Finite Group Databasebeta

← #31893678

Subgroups — #31893678

Order 18176.

562 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986276C211yesmaximal, characteristic
2C2×#17986078C211yesmaximal, characteristic
3#22588337C211yesmaximal, characteristic
4C2×#17985745C211yesmaximal, characteristic, complemented
5#22586223C211yesmaximal, characteristic, complemented
6#22586175C211yesmaximal, characteristic, complemented
7C71×#1956C211yesmaximal, characteristic, complemented
8C22×#11765848—21no—
9C2×C4×D284C2211yescharacteristic
10#17986276—22no—
11#17986276C414yescomplemented
12#17986078—22no—
13#17986078—22no—
14C2×#14556061—21no—
15C2×C4⋊(C71⋊C8)C2211yescharacteristic
16C2×C4⋊(C71⋊C8)C2211yescharacteristic
17C2×C4⋊(C71⋊C8)C2211yescharacteristic
18C2×#14556048—21no—
19C2×#14556048—21no—
20C2×(C2×D71)⋊C8—21no—
21#17985761—21no—
22#17985745—22no—
23C2×C4⋊Q8×C71C2211yescharacteristic
24C2×#103×C71—21no—
25C2×#103×C71—21no—
26C2×C4×C8×C71C2211yescharacteristic
27#384×C71C2211yescharacteristic
28C22×D284D411yescharacteristic
29C22×C4×D71—21no—
30C2×#11765848—24no—
31C2×#11765848—42no—
32C2×#11765848—42no—
33C22×C71⋊C8—21no—
34C22×C71⋊C8—21no—
35C22×C71⋊C8—21no—
36C4×D284—21no—
37C4×D284—21no—
38C4×D284C2×C412yes—
39C4×D284D412yes—
40C2×C4⋊Dic71D411yescharacteristic
41C2×(C2×D71)⋊C4—21no—
42#14556061—24no—
43C4⋊(C71⋊C8)—21no—
44C4⋊(C71⋊C8)—21no—
45C4⋊(C71⋊C8)—21no—
46C4⋊(C71⋊C8)—21no—
47C4⋊(C71⋊C8)C2×C412yes—
48C4⋊(C71⋊C8)D412yes—
49#14556048—24no—
50#14556048—42no—
51(C2×D71)⋊C8—24no—
52C22×Q8×C71—21no—
53C22×C8×C71D411yescharacteristic
54C22×C8×C71D411yescharacteristic
55C4⋊Q8×C71—21no—
56C4⋊Q8×C71—21no—
57C4⋊Q8×C71—21no—
58C4⋊Q8×C71C2×C412yes—
59C2×C4⋊C4×C71—21no—
60C2×C4⋊C4×C71—21no—
61C2×C4⋊C4×C71—21no—
62C2×C4⋊C4×C71—21no—
63C2×C4⋊C4×C71—21no—
64C2×C42×C71C2311yescharacteristic
65#103×C71—42no—
66#103×C71—42no—
67C4×C8×C71—21no—
68C4×C8×C71—21no—
69C23×D71—21no—
70C2×D284—21no—
71C2×D284—21no—
72C2×D284—21no—
73C2×D284—21no—
74C2×D284—21no—
75C2×D284C22⋊C412yes—
76C22×Dic71—21no—
77C2×C4×D71—22no—
78C2×C4×D71—22no—
79C2×C4×D71—41no—
80C2×C4×D71—41no—
81C2×C4×D71—41no—
82C2×C4×D71—41no—
83#11765848—48no—
84C2×C71⋊C8—22no—
85C2×C71⋊C8—22no—
86C2×C71⋊C8—41no—
87C2×C71⋊C8—41no—
88C2×C71⋊C8—41no—
89C2×C71⋊C8—41no—
90C2×C71⋊C8—41no—
91C2×C71⋊C8—41no—
92C2×C71⋊C8—41no—
93C4⋊Dic71—21no—
94C4⋊Dic71C22⋊C412yes—
95(C2×D71)⋊C4—22no—
96(C2×D71)⋊C4—22no—
97C2×Q8×C71—22no—
98C2×Q8×C71—41no—
99C2×Q8×C71—41no—
100C2×Q8×C71—41no—
101C2×Q8×C71—41no—
102C2×Q8×C71—41no—
103C22×C4×C71—41no—
104C22×C4×C71—41no—
105C22×C4×C71C2×D411yescharacteristic
106C22×C4×C71C2×D411yescharacteristic
107C22×C4×C71#4111yescharacteristic
108C2×C8×C71—21no—
109C2×C8×C71—21no—
110C2×C8×C71—21no—
111C2×C8×C71—21no—
112C2×C8×C71—21no—
113C2×C8×C71D812yes—
114C4⋊C4×C71—22no—
115C4⋊C4×C71—22no—
116C4⋊C4×C71—41no—
117C4⋊C4×C71—41no—
118C4⋊C4×C71—41no—
119C4⋊C4×C71—41no—
120C4⋊C4×C71—41no—
121C4⋊C4×C71—41no—
122C4⋊C4×C71—81no—
123C42×C71C22×C411yescharacteristic
124C42×C71C2×D411yescharacteristic
125C42×C71C2×D411yescharacteristic
126C42×C71#4111yescharacteristic
127C23×C71#12111yescharacteristic
128C22×D71—22no—
129C22×D71—41no—
130C22×D71—41no—
131C22×D71—41no—
132C22×D71—41no—
133C22×D71—41no—
134C22×D71—41no—
135Q8×C71—42no—
136Q8×C71—42no—
137Q8×C71—81no—
