512yes—
380C8—5682no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×#1455604311yescharacteristic
401C22#1798599511yescharacteristic
402C22C2×#1455605511yescharacteristic
403C4—21no—
404C4—22no—
405C4—42no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—5684no—
412C4#1798578612yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—82no—
417C2—81no—
418C2#2258683111yescharacteristic
419C2C2×#1798578611yescharacteristic
420C2#2259000511yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
Fitting, prime radicalC71×#2388C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#2388D71
hypocenterC71#11725
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. C4×#14556043C2semidirect
2. #22586661C2semidirect
3. #22586361C2semidirect
4. #22586337C2semidirect
12. C22×#41×C71C4semidirect
15. #395×C71C4semidirect
17. #394×C71C4semidirect
22. C4×C22⋊C4×C71C4semidirect
216. C71#11725semidirect
217. C22×#41Dic71semidirect
222. #395Dic71semidirect
224. #394Dic71semidirect
230. C4×C22⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×#145560432
#225866612
#225863612
#225863372
C71×#23882
#1172571