Finite Group Databasebeta

← #31895835

Subgroups — #31895835

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589267C211yesmaximal, characteristic
2#22589146C211yesmaximal, characteristic
3#22589098C211yesmaximal, characteristic, complemented
4C2×#17985742C211yesmaximal, characteristic
5#22586224C211yesmaximal, characteristic, complemented
6#22586175C211yesmaximal, characteristic, complemented
7C71×#1770C211yesmaximal, characteristic, complemented
8C2×#14556125C2211yescharacteristic
9C2×#14556061—21no—
10C2×C4⋊(C71⋊C8)C2211yescharacteristic
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12C2×#14556050—21no—
13C2×#14556050—21no—
14C2×#14556048—21no—
15C2×#14556048—21no—
16#17985869C2211yescharacteristic
17#17985761—21no—
18#17985742C414yescomplemented
19C2×C4⋊C8×C71C2211yescharacteristic
20C2×#97×C71—21no—
21C2×C4×C8×C71C2211yescharacteristic
22C4×C4⋊C4×C71C2211yescharacteristic
23#336×C71—21no—
24C22×D284—21no—
25C22×D284—21no—
26C22×D284—21no—
27C22×C71⋊C8—21no—
28C22×C71⋊C8—21no—
29#14556125—21no—
30#14556125—21no—
31#14556125—21no—
32#14556125C2×C412yes—
33C2×C4⋊Dic71—21no—
34C2×C4⋊Dic71—21no—
35C2×C4⋊Dic71—21no—
36#14556061—42no—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)C2×C412yes—
41#14556050—42no—
42#14556050—42no—
43#14556048—42no—
44#14556048—42no—
45C22×C8×C71—21no—
46C22×C8×C71D411yescharacteristic
47C22×C8×C71D411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C42×C71—21no—
51C2×C42×C71C2311yescharacteristic
52C4⋊C8×C71—21no—
53C4⋊C8×C71C2×C412yes—
54#97×C71—42no—
55C4×C8×C71—21no—
56C4×C8×C71—21no—
57#95×C71—21no—
58C23×D71—41no—
59C2×D284—22no—
60C2×D284—22no—
61C2×D284—22no—
62C2×D284—41no—
63C2×D284—41no—
64C2×D284—41no—
65C2×D284—41no—
66C2×D284—41no—
67C2×D284—41no—
68C2×D284—41no—
69C2×D284—41no—
70C2×D284—41no—
71C2×D284—41no—
72C2×D284—41no—
73C2×D284—81no—
74C2×D284—81no—
75C22×Dic71—41no—
76C2×C71⋊C8—22no—
77C2×C71⋊C8—41no—
78C2×C71⋊C8—41no—
79C2×C71⋊C8—41no—
80C2×C71⋊C8—41no—
81C2×C71⋊C8—41no—
82C4⋊Dic71—41no—
83C4⋊Dic71—41no—
84C4⋊Dic71—41no—
85C4⋊Dic71—41no—
86C4⋊Dic71—81no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71#4111yescharacteristic
92C2×C8×C71—21no—
93C2×C8×C71—21no—
94C2×C8×C71—22no—
95C2×C8×C71—21no—
96C2×C8×C71—21no—
97C2×C8×C71—21no—
98C2×C8×C71—41no—
99C2×C8×C71—41no—
100C2×C8×C71D812yes—
101C4⋊C4×C71—21no—
102C4⋊C4×C71—21no—
103C4⋊C4×C71—21no—
104C4⋊C4×C71—21no—
105C42×C71—41no—
106C42×C71—41no—
107C42×C71C22×C411yescharacteristic
108C42×C71#4111yescharacteristic
109C42×C71#4111yescharacteristic
110C42×C71#4111yescharacteristic
111C23×C71#12111yescharacteristic
112C22×D71—42no—
113C22×D71—81no—
114C22×D71—81no—
115C22×D71—81no—
116C22×D71—81no—
117C22×D71—81no—
118C22×D71—81no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71#11711yescharacteristic
128C2×C4×C71C4×D411yescharacteristic
129C2×C4×C71C4×D411yescharacteristic
130C2×C4×C71#12311yescharacteristic
131C2×C4×C71#12411yescharacteristic
132C2×C4×C71#12611yescharacteristic
133C2×C4×C71C2×D811yescharacteristic
134C2×C4×C71#13511yescharacteristic