138Q8×C71—81no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—42no—
142C2×C4×C71—81no—
143C2×C4×C71—81no—
144C2×C4×C71—81no—
145C2×C4×C71—81no—
146C2×C4×C71—81no—
147C2×C4×C71C2×C22⋊C411yescharacteristic
148C2×C4×C71C4×D411yescharacteristic
149C2×C4×C71C4×D411yescharacteristic
150C2×C4×C71#12011yescharacteristic
151C2×C4×C71#12111yescharacteristic
152C2×C4×C71#12311yescharacteristic
153C2×C4×C71C2×D811yescharacteristic
154C2×C4×C71#13511yescharacteristic
155C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C71#13711yescharacteristic
159C2×C4×C71#13711yescharacteristic
160C2×C4×C71#13711yescharacteristic
161C2×Dic71—22no—
162C2×Dic71—41no—
163C2×Dic71—41no—
164D284—22no—
165D284—22no—
166D284—41no—
167D284—41no—
168C4×D71—42no—
169C4×D71—42no—
170C4×D71—42no—
171C4×D71—42no—
172C8×C71—22no—
173C8×C71—41no—
174C71⋊C8—42no—
175C71⋊C8—42no—
176C71⋊C8—81no—
177C22×C71#38611yescharacteristic
178C22×C71#45911yescharacteristic
179C22×C71#46311yescharacteristic
180C22×C71#46811yescharacteristic
181C22×C71#47011yescharacteristic
182C22×C71#47111yescharacteristic
183C22×C71#47311yescharacteristic
184C2×D71—42no—
185C2×D71—42no—
186C2×D71—42no—
187C2×D71—81no—
188C2×D71—81no—
189C2×D71—81no—
190C2×D71—81no—
191C4×C71—21no—
192C4×C71—21no—
193C4×C71—82no—
194C4×C71—161no—
195C4×C71#41711yescharacteristic
196C4×C71#41911yescharacteristic
197C4×C71C4×D811yescharacteristic
198C4×C71#44211yescharacteristic
199C4×C71#44711yescharacteristic
200C4×C71#44911yescharacteristic
201C4×C71#46811yescharacteristic
202C4×C71#47011yescharacteristic
203Dic71—42no—
204#9227—711nomaximal, Sylow
205C2×C71#168511yescharacteristic
206C2×C71#168911yescharacteristic
207C2×C71#169911yescharacteristic
208C2×C71#170311yescharacteristic
209C2×C71#187911yescharacteristic
210C2×C71#189411yescharacteristic
211C2×C71#190811yescharacteristic
212D71—82no—
213C2×#474—711no—
214C2×#436—711no—
215#1956D7111yescharacteristic, complemented
216#1802—711no—
217C2×#332—711no—
218#1296—711no—
219#1277—711no—
220C71#922711yescharacteristic, complemented, Sylow
221C22×M16—1421no—
222C2×C4⋊Q8C2×D7111yescharacteristic
223C2×C4×D4—711no—
224#474—714no—
225#474—1422no—
226#436—1422no—
227#436—1422no—
228C2×#107—1421no—
229C2×C4⋊C8—711no—
230C2×C4⋊C8—711no—
231C2×C4⋊C8—711no—
232C2×#103—21no—
233C2×#103—21no—
234C2×#102—1421no—
235C2×#102—1421no—
236C2×C22⋊C8—1421no—
237C2×C4×C8C2×D7111yescharacteristic
238#384C2×D7111yescharacteristic
239#340—1421no—
240#332—1422no—
241C22×Q8—21no—
242C22×D4—711no—
243C23×C4—1421no—
244C2×M16—2842no—
245C2×M16—2842no—
246C2×M16—1424no—
247C22×C8—1421no—
248C22×C8—1421no—
249C22×C8—1421no—
250C22×C8C22⋊D7111yescharacteristic
251C22×C8C22⋊D7111yescharacteristic
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8C4×D7112yes—
256C4×D4—712no—
257C4×D4—712no—
258C4×D4—1421no—
259C4×D4—1421no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—711no—
266C2×C22⋊C4—1421no—
267C2×C42C22×D7111yescharacteristic
268#107—1424no—
269C4⋊C8—712no—
270C4⋊C8—712no—
271C4⋊C8—1421no—
272C4⋊C8—1421no—
273C4⋊C8—1421no—
274C4⋊C8—1421no—
275#103—42no—
276#103—42no—
277#102—2842no—
278#102—1424no—
279C22⋊C8—1424no—
280C4×C8—21no—
281C4×C8—21no—
282C24—1421no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—712no—
290C2×D4—1421no—
291C2×D4—1421no—
292C2×D4—1421no—
293C2×D4—1421no—
