Finite Group Databasebeta

← #31916607

Subgroups — #31916607

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986235C211yesmaximal, characteristic
2C2×#17985824C211yesmaximal, characteristic, complemented
3C2×#17985799C211yesmaximal, characteristic, complemented
4C2×#17985799C211yesmaximal, characteristic, complemented
5C2×C22⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
6#22586431C211yesmaximal, characteristic, complemented
7C71×#1808C211yesmaximal, characteristic, complemented
8C22×D568C2211yescharacteristic
9C2×#14556102—21no—
10C22×C71⋊C16C2211yescharacteristic
11C22×C71⋊C16C2211yescharacteristic
12#17986235—22no—
13#17986235—22no—
14C2×#14556062C2211yescharacteristic
15C2×#14556050—21no—
16#17985824—22no—
17#17985799—22no—
18#17985799—22no—
19C22⋊(C71⋊C16)—22no—
20C23×C8×C71C2211yescharacteristic
21C2×#107×C71C2211yescharacteristic
22C2×#107×C71C2211yescharacteristic
23#380×C71—21no—
24#340×C71—21no—
25C22×D284—21no—
26C22×C22⋊D71—41no—
27C2×D568—21no—
28C2×D568—21no—
29C2×D568—21no—
30C2×D568—21no—
31C2×D568—21no—
32C2×D568D412yes—
33#14556102—24no—
34#14556102—42no—
35C2×C4⋊Dic71—21no—
36C2×(C2×D71)⋊C4—41no—
37C2×C71⋊C16—21no—
38C2×C71⋊C16—21no—
39C2×C71⋊C16—21no—
40C2×C71⋊C16—21no—
41C2×C71⋊C16—21no—
42C2×C71⋊C16D412yescomplemented
43#14556062—21no—
44#14556062D412yes—
45#14556050—24no—
46C23×C4×C71D411yescharacteristic
47C22×C8×C71—21no—
48C22×C8×C71—21no—
49C22×C8×C71—21no—
50C22×C8×C71—21no—
51C22×C8×C71—21no—
52C22×C8×C71—21no—
53C22×C8×C71D411yescharacteristic
54C22×C8×C71C2311yescharacteristic
55C2×C4⋊C4×C71—21no—
56C2×C4⋊C4×C71—21no—
57C2×C22⋊C4×C71—41no—
58#107×C71—21no—
59#107×C71—21no—
60#107×C71—21no—
61#107×C71—21no—
62#95×C71—41no—
63C23×D71—41no—
64C2×D284—22no—
65C2×D284—41no—
66C2×D284—41no—
67C2×D284—41no—
68C2×D284—41no—
69C2×D284—41no—
70C2×C22⋊D71—44no—
71C2×C22⋊D71—82no—
72C2×C22⋊D71—82no—
73C22×Dic71—41no—
74D568—22no—
75D568—22no—
76D568—41no—
77D568—41no—
78C4⋊Dic71—22no—
79C4⋊Dic71—41no—
80(C2×D71)⋊C4—44no—
81C71⋊C16—22no—
82C71⋊C16—41no—
83C24×C71D811yescharacteristic
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—21no—
90C22×C4×C71—41no—
91C22×C4×C71—41no—
92C22×C4×C71D811yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
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×C71C2×D411yescharacteristic
102C2×C8×C71C2×D411yescharacteristic
103C2×C8×C71C2×D411yescharacteristic
104C2×C8×C71#4111yescharacteristic
105C2×C8×C71#4111yescharacteristic
106C2×C8×C71#4111yescharacteristic
107C4⋊C4×C71—41no—
108C4⋊C4×C71—41no—
109C4⋊C4×C71—41no—
110C4⋊C4×C71—41no—
111C22⋊C4×C71—82no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71D1612yes—
119C23×C71C2×D811yescharacteristic
120C22×D71—42no—
121C22×D71—81no—
122C22×D71—81no—
123C22×D71—81no—
124C22×D71—81no—
125C22×D71—81no—
126C22×D71—81no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—81no—
135C2×C4×C71—81no—
136C2×C4×C71—81no—
137C2×C4×C71—81no—
138C2×C4×C71—81no—
139C2×C4×C71—81no—
140C2×C4×C71#12011yescharacteristic
141C2×C4×C71#12111yescharacteristic
142C2×C4×C71#12311yescharacteristic
143C2×C4×C71#13511yescharacteristic
144C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C71#13711yescharacteristic
146C22⋊D71—88no—
147C2×Dic71—42no—
148C2×Dic71—81no—
149C2×Dic71—81no—
150D284—42no—
151D284—42no—
152D284—81no—
153D284—81no—
154C8×C71—41no—
155C8×C71#12111yescharacteristic
156C8×C71#12111yescharacteristic
157C8×C71#12311yescharacteristic
158C8×C71#12411yescharacteristic
159C22×C71—22no—
160C22×C71—22no—
161C22×C71—22no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71#44711yescharacteristic
167C22×C71#46611yescharacteristic
168C22×C71#48011yescharacteristic
169C22×C71C2×D1611yescharacteristic
170C22×C71#50811yescharacteristic
171C22×C71#50911yescharacteristic
172C22×C71#51011yescharacteristic
173C2×D71—82no—
174C2×D71—82no—
175C2×D71—82no—
176C2×D71—161no—
177C2×D71—161no—
178C2×D71—161no—
179C2×D71—161no—
180C4×C71—41no—
181C4×C71—161no—
182C4×C71—161no—
183C4×C71#39411yescharacteristic
184C4×C71#45211yescharacteristic
185C4×C71#46811yescharacteristic
186C4×C71#48211yescharacteristic
187Dic71—82no—
188#13380—711nomaximal, Sylow
189C2×C71—42no—
190C2×C71#203311yescharacteristic
191C2×C71#218411yescharacteristic
192C2×C71#218911yescharacteristic
193C2×C71#221511yescharacteristic
