Finite Group Databasebeta

← #31914813

Subgroups — #31914813

Order 18176.

578 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
1C22×C16×D71C211yesmaximal, characteristic, complemented
2C2×#17986236C211yesmaximal, characteristic
3C2×#17985801C211yesmaximal, characteristic, complemented
4C2×#17985799C211yesmaximal, characteristic, complemented
5#22588366C211yesmaximal, characteristic, complemented
6#22586431C211yesmaximal, characteristic, complemented
7C71×#1380C211yesmaximal, characteristic, complemented
8C22×D568C2211yescharacteristic
9C22×C8×D71C2211yescharacteristic
10C2×#14556103—21no—
11C22×C71⋊C16C2211yescharacteristic
12C2×C16×D71—22no—
13C2×C16×D71—22no—
14C2×C16×D71—22no—
15#17986236—22no—
16#17986236C414yescomplemented
17C2×#14556062C2211yescharacteristic
18C2×#14556048—21no—
19#17985856—21no—
20#17985801—22no—
21#17985799—22no—
22#17985761—21no—
23C22×C16×C71C2211yescharacteristic
24C2×#107×C71C2211yescharacteristic
25C2×#107×C71C2211yescharacteristic
26C22×D284—21no—
27C22×D284—41no—
28C22×C4×D71D411yescharacteristic
29C2×D568—21no—
30C2×D568—21no—
31C2×D568—21no—
32C2×D568—21no—
33C2×D568—21no—
34C2×D568C2×C412yes—
35C22×C71⋊C8D411yescharacteristic
36C2×C8×D71—21no—
37C2×C8×D71—21no—
38C2×C8×D71—21no—
39C2×C8×D71—21no—
40C2×C8×D71C2×C412yes—
41C2×C8×D71D412yes—
42#14556103—24no—
43#14556103—42no—
44C2×C4⋊Dic71—21no—
45C2×(C2×D71)⋊C4—41no—
46C2×(C2×D71)⋊C4—41no—
47C2×C71⋊C16—21no—
48C2×C71⋊C16—21no—
49C2×C71⋊C16D412yescomplemented
50C16×D71—28no—
51#14556062—21no—
52#14556062—21no—
53#14556048—24no—
54C71⋊#95—41no—
55C22×C8×C71C2311yescharacteristic
56C2×C4⋊C4×C71—21no—
57C2×C4⋊C4×C71—21no—
58C2×C16×C71—21no—
59C2×C16×C71—21no—
60C2×C16×C71D412yescomplemented
61#107×C71—21no—
62#107×C71—21no—
63#107×C71—21no—
64#107×C71C2×C412yes—
65C23×D71—41no—
66C23×D71D811yescharacteristic
67C2×D284—22no—
68C2×D284—41no—
69C2×D284—41no—
70C2×D284—41no—
71C2×D284—41no—
72C2×D284—44no—
73C2×D284—41no—
74C2×D284—82no—
75C2×D284—82no—
76C22×Dic71—41no—
77C22×Dic71D811yescharacteristic
78C2×C4×D71—21no—
79C2×C4×D71—21no—
80C2×C4×D71—21no—
81C2×C4×D71—21no—
82C2×C4×D71C22⋊C412yes—
83C2×C4×D71SD1612yes—
84D568—22no—
85D568—22no—
86D568—41no—
87D568—41no—
88C2×C71⋊C8—21no—
89C2×C71⋊C8—21no—
90C2×C71⋊C8C22⋊C412yes—
91C8×D71—22no—
92C8×D71—22no—
93C8×D71—22no—
94C8×D71—22no—
95C4⋊Dic71—41no—
96C4⋊Dic71—41no—
97(C2×D71)⋊C4—44no—
98(C2×D71)⋊C4—82no—
99C71⋊C16—22no—
100C22×C4×C71—41no—
101C22×C4×C71—41no—
102C22×C4×C71C2×D411yescharacteristic
103C2×C8×C71C22×C411yescharacteristic
104C2×C8×C71C2×D411yescharacteristic
105C2×C8×C71C2×D411yescharacteristic
106C2×C8×C71C2×D411yescharacteristic
107C2×C8×C71#4111yescharacteristic
108C2×C8×C71#4111yescharacteristic
109C4⋊C4×C71—22no—
110C4⋊C4×C71—41no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71—41no—
113C16×C71—22no—
114C23×C71C2×D811yescharacteristic
115C22×D71—21no—
116C22×D71—21no—
117C22×D71—21no—
118C22×D71—21no—
119C22×D71—42no—
120C22×D71—81no—
121C22×D71—81no—
122C22×D71—81no—
123C22×D71—81no—
124C22×D71—81no—
125C22×D71—81no—
126C22×D71#10212yes—
127C22×D71D1612yes—
128C22×D71SD3212yes—
129C2×C4×C71—42no—
130C2×C4×C71—81no—
131C2×C4×C71—81no—
132C2×C4×C71—81no—
133C2×C4×C71—81no—
