Finite Group Databasebeta

← #31894783

Subgroups — #31894783

Order 18176.

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

normal subgroups only

This group has 618 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, complemented
2#22588328C211yesmaximal, characteristic, complemented
3#22586908C212yesmaximal, complemented
4#22586219C212yesmaximal, complemented
5C71×#2543C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8C2×#14556043—21no—
9#17985963—21no—
10C2×C22⋊(C71⋊C8)C2212yes—
11#17985859C2212yes—
12#17985766—21no—
13#17985733—24no—
14C2×#121×C71—21no—
15C2×#121×C71C2211yescharacteristic
16C2×C4×D4×C71C2211yescharacteristic
17#398×C71C411yescharacteristic, complemented
18#397×C71C411yescharacteristic, complemented
19#394×C71C411yescharacteristic, complemented
20#392×C71C411yescharacteristic, complemented
21#388×C71—21no—
22#386×C71—21no—
23#386×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)D414yescomplemented
34#14556039—22no—
35C71⋊#95—22no—
36C22×D4×C71—21no—
37C22×D4×C71—21no—
38C22×D4×C71C2×C411yescharacteristic
39C22×D4×C71D411yescharacteristic
40C23×C4×C71—21no—
41C23×C4×C71D411yescharacteristic
42C23×C4×C71C2311yescharacteristic
43#121×C71—22no—
44#121×C71—22no—
45#121×C71—44no—
46C4×D4×C71—22no—
47C4×D4×C71—22no—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71C2×C411yescharacteristic
50C2×C4⋊C4×C71C2×C411yescharacteristic
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C22⋊C4×C71—21no—
56C2×C22⋊C4×C71C2×C411yescharacteristic
57C2×C22⋊C4×C71D411yescharacteristic
58C2×C42×C71D411yescharacteristic
59#95×C71—21no—
60#95×C71—21no—
61#95×C71—21no—
62#95×C71C2×C411yescharacteristic
63#95×C71C2×C411yescharacteristic
64C22×Dic71—42no—
65#11765849—82no—
66C2×C71⋊C8—24no—
67C2×C71⋊C8—82no—
68C4⋊Dic71—82no—
69C22⋊Dic71—42no—
70C22⋊Dic71—42no—
71C24×C71—21no—
72C24×C71C22⋊C411yescharacteristic
73C24×C71C2×D411yescharacteristic
74C2×D4×C71—22no—
75C2×D4×C71—22no—
76C2×D4×C71—21no—
77C2×D4×C71—21no—
78C2×D4×C71—22no—
79C2×D4×C71—42no—
80C2×D4×C71—42no—
81C2×D4×C71—42no—
82C2×D4×C71—42no—
83C2×D4×C71—41no—
84C2×D4×C71—41no—
85C2×D4×C71—81no—
86C2×D4×C71—81no—
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—81no—
95C22×C4×C71—81no—
96C22×C4×C71—81no—
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22×C411yescharacteristic
101C22×C4×C71C2×D412yes—
102C22×C4×C71C2×D411yescharacteristic
103C22×C4×C71#4112yes—
104C4⋊C4×C71—22no—
105C4⋊C4×C71—22no—
106C4⋊C4×C71—42no—
107C22⋊C4×C71—22no—
108C22⋊C4×C71—22no—
109C22⋊C4×C71—42no—
110C22⋊C4×C71—42no—
111C22⋊C4×C71—42no—
112C22⋊C4×C71—41no—
113C22⋊C4×C71—41no—
114C22⋊C4×C71—81no—
115C42×C71—21no—
116C42×C71—21no—
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×C71—41no—
131C23×C71C2×C22⋊C411yescharacteristic
132C23×C71#12112yes—
133C23×C71AGL(1,ℤ/8)11yescharacteristic
134C23×C71AGL(1,ℤ/8)12yes—
135C23×C71#13711yescharacteristic
136D4×C71—42no—
137D4×C71—42no—
138D4×C71—82no—
139D4×C71—84no—
140D4×C71—82no—
141D4×C71—82no—
142C2×C4×C71—22no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—42no—
151C2×C4×C71—42no—
152C2×C4×C71—42no—
153C2×C4×C71—41no—
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—81no—
164C2×C4×C71—84no—
165C2×C4×C71C4×D412yes—
166C2×C4×C71#12011yescharacteristic
167C2×C4×C71#12311yescharacteristic
168C2×Dic71—82no—
169C2×Dic71—82no—
170C2×Dic71—82no—
171C71⋊C8—44no—
172C22×C71—22no—
173C22×C71—21no—
174C22×C71—22no—
175C22×C71—21no—
176C22×C71—42no—
177C22×C71—41no—
178C22×C71—41no—
179C22×C71—42no—
180C22×C71—41no—
181C22×C71—41no—
182C22×C71—41no—
183C22×C71—81no—
184C22×C71—81no—
185C22×C71—81no—
186C22×C71—81no—
187C22×C71—81no—
188C22×C71—81no—
189C22×C71—81no—
190C22×C71—81no—
191C22×C71#38611yescharacteristic
192C22×C71#41711yescharacteristic
193C22×C71#41811yescharacteristic
194C22×C71#46812yes—
195C22×C71#46912yes—
196C4×C71—21no—
197C4×C71—21no—
198C4×C71—22no—
199C4×C71—42no—
200C4×C71—41no—
201C4×C71—41no—
202C4×C71—41no—
203C4×C71—81no—
204C4×C71—81no—
205Dic71—162no—
