Finite Group Databasebeta

← #32097388

Subgroups — #32097388

Order 18176.

798 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22591065C211yesmaximal, characteristic
2#22589483C212yesmaximal, complemented
3#22589473C212yesmaximal, complemented
4#22589163C211yesmaximal, characteristic
5#22589145C211yesmaximal, characteristic
6#22588475C211yesmaximal, characteristic, complemented
7#22588470C211yesmaximal, characteristic, complemented
8#22588347C212yesmaximal, complemented
9#22588337C212yesmaximal, complemented
10#22588197C211yesmaximal, characteristic, complemented
11C71×#2660C211yesmaximal, characteristic, complemented
12C2×#14556125C2211yescharacteristic
13C2×C4×D284C2212yes—
14C2×C4⋊(C71⋊C8)C2212yes—
15C2×#14556048C2212yes—
16C2×#14556048C2212yes—
17C2×#14556048C2212yes—
18C2×#14556048C2212yes—
19C2×#14556039C2211yescharacteristic
20C2×C4×C71⋊C8C2211yescharacteristic
21#17985901C2212yes—
22#17985877—22no—
23#17985877—22no—
24#17985865C2211yescharacteristic
25#17985761C2212yes—
26#17985761C2212yes—
27#17985761C2212yes—
28#17985761C2212yes—
29#17985754C2212yes—
30C2×C4⋊Q8×C71C2211yescharacteristic
31C2×#125×C71C2211yescharacteristic
32#398×C71—22no—
33#395×C71—22no—
34#384×C71C2211yescharacteristic
35#384×C71C2211yescharacteristic
36#382×C71C2211yescharacteristic
37#382×C71C2211yescharacteristic
38#376×C71C2211yescharacteristic
39C22×D284—22no—
40C22×D284C2312yes—
41C22×C4×D71—22no—
42C22×C71⋊C8C2312yes—
43C22×C71⋊C8C2312yes—
44#14556125—22no—
45#14556125Q814yescomplemented
46C4×D284—24no—
47C4×D284—24no—
48C2×C4⋊Dic71C2312yes—
49C2×(C2×D71)⋊C4—22no—
50C2×(C2×D71)⋊C4—22no—
51C2×(C2×D71)⋊C4—22no—
52C4⋊(C71⋊C8)—24no—
53#14556048—42no—
54#14556048—42no—
55#14556048—42no—
56#14556048—42no—
57#14556039—22no—
58C4×C71⋊C8—22no—
59C71⋊#95—22no—
60C22×Q8×C71—21no—
61C4⋊Q8×C71—42no—
62#125×C71—42no—
63C2×C4⋊C4×C71—21no—
64C2×C4⋊C4×C71—21no—
65C2×C4⋊C4×C71—21no—
66C2×C4⋊C4×C71—22no—
67C2×C4⋊C4×C71—22no—
68C2×C4⋊C4×C71C2311yescharacteristic
69C2×C4⋊C4×C71C2311yescharacteristic
70C2×C4⋊C4×C71C2311yescharacteristic
71C2×C4⋊C4×C71C2311yescharacteristic
72C2×C42×C71C2311yescharacteristic
73C2×C42×C71C2311yescharacteristic
74C2×C42×C71C2311yescharacteristic
75#95×C71—22no—
76#95×C71—22no—
77#95×C71D412yes—
78#95×C71D412yes—
79C23×D71—22no—
80C2×D284—22no—
81C2×D284—22no—
82C2×D284—22no—
83C2×D284—22no—
84C2×D284—22no—
85C2×D284—28no—
86C2×D284—82no—
87C2×D284—82no—
88C2×D284C2×Q814yes—
89C22×Dic71—22no—
90C2×C4×D71—44no—
91C2×C4×D71—82no—
92C2×C4×D71—82no—
93C2×C71⋊C8—22no—
94C2×C71⋊C8—22no—
95C2×C71⋊C8—22no—
96C2×C71⋊C8—22no—
97C2×C71⋊C8—22no—
98C2×C71⋊C8—22no—
99C4⋊Dic71—22no—
100C4⋊Dic71—22no—
101(C2×D71)⋊C4—44no—
102(C2×D71)⋊C4—82no—
103(C2×D71)⋊C4—82no—
104C2×Q8×C71—42no—
105C2×Q8×C71—82no—
106C22×C4×C71—21no—
107C22×C4×C71—21no—
108C22×C4×C71—21no—
109C22×C4×C71—21no—
110C22×C4×C71C2×D411yescharacteristic
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71C2×D412yes—
113C22×C4×C71C2×D412yes—