135C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C71#13711yescharacteristic
140C2×C4×C71#13711yescharacteristic
141C2×Dic71—81no—
142C2×Dic71—81no—
143C2×Dic71—81no—
144D284—42no—
145D284—42no—
146D284—42no—
147D284—42no—
148D284—82no—
149D284—82no—
150D284—81no—
151D284—81no—
152D284—81no—
153D284—81no—
154C8×C71—22no—
155C8×C71—42no—
156C8×C71—41no—
157C71⋊C8—42no—
158C71⋊C8—81no—
159C22×C71#38811yescharacteristic
160C22×C71#45911yescharacteristic
161C22×C71#46411yescharacteristic
162C22×C71#46811yescharacteristic
163C22×C71#46911yescharacteristic
164C22×C71#47111yescharacteristic
165C22×C71#47211yescharacteristic
166C2×D71—82no—
167C2×D71—82no—
168C2×D71—82no—
169C2×D71—161no—
170C2×D71—161no—
171C2×D71—161no—
172C2×D71—161no—
173C4×C71—21no—
174C4×C71—21no—
175C4×C71—81no—
176C4×C71—81no—
177C4×C71C4×D811yescharacteristic
178C4×C71#44011yescharacteristic
179C4×C71#44011yescharacteristic
180C4×C71#44211yescharacteristic
181C4×C71#48011yescharacteristic
182C4×C71#48211yescharacteristic
183C4×C71#48911yescharacteristic
184C4×C71#49011yescharacteristic
185Dic71—161no—
186#9571—711nomaximal, Sylow
187C2×C71#168511yescharacteristic
188C2×C71#168711yescharacteristic
189C2×C71#169811yescharacteristic
190C2×C71#170211yescharacteristic
191C2×C71#192511yescharacteristic
192C2×C71#193111yescharacteristic
193C2×C71#193611yescharacteristic
194D71—162no—
195#1982—711no—
196#1957—711no—
197#1947—711no—
198#1770D7111yescharacteristic, complemented
199C2×#331—711no—
200#1296—711no—
201#1277—711no—
202C71#957111yescharacteristic, complemented, Sylow
203C2×#127—711no—
204C2×#107—1421no—
205C2×C4⋊C8—711no—
206C2×C4⋊C8—711no—
207C2×C4⋊C8C2×D7111yescharacteristic
208C2×#102—1421no—
209C2×#102—1421no—
210C2×#102—1421no—
211C2×#102—1421no—
212C2×#97—21no—
213C2×C4×C8C2×D7111yescharacteristic
214#384—711no—
215C4×C4⋊C4C2×D7111yescharacteristic
216#340—1421no—
217#336—21no—
218#331—714no—
219C22×D4—1421no—
220C22×D4—1421no—
221C22×D4—1421no—
222C22×C8—21no—
223C22×C8—1421no—
224C22×C8—1421no—
225C22×C8C22⋊D7111yescharacteristic
226C22×C8C22⋊D7111yescharacteristic
227#127—712no—
228#127—1421no—
229#127—1421no—
230#127—1421no—
231C2×C4⋊C4—1421no—
232C2×C4⋊C4—1421no—
233C2×C4⋊C4—1421no—
234C2×C4⋊C4D28411yescharacteristic
235C2×C4⋊C4D28411yescharacteristic
236C2×C42—21no—
237C2×C42C22×D7111yescharacteristic
238#107—2842no—
239C4⋊C8—21no—
240C4⋊C8—712no—
241C4⋊C8—1421no—
242C4⋊C8—1421no—
243C4⋊C8—1421no—
244C4⋊C8C4×D7112yes—
245#102—2842no—
246#102—2842no—
247#102—2842no—
248#102—2842no—
249#97—42no—
250C4×C8—21no—
251C4×C8—21no—
252#95—21no—
253C24—2841no—
254C2×D4—2841no—
255C2×D4—2841no—
256C2×D4—2841no—
257C2×D4—2841no—
258C2×D4—2841no—
259C2×D4—2841no—
260C2×D4—2841no—
261C2×D4—2841no—
262C2×D4—2841no—
263C2×D4—2841no—
264C2×D4—2841no—
265C2×D4—5681no—
266C2×D4—5681no—
267C2×D4—1422no—
268C2×D4—1422no—
269C2×D4—1422no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—2841no—
273C22×C4C2×C22⋊D7111yescharacteristic
274C22×C4C2×D28411yescharacteristic