134C2×C4×C71—81no—
135C2×C4×C71C2×C22⋊C411yescharacteristic
136C2×C4×C71#12111yescharacteristic
137C2×C4×C71#12111yescharacteristic
138C2×C4×C71C2×SD1611yescharacteristic
139C2×C4×C71#13511yescharacteristic
140C2×C4×C71#13511yescharacteristic
141C2×Dic71—21no—
142C2×Dic71—21no—
143C2×Dic71—81no—
144C2×Dic71—81no—
145C2×Dic71—81no—
146C2×Dic71#10212yes—
147D284—42no—
148D284—42no—
149D284—88no—
150D284—81no—
151D284—81no—
152C4×D71—22no—
153C4×D71—22no—
154C4×D71—22no—
155C4×D71—22no—
156C8×C71C4×D411yescharacteristic
157C8×C71C4×D411yescharacteristic
158C8×C71#12011yescharacteristic
159C8×C71#12311yescharacteristic
160C71⋊C8—22no—
161C22×C71C2×#10211yescharacteristic
162C22×C71#46611yescharacteristic
163C22×C71#46611yescharacteristic
164C22×C71C2×D1611yescharacteristic
165C22×C71C2×SD3211yescharacteristic
166C22×C71#50811yescharacteristic
167C22×C71#50811yescharacteristic
168C2×D71—22no—
169C2×D71—22no—
170C2×D71—22no—
171C2×D71—22no—
172C2×D71—22no—
173C2×D71—22no—
174C2×D71—82no—
175C2×D71—82no—
176C2×D71—82no—
177C2×D71—161no—
178C2×D71—161no—
179C2×D71—161no—
180C2×D71—161no—
181C2×D71#35714yescomplemented
182C4×C71—82no—
183C4×C71—161no—
184C4×C71#38611yescharacteristic
185C4×C71#41611yescharacteristic
186C4×C71#46511yescharacteristic
187C4×C71#46511yescharacteristic
188Dic71—22no—
189Dic71—161no—
190#13059—711nomaximal, Sylow
191C2×C71#189311yescharacteristic
192C2×C71C2×#35711yescharacteristic
193C2×C71#213811yescharacteristic
194C2×C71#221511yescharacteristic
195C2×C71#221511yescharacteristic
196C2×C71#221711yescharacteristic
197C2×C71#221711yescharacteristic
198D71—24no—
199D71—162no—
200C23×C16—711no—
201C2×#466—711no—
202C2×#357—711no—
203C2×#357—711no—
204#1808—711no—
205#1380—711no—
206#1380D7111yescharacteristic, complemented
207C71#1305911yescharacteristic, complemented, Sylow
208C22×D8—711no—
209C23×C8—711no—
210C2×#121—1421no—
211C22×C16—711no—
212C22×C16—1422no—
213C22×C16—1422no—
214C22×C16—1422no—
215C22×C16C2×D7111yescharacteristic
216#466—714no—
217#466—1422no—
218C2×#107—711no—
219C2×#107C2×D7111yescharacteristic
220C2×#107C2×D7111yescharacteristic
221C2×#102—1421no—
222#380—1421no—
223#357—1422no—
224#357—1422no—
225#340—1421no—
226C22×D4—2841no—
227C22×D4—1421no—
228C23×C4—711no—
229C2×D8—712no—
230C2×D8—1421no—
231C2×D8—1421no—
232C2×D8—1421no—
233C2×D8—1421no—
234C2×D8—1421no—
235C22×C8—711no—
236C22×C8—712no—
237C22×C8—712no—
238C22×C8—1421no—
239C22×C8—1421no—
240C22×C8—1421no—
241C22×C8—1421no—
242C22×C8C22×D7111yescharacteristic
243#121—2842no—
244#121—1424no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1421no—
248C2×C22⋊C4—2841no—
249C2×C22⋊C4—2841no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—712no—
253C2×C16—1428no—
254C2×C16—1421no—
255C2×C16—1421no—
256C2×C16D28412yescomplemented
257#107—21no—
258#107—21no—
259#107—21no—
260#107—1421no—
261#107—1421no—
262#107C4×D7112yes—
263#102—1424no—
264#95—2841no—
265C24—2841no—
266C24—711no—
267C2×D4—2844no—
268C2×D4—2841no—
269C2×D4—2841no—
270C2×D4—2841no—
271C2×D4—2841no—
272C2×D4—2841no—
273C2×D4—5682no—
274C2×D4—5682no—
275C2×D4—1422no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—2841no—
279C22×C4—711no—
280C22×C4—712no—
281C22×C4—712no—
282C22×C4—1421no—
