Finite Group Databasebeta

← #31906675

Subgroups — #31906675

Order 18176.

454 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#17986064C211yesmaximal, characteristic
2C22⋊C16×D71C212yesmaximal, complemented
3#22586280C211yesmaximal, characteristic, complemented
4#22586263C212yesmaximal, complemented
5C71×#1322C211yesmaximal, characteristic, complemented
6C22×C8×D71C411yescharacteristic
7C22×C8×D71C2211yescharacteristic
8C2×C4×D284C411yescharacteristic
9C2×C16×D71—22no—
10#17986064—22no—
11#17986064—22no—
12C2×(C2×D71)⋊C8C411yescharacteristic
13C2×(C2×D71)⋊C8C2211yescharacteristic
14C2×C4×C71⋊C8C411yescharacteristic
15C22⋊(C71⋊C16)C2212yes—
16(C2×D71)⋊C16—22no—
17C2×C4⋊C8×C71C2211yescharacteristic
18C22⋊C16×C71C2212yes—
19C22×D284C811yescharacteristic
20C22×C4×D71C2×C411yescharacteristic
21C22×C4×D71C2×C411yescharacteristic
22C22×C71⋊C8C2×C411yescharacteristic
23C22×C71⋊C8C2×C411yescharacteristic
24C2×C8×D71—21no—
25C2×C8×D71—21no—
26C2×C8×D71—42no—
27C2×C8×D71—42no—
28C2×C8×D71D412yes—
29C2×C8×D71D412yes—
30C4×D284—22no—
31C4×D284—22no—
32C2×C4⋊Dic71C811yescharacteristic
33C2×(C2×D71)⋊C4C811yescharacteristic
34C2×(C2×D71)⋊C4C811yescharacteristic
35C2×C71⋊C16—22no—
36C16×D71—44no—
37(C2×D71)⋊C8—22no—
38(C2×D71)⋊C8—22no—
39C4×C71⋊C8—21no—
40C4×C71⋊C8—21no—
41C22×C8×C71C2×C411yescharacteristic
42C22×C8×C71C2311yescharacteristic
43C2×C42×C71C2×C411yescharacteristic
44C2×C16×C71—22no—
45C4⋊C8×C71—22no—
46C23×D71C2×C811yescharacteristic
47C23×D71C2×C811yescharacteristic
48C2×D284—22no—
49C2×D284—22no—
50C2×D284C1614yescomplemented
51C22×Dic71C2×C811yescharacteristic
52C22×Dic71C2×C811yescharacteristic
53C2×C4×D71—21no—
54C2×C4×D71—21no—
55C2×C4×D71—41no—
56C2×C4×D71—41no—
57C2×C4×D71—41no—
58C2×C4×D71—41no—
59C2×C4×D71C22⋊C412yes—
60C2×C4×D71M1612yes—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—21no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8C22⋊C411yescharacteristic
65C2×C71⋊C8C22⋊C411yescharacteristic
66C8×D71—24no—
67C8×D71—42no—
68C8×D71—42no—
69C8×D71—42no—
70C4⋊Dic71—22no—
71(C2×D71)⋊C4—22no—
72(C2×D71)⋊C4—22no—
73C71⋊C16—42no—
74C22×C4×C71C2×C811yescharacteristic
75C22×C4×C71C2×C811yescharacteristic
76C22×C4×C71C22×C411yescharacteristic
77C2×C8×C71—21no—
78C2×C8×C71—41no—
79C2×C8×C71—41no—
80C2×C8×C71C2×D412yes—
81C42×C71—21no—
82C42×C71—21no—
83C16×C71—42no—
84C23×C71C22×C811yescharacteristic
85C22×D71—21no—
86C22×D71—21no—
87C22×D71—41no—
88C22×D71—41no—
89C22×D71—41no—
90C22×D71—41no—
91C22×D71C22⋊C812yes—
92C22×D71C2×C1612yes—
93C22×D71C2×C1612yescomplemented
94C22×D71M3212yes—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—41no—
101C2×C4×C71C2×C1612yescomplemented
102C2×C4×C71C2×C22⋊C411yescharacteristic
103C2×C4×C71C2×M1611yescharacteristic
104C2×Dic71—21no—
105C2×Dic71—41no—
106C2×Dic71—41no—
107C2×Dic71C22⋊C812yes—
108D284—28no—
109C4×D71—22no—
110C4×D71—22no—
111C4×D71—42no—
112C4×D71—42no—
113C4×D71—41no—
114C4×D71—41no—
115C8×C71—22no—
116C8×C71—41no—
117C71⋊C8—21no—
118C71⋊C8—41no—
119C71⋊C8#9912yes—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71C2×C22⋊C811yescharacteristic
123C22×C71C22×C1611yescharacteristic
124C22×C71C2×M3211yescharacteristic
125C2×D71—22no—
126C2×D71—22no—
127C2×D71—22no—
128C2×D71—42no—
129C2×D71—42no—
130C2×D71—42no—
131C2×D71—41no—
132C2×D71—41no—
133C2×D71—42no—
134C2×D71C22⋊C1614yes—
135C4×C71—22no—
136C4×C71—21no—
137C4×C71—41no—
138C4×C71C2×#9911yescharacteristic
139C4×C71#41311yescharacteristic
140Dic71—22no—
141Dic71—41no—
142#11465—711nomaximal, Sylow
143C2×C71—21no—
144C2×C71—21no—
