Finite Group Databasebeta

← #31885600

Subgroups — #31885600

Order 18176.

612 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588408C211yesmaximal, characteristic
2#22588034C211yesmaximal, characteristic
3#22586163C214yesmaximal, complemented
4C2×#389×C71C211yesmaximal, characteristic
5C22×#11765849—21no—
6C2×C22⋊(C71⋊C8)—21no—
7#17985754—21no—
8#17985754—21no—
9#17985754C2212yes—
10#17985754C2212yes—
11C22×C4⋊C4×C71—21no—
12C22×C4⋊C4×C71C411yescharacteristic
13C22×C4⋊C4×C71C411yescharacteristic
14C22×C42×C71C2211yescharacteristic
15#389×C71—21no—
16#389×C71—21no—
17#389×C71C412yes—
18#389×C71C2212yes—
19C2×#95×C71—21no—
20C2×#11765849—24no—
21C2×#11765849—42no—
22C2×#11765849—42no—
23C22×C71⋊C8—22no—
24C22×C71⋊C8—22no—
25C22⋊(C71⋊C8)—42no—
26C23×C4×C71—21no—
27C23×C4×C71—21no—
28C23×C4×C71C2×C411yescharacteristic
29C23×C4×C71D411yescharacteristic
30C23×C4×C71D411yescharacteristic
31C2×C4⋊C4×C71—21no—
32C2×C4⋊C4×C71—24no—
33C2×C4⋊C4×C71—22no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—42no—
37C2×C4⋊C4×C71—42no—
38C2×C4⋊C4×C71—42no—
39C2×C4⋊C4×C71C2×C412yes—
40C2×C4⋊C4×C71C2×C412yes—
41C2×C42×C71—42no—
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71C2311yescharacteristic
46#95×C71—22no—
47#95×C71—22no—
48#11765849—48no—
49C2×C71⋊C8—24no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—42no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8—42no—
54C2×C71⋊C8—42no—
55C24×C71C22⋊C411yescharacteristic
56C22×C4×C71—22no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71—24no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—41no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71D812yes—
76C22×C4×C71SD1612yes—
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C4⋊C4×C71—24no—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—44no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71M1614yescomplemented
88C42×C71—41no—
89C42×C71—41no—
90C42×C71—41no—
91C42×C71—41no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71#9912yes—
97C23×C71#10212yes—
98C23×C71#10412yes—
99C23×C71C2×C22⋊C411yescharacteristic
100C2×C4×C71—28no—
101C2×C4×C71—24no—
102C2×C4×C71—24no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
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—24no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—44no—
118C2×C4×C71—48no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—84no—
125C2×C4×C71#10112yes—
126C2×C4×C71#10212yes—
127C2×C4×C71C2×M1612yes—
128C2×C4×C71C2×D811yescharacteristic
129C2×C4×C71C2×SD1611yescharacteristic
130C2×C4×C71#13712yes—
131C71⋊C8—44no—
132C71⋊C8—82no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71—22no—
136C22×C71—22no—
137C22×C71—22no—
138C22×C71—22no—
139C22×C71#32714yes—
140C22×C71#40711yescharacteristic
141C22×C71C2×#9911yescharacteristic
142C22×C71C2×#10111yescharacteristic
143C22×C71C2×#10211yescharacteristic
144C22×C71#41911yescharacteristic
145C22×C71C2×#10411yescharacteristic
146C22×C71#42111yescharacteristic
147C4×C71—22no—
148C4×C71—22no—
149C4×C71—21no—
150C4×C71—22no—
151C4×C71—21no—
152C4×C71—22no—
153C4×C71—44no—
154C4×C71—41no—
155C4×C71—82no—
156#7610—711nomaximal, Sylow
157C2×C71—24no—
158C2×C71#146311yescharacteristic
159C2×C71#148611yescharacteristic
160C2×C71#149111yescharacteristic
161C2×C71C2×#32711yescharacteristic
162C2×C71#151111yescharacteristic
163C2×C71#152811yescharacteristic
164C2×C71#154611yescharacteristic
165C2×#389D7111yescharacteristic
166#1817—711no—
167#1728—711no—
168#1271—714no—
169C71#761011yescharacteristic, complemented, Sylow
170C22×M16—1421no—
171C22×C4⋊C4—21no—
172C22×C4⋊C4Dic7111yescharacteristic
173C22×C4⋊C4Dic7111yescharacteristic
174C22×C42C2×D7111yescharacteristic
175C2×C22⋊C8—1421no—
176#389—21no—
177#389—21no—
178#389Dic7112yes—
179#389C2×D7112yes—
180C2×#95—21no—
181#336—712no—
182#336—712no—
183#336—1421no—
184#336—1421no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×Dic7111yescharacteristic
188C23×C4C22⋊D7111yescharacteristic
189C23×C4C22⋊D7111yescharacteristic
190C2×M16—2842no—
191C2×M16—2842no—
192C2×M16—1424no—
193C22×C8—1422no—
194C22×C8—1422no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×Dic7112yes—
204C2×C4⋊C4C2×Dic7112yes—
205C2×C42—42no—
206C2×C42C2×Dic7111yescharacteristic
207C2×C42C22⋊D7111yescharacteristic
208C2×C42C22⋊D7111yescharacteristic
209C2×C42C22×D7111yescharacteristic
210C22⋊C8—2842no—
211#95—22no—
212#95—22no—
213C24C22⋊Dic7111yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊Dic7111yescharacteristic
231C22×C4C22⋊Dic7111yescharacteristic
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4#1176585012yes—
234C22×C4#1176585312yes—
235C22×C4C22×Dic7111yescharacteristic
236C22×C4C2×C22⋊D7111yescharacteristic
237C22×C4C2×C22⋊D7111yescharacteristic
238M16—2848no—
239C2×C8—2842no—
240C2×C8—2842no—
241C2×C8—2842no—
242C2×C8—2842no—
243C2×C8—2842no—
244C2×C8—1424no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4#1176584914yescomplemented
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#1455604312yes—
262C23#1455604912yes—
263C23#1455605512yes—
264C23C2×C22⋊Dic7111yescharacteristic
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4#1455604712yes—
291C2×C4#1455604912yes—
292C2×C4C2×#1176584912yes—
293C2×C4C2×#1176585011yescharacteristic
294C2×C4C2×#1176585311yescharacteristic
295C2×C4#1455615912yes—
296C8—2844no—
297C8—5682no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#1798573314yes—
305C22#1798596311yescharacteristic
306C22C2×#1455604311yescharacteristic
307C22C2×#1455604711yescharacteristic
308C22C2×#1455604911yescharacteristic
309C22#1798600811yescharacteristic
310C22C2×#1455605511yescharacteristic
311C22#1798601611yescharacteristic
312C4—22no—
313C4—22no—
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—44no—
319C4—41no—
320C4—82no—
321C2—24no—
322C2#2258667011yescharacteristic
323C2#2258676511yescharacteristic
324C2#2258678711yescharacteristic
325C2C2×#1798573311yescharacteristic
326C2#2258687311yescharacteristic
327C2#2258694211yescharacteristic
328C2#2258701211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#389×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC2×#389D71
hypocenterC71#7610
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. #22586163C2semidirect
87. C4⋊C4×C71M16semidirect
169. C71#7610semidirect
252. C4⋊C4#11765849semidirect

Maximal subgroups

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

subgroupindex
#225884082
#225880342
#225861632
C2×#389×C712
#761071