283C22×C4—1421no—
284C22×C4—1421no—
285C22×C4—1421no—
286C22×C4D4×D7111yescharacteristic
287D8—2841no—
288D8—2841no—
289D8—1422no—
290D8—1422no—
291C2×C8—712no—
292C2×C8—1421no—
293C2×C8—1421no—
294C2×C8—1422no—
295C2×C8—1422no—
296C2×C8—1422no—
297C2×C8—1422no—
298C2×C8C2×C4×D7111yescharacteristic
299C2×C8D4×D7111yescharacteristic
300C2×C8D4×D7111yescharacteristic
301C2×C8C2×D28411yescharacteristic
302C2×C8#1176586511yescharacteristic
303C2×C8#1176586611yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—2841no—
309C4⋊C4—2841no—
310C22⋊C4—2844no—
311C22⋊C4—5682no—
312C16—22no—
313C16—1422no—
314C23—2842no—
315C23—5681no—
316C23—5681no—
317C23—5681no—
318C23—5681no—
319C23—5681no—
320C23—5681no—
321C23—712no—
322C23—712no—
323C23—712no—
324C23—1421no—
325C23—1421no—
326C23—1421no—
327C23—1421no—
328C23D8×D7111yescharacteristic
329D4—2842no—
330D4—2842no—
331D4—5688no—
332D4—5681no—
333D4—5681no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—5681no—
341C2×C4—5681no—
342C2×C4—5681no—
343C2×C4—712no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4—1422no—
347C2×C4—1422no—
348C2×C4—1422no—
349C2×C4—1422no—
350C2×C4C22⋊C4×D7111yescharacteristic
351C2×C4#1455610111yescharacteristic
352C2×C4#1455610311yescharacteristic
353C2×C4SD16×D7111yescharacteristic
354C2×C4#1455614911yescharacteristic
355C2×C4#1455615111yescharacteristic
356C8—1422no—
357C8#1455609011yescharacteristic
358C8C4×D28411yescharacteristic
359C8#1455609811yescharacteristic
360C8#1455611311yescharacteristic
361C22—5682no—
362C22—5682no—
363C22—5682no—
364C22—714no—
365C22—11361no—
366C22—11361no—
367C22—11361no—
368C22—11361no—
369C22—1422no—
370C22—1422no—
371C22—1422no—
372C22—1422no—
373C22—1422no—
374C22—1422no—
375C22#102×D7111yescharacteristic
376C22#1798623411yescharacteristic
377C22#1798623611yescharacteristic
378C22D16×D7111yescharacteristic
379C22SD32×D7111yescharacteristic
380C22#1798643011yescharacteristic
381C22#1798643211yescharacteristic
382C4—82no—
383C4—161no—
384C4—11361no—
385C4—1422no—
386C4#1798587711yescharacteristic
387C4#1798599411yescharacteristic
388C4#1798622911yescharacteristic
389C4#1798623111yescharacteristic
390C2—11362no—
391C2—1424no—
392C2#2258876711yescharacteristic
393C2#357×D7111yescharacteristic
394C2#2259004611yescharacteristic
395C2#2259042511yescharacteristic
396C2#2259042711yescharacteristic
397C2#2259043711yescharacteristic
398C2#2259043911yescharacteristic
subgroupG/H
centerC23D8×D71
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC71×#1380C2
socleC23×C71C2×D8
prime radicalC2×#17986236C2
cosocleC22×C8×C71C23
hypercenter#1380D71
hypocenterC71#13059
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
1. C22×C16×D71C2semidirect
3. C2×#17985801C2semidirect
4. C2×#17985799C2semidirect
5. #22588366C2semidirect
6. #22586431C2semidirect
7. C71×#1380C2semidirect
16. #17986236C4semidirect
49. C2×C71⋊C16D4semidirect
60. C2×C16×C71D4semidirect
181. C2×D71#357semidirect
206. #1380D71semidirect
207. C71#13059semidirect
256. C2×C16D284semidirect

Maximal subgroups

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

subgroupindex
C22×C16×D712
C2×#179862362
C2×#179858012
C2×#179857992
#225883662
#225864312
C71×#13802
#1305971