Finite Group Databasebeta

← #31908541

Subgroups — #31908541

Order 18176.

416 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589951C211yesmaximal, characteristic, complemented
2#22588369C211yesmaximal, characteristic, complemented
3#22586352C212yesmaximal, complemented
4#22586349C212yesmaximal, complemented
5C71×#3050C211yesmaximal, characteristic
6C2×#14556068—21no—
7C2×#14556062—21no—
8#17985859—21no—
9C22⋊(C71⋊C16)—21no—
10C22⋊(C71⋊C16)C2212yes—
11#17985762C2212yes—
12C2×#135×C71C411yescharacteristic, complemented
13C23×C8×C71C2211yescharacteristic
14#525×C71—21no—
15#467×C71C411yescharacteristic, complemented
16#467×C71C2211yescharacteristic
17#466×C71C411yescharacteristic, complemented
18#466×C71C2211yescharacteristic
19#465×C71—21no—
20#465×C71—21no—
21#427×C71C411yescharacteristic, complemented
22#416×C71—21no—
23C2×C4⋊Dic71—22no—
24C2×C22⋊Dic71—41no—
25#14556068—42no—
26C2×C71⋊C16—22no—
27#14556062—42no—
28C71⋊#95—41no—
29C2×#41×C71—21no—
30C23×C4×C71D411yescharacteristic
31#135×C71—42no—
32C2×Q16×C71C2×C411yescharacteristic
33C2×SD16×C71—21no—
34C2×D8×C71C2×C411yescharacteristic
35C22×C8×C71—42no—
36C22×C8×C71C2×C411yescharacteristic
37C22×C8×C71C2×C411yescharacteristic
38C22×C8×C71D411yescharacteristic
39C22×C8×C71D411yescharacteristic
40C22×C8×C71D411yescharacteristic
41C22×C8×C71C2311yescharacteristic
42#123×C71—41no—
43C22⋊Q8×C71—21no—
44C22⋊Q8×C71—21no—
45#121×C71—21no—
46#121×C71—21no—
47#120×C71—41no—
48C4×D4×C71—41no—
49C4×D4×C71—41no—
50#117×C71—21no—
51#107×C71C2×C411yescharacteristic
52#107×C71C2×C411yescharacteristic
53#106×C71—21no—
54#103×C71—21no—
55#103×C71—21no—
56#102×C71—21no—
57#102×C71—21no—
58C22×Dic71—42no—
59C4⋊Dic71—42no—
60C4⋊Dic71—42no—
61C22⋊Dic71—82no—
62C71⋊C16—42no—
63C24×C71D811yescharacteristic
64#41×C71—42no—
65#41×C71—42no—
66C2×Q8×C71—21no—
67C2×D4×C71—21no—
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C22×C4×C71—41no—
72C22×C4×C71—41no—
73C22×C4×C71—41no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71D811yescharacteristic
77C22×C4×C71SD1611yescharacteristic
78C22×C4×C71SD1611yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80Q16×C71—41no—
81SD16×C71—42no—
82D8×C71—41no—
83C2×C8×C71—22no—
84C2×C8×C71—22no—
85C2×C8×C71—22no—
86C2×C8×C71—41no—
87C2×C8×C71—41no—
88C2×C8×C71—41no—
89C2×C8×C71C22⋊C411yescharacteristic
90C2×C8×C71C22⋊C411yescharacteristic
91C2×C8×C71C22×C411yescharacteristic
92C2×C8×C71C2×D411yescharacteristic
93C4⋊C4×C71—21no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C22⋊C4×C71—41no—
98C22⋊C4×C71—41no—
99C22⋊C4×C71—41no—
100C22⋊C4×C71—41no—
101C22⋊C4×C71—41no—
102C42×C71—41no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71#10211yescharacteristic
109C23×C71#10211yescharacteristic
110C23×C71C2×D811yescharacteristic
111Q8×C71—41no—
112Q8×C71—41no—
113D4×C71—41no—
114D4×C71—41no—
115D4×C71—82no—
116D4×C71—82no—
117D4×C71—81no—
118D4×C71—81no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—82no—
132C2×C4×C71#10211yescharacteristic
133C2×C4×C71#10211yescharacteristic
134C2×C4×C71C2×C22⋊C411yescharacteristic
135C2×C4×C71C2×SD1611yescharacteristic
136C2×Dic71—82no—
137C2×Dic71—82no—
138C2×Dic71—82no—
139C8×C71—21no—
140C8×C71—21no—
141C8×C71—21no—
142C8×C71#9912yes—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—21no—
150C22×C71—21no—
151C22×C71—42no—
152C22×C71—42no—
153C22×C71—81no—
154C22×C71—81no—
155C22×C71—81no—
156C22×C71C2×#10211yescharacteristic
157C22×C71#50911yescharacteristic
158C22×C71#51011yescharacteristic
159C4×C71—21no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—81no—
163C4×C71—81no—
164C4×C71—81no—
165C4×C71C2×#9911yescharacteristic
166C4×C71#42111yescharacteristic
167Dic71—162no—
168#11890—711nomaximal, Sylow
169C2×C71—21no—
170C2×C71—21no—
171C2×C71—41no—
172C2×C71—41no—
173C2×C71—161no—
174C2×C71#151911yescharacteristic
175C2×C71#213911yescharacteristic
176C2×C71#214511yescharacteristic