294C2×D4—1421no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—2841no—
298C22×C4—2841no—
299C22×C4—2841no—
300C22×C4—2841no—
301C22×C4—1421no—
302C22×C4—1422no—
303C22×C4—1422no—
304C22×C4D4×D7111yescharacteristic
305C22×C4C2×C22⋊D7111yescharacteristic
306C22×C4#1176586411yescharacteristic
307M16—2848no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—2841no—
314C2×C8—2841no—
315C2×C8—2841no—
316C2×C8—2841no—
317C2×C8—2841no—
318C2×C8—2841no—
319C2×C8—2841no—
320C2×C8—1422no—
321C2×C8—1422no—
322C2×C8#1176585012yes—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—81no—
332C4⋊C4—712no—
333C4⋊C4—1421no—
334C22⋊C4—1422no—
335C22⋊C4—1422no—
336C42C2×C4×D7111yescharacteristic
337C42D4×D7111yescharacteristic
338C42D4×D7111yescharacteristic
339C42#1176586511yescharacteristic
340C23—2841no—
341C23—2841no—
342C23—2841no—
343C23—2841no—
344C23—2841no—
345C23—2841no—
346C23—1422no—
347C23#1455609911yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352D4—2841no—
353D4—2841no—
354D4—1422no—
355D4—1422no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—2841no—
365C2×C4—2841no—
366C2×C4—2842no—
367C2×C4—2842no—
368C2×C4—2842no—
369C2×C4—2842no—
370C2×C4—1422no—
371C2×C4C22⋊C4×D7111yescharacteristic
372C2×C4(C22⋊D71)⋊C411yescharacteristic
373C2×C4C4×C22⋊D7111yescharacteristic
374C2×C4#1455609611yescharacteristic
375C2×C4#1455610111yescharacteristic
376C2×C4#1455611011yescharacteristic
377C2×C4C2×#1176585011yescharacteristic
378C2×C4#1455614811yescharacteristic
379C2×C4#1455615311yescharacteristic
380C2×C4#1455615411yescharacteristic
381C2×C4#1455615511yescharacteristic
382C2×C4#1455615711yescharacteristic
383C2×C4#1455616011yescharacteristic
384C2×C4#1455616011yescharacteristic
385C8—22no—
386C8—41no—
387C8—2842no—
388C8—2842no—
389C8—5681no—
390C22—2842no—
391C22—2842no—
392C22—2842no—
393C22—5681no—
394C22—5681no—
395C22—5681no—
396C22—5681no—
397C22#1798587511yescharacteristic
398C22#1798619111yescharacteristic
399C22#1798621111yescharacteristic
400C22#1798624211yescharacteristic
401C22#1798625411yescharacteristic
402C22#1798625911yescharacteristic
403C22#1798627111yescharacteristic
404C4—21no—
405C4—21no—
406C4—82no—
407C4—161no—
408C4—2842no—
409C4#1798599911yescharacteristic
410C4#1798600611yescharacteristic
411C4C4×#1176585011yescharacteristic
412C4#1798610911yescharacteristic
413C4#1798613211yescharacteristic
414C4#1798614211yescharacteristic
415C4#1798624811yescharacteristic
416C4#1798625611yescharacteristic
417C2—5682no—
418C2#2258780211yescharacteristic
419C2#2258782611yescharacteristic
420C2#2258788411yescharacteristic
421C2#2258791211yescharacteristic
422C2#2258869111yescharacteristic
423C2#2258877011yescharacteristic
424C2#2258885811yescharacteristic
subgroupG/H
centerC23#14556099
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1956C2
socleC23×C71#121
prime radicalC22×D284D4
cosocleC2×C42×C71C23
hypercenter#1956D71
hypocenterC71#9227
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
4. C2×#17985745C2semidirect
5. #22586223C2semidirect
6. #22586175C2semidirect
7. C71×#1956C2semidirect
11. #17986276C4semidirect
215. #1956D71semidirect
220. C71#9227semidirect

Maximal subgroups

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

subgroupindex
C2×#179862762
C2×#179860782
#225883372
C2×#179857452
#225862232
#225861752
C71×#19562
#922771