Finite Group Databasebeta

← #31907768

Subgroups — #31907768

Order 18176.

432 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 107 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#22594161C211yesmaximal, characteristic
2#22588079C211yesmaximal, characteristic
3#22586368C214yesmaximal, complemented
4C71×#1458C211yesmaximal, characteristic
5C22×#11765849—21no—
6C2×#14556128C411yescharacteristic
7#17985938C2212yes—
8#17985935C412yes—
9#17985919—21no—
10#17985911—21no—
11#17985897—21no—
12#17985865C411yescharacteristic
13#17985859—21no—
14#17985768—21no—
15#17985768C2212yes—
16C2×#117×C71C2211yescharacteristic
17#431×C71—21no—
18#324×C71C2212yes—
19#323×C71—21no—
20C22×Dic142—21no—
21C2×#11765849—24no—
22C2×#11765849—22no—
23C2×#11765849—41no—
24C2×#11765849—42no—
25C22×C71⋊C8—41no—
26#14556128—41no—
27#14556128—41no—
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71—21no—
30C2×C4⋊Dic71C2×C412yes—
31C2×Dic71⋊C4—22no—
32C2×C22⋊Dic71—22no—
33C71⋊#95—21no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95C2×C412yes—
37C23×C4×C71D411yescharacteristic
38#117×C71—41no—
39#117×C71—41no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C22⋊C4×C71C2×C411yescharacteristic
43C2×C22⋊C4×C71D411yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C4⋊C8×C71—41no—
47C22⋊C8×C71—22no—
48C4×C8×C71—41no—
49C2×Dic142—42no—
50C2×Dic142—82no—
51C22×Dic71—22no—
52C22×Dic71—22no—
53#11765849—24no—
54#11765849—44no—
55#11765849—42no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—44no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60C4⋊Dic71—42no—
61C4⋊Dic71—42no—
62Dic71⋊C4—82no—
63C22⋊Dic71—82no—
64C24×C71C22⋊C411yescharacteristic
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71SD1612yes—
72C22×C4×C71Q1612yes—
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C2×C8×C71—42no—
77C4⋊C4×C71—41no—
78C4⋊C4×C71—41no—
79C22⋊C4×C71—41no—
80C22⋊C4×C71—41no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—21no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71—41no—
88C23×C71—41no—
89C23×C71#10312yes—
90C23×C71C2×C22⋊C411yescharacteristic
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71#9911yescharacteristic
106C2×C4×C71#9911yescharacteristic
107C2×C4×C71#10312yes—
108C2×C4×C71#10412yes—
109C2×C4×C71C2×SD1611yescharacteristic
110C2×C4×C71C2×Q1611yescharacteristic
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C2×Dic71—42no—
117Dic142—84no—
118C8×C71—82no—
119C71⋊C8—44no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—22no—
123C22×C71—42no—
124C22×C71—41no—
125C22×C71—41no—
126C22×C71—81no—
127C22×C71C2×#9911yescharacteristic
128C22×C71C2×#10311yescharacteristic
129C22×C71C2×#10411yescharacteristic
130C4×C71—21no—
131C4×C71—21no—
132C4×C71—41no—
133C4×C71—41no—
134C4×C71—41no—
135C4×C71—41no—
136C4×C71#32812yes—
137C4×C71#32812yes—
138Dic71—82no—
139Dic71—82no—
140#11723—711nomaximal, Sylow
141C2×C71—21no—
142C2×C71—21no—
143C2×C71—42no—
144C2×C71C2×#32811yescharacteristic
145C2×C71#212311yescharacteristic
146C2×C71#213411yescharacteristic
147#2628—711no—
148#1742—711no—
149#1458D7111yescharacteristic
150#1353—714no—
151C71#1172311yescharacteristic, complemented, Sylow
152C22×M16—1421no—
153C2×C4⋊Q8—711no—
154C2×#117C2×D7111yescharacteristic
155#431—21no—
156#400—712no—
157#399—712no—
158#395—1421no—
159#393—1421no—
160#389—1421no—
161#382—711no—
162#380—1421no—
163#343—712no—
164#343—1421no—
165#324C2×D7112yes—
166#323—21no—
167C22×Q8—1421no—
168C23×C4D28411yescharacteristic
169C2×M16—2841no—
170C2×M16—2842no—
171C2×M16—1424no—
172C2×M16—1422no—
173C22×C8—2841no—
174C4⋊Q8—2841no—
175C4⋊Q8—2841no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—712no—
179C2×C4⋊C4—1421no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4—1422no—
182C2×C4⋊C4D28411yescharacteristic
183C2×C4⋊C4C22×D7111yescharacteristic
184C2×C22⋊C4—1422no—
185C2×C22⋊C4C4×D7111yescharacteristic
186C2×C22⋊C4C22⋊D7111yescharacteristic
187C2×C42C4×D7111yescharacteristic
188C2×C42C22⋊D7111yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191C4×C8—41no—
192#95—712no—
193#95—1421no—
194#95—1422no—
195#95—1422no—
196C24(C2×D71)⋊C411yescharacteristic
197C2×Q8—2842no—
198C2×Q8—5682no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—1422no—
203C22×C4—1422no—
204C22×C4(C2×D71)⋊C411yescharacteristic
205C22×C4(C2×D71)⋊C411yescharacteristic
206C22×C4(C2×D71)⋊C411yescharacteristic
207C22×C4#1176585412yes—
208C22×C4Dic28412yes—
209C22×C4C2×C4×D7111yescharacteristic
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4C2×D28411yescharacteristic
212M16—2844no—
213M16—2842no—
214M16—1424no—
215C2×C8—42no—
216C2×C8—2842no—
217C2×C8—2844no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—2842no—
221C4⋊C4—2841no—
222C4⋊C4—2841no—
223C4⋊C4—2842no—
224C4⋊C4—5682no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—5682no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1455605312yes—
237C23C2×(C2×D71)⋊C411yescharacteristic
238Q8—5684no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—2842no—
254C2×C4—2842no—
255C2×C4—2842no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—2842no—
259C2×C4#1455604211yescharacteristic
260C2×C4#1455604211yescharacteristic
261C2×C4#1455605312yes—
262C2×C4#1455605412yes—
263C2×C4C2×#1176585411yescharacteristic
264C2×C4C2×Dic28411yescharacteristic
265C8—82no—
266C8—2844no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1455604211yescharacteristic
275C22C2×#1455605311yescharacteristic
276C22C2×#1455605411yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—5682no—
284C4—5682no—
285C4#1798573712yes—
286C4#1798573712yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1798573711yescharacteristic
291C2#2258999311yescharacteristic
292C2#2259003111yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1458C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1458D71
hypocenterC71#11723
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #22586368C2semidirect
151. C71#11723semidirect

Maximal subgroups

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

subgroupindex
#225941612
#225880792
#225863682
C71×#14582
#1172371