206#9386—711nomaximal, Sylow
207C2×C71—22no—
208C2×C71—21no—
209C2×C71—21no—
210C2×C71—22no—
211C2×C71—81no—
212C2×C71—161no—
213C2×C71#188411yescharacteristic
214C2×C71#189511yescharacteristic
215C2×C71#191311yescharacteristic
216#2973—711no—
217#2543D7111yescharacteristic
218#1799—711no—
219#1519—712no—
220#1294—712no—
221C71#938611yescharacteristic, complemented, Sylow
222C2×#121—21no—
223C2×#121C2×D7111yescharacteristic
224C2×C4×D4C2×D7111yescharacteristic
225#435—1424no—
226#432—1421no—
227C2×#99—1421no—
228#407—1421no—
229C2×C22⋊C8—712no—
230#398Dic7111yescharacteristic, complemented
231#397Dic7111yescharacteristic, complemented
232#394Dic7111yescharacteristic, complemented
233#392Dic7111yescharacteristic, complemented
234#388—21no—
235#386—21no—
236#386—21no—
237#380—712no—
238C2×#95C2×D7111yescharacteristic
239#342—1421no—
240#327—1424no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×Dic7111yescharacteristic
244C22×D4C22⋊D7111yescharacteristic
245C23×C4—21no—
246C23×C4C22⋊D7111yescharacteristic
247C23×C4C22×D7111yescharacteristic
248C2×M16—1422no—
249C22×C8—1422no—
250#121—22no—
251#121—22no—
252#121—44no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—1422no—
257C2×C4⋊C4C2×Dic7111yescharacteristic
258C2×C4⋊C4C2×Dic7111yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—1422no—
265C2×C22⋊C4C2×Dic7111yescharacteristic
266C2×C22⋊C4C22⋊D7111yescharacteristic
267C2×C42C22⋊D7111yescharacteristic
268C4⋊C8—1422no—
269#99—2842no—
270C22⋊C8—714no—
271C22⋊C8—1422no—
272C22⋊C8—1422no—
273#97—1422no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—1422no—
278#95C2×Dic7111yescharacteristic
279#95C2×Dic7111yescharacteristic
280C24—21no—
281C24C22⋊Dic7111yescharacteristic
282C24C2×C22⋊D7111yescharacteristic
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—81no—
295C2×D4—81no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—2842no—
307C22×C4C22⋊Dic7111yescharacteristic
308C22×C4C22⋊Dic7111yescharacteristic
309C22×C4C22⋊Dic7111yescharacteristic
310C22×C4C22×Dic7111yescharacteristic
311C22×C4D4×D7112yes—
312C22×C4C2×C22⋊D7111yescharacteristic
313C22×C4#1176586412yes—
314M16—5682no—
315C2×C8—5682no—
316C2×C8—1424no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—5682no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—81no—
329C22⋊C4—2842no—
330C22⋊C4—2842no—
331C42—21no—
332C42—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C2×C22⋊Dic7111yescharacteristic
348C23#1455610012yes—
349C23#1455615311yescharacteristic
350C23#1455615412yes—
351C23#1455615811yescharacteristic
352D4—42no—
353D4—42no—
354D4—82no—
355D4—84no—
356D4—82no—
357D4—82no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—84no—
381C2×C4—5682no—
382C2×C4—5682no—
383C2×C4—5682no—
384C2×C4D4×Dic7112yes—
385C2×C4#1455609611yescharacteristic
386C2×C4#1455610911yescharacteristic
387C8—2844no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—22no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22#1798587911yescharacteristic
408C22#1798600011yescharacteristic
409C22#1798600411yescharacteristic
410C22#1798624312yes—
411C22#1798625012yes—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—11362no—
422C2—21no—
423C2—22no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—161no—
428C2#2258872411yescharacteristic
429C2#2258877911yescharacteristic
430C2#2258888311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
Fitting, prime radicalC71×#2543C2
socleC22×C71#386
cosocleC23×C4×C71C23
hypercenter#2543D71
hypocenterC71#9386
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. #22597369C2semidirect
2. #22588328C2semidirect
3. #22586908C2semidirect
4. #22586219C2semidirect
17. #398×C71C4semidirect
18. #397×C71C4semidirect
19. #394×C71C4semidirect
20. #392×C71C4semidirect
33. C22⋊(C71⋊C8)D4semidirect
221. C71#9386semidirect
230. #398Dic71semidirect
231. #397Dic71semidirect
232. #394Dic71semidirect
233. #392Dic71semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225883282
#225869082
#225862192
C71×#25432
#938671