Finite Group Databasebeta

← #31908152

Subgroups — #31908152

Order 18176.

488 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588423C211yesmaximal, characteristic, complemented
2#22586674C211yesmaximal, characteristic, complemented
3#22586343C214yesmaximal, complemented
4C71×#2619C211yesmaximal, characteristic
5#17986056—21no—
6#17985963—21no—
7#17985768—21no—
8#17985768C2212yes—
9#17985725C2212yes—
10#17985723—21no—
11C2×#127×C71C411yescharacteristic
12C2×#121×C71—21no—
13C2×#117×C71C2211yescharacteristic
14#394×C71—21no—
15#394×C71C2212yes—
16#392×C71—21no—
17#392×C71C412yes—
18#390×C71—21no—
19#382×C71C411yescharacteristic
20C2×#11765849—22no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—41no—
24C22⋊(C71⋊C8)—41no—
25#14556039—41no—
26C22×D4×C71—22no—
27C22×D4×C71—21no—
28C22×D4×C71—21no—
29C22×D4×C71C2×C412yes—
30C23×C4×C71D411yescharacteristic
31#127×C71—41no—
32#127×C71—41no—
33#121×C71—44no—
34#117×C71—41no—
35#117×C71—41no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C4⋊C4×C71C2311yescharacteristic
39C2×C22⋊C4×C71—22no—
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71C2×C411yescharacteristic
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46#95×C71—21no—
47#95×C71C2×C412yes—
48#11765849—42no—
49#11765849—42no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—42no—
52C24×C71—22no—
53C24×C71C22⋊C411yescharacteristic
54C2×D4×C71—42no—
55C2×D4×C71—42no—
56C2×D4×C71—42no—
57C2×D4×C71—42no—
58C2×D4×C71—44no—
59C2×D4×C71—41no—
60C2×D4×C71—41no—
61C2×D4×C71—82no—
62C2×D4×C71—82no—
63C2×D4×C71—82no—
64C2×D4×C71—82no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—41no—
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C4⋊C4×C71—41no—
77C4⋊C4×C71—41no—
78C4⋊C4×C71—42no—
79C22⋊C4×C71—41no—
80C22⋊C4×C71—41no—
81C22⋊C4×C71—44no—
82C22⋊C4×C71—82no—
83C22⋊C4×C71—82no—
84C42×C71—21no—
85C42×C71—21no—
86C42×C71—21no—
87C42×C71—21no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71—42no—
94C23×C71—42no—
95C23×C71—42no—
96C23×C71—42no—
97C23×C71—42no—
98C23×C71—42no—
99C23×C71—42no—
100C23×C71C2×C22⋊C411yescharacteristic
101D4×C71—82no—
102D4×C71—82no—
103D4×C71—84no—
104D4×C71—84no—
105D4×C71—88no—
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—42no—
115C2×C4×C71—42no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71#9911yescharacteristic
124C2×C4×C71#9911yescharacteristic
125C2×C4×C71AGL(1,ℤ/8)12yes—
126C71⋊C8—82no—
127C71⋊C8—82no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—42no—
131C22×C71—41no—
132C22×C71—41no—
133C22×C71—41no—
134C22×C71—81no—
135C22×C71—82no—
136C22×C71—82no—
137C22×C71—82no—
138C22×C71—82no—
139C22×C71—82no—
140C22×C71—82no—
141C22×C71—82no—
142C22×C71C2×#9911yescharacteristic
143C22×C71#41811yescharacteristic
144C22×C71#42111yescharacteristic
145C4×C71—22no—
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—41no—
152C4×C71—82no—
153#11806—711nomaximal, Sylow
154C2×C71—21no—
155C2×C71—21no—
156C2×C71—81no—
157C2×C71—162no—
158C2×C71#151011yescharacteristic
159C2×C71#212211yescharacteristic
160C2×C71#213411yescharacteristic
161#2619D7111yescharacteristic
162#1822—711no—
163#1464—711no—
164#1345—714no—
165C71#1180611yescharacteristic, complemented, Sylow
166C2×#127Dic7111yescharacteristic
167C2×#121—21no—
168C2×#117C2×D7111yescharacteristic
169#432—1421no—
170#407—1421no—
171#394—21no—
172#394C2×D7112yes—
173#392—21no—
174#392Dic7112yes—
175#390—21no—
176#382Dic7111yescharacteristic
177#343—712no—
178#343—1421no—
179#324—712no—
180#323—1421no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C2×Dic7112yes—
185C23×C4C22⋊D7111yescharacteristic
186C2×M16—1422no—
187#127—41no—
188#127—41no—
189#121—44no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22⋊D7111yescharacteristic
194C2×C4⋊C4C22×D7111yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C2×Dic7111yescharacteristic
199C2×C22⋊C4C22⋊D7111yescharacteristic
200C2×C42C2×Dic7111yescharacteristic
201C2×C42C22⋊D7111yescharacteristic
202C4⋊C8—2841no—
203C22⋊C8—2841no—
204C22⋊C8—2841no—
205C22⋊C8—1422no—
206#97—2841no—
207#95—21no—
208#95C2×Dic7112yes—
209C24—22no—
210C24C22⋊Dic7111yescharacteristic
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4C22⋊Dic7111yescharacteristic
228C22×C4C22⋊Dic7111yescharacteristic
229C22×C4C22⋊Dic7111yescharacteristic
230C22×C4C22×Dic7111yescharacteristic
231C22×C4C2×C22⋊D7111yescharacteristic
232C22×C4C2×C22⋊D7111yescharacteristic
233M16—2842no—
234M16—2842no—
235C2×C8—2842no—
236C2×C8—2842no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—44no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23C2×C22⋊Dic7111yescharacteristic
262D4—82no—
263D4—82no—
264D4—84no—
265D4—84no—
266D4—88no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4#1455604311yescharacteristic
285C2×C4#1455604311yescharacteristic
286C2×C4#1455615412yes—
287C8—5682no—
288C8—5682no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22C2×#1455604311yescharacteristic
304C22#1798600411yescharacteristic
305C22#1798601611yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—162no—
318C2#2258686911yescharacteristic
319C2#2258999011yescharacteristic
320C2#2259003211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
Fitting, prime radicalC71×#2619C2
socleC22×C71C2×#99
cosocleC2×C4⋊C4×C71C23
hypercenter#2619D71
hypocenterC71#11806
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #22588423C2semidirect
2. #22586674C2semidirect
3. #22586343C2semidirect
165. C71#11806semidirect

Maximal subgroups

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

subgroupindex
#225884232
#225866742
#225863432
C71×#26192
#1180671