Finite Group Databasebeta

← #31892482

Subgroups — #31892482

Order 18176.

812 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590954C211yesmaximal, characteristic
2#22588430C212yesmaximal, complemented
3C2×#17985723C211yesmaximal, characteristic
4#22586219C212yesmaximal, complemented
5C2×#399×C71C211yesmaximal, characteristic
6C22×C22⋊Dic71—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985874—24no—
9#17985859—21no—
10#17985859C2212yes—
11#17985761—24no—
12#17985723—22no—
13C22×C4⋊C4×C71—21no—
14C22×C4⋊C4×C71C2211yescharacteristic
15C22×C22⋊C4×C71—21no—
16C22×C22⋊C4×C71C2211yescharacteristic
17#399×C71—22no—
18#399×C71C414yescomplemented
19C2×#95×C71C2211yescharacteristic
20C23×Dic71—22no—
21C22×C71⋊C8—22no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71D414yes—
24C2×C22⋊Dic71—22no—
25C2×C22⋊Dic71—22no—
26C2×C22⋊Dic71—42no—
27C2×C22⋊Dic71—42no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C25×C71D411yescharacteristic
33C23×C4×C71—21no—
34C23×C4×C71—21no—
35C23×C4×C71D411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—24no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—44no—
42C2×C4⋊C4×C71C2×C412yescomplemented
43C2×C22⋊C4×C71—22no—
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71—24no—
46C2×C22⋊C4×C71—44no—
47C2×C22⋊C4×C71C2×C412yes—
48C2×C22⋊C4×C71D412yes—
49#95×C71—21no—
50#95×C71C2×C412yescomplemented
51C22×Dic71—24no—
52C22×Dic71—44no—
53C22×Dic71—82no—
54C22×Dic71—82no—
55C2×C71⋊C8—44no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—44no—
58C4⋊Dic71—82no—
59C22⋊Dic71—44no—
60C22⋊Dic71—42no—
61C22⋊Dic71—42no—
62C24×C71—21no—
63C24×C71—21no—
64C24×C71—21no—
65C24×C71—21no—
66C24×C71—42no—
67C24×C71—42no—
68C24×C71C22⋊C412yes—
69C24×C71SD1612yes—
70C24×C71Q1612yes—
71C24×C71C2×D411yescharacteristic
72C22×C4×C71—24no—
73C22×C4×C71—22no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—22no—
77C22×C4×C71—42no—
78C22×C4×C71—42no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—42no—
82C22×C4×C71—42no—
83C22×C4×C71—42no—
84C22×C4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71C22⋊C412yes—
87C22×C4×C71C22×C411yescharacteristic
88C22×C4×C71C2×D412yes—
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71#4112yes—
91C4⋊C4×C71—24no—
92C4⋊C4×C71—48no—
93C4⋊C4×C71—42no—
94C22⋊C4×C71—24no—
95C22⋊C4×C71—24no—
96C22⋊C4×C71—48no—
97C23×C71—22no—
98C23×C71—22no—
99C23×C71—22no—
100C23×C71—22no—
101C23×C71—22no—
102C23×C71—22no—
103C23×C71—22no—
104C23×C71—22no—
105C23×C71—42no—
106C23×C71—42no—
107C23×C71—44no—
108C23×C71—44no—
109C23×C71—42no—
110C23×C71—42no—
111C23×C71—82no—
112C23×C71—82no—
113C23×C71—81no—
114C23×C71—81no—
115C23×C71#10314yes—
116C23×C71C2×C22⋊C411yescharacteristic
117C23×C71#12011yescharacteristic
118C23×C71#12311yescharacteristic
119C23×C71C2×SD1611yescharacteristic
120C23×C71C2×Q1611yescharacteristic
121C23×C71AGL(1,ℤ/8)11yescharacteristic
122C23×C71#13711yescharacteristic
123C2×C4×C71—24no—
124C2×C4×C71—24no—
125C2×C4×C71—22no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—42no—
129C2×C4×C71—44no—
130C2×C4×C71—44no—
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—81no—
139C2×C4×C71—82no—
140C2×C4×C71—82no—
141C2×C4×C71C4×D412yes—
142C2×C4×C71#12112yes—
143C2×Dic71—44no—
144C2×Dic71—44no—
145C2×Dic71—44no—
146C2×Dic71—88no—
147C71⋊C8—82no—
148C22×C71—22no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—22no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—24no—
157C22×C71—44no—
158C22×C71—44no—
159C22×C71—44no—
160C22×C71—42no—
161C22×C71—42no—
162C22×C71—82no—
163C22×C71—82no—
164C22×C71—82no—
165C22×C71#38611yescharacteristic
166C22×C71C2×#10311yescharacteristic
167C22×C71#41711yescharacteristic
168C22×C71#45011yescharacteristic
169C22×C71#45111yescharacteristic
170C22×C71#45312yes—
171C22×C71#45812yes—
172C22×C71#48311yescharacteristic
173C22×C71#48411yescharacteristic
174C4×C71—22no—
175C4×C71—22no—
