Finite Group Databasebeta

← #31893496

Subgroups — #31893496

Order 18176.

684 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 135 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#22590954C211yesmaximal, characteristic
2#22586912C212yesmaximal, complemented
3C2×#17985723C211yesmaximal, characteristic
4#22586207C212yesmaximal, complemented
5C71×#2626C211yesmaximal, characteristic
6C22×C22⋊Dic71—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985874—24no—
9#17985859—21no—
10#17985859C2212yes—
11#17985736—24no—
12#17985723—22no—
13C2×C22⋊Q8×C71C2211yescharacteristic
14C22×C22⋊C4×C71C2211yescharacteristic
15#399×C71C412yescomplemented
16#396×C71—22no—
17#393×C71C412yescomplemented
18#385×C71—21no—
19#380×C71C2211yescharacteristic
20#379×C71—21no—
21C23×Dic71—22no—
22C22×C71⋊C8—22no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71—22no—
25C2×C22⋊Dic71—22no—
26C2×C22⋊Dic71—22no—
27C2×C22⋊Dic71—42no—
28C2×C22⋊Dic71—42no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C71⋊#95—22no—
32C71⋊#95D414yes—
33C25×C71D411yescharacteristic
34C22×Q8×C71C2×C411yescharacteristic, complemented
35C23×C4×C71D411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37C22⋊Q8×C71—22no—
38C22⋊Q8×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71C2×C411yescharacteristic, complemented
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—22no—
48C2×C22⋊C4×C71—22no—
49C2×C22⋊C4×C71C2×C412yes—
50C2×C22⋊C4×C71D412yes—
51#95×C71—22no—
52#95×C71C2×C412yescomplemented
53C22×Dic71—24no—
54C22×Dic71—44no—
55C22×Dic71—82no—
56C22×Dic71—82no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—82no—
59C4⋊Dic71—82no—
60C4⋊Dic71—82no—
61C22⋊Dic71—44no—
62C22⋊Dic71—42no—
63C22⋊Dic71—42no—
64C24×C71—21no—
65C24×C71—21no—
66C24×C71—21no—
67C24×C71—21no—
68C24×C71—81no—
69C24×C71—81no—
70C24×C71C22⋊C412yes—
71C24×C71SD1612yes—
72C24×C71Q1612yes—
73C24×C71C2×D411yescharacteristic
74C2×Q8×C71—22no—
75C2×Q8×C71—81no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—22no—
80C22×C4×C71—21no—
81C22×C4×C71—81no—
82C22×C4×C71—81no—
83C22×C4×C71C22⋊C412yes—
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D412yes—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71#4112yes—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—81no—
91C4⋊C4×C71—82no—
92C22⋊C4×C71—24no—
93C22⋊C4×C71—24no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C22⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—82no—
99C22⋊C4×C71—82no—
100C23×C71—22no—
101C23×C71—22no—
102C23×C71—22no—
103C23×C71—22no—
104C23×C71—22no—
105C23×C71—22no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—42no—
109C23×C71—42no—
110C23×C71—42no—
111C23×C71—42no—
112C23×C71—81no—
113C23×C71—81no—
114C23×C71—82no—
115C23×C71—82no—
116C23×C71—82no—
117C23×C71—82no—
118C23×C71#10314yes—
119C23×C71C2×C22⋊C411yescharacteristic
120C23×C71#12011yescharacteristic
121C23×C71#12311yescharacteristic
122C23×C71C2×SD1611yescharacteristic
123C23×C71C2×Q1611yescharacteristic
124C23×C71AGL(1,ℤ/8)11yescharacteristic
125C23×C71#13711yescharacteristic
126Q8×C71—82no—
127C2×C4×C71—22no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—42no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—42no—
141C2×C4×C71—81no—
142C2×C4×C71C4×D412yes—
143C2×C4×C71#12112yes—
144C2×Dic71—44no—
145C2×Dic71—44no—
146C2×Dic71—44no—
147C2×Dic71—88no—
148C71⋊C8—82no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—24no—
156C22×C71—41no—
157C22×C71—41no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—44no—
161C22×C71—44no—
162C22×C71—44no—
163C22×C71—42no—
164C22×C71—42no—
165C22×C71—82no—
166C22×C71—82no—
167C22×C71—82no—
168C22×C71#38611yescharacteristic
169C22×C71C2×#10311yescharacteristic
170C22×C71#41711yescharacteristic
171C22×C71#45011yescharacteristic
172C22×C71#45111yescharacteristic
173C22×C71#48311yescharacteristic