114C22×C4×C71C2411yescharacteristic
115C4⋊C4×C71—22no—
116C4⋊C4×C71—22no—
117C4⋊C4×C71—22no—
118C4⋊C4×C71—22no—
119C4⋊C4×C71—42no—
120C4⋊C4×C71—42no—
121C4⋊C4×C71—42no—
122C4⋊C4×C71—82no—
123C4⋊C4×C71—82no—
124C42×C71—22no—
125C42×C71C2×Q812yes—
126C42×C71#4112yes—
127C42×C71#4114yes—
128C23×C71C22×D411yescharacteristic
129C22×D71—24no—
130C22×D71—42no—
131C22×D71—42no—
132C22×D71—42no—
133C22×D71—42no—
134C22×D71—42no—
135C22×D71—42no—
136Q8×C71—84no—
137C2×C4×C71—22no—
138C2×C4×C71—22no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—22no—
144C2×C4×C71—22no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—42no—
150C2×C4×C71—42no—
151C2×C4×C71—42no—
152C2×C4×C71—42no—
153C2×C4×C71C22⋊Q814yes—
154C2×C4×C71#12414yes—
155C2×C4×C71C22×Q811yescharacteristic
156C2×C4×C71C2×#4112yes—
157C2×C4×C71C2×#4111yescharacteristic
158C2×C4×C71H(5,𝔽2)12yes—
159C2×Dic71—42no—
160C2×Dic71—42no—
161C2×Dic71—42no—
162D284—24no—
163D284—24no—
164D284—42no—
165D284—42no—
166D284—88no—
167C4×D71—88no—
168C71⋊C8—42no—
169C71⋊C8—42no—
170C22×C71C2×C22⋊Q811yescharacteristic
171C22×C71C2×#12411yescharacteristic
172C22×C71#53511yescharacteristic
173C22×C71#57612yes—
174C22×C71#57712yes—
175C2×D71—44no—
176C2×D71—44no—
177C2×D71—44no—
178C2×D71—82no—
179C2×D71—82no—
180C2×D71—82no—
181C2×D71—82no—
182C4×C71—24no—
183C4×C71—24no—
184C4×C71—41no—
185C4×C71—41no—
186C4×C71—81no—
187C4×C71—81no—
188C4×C71—81no—
189C4×C71—81no—
190C4×C71#55412yes—
191C4×C71#55512yes—
192Dic71—82no—
193#29152—711nomaximal, Sylow
194C2×C71#256311yescharacteristic
195C2×C71#308112yes—
196C2×C71#313612yes—
197C2×C71#322011yescharacteristic
198C2×C71#322111yescharacteristic
199D71—84no—
200#2660D7111yescharacteristic, complemented
201#2328—711no—
202#2025—712no—
203#2023—712no—
204#1961—711no—
205#1957—711no—
206#1836—711no—
207#1835—711no—
208#1804—712no—
209#1802—712no—
210#1768—711no—
211C71#2915211yescharacteristic, complemented, Sylow
212C2×C4⋊Q8C2×D7111yescharacteristic
213C2×#127—711no—
214C2×#125C2×D7111yescharacteristic
215C2×C4×D4—712no—
216C2×C4⋊C8—712no—
217C2×#102—712no—
218C2×#102—712no—
219C2×#102—712no—
220C2×#102—712no—
221C2×#97—711no—
222C2×C4×C8—711no—
223#398—22no—
224#395—22no—
225#390—712no—
226#386—1422no—
227#386—1422no—
228#384C2×D7111yescharacteristic
229#384C2×D7111yescharacteristic
230#382—711no—
231#382C2×D7111yescharacteristic
232#382C2×D7111yescharacteristic
233#376C2×D7111yescharacteristic
234#340—712no—
235#340—712no—
236#340—712no—
237#340—712no—
238#336—712no—
239C22×Q8—21no—
240C22×D4—712no—
241C22×D4—1422no—
242C23×C4—1422no—
243C22×C8—712no—
244C22×C8—712no—
245C4⋊Q8—42no—
246#127—714no—
247#127—1422no—
248#125—42no—
249C4×D4—1424no—
250C4×D4—1424no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—712no—
257C2×C4⋊C4C22×D7111yescharacteristic
258C2×C4⋊C4C22×D7111yescharacteristic
259C2×C4⋊C4C22×D7111yescharacteristic
260C2×C4⋊C4C22×D7111yescharacteristic
261C2×C22⋊C4—1422no—
262C2×C22⋊C4—1422no—