194C2×C71#222011yescharacteristic
195C2×C71#223211yescharacteristic
196C2×C71#223411yescharacteristic
197D71—162no—
198C2×#466—711no—
199C2×#366—711no—
200C2×#357—711no—
201C2×#357—711no—
202C2×C22⋊C16—711no—
203#1808D7111yescharacteristic, complemented
204#1380—711no—
205C71#1338011yescharacteristic, complemented, Sylow
206C22×D8—711no—
207C23×C8C2×D7111yescharacteristic
208C2×#121—1421no—
209C22×C16—711no—
210C22×C16—711no—
211#466—1422no—
212#466—1422no—
213C2×#107—711no—
214C2×#107C2×D7111yescharacteristic
215C2×#107C2×D7111yescharacteristic
216C2×#102—1421no—
217#380—21no—
218#366—1422no—
219#357—1422no—
220#357—1422no—
221C22⋊C16—1422no—
222#340—21no—
223C22×D4—2841no—
224C22×D4—1421no—
225C23×C4C22⋊D7111yescharacteristic
226C2×D8—712no—
227C2×D8—1421no—
228C2×D8—1421no—
229C2×D8—1421no—
230C2×D8—1421no—
231C2×D8—1421no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8C22⋊D7111yescharacteristic
239C22×C8C22×D7111yescharacteristic
240#121—2842no—
241#121—1424no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1421no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—2841no—
247C2×C16—712no—
248C2×C16—1421no—
249C2×C16—1421no—
250C2×C16—1421no—
251C2×C16—1421no—
252C2×C16—1421no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—712no—
258#107—1421no—
259#102—1424no—
260#95—41no—
261C24—2841no—
262C24#1176585011yescharacteristic
263C2×D4—2844no—
264C2×D4—2841no—
265C2×D4—2841no—
266C2×D4—2841no—
267C2×D4—2841no—
268C2×D4—2841no—
269C2×D4—5682no—
270C2×D4—5682no—
271C2×D4—1422no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—2841no—
281C22×C4#1176585011yescharacteristic
282C22×C4C2×C22⋊D7111yescharacteristic
283D8—2841no—
284D8—2841no—
285D8—1422no—
286D8—1422no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8D4×D7111yescharacteristic
295C2×C8D4×D7111yescharacteristic
296C2×C8D4×D7111yescharacteristic
297C2×C8#1176586411yescharacteristic
298C2×C8#1176586511yescharacteristic
299C2×C8#1176586511yescharacteristic
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—2841no—
305C4⋊C4—1422no—
306C22⋊C4—82no—
307C22⋊C4—2844no—
308C16—2841no—
309C16—1422no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—2842no—
317C23—5681no—
318C23—5681no—
319C23—5681no—
320C23—5681no—
321C23—5681no—
322C23—5681no—
323C23#1455606912yes—
324C23C2×#1176585011yescharacteristic
325D4—2842no—
326D4—2842no—
327D4—5688no—
328D4—5681no—
329D4—5681no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—2842no—
344C2×C4—5681no—
345C2×C4—5681no—
346C2×C4#1455609611yescharacteristic
347C2×C4#1455610011yescharacteristic
348C2×C4#1455611111yescharacteristic
349C2×C4#1455614811yescharacteristic
350C2×C4#1455615311yescharacteristic
351C2×C4#1455615911yescharacteristic
352C8—41no—
353C8#1455610111yescharacteristic
354C8#1455610111yescharacteristic
355C8#1455611011yescharacteristic
356C8#1455611411yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—5682no—
365C22—5682no—
366C22—5682no—
367C22—11361no—
368C22—11361no—
369C22—11361no—
370C22—11361no—
371C22#1798613211yescharacteristic
372C22#1798623311yescharacteristic
373C22#1798631211yescharacteristic
374C22C2×#1455606911yescharacteristic
375C22#1798642911yescharacteristic
376C22#1798643411yescharacteristic
377C22#1798644011yescharacteristic
378C4—41no—
379C4—161no—
380C4—161no—
381C4—5682no—
382C4#1798591411yescharacteristic
383C4#1798615911yescharacteristic
384C4#1798624711yescharacteristic
385C4#1798632311yescharacteristic
386C2—42no—
387C2—11362no—
388C2#2258952211yescharacteristic
389C2#2259025111yescharacteristic
390C2#2259027811yescharacteristic
391C2#2259042411yescharacteristic
392C2#2259045511yescharacteristic
393C2#2259052011yescharacteristic
394C2#2259053111yescharacteristic
subgroupG/H
centerC23C2×#11765850
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC71×#1808C2
socleC23×C71C2×D8
prime radicalC2×#17986235C2
hypercenter#1808D71
hypocenterC71#13380
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#17985824C2semidirect
3. C2×#17985799C2semidirect
4. C2×#17985799C2semidirect
5. C2×C22⋊(C71⋊C16)C2semidirect
6. #22586431C2semidirect
7. C71×#1808C2semidirect
42. C2×C71⋊C16D4semidirect
203. #1808D71semidirect
205. C71#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#179862352
C2×#179858242
C2×#179857992
C2×C22⋊(C71⋊C16)2
#225864312
C71×#18082
#1338071