176C4×C71—44no—
177C4×C71—42no—
178C4×C71—82no—
179Dic71—84no—
180#9021—711nomaximal, Sylow
181C2×C71—21no—
182C2×C71—21no—
183C2×C71—21no—
184C2×C71—21no—
185C2×C71—44no—
186C2×C71#161312yes—
187C2×C71#161412yes—
188C2×C71#187211yescharacteristic
189C2×C71#188711yescharacteristic
190C2×C71#189011yescharacteristic
191C2×#399D7111yescharacteristic
192#2315—711no—
193#1824—712no—
194C2×#323—711no—
195#1294—712no—
196C71#902111yescharacteristic, complemented, Sylow
197C22×C4⋊C4—21no—
198C22×C4⋊C4C2×D7111yescharacteristic
199C22×C22⋊C4—21no—
200C22×C22⋊C4—1421no—
201C22×C22⋊C4C2×D7111yescharacteristic
202C2×C22⋊C8—712no—
203#399—22no—
204#399Dic7114yescomplemented
205#385—1424no—
206#380—712no—
207#380—1421no—
208C2×#95C2×D7111yescharacteristic
209#340—1424no—
210#323—1422no—
211C25C22⋊D7111yescharacteristic
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—1422no—
215C23×C4C22⋊D7111yescharacteristic
216C23×C4C22×D7111yescharacteristic
217C22×C8—1422no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—714no—
224C2×C4⋊C4—1422no—
225C2×C4⋊C4C2×Dic7112yescomplemented
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—2842no—
231C2×C22⋊C4—2842no—
232C2×C22⋊C4—1422no—
233C2×C22⋊C4—1422no—
234C2×C22⋊C4C2×Dic7112yes—
235C2×C22⋊C4C22⋊D7112yes—
236C22⋊C8—1422no—
237C22⋊C8—1422no—
238#95—21no—
239#95—1422no—
240#95—1422no—
241#95C2×Dic7112yescomplemented
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—42no—
247C24—42no—
248C24C22⋊Dic7112yes—
249C24#1176585212yes—
250C24#1176585512yes—
251C24C2×C22⋊D7111yescharacteristic
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—2844no—
267C22×C4—5682no—
268C22×C4—5682no—
269C22×C4—1424no—
270C22×C4C22⋊Dic7112yes—
271C22×C4C22×Dic7111yescharacteristic
272C22×C4D4×D7112yes—
273C22×C4C2×C22⋊D7111yescharacteristic
274C22×C4#1176586412yes—
275C2×C8—2844no—
276C2×C8—2842no—
277C4⋊C4—24no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—2844no—
281C4⋊C4—5682no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—48no—
285C22⋊C4—2844no—
286C22⋊C4—2842no—
287C22⋊C4—2842no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—81no—
303C23—81no—
304C23—82no—
305C23—82no—
306C23#1455605214yes—
307C23C2×C22⋊Dic7111yescharacteristic
308C23#1455609611yescharacteristic
309C23#1455610911yescharacteristic
310C23C2×#1176585211yescharacteristic
311C23C2×#1176585511yescharacteristic
312C23#1455615311yescharacteristic
313C23#1455615811yescharacteristic
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—2844no—
333C2×C4—2844no—
334C2×C4—2844no—
335C2×C4—5688no—
336C2×C4D4×Dic7112yes—
337C2×C4#1455610012yes—
338C8—5682no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—24no—
348C22—44no—
349C22—42no—
350C22—42no—
351C22—44no—
352C22—44no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22#1798587911yescharacteristic
357C22C2×#1455605211yescharacteristic
358C22#1798600011yescharacteristic
359C22#1798614911yescharacteristic
360C22#1798615411yescharacteristic
361C22#1798616612yes—
362C22#1798618812yes—
363C22#1798632711yescharacteristic
364C22#1798633211yescharacteristic
365C4—22no—
366C4—22no—
367C4—44no—
368C4—42no—
369C4—82no—
370C4—5684no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—44no—
376C2#2258735912yes—
377C2#2258736412yes—
378C2#2258864811yescharacteristic
379C2#2258873911yescharacteristic
380C2#2258875411yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#399×C71C2
socleC23×C71#120
prime radicalC25×C71D4
cosocleC23×C4×C71C23
hypercenterC2×#399D71
hypocenterC71#9021
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. #22588430C2semidirect
4. #22586219C2semidirect
18. #399×C71C4semidirect
42. C2×C4⋊C4×C71C2×C4semidirect
50. #95×C71C2×C4semidirect
196. C71#9021semidirect
204. #399Dic71semidirect
225. C2×C4⋊C4C2×Dic71semidirect
241. #95C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909542
#225884302
C2×#179857232
#225862192
C2×#399×C712
#902171