145C2×C71#145911yescharacteristic
146C2×C71C2×C22⋊C1611yescharacteristic
147C2×C71#211511yescharacteristic
148D71—24no—
149D71—42no—
150C2×C8×D4—711no—
151C2×C22⋊C16—712no—
152#1322—711no—
153#1322D7111yescharacteristic, complemented
154#1314—712no—
155C71#1146511yescharacteristic, complemented, Sylow
156C23×C8—711no—
157C23×C8—711no—
158C2×C4×D4—711no—
159C22×C16—1422no—
160C8×D4—1422no—
161C8×D4—1422no—
162C2×C4⋊C8C2×D7111yescharacteristic
163C2×C22⋊C8—711no—
164C2×C22⋊C8—711no—
165C2×C4×C8—711no—
166C22⋊C16—712no—
167C22⋊C16—1422no—
168C22⋊C16C2×D7112yes—
169C22×D4—711no—
170C23×C4—711no—
171C23×C4—711no—
172C22×C8—2842no—
173C22×C8—2842no—
174C22×C8—712no—
175C22×C8—711no—
176C22×C8—711no—
177C22×C8—712no—
178C22×C8—1421no—
179C22×C8—1421no—
180C22×C8C4×D7111yescharacteristic
181C22×C8C22×D7111yescharacteristic
182C4×D4—1422no—
183C4×D4—1422no—
184C2×C4⋊C4—711no—
185C2×C22⋊C4—711no—
186C2×C22⋊C4—711no—
187C2×C42C4×D7111yescharacteristic
188C2×C16—22no—
189C2×C16—2844no—
190C2×C16—1422no—
191C4⋊C8—22no—
192C22⋊C8—1422no—
193C22⋊C8—1422no—
194C4×C8—1421no—
195C4×C8—1421no—
196C24—711no—
197C24—711no—
198C2×D4—714no—
199C2×D4—1422no—
200C2×D4—1422no—
201C22×C4—2841no—
202C22×C4—2841no—
203C22×C4—2841no—
204C22×C4—2841no—
205C22×C4—711no—
206C22×C4—712no—
207C22×C4—711no—
208C22×C4—712no—
209C22×C4—1421no—
210C22×C4—1421no—
211C22×C4C8×D7111yescharacteristic
212C22×C4C8×D7111yescharacteristic
213C22×C4C2×C4×D7111yescharacteristic
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—2842no—
218C2×C8—2842no—
219C2×C8—2842no—
220C2×C8—2841no—
221C2×C8—711no—
222C2×C8—711no—
223C2×C8—1424no—
224C2×C8—1421no—
225C2×C8—1422no—
226C2×C8D4×D7112yes—
227C4⋊C4—1422no—
228C22⋊C4—1422no—
229C22⋊C4—1422no—
230C42—21no—
231C42—21no—
232C16—42no—
233C16—2842no—
234C23—2841no—
235C23—2841no—
236C23—2841no—
237C23—2841no—
238C23—712no—
239C23—712no—
240C23—712no—
241C23—712no—
242C23—1421no—
243C23—1421no—
244C23C2×C8×D7111yescharacteristic
245D4—1428no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—2842no—
253C2×C4—2842no—
254C2×C4—2841no—
255C2×C4—2841no—
256C2×C4—2841no—
257C2×C4—2841no—
258C2×C4—712no—
259C2×C4—1421no—
260C2×C4—1422no—
261C2×C4—1422no—
262C2×C4C16×D7112yescomplemented
263C2×C4C22⋊C4×D7111yescharacteristic
264C2×C4M16×D7111yescharacteristic
265C8—22no—
266C8—41no—
267C8—2841no—
268C8—712no—
269C8—1421no—
270C22—21no—
271C22—21no—
272C22—2842no—
273C22—2842no—
274C22—2842no—
275C22—2842no—
276C22—2841no—
277C22—2841no—
278C22—714no—
279C22—1422no—
280C22—1422no—
281C22—1422no—
282C22C22⋊C8×D7111yescharacteristic
283C22C2×C16×D7111yescharacteristic
284C22M32×D7111yescharacteristic
285C4—21no—
286C4—22no—
287C4—41no—
288C4—2841no—
289C4—1422no—
290C4#99×D7111yescharacteristic
291C4#1798598211yescharacteristic
292C2—21no—
293C2—21no—
294C2—2842no—
295C2—1424no—
296C2#2258665711yescharacteristic
297C2C22⋊C16×D7111yescharacteristic
298C2#2258995911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC22×C71C22×C16
FrattiniC22×C8C22×D71
FittingC71×#1322C2
socleC22×C71C2×C22⋊C8
prime radicalC22×D284C8
cosocleC22×C8×C71C23
hypercenter#1322D71
hypocenterC71#11465
LCSG ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D71C2semidirect
3. #22586280C2semidirect
4. #22586263C2semidirect
5. C71×#1322C2semidirect
50. C2×D284C16semidirect
93. C22×D71C2×C16semidirect
101. C2×C4×C71C2×C16semidirect
153. #1322D71semidirect
155. C71#11465semidirect
262. C2×C4C16×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860642
C22⋊C16×D712
#225862802
#225862632
C71×#13222
#1146571