Finite Group Databasebeta

← #31884774

Subgroups — #31884774

Order 18176.

374 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 71 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#22586670C211yesmaximal, characteristic
2#22586608C211yesmaximal, characteristic
3C71×#1463C211yesmaximal, characteristic
4#17985963—21no—
5C2×C71⋊#95—21no—
6C2×C71⋊#95—21no—
7C2×C71⋊#95C411yescharacteristic
8C2×C71⋊#95C411yescharacteristic
9#17985725—21no—
10#17985725C412yescomplemented
11C22×C4⋊C4×C71C2211yescharacteristic
12#407×C71—21no—
13#324×C71—21no—
14#324×C71C412yescomplemented
15C23×Dic71—21no—
16C23×Dic71—21no—
17C2×#11765849—41no—
18C22⋊(C71⋊C8)—22no—
19C22⋊(C71⋊C8)—41no—
20C71⋊#95—41no—
21C71⋊#95—41no—
22C71⋊#95—42no—
23C71⋊#95—42no—
24C23×C4×C71C2×C411yescharacteristic
25C23×C4×C71D411yescharacteristic
26C23×C4×C71D411yescharacteristic
27C2×M16×C71—41no—
28C2×C4⋊C4×C71—41no—
29C2×C4⋊C4×C71—41no—
30C2×C4⋊C4×C71C2×C411yescharacteristic
31C2×C4⋊C4×C71C2×C411yescharacteristic
32C2×C4⋊C4×C71D411yescharacteristic
33C2×C4⋊C4×C71Q811yescharacteristic
34C22⋊C8×C71—22no—
35C22⋊C8×C71—41no—
36C22×Dic71—22no—
37C22×Dic71—22no—
38C22×Dic71—41no—
39C22×Dic71—41no—
40C22×Dic71—41no—
41C22×Dic71—41no—
42C22×Dic71—41no—
43C22×Dic71—41no—
44C22×Dic71—41no—
45C22×Dic71—41no—
46C22×Dic71—41no—
47C22×Dic71—41no—
48C22×Dic71—41no—
49C22×Dic71—41no—
50#11765849—81no—
51#11765849—81no—
52C2×C71⋊C8—42no—
53C24×C71C22⋊C411yescharacteristic
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—21no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71—41no—
61C22×C4×C71—41no—
62C22×C4×C71—41no—
63C22×C4×C71—41no—
64C22×C4×C71—41no—
65C22×C4×C71—41no—
66C22×C4×C71C4211yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C4⋊C411yescharacteristic
70C22×C4×C71C4⋊C411yescharacteristic
71C22×C4×C71C4⋊C411yescharacteristic
72M16×C71—81no—
73M16×C71—81no—
74C2×C8×C71—42no—
75C4⋊C4×C71—41no—
76C4⋊C4×C71—41no—
77C4⋊C4×C71—41no—
78C4⋊C4×C71—41no—
79C23×C71—21no—
80C23×C71—21no—
81C23×C71—21no—
82C23×C71—21no—
83C23×C71—21no—
84C23×C71—21no—
85C23×C71#9511yescharacteristic
86C23×C71#9911yescharacteristic
87C23×C71#9911yescharacteristic
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—81no—
109C2×C4×C71—81no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C2×Dic71—81no—
117C2×Dic71—81no—
118C2×Dic71—81no—
119C2×Dic71—81no—
120C2×Dic71—81no—
121C2×Dic71—81no—
122C2×Dic71—81no—
123C2×Dic71—81no—
124C8×C71—82no—
125C71⋊C8—82no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71#34211yescharacteristic
141C22×C71#34411yescharacteristic
142C22×C71#34411yescharacteristic
143C22×C71#35311yescharacteristic
144C22×C71#35411yescharacteristic
145C22×C71#35511yescharacteristic
146C22×C71#35611yescharacteristic
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—41no—
152C4×C71—41no—
153Dic71—82no—
154Dic71—82no—
155#7326—711nomaximal, Sylow
156C2×C71—21no—
157C2×C71—21no—
158C2×C71—21no—
159C2×C71—21no—
160C2×C71—21no—
161C2×C71—21no—
162C2×C71#130611yescharacteristic
