Finite Group Databasebeta

← #31902174

Subgroups — #31902174

Order 18176.

418 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#22589134C211yesmaximal, characteristic
2#22588497C211yesmaximal, characteristic, complemented
3#22588341C211yesmaximal, characteristic
4#22588341C211yesmaximal, characteristic
5#22586224C211yesmaximal, characteristic, complemented
6#22586224C211yesmaximal, characteristic, complemented
7C71×#1277C211yesmaximal, characteristic, complemented
8C2×C4×D284C2211yescharacteristic
9C2×#14556060—21no—
10C2×#14556060—21no—
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12C2×#14556050—21no—
13C2×#14556050—21no—
14C2×(C2×D71)⋊C8—21no—
15C2×(C2×D71)⋊C8—21no—
16#17985869C2211yescharacteristic
17#17985869C2211yescharacteristic
18#17985762—21no—
19#17985762—21no—
20C2×C4⋊C8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22C2×C4×C8×C71C2211yescharacteristic
23C22×D284D411yescharacteristic
24C22×C4×D71—21no—
25C22×C71⋊C8—21no—
26C4×D284—21no—
27C4×D284—21no—
28C4×D284—21no—
29C4×D284—21no—
30C2×C4⋊Dic71—21no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33C2×C4⋊Dic71—21no—
34C2×C4⋊Dic71—21no—
35C2×C4⋊Dic71—21no—
36C2×C4⋊Dic71D411yescharacteristic
37C2×(C2×D71)⋊C4—21no—
38#14556060—42no—
39#14556060—42no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)—21no—
42#14556050—42no—
43#14556050—42no—
44(C2×D71)⋊C8—42no—
45(C2×D71)⋊C8—42no—
46C22×C8×C71—21no—
47C22×C8×C71—21no—
48C22×C8×C71D411yescharacteristic
49C22×C8×C71D411yescharacteristic
50C2×C42×C71C2311yescharacteristic
51C4⋊C8×C71—21no—
52C4⋊C8×C71—21no—
53C4⋊C8×C71—21no—
54C4⋊C8×C71—21no—
55C4×C8×C71—21no—
56C4×C8×C71—21no—
57C23×D71—21no—
58C2×D284—21no—
59C2×D284—21no—
60C2×D284—21no—
61C2×D284—21no—
62C2×D284—21no—
63C2×D284SD1612yes—
64C22×Dic71—21no—
65C22×Dic71—41no—
66C22×Dic71—41no—
67C2×C4×D71—41no—
68C2×C4×D71—41no—
69C2×C4×D71—41no—
70C2×C4×D71—41no—
71C2×C4×D71—41no—
72C2×C4×D71—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8—41no—
75C2×C71⋊C8—41no—
76C4⋊Dic71—21no—
77C4⋊Dic71—21no—
78C4⋊Dic71—41no—
79C4⋊Dic71—41no—
80C4⋊Dic71—41no—
81C4⋊Dic71—41no—
82C4⋊Dic71—41no—
83C4⋊Dic71—41no—
84C4⋊Dic71—41no—
85C4⋊Dic71—41no—
86C4⋊Dic71—81no—
87C4⋊Dic71—81no—
88(C2×D71)⋊C4—41no—
89(C2×D71)⋊C4—41no—
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71#4111yescharacteristic
93C2×C8×C71—21no—
94C2×C8×C71—21no—
95C2×C8×C71—21no—
96C2×C8×C71—21no—
97C2×C8×C71—41no—
98C2×C8×C71—41no—
99C2×C8×C71—41no—
100C2×C8×C71—41no—
101C2×C8×C71—41no—
102C2×C8×C71—41no—
103C2×C8×C71D812yes—
104C2×C8×C71SD1612yes—
105C42×C71#4111yescharacteristic
106C42×C71#4111yescharacteristic
107C42×C71#4111yescharacteristic
108C42×C71#4111yescharacteristic
109C23×C71#12111yescharacteristic
110C22×D71—22no—
111C22×D71—41no—
112C22×D71—41no—
113C22×D71—41no—
114C22×D71—41no—
115C22×D71—41no—
116C22×D71—41no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71#12311yescharacteristic
120C2×C4×C71#12311yescharacteristic
121C2×C4×C71#12311yescharacteristic
122C2×C4×C71#12411yescharacteristic
123C2×C4×C71#12611yescharacteristic
124C2×C4×C71#12611yescharacteristic
