Finite Group Databasebeta

← #31894768

Subgroups — #31894768

Order 18176.

570 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597364C211yesmaximal, characteristic
2#22588328C211yesmaximal, characteristic
3#22586912C212yesmaximal, complemented
4#22586219C212yesmaximal, complemented
5C71×#2564C211yesmaximal, characteristic
6D4×C71⋊C8—24no—
7#17986052—21no—
8C2×#14556043—21no—
9C2×C22⋊(C71⋊C8)—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
25C22×C71⋊C8—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—
34C4×C71⋊C8—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—
64C2×C71⋊C8—24no—
65C2×C71⋊C8—82no—
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—22no—
76C2×D4×C71SD1612yes—
77C2×D4×C71Q1612yes—
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—42no—
86C22×C4×C71—81no—
87C22×C4×C71—81no—
88C22×C4×C71—81no—
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71C22⋊C411yescharacteristic
91C22×C4×C71C22⋊C411yescharacteristic
92C22×C4×C71C22×C411yescharacteristic
93C22×C4×C71C2×D412yes—
94C22×C4×C71C2×D411yescharacteristic
95C22×C4×C71#4112yes—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—42no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—81no—
102C22⋊C4×C71—22no—
103C22⋊C4×C71—22no—
104C22⋊C4×C71—42no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—81no—
107C42×C71—21no—
108C42×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71#10312yes—
116C23×C71C2×C22⋊C411yescharacteristic
117C23×C71#12112yes—
118C23×C71C2×SD1611yescharacteristic
119C23×C71C2×Q1611yescharacteristic
120C23×C71#13512yes—
121Q8×C71—82no—
122D4×C71—24no—
123D4×C71—24no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—22no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
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—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—81no—
148C2×C4×C71—84no—
149C2×C4×C71#10312yes—
150C2×C4×C71C4×D412yes—
151C2×C4×C71#12011yescharacteristic
152C2×C4×C71#12311yescharacteristic
153C2×Dic71—82no—
154C2×Dic71—82no—
155C2×Dic71—82no—
156C71⋊C8—44no—
157C22×C71—22no—
158C22×C71—21no—
159C22×C71—22no—
160C22×C71—21no—
161C22×C71—22no—
162C22×C71—42no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—42no—
166C22×C71—41no—
167C22×C71—41no—
168C22×C71—81no—
169C22×C71#38611yescharacteristic
170C22×C71C2×#10311yescharacteristic
171C22×C71#41611yescharacteristic
172C22×C71#46512yes—
173C22×C71#46712yes—
174C4×C71—22no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—22no—
178C4×C71—42no—
179C4×C71—41no—
180C4×C71—41no—
181C4×C71—81no—
182C4×C71—81no—
183C4×C71—81no—
184Dic71—162no—
185#9383—711nomaximal, Sylow
186C2×C71—22no—
187C2×C71—21no—
188C2×C71—21no—
189C2×C71—22no—
190C2×C71—42no—
191C2×C71#188411yescharacteristic
192C2×C71#189211yescharacteristic
193C2×C71#191311yescharacteristic
194#2972—711no—
195#2564D7111yescharacteristic
196#1799—711no—
197#1520—712no—
198#1294—712no—
199C71#938311yescharacteristic, complemented, Sylow
200C2×C22⋊Q8C2×D7111yescharacteristic
201C2×C4×D4C2×D7111yescharacteristic
202C8×D4—1424no—
203#431—1421no—
204C2×#99—1421no—
205C2×C22⋊C8—712no—
206C2×C22⋊C8—1421no—
207#400Dic7111yescharacteristic, complemented
208#399Dic7111yescharacteristic, complemented
209#395Dic7111yescharacteristic, complemented
210#393Dic7111yescharacteristic, complemented
211#387—21no—
212#386—21no—
213#385—21no—
214#380—21no—
215#380—712no—
216C2×#95C2×D7111yescharacteristic
217#342—1421no—
218#328—1424no—
219C22×Q8C2×Dic7111yescharacteristic
220C22×D4C22⋊D7111yescharacteristic
221C23×C4—21no—
222C23×C4C22⋊D7111yescharacteristic
223C23×C4C22×D7111yescharacteristic
224C22×C8—1422no—
225C22×C8—1422no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1422no—
234C2×C4⋊C4C2×Dic7111yescharacteristic
235C2×C4⋊C4C2×Dic7111yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1422no—
240C2×C22⋊C4C2×Dic7111yescharacteristic
241C2×C22⋊C4C22⋊D7111yescharacteristic
242C2×C42C22⋊D7111yescharacteristic
243C4⋊C8—1422no—
244#99—2842no—
245C22⋊C8—714no—
246C22⋊C8—1422no—
247C22⋊C8—1422no—
248C4×C8—1422no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—1422no—
255#95C2×Dic7111yescharacteristic
256#95C2×Dic7111yescharacteristic
257C24C22⋊Dic7111yescharacteristic
258C24C2×C22⋊D7111yescharacteristic
259C2×Q8—22no—
260C2×Q8—81no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4#1176585212yes—
264C2×D4#1176585512yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—2842no—
277C22×C4C22⋊Dic7111yescharacteristic
278C22×C4C22⋊Dic7111yescharacteristic
279C22×C4C22⋊Dic7111yescharacteristic
280C22×C4C22×Dic7111yescharacteristic
281C22×C4D4×D7112yes—
282C22×C4C2×C22⋊D7111yescharacteristic
283C22×C4#1176586412yes—
284C2×C8—5682no—
285C2×C8—5682no—
286C2×C8—1424no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—81no—
293C4⋊C4—5682no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—81no—
299C22⋊C4—2842no—
300C22⋊C4—2842no—
301C42—21no—
302C42—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1455605212yes—
310C23C2×C22⋊Dic7111yescharacteristic
311C23#1455610012yes—
312C23C2×#1176585211yescharacteristic
313C23C2×#1176585511yescharacteristic
314C23#1455614812yes—
315Q8—82no—
316D4—24no—
317D4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
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—81no—
342C2×C4—84no—
343C2×C4—5682no—
344C2×C4—5682no—
345C2×C4—5682no—
346C2×C4#1455605212yes—
347C2×C4D4×Dic7112yes—
348C2×C4#1455609611yescharacteristic
349C2×C4#1455610911yescharacteristic
350C8—2844no—
351C22—21no—
352C22—22no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22#1798587911yescharacteristic
364C22C2×#1455605211yescharacteristic
365C22#1798599511yescharacteristic
366C22#1798623012yes—
367C22#1798623812yes—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—11362no—
379C2—21no—
380C2—22no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2#2258872411yescharacteristic
385C2#2258876411yescharacteristic
386C2#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#9383
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
199. C71#9383semidirect
207. #400Dic71semidirect
208. #399Dic71semidirect
209. #395Dic71semidirect
210. #393Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973642
#225883282
#225869122
#225862192
C71×#25642
#938371