177#3050D7111yescharacteristic
178#2112—711no—
179#1808—711no—
180#1348—712no—
181#1347—712no—
182C71#1189011yescharacteristic, complemented, Sylow
183C2×#135Dic7111yescharacteristic, complemented
184C23×C8C2×D7111yescharacteristic
185#525—21no—
186C2×M32—1421no—
187#467Dic7111yescharacteristic, complemented
188#467C2×D7111yescharacteristic
189#466Dic7111yescharacteristic, complemented
190#466C2×D7111yescharacteristic
191#465—21no—
192#465—21no—
193#427Dic7111yescharacteristic, complemented
194C2×#107—1421no—
195#416—21no—
196#380—1421no—
197C22⋊C16—712no—
198C22⋊C16—1421no—
199#340—712no—
200C2×#41—21no—
201C23×C4C22⋊D7111yescharacteristic
202#135—42no—
203C2×Q16C2×Dic7111yescharacteristic
204C2×SD16—21no—
205C2×D8C2×Dic7111yescharacteristic
206C22×C8—42no—
207C22×C8C2×Dic7111yescharacteristic
208C22×C8C2×Dic7111yescharacteristic
209C22×C8C22⋊D7111yescharacteristic
210C22×C8C22⋊D7111yescharacteristic
211C22×C8C22⋊D7111yescharacteristic
212C22×C8C22×D7111yescharacteristic
213#123—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#121—21no—
217#121—21no—
218#120—41no—
219C4×D4—41no—
220C4×D4—41no—
221#117—21no—
222C2×C4⋊C4—1422no—
223C2×C22⋊C4—2841no—
224M32—2842no—
225C2×C16—1422no—
226#107—2842no—
227#107C2×Dic7111yescharacteristic
228#107C2×Dic7111yescharacteristic
229#106—21no—
230#103—21no—
231#103—21no—
232#102—21no—
233#102—21no—
234#95—2841no—
235C24#1176585011yescharacteristic
236#41—42no—
237#41—42no—
238C2×Q8—21no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—2842no—
247C22×C4C22⋊Dic7111yescharacteristic
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4#1176585011yescharacteristic
250C22×C4#1176585311yescharacteristic
251C22×C4#1176585311yescharacteristic
252C22×C4C2×C22⋊D7111yescharacteristic
253Q16—41no—
254SD16—42no—
255D8—41no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8C22⋊Dic7111yescharacteristic
263C2×C8C22⋊Dic7111yescharacteristic
264C2×C8C22×Dic7111yescharacteristic
265C2×C8C2×C22⋊D7111yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—2842no—
271C4⋊C4—2842no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—5682no—
278C42—41no—
279C16—2842no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#1455604911yescharacteristic
286C23#1455604911yescharacteristic
287C23C2×#1176585011yescharacteristic
288Q8—41no—
289Q8—41no—
290D4—41no—
291D4—41no—
292D4—82no—
293D4—82no—
294D4—81no—
295D4—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—82no—
309C2×C4—5682no—
310C2×C4—5682no—
311C2×C4—5682no—
312C2×C4#1455604911yescharacteristic
313C2×C4#1455604911yescharacteristic
314C2×C4C2×C22⋊Dic7111yescharacteristic
315C2×C4C2×#1176585311yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8#1455604312yes—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C2×#1455604911yescharacteristic
334C22#1798643411yescharacteristic
335C22#1798643911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—11362no—
343C4C2×#1455604311yescharacteristic
344C4#1798601611yescharacteristic
345C2—21no—
346C2—21no—
347C2—41no—
348C2—41no—
349C2—161no—
350C2#2258690811yescharacteristic
351C2#2259005211yescharacteristic
352C2#2259007611yescharacteristic
subgroupG/H
centerC22C2×#14556049
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
Fitting, prime radicalC71×#3050C2
socleC22×C71C2×#102
cosocleC22×C8×C71C23
hypercenter#3050D71
hypocenterC71#11890
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. #22589951C2semidirect
2. #22588369C2semidirect
3. #22586352C2semidirect
4. #22586349C2semidirect
12. C2×#135×C71C4semidirect
15. #467×C71C4semidirect
17. #466×C71C4semidirect
21. #427×C71C4semidirect
182. C71#11890semidirect
183. C2×#135Dic71semidirect
187. #467Dic71semidirect
189. #466Dic71semidirect
193. #427Dic71semidirect

Maximal subgroups

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

subgroupindex
#225899512
#225883692
#225863522
#225863492
C71×#30502
#1189071