125C2×C4×C71C2×D811yescharacteristic
126C2×C4×C71C2×SD1611yescharacteristic
127C2×C4×C71C2×SD1611yescharacteristic
128C2×C4×C71#13511yescharacteristic
129C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C71#13711yescharacteristic
132C2×C4×C71#13711yescharacteristic
133C2×Dic71—41no—
134C2×Dic71—41no—
135C2×Dic71—41no—
136C2×Dic71—81no—
137C2×Dic71—81no—
138C2×Dic71—81no—
139C2×Dic71—81no—
140C2×Dic71—81no—
141C2×Dic71—81no—
142D284—22no—
143D284—22no—
144D284—41no—
145D284—41no—
146C4×D71—81no—
147C4×D71—81no—
148C4×D71—81no—
149C4×D71—81no—
150C8×C71—22no—
151C8×C71—22no—
152C8×C71—81no—
153C8×C71—81no—
154C71⋊C8—81no—
155C22×C71#39711yescharacteristic
156C22×C71#45911yescharacteristic
157C22×C71#46111yescharacteristic
158C22×C71#46511yescharacteristic
159C22×C71#46811yescharacteristic
160C22×C71#47111yescharacteristic
161C22×C71#47111yescharacteristic
162C2×D71—42no—
163C2×D71—42no—
164C2×D71—42no—
165C2×D71—81no—
166C2×D71—81no—
167C2×D71—81no—
168C2×D71—81no—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71#48011yescharacteristic
172C4×C71#48111yescharacteristic
173C4×C71#48111yescharacteristic
174C4×C71#48211yescharacteristic
175C4×C71#48311yescharacteristic
176C4×C71#48511yescharacteristic
177C4×C71#48611yescharacteristic
178C4×C71#48911yescharacteristic
179Dic71—81no—
180Dic71—161no—
181Dic71—161no—
182#10619—711nomaximal, Sylow
183C2×C71#166511yescharacteristic
184C2×C71#167111yescharacteristic
185C2×C71#168511yescharacteristic
186C2×C71#169111yescharacteristic
187C2×C71#207111yescharacteristic
188C2×C71#207711yescharacteristic
189C2×C71#207911yescharacteristic
190D71—82no—
191#1954—711no—
192#1844—711no—
193#1802—711no—
194#1802—711no—
195#1296—711no—
196#1296—711no—
197#1277D7111yescharacteristic, complemented
198C71#1061911yescharacteristic, complemented, Sylow
199C2×C4×D4—711no—
200C2×#106—1421no—
201C2×#106—1421no—
202C2×C4⋊C8—711no—
203C2×C4⋊C8C2×D7111yescharacteristic
204C2×C4⋊C8C2×D7111yescharacteristic
205C2×#102—1421no—
206C2×#102—1421no—
207C2×C22⋊C8—1421no—
208C2×C22⋊C8—1421no—
209C2×C4×C8C2×D7111yescharacteristic
210#384—711no—
211#384—711no—
212#340—1421no—
213#340—1421no—
214C22×D4—711no—
215C23×C4—1421no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—1421no—
219C22×C8D28411yescharacteristic
220C22×C8D28411yescharacteristic
221C4×D4—1421no—
222C4×D4—1421no—
223C4×D4—1421no—
224C4×D4—1421no—
225C2×C4⋊C4—711no—
226C2×C4⋊C4—1421no—
227C2×C4⋊C4—1421no—
228C2×C4⋊C4—1421no—
229C2×C4⋊C4—1421no—
230C2×C4⋊C4—1421no—
231C2×C4⋊C4—1421no—
232C2×C22⋊C4—1421no—
233C2×C42C22×D7111yescharacteristic
234#106—2842no—
235#106—2842no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—1421no—
241C4⋊C8—1421no—
242#102—2842no—
243#102—2842no—
244C22⋊C8—2842no—
245C22⋊C8—2842no—
246C4×C8—21no—
247C4×C8—21no—
248C24—1421no—
249C2×D4—712no—
250C2×D4—1421no—
251C2×D4—1421no—
252C2×D4—1421no—
253C2×D4—1421no—
254C2×D4—1421no—
255C22×C4—2841no—
256C22×C4—2841no—
257C22×C4—2841no—
258C22×C4—2841no—
259C22×C4—2841no—
260C22×C4—2841no—
261C22×C4—2841no—
262C22×C4—2841no—
263C22×C4—1421no—
264C22×C4D4×D7111yescharacteristic
265C22×C4C2×D28411yescharacteristic
