Finite Group Databasebeta

← #31894778

Subgroups — #31894778

Order 18176.

548 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 119 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#22588328C211yesmaximal, characteristic
3#22586912C212yesmaximal, complemented
4#22586219C212yesmaximal, complemented
5C71×#2564C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8C2×#14556043—21no—
9#17985963—21no—
10C2×C22⋊(C71⋊C8)C2212yes—
11#17985859C2212yes—
12#17985766—21no—
13#17985736—24no—
14C2×C22⋊Q8×C71C2211yescharacteristic
15C2×C4×D4×C71C2211yescharacteristic
16#400×C71C411yescharacteristic, complemented
17#399×C71C411yescharacteristic, complemented
18#395×C71C411yescharacteristic, complemented
19#393×C71C411yescharacteristic, complemented
20#387×C71—21no—
21#386×C71—21no—
22#385×C71—21no—
23#380×C71—21no—
24C2×#95×C71C2211yescharacteristic
25C2×#11765849—22no—
26C22×C71⋊C8—22no—
27C2×C4⋊Dic71—22no—
28C2×C22⋊Dic71—22no—
29C4⋊(C71⋊C8)—22no—
30#14556043—42no—
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×C71—21no—
39C23×C4×C71D411yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C22⋊Q8×C71—22no—
42C22⋊Q8×C71—22no—
43C4×D4×C71—22no—
44C4×D4×C71—22no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C4⋊C4×C71C2×C411yescharacteristic
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—
67C4⋊Dic71—82no—
68C22⋊Dic71—42no—
69C22⋊Dic71—42no—
70C24×C71C22⋊C411yescharacteristic
71C24×C71C2×D411yescharacteristic
72C2×Q8×C71—22no—
73C2×Q8×C71—81no—
74C2×D4×C71—22no—
75C2×D4×C71—21no—
76C2×D4×C71—21no—
77C2×D4×C71—22no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—42no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71—81no—
88C22×C4×C71—81no—
89C22×C4×C71—81no—
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22⋊C411yescharacteristic
93C22×C4×C71C22×C411yescharacteristic
94C22×C4×C71C2×D412yes—
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71#4112yes—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—22no—
99C4⋊C4×C71—42no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—41no—
102C4⋊C4×C71—81no—
103C22⋊C4×C71—22no—
104C22⋊C4×C71—22no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—42no—
107C22⋊C4×C71—81no—
108C42×C71—21no—
109C42×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71C2×C22⋊C411yescharacteristic
118C23×C71#12112yes—
119C23×C71AGL(1,ℤ/8)11yescharacteristic
120C23×C71#13711yescharacteristic
121C23×C71#13712yes—
122Q8×C71—82no—
123D4×C71—42no—
124D4×C71—42no—
125C2×C4×C71—22no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—42no—
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—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—81no—
150C2×C4×C71—82no—
151C2×C4×C71—82no—
152C2×C4×C71C4×D412yes—
153C2×C4×C71#12011yescharacteristic
154C2×C4×C71#12311yescharacteristic
155C2×Dic71—82no—
156C2×Dic71—82no—
157C2×Dic71—82no—
158C71⋊C8—44no—
159C22×C71—22no—
160C22×C71—21no—
161C22×C71—22no—
162C22×C71—21no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—42no—
166C22×C71—42no—
167C22×C71—41no—
168C22×C71—41no—
169C22×C71—41no—
170C22×C71—81no—
171C22×C71#38611yescharacteristic
172C22×C71#41711yescharacteristic
173C22×C71#41911yescharacteristic
174C22×C71#46812yes—
175C22×C71#47012yes—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—22no—
179C4×C71—42no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183C4×C71—81no—
184C4×C71—81no—
185C4×C71—81no—
186Dic71—162no—
187#9385—711nomaximal, Sylow
188C2×C71—22no—
189C2×C71—21no—
190C2×C71—21no—
191C2×C71—22no—
