Finite Group Databasebeta

← #31912699

Subgroups — #31912699

Order 18176.

466 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597111C211yesmaximal, characteristic
2C2×#17985810C211yesmaximal, characteristic
3C2×#17985808C211yesmaximal, characteristic
4M32×Dic71C211yesmaximal, characteristic, complemented
5#106×Dic71C211yesmaximal, characteristic, complemented
6#22586445C211yesmaximal, characteristic, complemented
7C71×#1386C211yesmaximal, characteristic, complemented
8C2×#14556150C2211yescharacteristic
9#17986467—21no—
10C2×#14556068C2211yescharacteristic
11#17986093C411yescharacteristic, complemented
12#17986093C411yescharacteristic, complemented
13#17986086—21no—
14#17986086—21no—
15#17986081C411yescharacteristic, complemented
16#17986081C411yescharacteristic, complemented
17C2×#14556060C2211yescharacteristic
18#17985993—21no—
19C2×C8×Dic71C2211yescharacteristic
20C4⋊C4×Dic71—21no—
21#17985810C414yescomplemented
22#17985808C414yescomplemented
23#17985774—21no—
24C16×Dic71—21no—
25#17985761—21no—
26C2×M32×C71C2211yescharacteristic
27#427×C71C2211yescharacteristic
28C2×#106×C71C2211yescharacteristic
29C2×#11765865—21no—
30#14556150—21no—
31#14556150—21no—
32#14556150C2×C412yes—
33#14556150C2×C412yes—
34C2×Dic284C2×C411yescharacteristic
35C2×#11765854—21no—
36C2×D568C2×C411yescharacteristic
37C22×C71⋊C8D411yescharacteristic
38#14556093—21no—
39#14556093—21no—
40#14556090—21no—
41#14556090—21no—
42(C22⋊D71)⋊C4—41no—
43(C22⋊D71)⋊C4—41no—
44#14556085—41no—
45C2×C4⋊Dic71—21no—
46C2×C4×Dic71—41no—
47C2×C4×Dic71D411yescharacteristic
48C42×D71—41no—
49#14556068—21no—
50#14556068C2×C412yescomplemented
51C2×C71⋊C16—21no—
52#14556060—21no—
53#14556060C2×C412yescomplemented
54#14556051—21no—
55#14556051—21no—
56#14556048—21no—
57#14556048—21no—
58C8×Dic71C2×C411yescharacteristic
59C8×Dic71C2×C411yescharacteristic
60C8×Dic71D411yescharacteristic
61C8×Dic71D411yescharacteristic
62C71⋊#95—41no—
63C22×C8×C71C2311yescharacteristic
64#117×C71—21no—
65C2×C4⋊C4×C71—21no—
66M32×C71—21no—
67M32×C71C2×C412yescomplemented
68C2×C16×C71—21no—
69#107×C71C2×C411yescharacteristic
70#107×C71C2×C411yescharacteristic
71#106×C71—21no—
72#106×C71—21no—
73#106×C71C2×C412yescomplemented
74#11765865—22no—
75#11765865—22no—
76#11765865—41no—
77#11765865—41no—
78C2×Dic142—21no—
79C2×D284—21no—
80C2×C22⋊D71—41no—
81C22×Dic71—41no—
82C22×Dic71D811yescharacteristic
83C22×Dic71D811yescharacteristic
84C2×C4×D71—41no—
85C2×C4×D71—41no—
86C2×C4×D71—41no—
87Dic284—21no—
88Dic284C4212yescomplemented
89#11765854—22no—
90#11765854—22no—
91D568—21no—
92D568C4212yescomplemented
93C2×C71⋊C8—21no—
94C2×C71⋊C8—21no—
95C2×C71⋊C8C22⋊C411yescharacteristic
96C2×C71⋊C8C22⋊C411yescharacteristic
97C4⋊Dic71—22no—
98C4⋊Dic71—41no—
99Dic71⋊C4—41no—
100Dic71⋊C4—41no—
101(C2×D71)⋊C4—41no—
102(C2×D71)⋊C4—41no—
103C4×Dic71—41no—
104C4×Dic71—41no—
105C4×Dic71—82no—
106C4×Dic71C22⋊C411yescharacteristic
107C4×Dic71C22⋊C411yescharacteristic
108C4×Dic71SD1611yescharacteristic
109C4×Dic71SD1611yescharacteristic
110C71⋊C16—22no—
111C22×C4×C71—41no—
112C22×C4×C71C2×D411yescharacteristic
113C2×C8×C71C22×C411yescharacteristic
114C2×C8×C71C22×C411yescharacteristic
115C2×C8×C71C22×C411yescharacteristic
116C2×C8×C71C2×D411yescharacteristic
117C2×C8×C71#4111yescharacteristic