275C22×C4#1176586511yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—2841no—
285C2×C8—2841no—
286C2×C8—2841no—
287C2×C8—2841no—
288C2×C8—2841no—
289C2×C8—1422no—
290C2×C8#1176585012yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—2841no—
296C4⋊C4—2841no—
297C4⋊C4—2841no—
298C4⋊C4—2841no—
299C4⋊C4—5681no—
300C42—41no—
301C42—41no—
302C42C2×C4×D7111yescharacteristic
303C42#1176586511yescharacteristic
304C42#1176586511yescharacteristic
305C42#1176586511yescharacteristic
306C23—2842no—
307C23—5681no—
308C23—5681no—
309C23—5681no—
310C23—5681no—
311C23—5681no—
312C23—5681no—
313C23#1455610211yescharacteristic
314D4—2842no—
315D4—2842no—
316D4—2842no—
317D4—2842no—
318D4—5682no—
319D4—5682no—
320D4—5681no—
321D4—5681no—
322D4—5681no—
323D4—5681no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—5681no—
333C2×C4—5681no—
334C2×C4—5681no—
335C2×C4#1455608511yescharacteristic
336C2×C4C4×C22⋊D7111yescharacteristic
337C2×C4C4×D28411yescharacteristic
338C2×C4#1455611111yescharacteristic
339C2×C4#1455611611yescharacteristic
340C2×C4#1455612211yescharacteristic
341C2×C4C2×#1176585011yescharacteristic
342C2×C4#1455614811yescharacteristic
343C2×C4#1455615311yescharacteristic
344C2×C4#1455615611yescharacteristic
345C2×C4#1455615611yescharacteristic
346C2×C4#1455615611yescharacteristic
347C2×C4#1455615911yescharacteristic
348C2×C4#1455616111yescharacteristic
349C8—22no—
350C8—41no—
351C8—42no—
352C8—2842no—
353C8—5681no—
354C22—5682no—
355C22—5682no—
356C22—5682no—
357C22—11361no—
358C22—11361no—
359C22—11361no—
360C22—11361no—
361C22#1798589011yescharacteristic
362C22#1798619411yescharacteristic
363C22#1798622111yescharacteristic
364C22#1798624511yescharacteristic
365C22#1798625211yescharacteristic
366C22#1798626211yescharacteristic
367C22#1798626911yescharacteristic
368C4—21no—
369C4—21no—
370C4—81no—
371C4—81no—
372C4—11361no—
373C4C4×#1176585011yescharacteristic
374C4#1798609711yescharacteristic
375C4#1798610111yescharacteristic
376C4#1798611211yescharacteristic
377C4#1798631211yescharacteristic
378C4#1798632311yescharacteristic
379C4#1798635911yescharacteristic
380C4#1798636411yescharacteristic
381C2—11362no—
382C2#2258780511yescharacteristic
383C2#2258781911yescharacteristic
384C2#2258788011yescharacteristic
385C2#2258790811yescharacteristic
386C2#2258896011yescharacteristic
387C2#2258900211yescharacteristic
388C2#2258903711yescharacteristic
subgroupG/H
centerC23#14556102
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1770C2
socleC23×C71#121
prime radicalC2×#14556125C22
cosocleC2×C42×C71C23
hypercenter#1770D71
hypocenterC71#9571
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1770

Complements

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

NH ≅ G/Ntype
3. #22589098C2semidirect
5. #22586224C2semidirect
6. #22586175C2semidirect
7. C71×#1770C2semidirect
18. #17985742C4semidirect
198. #1770D71semidirect
202. C71#9571semidirect

Maximal subgroups

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

subgroupindex
#225892672
#225891462
#225890982
C2×#179857422
#225862242
#225861752
C71×#17702
#957171