163C2×C71#139311yescharacteristic
164C2×C71#139411yescharacteristic
165#1463—711no—
166#1463D7111yescharacteristic
167#1445—711no—
168C71#732611yescharacteristic, complemented, Sylow
169C22×C4⋊C4C2×D7111yescharacteristic
170#407—21no—
171#407—1421no—
172C2×#95—711no—
173C2×#95—711no—
174C2×#95—1421no—
175C2×#95—1421no—
176#324—21no—
177#324—712no—
178#324—1421no—
179#324Dic7112yescomplemented
180C23×C4—1421no—
181C23×C4—1421no—
182C23×C4C4×D7111yescharacteristic
183C23×C4D28411yescharacteristic
184C23×C4C22⋊D7111yescharacteristic
185C2×M16—41no—
186C2×M16—2841no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4Dic14211yescharacteristic
190C2×C4⋊C4C4×D7111yescharacteristic
191C2×C4⋊C4C2×Dic7111yescharacteristic
192C2×C4⋊C4C22⋊D7111yescharacteristic
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—2841no—
196C22⋊C8—1422no—
197#95—2841no—
198#95—2841no—
199#95—2842no—
200#95—2842no—
201C24(C2×D71)⋊C411yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—2841no—
215C22×C4—2841no—
216C22×C4—2841no—
217C22×C4—2841no—
218C22×C4—2841no—
219C22×C4—2841no—
220C22×C4—2841no—
221C22×C4—2841no—
222C22×C4—2841no—
223C22×C4—2841no—
224C22×C4—2841no—
225C22×C4—2841no—
226C22×C4—1422no—
227C22×C4—1422no—
228C22×C4C4×Dic7111yescharacteristic
229C22×C4(C2×D71)⋊C411yescharacteristic
230C22×C4C22⋊Dic7111yescharacteristic
231C22×C4Dic71⋊C411yescharacteristic
232C22×C4Dic71⋊C411yescharacteristic
233C22×C4C4⋊Dic7111yescharacteristic
234M16—81no—
235M16—81no—
236M16—5681no—
237M16—5681no—
238C2×C8—42no—
239C2×C8—2842no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23C71⋊#9511yescharacteristic
251C23#1455604211yescharacteristic
252C23#1455604211yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—2842no—
276C2×C4—2842no—
277C2×C4—2842no—
278C2×C4—2842no—
279C2×C4—2842no—
280C2×C4—2842no—
281C2×C4—5681no—
282C2×C4—5681no—
283C2×C4—5681no—
284C2×C4—5681no—
285C2×C4—5681no—
286C2×C4—5681no—
287C2×C4—5681no—
288C2×C4—5681no—
289C8—82no—
290C8—5682no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1798576511yescharacteristic
306C22#1798577011yescharacteristic
307C22#1798577111yescharacteristic
308C22#1798579111yescharacteristic
309C22#1798579311yescharacteristic
310C22#1798579511yescharacteristic
311C22#1798579711yescharacteristic
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—5682no—
319C4—5682no—
320C2—21no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2#2258624911yescharacteristic
327C2#2258646411yescharacteristic
328C2#2258646611yescharacteristic
subgroupG/H
centerC22#17985765
commutatorC22×C4×C71C42
FrattiniC22×C4⋊C4C2×D71
FittingC71×#1463C2
socleC22×C71#342
prime radicalC24×C71C22⋊C4
cosocleC22×C4⋊C4×C71C22
hypercenter#1463D71
hypocenterC71#7326
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
10. #17985725C4semidirect
14. #324×C71C4semidirect
168. C71#7326semidirect
179. #324Dic71semidirect

Maximal subgroups

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

subgroupindex
#225866702
#225866082
C71×#14632
#732671