174C22×C71#48411yescharacteristic
175C4×C71—22no—
176C4×C71—22no—
177C4×C71—42no—
178C4×C71—82no—
179C4×C71—81no—
180C4×C71—81no—
181Dic71—84no—
182#9199—711nomaximal, Sylow
183C2×C71—21no—
184C2×C71—21no—
185C2×C71—21no—
186C2×C71—21no—
187C2×C71—21no—
188C2×C71—21no—
189C2×C71—44no—
190C2×C71#187211yescharacteristic
191C2×C71#191111yescharacteristic
192C2×C71#191311yescharacteristic
193#2626D7111yescharacteristic
194#2315—711no—
195#1520—712no—
196C2×#323—711no—
197#1289—712no—
198C71#919911yescharacteristic, complemented, Sylow
199C2×C22⋊Q8C2×D7111yescharacteristic
200C22×C22⋊C4—1421no—
201C22×C22⋊C4C2×D7111yescharacteristic
202C2×C22⋊C8—712no—
203#399Dic7112yescomplemented
204#396—22no—
205#393Dic7112yescomplemented
206#385—21no—
207#385—1424no—
208#380—712no—
209#380—1421no—
210#380C2×D7111yescharacteristic
211#379—21no—
212#328—1424no—
213#323—1422no—
214C25C22⋊D7111yescharacteristic
215C22×Q8C2×Dic7111yescharacteristic, complemented
216C23×C4—1422no—
217C23×C4C22⋊D7111yescharacteristic
218C23×C4C22×D7111yescharacteristic
219C22×C8—1422no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1422no—
226C2×C4⋊C4—1422no—
227C2×C4⋊C4C2×Dic7111yescharacteristic, complemented
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—2842no—
235C2×C22⋊C4—2842no—
236C2×C22⋊C4—1422no—
237C2×C22⋊C4—1422no—
238C2×C22⋊C4C2×Dic7112yes—
239C2×C22⋊C4C22⋊D7112yes—
240C22⋊C8—1422no—
241C22⋊C8—1422no—
242#95—22no—
243#95—714no—
244#95—1422no—
245#95C2×Dic7112yescomplemented
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24C22⋊Dic7112yes—
253C24#1176585212yes—
254C24#1176585512yes—
255C24C2×C22⋊D7111yescharacteristic
256C2×Q8—22no—
257C2×Q8—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—2844no—
266C22×C4—5682no—
267C22×C4—5682no—
268C22×C4—1424no—
269C22×C4C22⋊Dic7112yes—
270C22×C4C22×Dic7111yescharacteristic
271C22×C4D4×D7112yes—
272C22×C4C2×C22⋊D7111yescharacteristic
273C22×C4#1176586412yes—
274C2×C8—2842no—
275C2×C8—5682no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4—82no—
280C4⋊C4—5682no—
281C4⋊C4—5682no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C22⋊C4—2844no—
291C22⋊C4—2842no—
292C22⋊C4—2842no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1455605214yes—
312C23C2×C22⋊Dic7111yescharacteristic
313C23#1455609611yescharacteristic
314C23#1455610911yescharacteristic
315C23C2×#1176585211yescharacteristic
316C23C2×#1176585511yescharacteristic
317C23#1455615311yescharacteristic
318C23#1455615811yescharacteristic
319Q8—82no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—2844no—
336C2×C4—2844no—
337C2×C4—2844no—
338C2×C4—5688no—
339C2×C4D4×Dic7112yes—
340C2×C4#1455610012yes—
341C8—5682no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1798587911yescharacteristic
362C22C2×#1455605211yescharacteristic
363C22#1798600011yescharacteristic
364C22#1798614911yescharacteristic
365C22#1798615411yescharacteristic
366C22#1798632711yescharacteristic
367C22#1798633211yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—82no—
372C4—81no—
373C4—81no—
374C4—5684no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2258864811yescharacteristic
383C2#2258887311yescharacteristic
384C2#2258888311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2626C2
socleC22×C71#386
prime radicalC25×C71D4
cosocleC23×C4×C71C23
hypercenter#2626D71
hypocenterC71#9199
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #22586912C2semidirect
4. #22586207C2semidirect
15. #399×C71C4semidirect
17. #393×C71C4semidirect
34. C22×Q8×C71C2×C4semidirect
42. C2×C4⋊C4×C71C2×C4semidirect
52. #95×C71C2×C4semidirect
198. C71#9199semidirect
203. #399Dic71semidirect
205. #393Dic71semidirect
215. C22×Q8C2×Dic71semidirect
227. C2×C4⋊C4C2×Dic71semidirect
245. #95C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909542
#225869122
C2×#179857232
#225862072
C71×#26262
#919971