118C2×C8×C71#4111yescharacteristic
119C4⋊C4×C71—21no—
120C4⋊C4×C71—21no—
121C4⋊C4×C71—22no—
122C4⋊C4×C71—41no—
123C22⋊C4×C71—41no—
124C42×C71—41no—
125C16×C71—22no—
126C23×C71C2×D811yescharacteristic
127C22×D71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—42no—
131C2×C4×C71—81no—
132C2×C4×C71—81no—
133C2×C4×C71C2×C22⋊C411yescharacteristic
134C2×C4×C71C4×D411yescharacteristic
135C2×C4×C71C4×D411yescharacteristic
136C2×C4×C71C2×SD1611yescharacteristic
137C2×C4×C71#13511yescharacteristic
138C2×C4×C71#13511yescharacteristic
139C22⋊D71—42no—
140C22⋊D71—42no—
141C2×Dic71—22no—
142C2×Dic71—41no—
143C2×Dic71—41no—
144C2×Dic71—42no—
145C2×Dic71—81no—
146C2×Dic71—81no—
147C2×Dic71#10211yescharacteristic
148C2×Dic71#10211yescharacteristic
149C2×Dic71#10211yescharacteristic
150C2×Dic71#10211yescharacteristic
151D284—22no—
152D284—41no—
153C4×D71—42no—
154C4×D71—42no—
155C4×D71—82no—
156C4×D71—82no—
157Dic142—22no—
158Dic142—41no—
159C8×C71C2×C4211yescharacteristic
160C8×C71#11711yescharacteristic
161C8×C71C4×D411yescharacteristic
162C8×C71C4×D411yescharacteristic
163C71⋊C8—21no—
164C71⋊C8—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71C2×#10211yescharacteristic
168C22×C71C4×D811yescharacteristic
169C22×C71C4×D811yescharacteristic
170C2×D71—42no—
171C2×D71—81no—
172C2×D71—81no—
173C4×C71—82no—
174C4×C71—81no—
175C4×C71—81no—
176C4×C71C4×C22⋊C411yescharacteristic
177C4×C71#41611yescharacteristic
178C4×C71C4×SD1611yescharacteristic
179C4×C71C4×SD1611yescharacteristic
180Dic71—21no—
181Dic71—21no—
182Dic71—21no—
183Dic71—42no—
184Dic71—82no—
185Dic71#36012yescomplemented
186C4×#360—711nomaximal, Sylow
187C2×C71—21no—
188C2×C71—21no—
189C2×C71C4×#10211yescharacteristic
190C2×C71C2×#36011yescharacteristic
191C2×C71#214511yescharacteristic
192D71—82no—
193C4×#135—711no—
194C2×#360—711no—
195C2×#360—711no—
196C4×M32—711no—
197C4×#106—711no—
198#1386—711no—
199#1386D7111yescharacteristic, complemented
200C71C4×#36011yescharacteristic, complemented, Sylow
201C2×#135—711no—
202C4×#41—1421no—
203C2×M32—711no—
204C2×M32C2×D7111yescharacteristic
205C4×Q16—711no—
206C4×Q16—711no—
207C4×SD16—1421no—
208C4×SD16—1421no—
209C4×D8—711no—
210C4×D8—711no—
211#427C2×D7111yescharacteristic
212C2×#106—711no—
213C2×#106C2×D7111yescharacteristic
214#416—1421no—
215C2×C4×C8—711no—
216C4×C4⋊C4—1421no—
217#360—714no—
218#360—714no—
219#346—1421no—
220C4×C16—1421no—
221#340—1421no—
222C2×#41—1421no—
223#135—712no—
224#135—712no—
225#135—1421no—
226#135—1421no—
227C2×Q16—711no—
228C2×SD16—1421no—
229C2×D8—711no—
230C22×C8—711no—
231C22×C8C22×D7111yescharacteristic
232C4×Q8—1421no—
233C4×Q8—1421no—
234C4×D4—2841no—
235C4×D4—2841no—
236C4×D4—1421no—
237C4×D4—1421no—
238#117—21no—
239#117—2841no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—1421no—
242C2×C42—2841no—
243C2×C42—2841no—
244C2×C42—711no—
245M32—21no—
246M32—712no—
247M32—1421no—
248M32C4×D7112yescomplemented
249C2×C16—21no—
250C2×C16—1421no—
251#107C4×D7111yescharacteristic
252#107C4×D7111yescharacteristic
253#106—21no—
254#106—21no—
255#106—712no—
256#106—1421no—
257#106C4×D7112yescomplemented
258#103—1421no—
259#103—1421no—
260#102—1421no—
261#102—1421no—
262C4×C8—711no—
263C4×C8—711no—
264C4×C8—711no—
265C4×C8—711no—
266#95—2841no—
267#41—2841no—
268#41—2841no—