266C22×C4#1176586611yescharacteristic
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—2841no—
278C2×C8—2841no—
279C2×C8—2841no—
280C2×C8D56812yes—
281C2×C8#1176585412yes—
282C4⋊C4—2841no—
283C4⋊C4—2841no—
284C4⋊C4—2841no—
285C4⋊C4—2841no—
286C4⋊C4—2841no—
287C4⋊C4—2841no—
288C4⋊C4—2841no—
289C4⋊C4—2841no—
290C4⋊C4—5681no—
291C4⋊C4—5681no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C22⋊C4—2841no—
295C22⋊C4—2841no—
296C42#1176586411yescharacteristic
297C42#1176586411yescharacteristic
298C42#1176586511yescharacteristic
299C42#1176586511yescharacteristic
300C23—2841no—
301C23—2841no—
302C23—2841no—
303C23—2841no—
304C23—2841no—
305C23—2841no—
306C23—1422no—
307C23#1455610311yescharacteristic
308D4—2841no—
309D4—2841no—
310D4—1422no—
311D4—1422no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—2841no—
315C2×C4—2841no—
316C2×C4—2841no—
317C2×C4—5681no—
318C2×C4—5681no—
319C2×C4—5681no—
320C2×C4—5681no—
321C2×C4—5681no—
322C2×C4—5681no—
323C2×C4—5681no—
324C2×C4—5681no—
325C2×C4—5681no—
326C2×C4—5681no—
327C2×C4#1455611011yescharacteristic
328C2×C4#1455611011yescharacteristic
329C2×C4#1455611211yescharacteristic
330C2×C4#1455611611yescharacteristic
331C2×C4#1455612311yescharacteristic
332C2×C4#1455612311yescharacteristic
333C2×C4C2×D56811yescharacteristic
334C2×C4SD16×D7111yescharacteristic
335C2×C4C2×#1176585411yescharacteristic
336C2×C4#1455614911yescharacteristic
337C2×C4#1455615211yescharacteristic
338C2×C4#1455615611yescharacteristic
339C2×C4#1455616011yescharacteristic
340C2×C4#1455616111yescharacteristic
341C8—22no—
342C8—22no—
343C8—81no—
344C8—81no—
345C8—5681no—
346C22—2842no—
347C22—2842no—
348C22—2842no—
349C22—5681no—
350C22—5681no—
351C22—5681no—
352C22—5681no—
353C22#1798592611yescharacteristic
354C22#1798619511yescharacteristic
355C22#1798620511yescharacteristic
356C22#1798622911yescharacteristic
357C22#1798624611yescharacteristic
358C22#1798626311yescharacteristic
359C22#1798626311yescharacteristic
360C4—21no—
361C4—21no—
362C4—5681no—
363C4—11361no—
364C4—11361no—
365C4#1798631311yescharacteristic
366C4#1798631611yescharacteristic
367C4#1798631811yescharacteristic
368C4#1798632211yescharacteristic
369C4#1798632811yescharacteristic
370C4#1798633811yescharacteristic
371C4#1798634611yescharacteristic
372C4#1798635911yescharacteristic
373C2—5682no—
374C2#2258768211yescharacteristic
375C2#2258771611yescharacteristic
376C2#2258780611yescharacteristic
377C2#2258784011yescharacteristic
378C2#2258974511yescharacteristic
379C2#2258977511yescharacteristic
380C2#2258978711yescharacteristic
subgroupG/H
centerC23#14556103
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1277C2
socleC23×C71#121
prime radicalC22×D284D4
hypercenter#1277D71
hypocenterC71#10619
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #22588497C2semidirect
5. #22586224C2semidirect
6. #22586224C2semidirect
7. C71×#1277C2semidirect
197. #1277D71semidirect
198. C71#10619semidirect

Maximal subgroups

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

subgroupindex
#225891342
#225884972
#225883412
#225862242
C71×#12772
#1061971