192C2×C71—81no—
193C2×C71#188411yescharacteristic
194C2×C71#189411yescharacteristic
195C2×C71#191311yescharacteristic
196#2973—711no—
197#2564D7111yescharacteristic
198#1799—711no—
199#1520—712no—
200#1294—712no—
201C71#938511yescharacteristic, complemented, Sylow
202C2×C22⋊Q8C2×D7111yescharacteristic
203C2×C4×D4C2×D7111yescharacteristic
204#435—1424no—
205#432—1421no—
206C2×#99—1421no—
207#407—1421no—
208C2×C22⋊C8—712no—
209#400Dic7111yescharacteristic, complemented
210#399Dic7111yescharacteristic, complemented
211#395Dic7111yescharacteristic, complemented
212#393Dic7111yescharacteristic, complemented
213#387—21no—
214#386—21no—
215#385—21no—
216#380—21no—
217#380—712no—
218C2×#95C2×D7111yescharacteristic
219#342—1421no—
220#328—1424no—
221C22×Q8C2×Dic7111yescharacteristic
222C22×D4C22⋊D7111yescharacteristic
223C23×C4—21no—
224C23×C4C22⋊D7111yescharacteristic
225C23×C4C22×D7111yescharacteristic
226C2×M16—1422no—
227C22×C8—1422no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—1422no—
236C2×C4⋊C4C2×Dic7111yescharacteristic
237C2×C4⋊C4C2×Dic7111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—1422no—
242C2×C22⋊C4C2×Dic7111yescharacteristic
243C2×C22⋊C4C22⋊D7111yescharacteristic
244C2×C42C22⋊D7111yescharacteristic
245C4⋊C8—1422no—
246#99—2842no—
247C22⋊C8—714no—
248C22⋊C8—1422no—
249C22⋊C8—1422no—
250#97—1422no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—1422no—
257#95C2×Dic7111yescharacteristic
258#95C2×Dic7111yescharacteristic
259C24C22⋊Dic7111yescharacteristic
260C24C2×C22⋊D7111yescharacteristic
261C2×Q8—22no—
262C2×Q8—81no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—2842no—
280C22×C4C22⋊Dic7111yescharacteristic
281C22×C4C22⋊Dic7111yescharacteristic
282C22×C4C22⋊Dic7111yescharacteristic
283C22×C4C22×Dic7111yescharacteristic
284C22×C4D4×D7112yes—
285C22×C4C2×C22⋊D7111yescharacteristic
286C22×C4#1176586412yes—
287M16—5682no—
288C2×C8—5682no—
289C2×C8—1424no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—5682no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—81no—
302C22⋊C4—2842no—
303C22⋊C4—2842no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C2×C22⋊Dic7111yescharacteristic
314C23#1455610012yes—
315C23#1455615311yescharacteristic
316C23#1455615811yescharacteristic
317C23#1455615912yes—
318Q8—82no—
319D4—42no—
320D4—42no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—5682no—
349C2×C4—5682no—
350C2×C4—5682no—
351C2×C4D4×Dic7112yes—
352C2×C4#1455609611yescharacteristic
353C2×C4#1455610911yescharacteristic
354C8—2844no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22#1798587911yescharacteristic
368C22#1798600011yescharacteristic
369C22#1798600811yescharacteristic
370C22#1798624712yes—
371C22#1798625512yes—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—11362no—
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#2258872411yescharacteristic
389C2#2258877411yescharacteristic
390C2#2258888311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2564C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC23×C4×C71C23
hypercenter#2564D71
hypocenterC71#9385
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #22586912C2semidirect
4. #22586219C2semidirect
16. #400×C71C4semidirect
17. #399×C71C4semidirect
18. #395×C71C4semidirect
19. #393×C71C4semidirect
201. C71#9385semidirect
209. #400Dic71semidirect
210. #399Dic71semidirect
211. #395Dic71semidirect
212. #393Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225883282
#225869122
#225862192
C71×#25642
#938571