269#41—1422no—
270#41—1422no—
271C2×Q8—1421no—
272C2×D4—2841no—
273C2×D4—1421no—
274C22×C4—41no—
275C22×C4—2841no—
276C22×C4—2841no—
277C22×C4—2841no—
278C22×C4—2841no—
279C22×C4—711no—
280C22×C4—711no—
281C22×C4D4×D7111yescharacteristic
282Q16—712no—
283Q16—1421no—
284SD16—1422no—
285SD16—1422no—
286D8—712no—
287D8—1421no—
288C2×C8—711no—
289C2×C8—711no—
290C2×C8—1421no—
291C2×C8—1421no—
292C2×C8C2×C4×D7111yescharacteristic
293C2×C8C2×C4×D7111yescharacteristic
294C2×C8C2×C4×D7111yescharacteristic
295C2×C8D4×D7111yescharacteristic
296C2×C8#1176586411yescharacteristic
297C2×C8#1176586611yescharacteristic
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—2841no—
303C4⋊C4—2841no—
304C4⋊C4—2841no—
305C4⋊C4—1422no—
306C22⋊C4—41no—
307C22⋊C4—2841no—
308C22⋊C4—2841no—
309C42—41no—
310C42—2841no—
311C42—2841no—
312C42—5682no—
313C42—711no—
314C42—711no—
315C42—711no—
316C42—711no—
317C16—22no—
318C16—1422no—
319C23—2841no—
320C23D8×D7111yescharacteristic
321Q8—2841no—
322Q8—1422no—
323D4—2842no—
324D4—2842no—
325D4—2841no—
326D4—1422no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—2841no—
333C2×C4—2842no—
334C2×C4—2842no—
335C2×C4—2842no—
336C2×C4—2841no—
337C2×C4—5682no—
338C2×C4—5682no—
339C2×C4—5681no—
340C2×C4—5681no—
341C2×C4—711no—
342C2×C4—711no—
343C2×C4—711no—
344C2×C4—711no—
345C2×C4—1422no—
346C2×C4C22⋊C4×D7111yescharacteristic
347C2×C4(C22⋊D71)⋊C411yescharacteristic
348C2×C4#1455609011yescharacteristic
349C2×C4SD16×D7111yescharacteristic
350C2×C4#1455614911yescharacteristic
351C2×C4#1455615111yescharacteristic
352C8—1421no—
353C8—1421no—
354C8C42×D7111yescharacteristic
355C8#1455608611yescharacteristic
356C8(C22⋊D71)⋊C411yescharacteristic
357C8#1455609011yescharacteristic
358C22—21no—
359C22—21no—
360C22—2842no—
361C22—5681no—
362C22—5681no—
363C22#102×D7111yescharacteristic
364C22#1798607911yescharacteristic
365C22#1798608111yescharacteristic
366C4—82no—
367C4—81no—
368C4—81no—
369C4—2842no—
370C4—5682no—
371C4—712no—
372C4—1421no—
373C4—1421no—
374C4—1421no—
375C4#1798584611yescharacteristic
376C4#1798599411yescharacteristic
377C4#1798608411yescharacteristic
378C4#1798608611yescharacteristic
379C2—21no—
380C2—21no—
381C2—5682no—
382C2#2258815411yescharacteristic
383C2#360×D7111yescharacteristic
384C2#2259007411yescharacteristic
subgroupG/H
centerC22#102×D71
commutatorC8×C71C2×C42
FrattiniC22×C8C22×D71
FittingC71×#1386C2
socleC22×C71C2×#102
prime radicalC2×#14556150C22
cosocleC22×C8×C71C23
hypercenter#1386D71
hypocenterC71C4×#360
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
4. M32×Dic71C2semidirect
5. #106×Dic71C2semidirect
6. #22586445C2semidirect
7. C71×#1386C2semidirect
11. #17986093C4semidirect
12. #17986093C4semidirect
15. #17986081C4semidirect
16. #17986081C4semidirect
21. #17985810C4semidirect
22. #17985808C4semidirect
50. #14556068C2×C4semidirect
53. #14556060C2×C4semidirect
67. M32×C71C2×C4semidirect
73. #106×C71C2×C4semidirect
88. Dic284C42semidirect
92. D568C42semidirect
185. Dic71#360semidirect
199. #1386D71semidirect
200. C71C4×#360semidirect
248. M32C4×D71semidirect
257. #106C4×D71semidirect

Maximal subgroups

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

subgroupindex
#225971112
C2×#179858102
C2×#179858082
M32×Dic712
#106×Dic712
#225864452
C71×#13862
C4×#36071