Finite Group Databasebeta

← #31886001

Subgroups — #31886001

Order 18176.

470 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#22588320C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic
3#22586166C212yesmaximal, complemented
4#22586163C212yesmaximal, complemented
5C71×#2855C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)—21no—
9C2×#14556039—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic
13C22×C42×C71C2211yescharacteristic
14#391×C71C411yescharacteristic
15#391×C71C2211yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C2211yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#385×C71—21no—
21#381×C71—21no—
22#380×C71C411yescharacteristic
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—42no—
26C22⋊(C71⋊C8)—24no—
27C22⋊(C71⋊C8)—42no—
28#14556039—42no—
29C22×Q8×C71C2×C411yescharacteristic
30C23×C4×C71C2×C411yescharacteristic
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C22⋊Q8×C71—42no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C42×C71—42no—
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71C2311yescharacteristic
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54C2×C71⋊C8—24no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C24×C71C22⋊C411yescharacteristic
61C2×Q8×C71—22no—
62C2×Q8×C71—41no—
63C2×Q8×C71—41no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C4⋊C4×C71—22no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C42×C71—22no—
94C42×C71—22no—
95C42×C71—41no—
96C42×C71—41no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71#10412yes—
104C23×C71C2×C22⋊C411yescharacteristic
105Q8×C71—82no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—44no—
121C2×C4×C71—44no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—41no—
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—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71#9912yes—
137C2×C4×C71#13112yes—
138C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C71#13711yescharacteristic
140C2×C4×C71#13712yes—
141C71⋊C8—44no—
142C71⋊C8—82no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—22no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71#40811yescharacteristic
151C22×C71C2×#9911yescharacteristic
152C22×C71#41311yescharacteristic
153C22×C71#41711yescharacteristic
154C22×C71#41911yescharacteristic
155C22×C71C2×#10411yescharacteristic
156C22×C71#42111yescharacteristic
157C4×C71—22no—
158C4×C71—22no—
159C4×C71—22no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—41no—
163C4×C71—41no—
164C4×C71—81no—
165C4×C71—81no—
166C4×C71—81no—
167C4×C71—81no—
168#7701—711nomaximal, Sylow
169C2×C71—42no—
170C2×C71#146811yescharacteristic
171C2×C71#148311yescharacteristic
172C2×C71#149711yescharacteristic
173C2×C71#150811yescharacteristic
174C2×C71#151311yescharacteristic
175C2×C71#153211yescharacteristic
176C2×C71#155911yescharacteristic
177#2855D7111yescharacteristic
178#1797—711no—
179#1749—711no—
180#1272—712no—
181#1271—712no—
182C71#770111yescharacteristic, complemented, Sylow
183C2×C22⋊Q8Dic7111yescharacteristic
184C22×C42C2×D7111yescharacteristic
185C2×C4⋊C8—1421no—
186C2×C22⋊C8—1421no—
187C2×C22⋊C8—1421no—
188C2×#97—1421no—
189#391Dic7111yescharacteristic
190#391C2×D7111yescharacteristic
191#389Dic7111yescharacteristic
192#389C2×D7111yescharacteristic
193#387—21no—
194#387—21no—
195#385—21no—
196#381—21no—
197#380Dic7111yescharacteristic
198#336—712no—
199#336—712no—
200C22×Q8C2×Dic7111yescharacteristic
201C23×C4C2×Dic7111yescharacteristic
202C23×C4C22⋊D7111yescharacteristic
203C23×C4C22⋊D7111yescharacteristic
204C22×C8—1422no—
205C22×C8—1422no—
206C22⋊Q8—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C2×Dic7111yescharacteristic
211C2×C4⋊C4C2×Dic7111yescharacteristic
212C2×C4⋊C4C2×Dic7111yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42—42no—
216C2×C42C2×Dic7111yescharacteristic
217C2×C42C22⋊D7111yescharacteristic
218C2×C42C22⋊D7111yescharacteristic
219C2×C42C22×D7111yescharacteristic
220C4⋊C8—2842no—
221C22⋊C8—2842no—
222C22⋊C8—1424no—
223#97—2842no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C24C22⋊Dic7111yescharacteristic
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C22⋊Dic7111yescharacteristic
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22×Dic7111yescharacteristic
253C22×C4C2×C22⋊D7111yescharacteristic
254C22×C4C2×C22⋊D7111yescharacteristic
255C2×C8—2842no—
256C2×C8—2842no—
257C2×C8—2842no—
258C2×C8—2842no—
259C2×C8—2842no—
260C2×C8—1424no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C42—22no—
271C42—22no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1455605512yes—
281C23C2×C22⋊Dic7111yescharacteristic
282Q8—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4#1455604312yes—
314C2×C4#1455613512yes—
315C2×C4#1455615311yescharacteristic
316C2×C4#1455615811yescharacteristic
317C2×C4#1455615912yes—
318C8—2844no—
319C8—5682no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1798596611yescharacteristic
328C22C2×#1455604311yescharacteristic
329C22#1798598311yescharacteristic
330C22#1798600011yescharacteristic
331C22#1798600811yescharacteristic
332C22C2×#1455605511yescharacteristic
333C22#1798601611yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C2—42no—
346C2#2258668811yescharacteristic
347C2#2258675311yescharacteristic
348C2#2258681311yescharacteristic
349C2#2258686011yescharacteristic
350C2#2258688311yescharacteristic
351C2#2258696011yescharacteristic
352C2#2258707011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2855C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2855D71
hypocenterC71#7701
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #22586166C2semidirect
4. #22586163C2semidirect
182. C71#7701semidirect

Maximal subgroups

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

subgroupindex
#225883202
#225881022
#225861662
#225861632
C71×#28552
#770171