263C2×C22⋊C4—1422no—
264C2×C42C22×D7111yescharacteristic
265C2×C42C22×D7111yescharacteristic
266C2×C42C22×D7111yescharacteristic
267C4⋊C8—1424no—
268#102—2842no—
269#102—2842no—
270#102—2842no—
271#102—2842no—
272#97—1422no—
273C4×C8—1422no—
274#95—22no—
275#95—22no—
276#95—1422no—
277#95C22⋊D7112yes—
278#95C22⋊D7112yes—
279C24—1422no—
280C2×Q8—42no—
281C2×Q8—82no—
282C2×D4—5682no—
283C2×D4—5682no—
284C2×D4—714no—
285C2×D4—1422no—
286C2×D4—1422no—
287C2×D4—1422no—
288C2×D4—1422no—
289C2×D4—1422no—
290C2×D4—1428no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—2844no—
296C22×C4—5682no—
297C22×C4—5682no—
298C22×C4—1422no—
299C22×C4C2×C22⋊D7111yescharacteristic
300C22×C4C2×C22⋊D7111yescharacteristic
301C22×C4C2×C22⋊D7112yes—
302C22×C4C2×C22⋊D7112yes—
303C22×C4C23×D7111yescharacteristic
304C2×C8—1422no—
305C2×C8—1422no—
306C2×C8—1422no—
307C2×C8—1422no—
308C2×C8—1422no—
309C2×C8—1422no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—1422no—
320C4⋊C4—1422no—
321C22⋊C4—2844no—
322C22⋊C4—5682no—
323C22⋊C4—5682no—
324C42—22no—
325C42Q8×D7112yes—
326C42#1176586612yes—
327C42#1176586614yes—
328C23—2842no—
329C23—2842no—
330C23—2842no—
331C23—2842no—
332C23—2842no—
333C23—2842no—
334C23—1424no—
335C23C22×C22⋊D7111yescharacteristic
336Q8—84no—
337D4—2842no—
338D4—2842no—
339D4—5688no—
340D4—1424no—
341D4—1424no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—2842no—
359C2×C4—2842no—
360C2×C4—2842no—
361C2×C4—5688no—
362C2×C4#1455610414yes—
363C2×C4#1455611714yes—
364C2×C4C2×Q8×D7111yescharacteristic
365C2×C4C2×#1176586612yes—
366C2×C4C2×#1176586611yescharacteristic
367C2×C4#1455617312yes—
368C8—2842no—
369C8—2842no—
370C22—2844no—
371C22—2844no—
372C22—2844no—
373C22—5682no—
374C22—5682no—
375C22—5682no—
376C22—5682no—
377C22C2×#1455610411yescharacteristic
378C22C2×#1455611711yescharacteristic
379C22#1798656611yescharacteristic
380C22#1798680212yes—
381C22#1798680712yes—
382C4—24no—
383C4—24no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—5682no—
391C4#1798671012yes—
392C4#1798671512yes—
393C2—5684no—
394C2#2259351411yescharacteristic
395C2#2259816512yes—
396C2#2259869912yes—
397C2#2259939011yescharacteristic
398C2#2259940111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2660C2
socleC23×C71C22×D4
prime radicalC2×#14556125C22
hypercenter#2660D71
hypocenterC71#29152
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2660

Complements

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

NH ≅ G/Ntype
2. #22589483C2semidirect
3. #22589473C2semidirect
6. #22588475C2semidirect
7. #22588470C2semidirect
8. #22588347C2semidirect
9. #22588337C2semidirect
10. #22588197C2semidirect
11. C71×#2660C2semidirect
45. #14556125Q8semidirect
200. #2660D71semidirect
211. C71#29152semidirect

Maximal subgroups

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

subgroupindex
#225910652
#225894832
#225894732
#225891632
#225891452
#225884752
#225884702
#225883472
#225883372
#225881972
C71×#26602
#2915271