Finite Group Databasebeta

← #31906584

Subgroups — #31906584

Order 18176.

450 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 105 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×C4⋊C8×D71C211yesmaximal, characteristic
2#22589949C212yesmaximal
3#22586280C211yesmaximal, characteristic, complemented
4#22586279C212yesmaximal, complemented
5C71×#1322C211yesmaximal, characteristic, complemented
6C22×C8×D71C411yescharacteristic
7C22×C8×D71C2211yescharacteristic
8C2×C42×D71C411yescharacteristic
9C2×#14556067—22no—
10C2×C4⋊(C71⋊C8)C411yescharacteristic
11C2×Dic71⋊C8C411yescharacteristic
12C2×Dic71⋊C8C2211yescharacteristic
13C4⋊C8×D71—22no—
14C4⋊C8×D71—22no—
15C22⋊(C71⋊C16)C2212yes—
16(C2×D71)⋊C16—22no—
17C2×C4⋊C8×C71C2211yescharacteristic
18C22⋊C16×C71C2212yes—
19C22×C4×D71C811yescharacteristic
20C22×C4×D71C811yescharacteristic
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—
30C2×C4×Dic71C811yescharacteristic
31C2×C4×Dic71C811yescharacteristic
32C2×C4×Dic71C2×C411yescharacteristic
33C42×D71—21no—
34C42×D71—21no—
35C42×D71—21no—
36C42×D71—21no—
37#14556067—44no—
38C2×C71⋊C16—22no—
39C4⋊(C71⋊C8)—22no—
40Dic71⋊C8—22no—
41Dic71⋊C8—22no—
42C22×C8×C71C2×C411yescharacteristic
43C22×C8×C71C2311yescharacteristic
44C2×C42×C71C2×C411yescharacteristic
45C2×C16×C71—22no—
46C4⋊C8×C71—22no—
47C23×D71C2×C811yescharacteristic
48C22×Dic71C2×C811yescharacteristic
49C22×Dic71C2×C811yescharacteristic
50C22×Dic71C2×C811yescharacteristic
51C2×C4×D71—21no—
52C2×C4×D71—21no—
53C2×C4×D71—22no—
54C2×C4×D71—22no—
55C2×C4×D71—22no—
56C2×C4×D71—22no—
57C2×C4×D71—21no—
58C2×C4×D71—21no—
59C2×C4×D71—42no—
60C2×C4×D71C22⋊C411yescharacteristic
61C2×C4×D71C22⋊C411yescharacteristic
62C2×C4×D71M1611yescharacteristic
63C2×C4×D71M1611yescharacteristic
64C2×C71⋊C8—21no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8C22⋊C412yes—
68C8×D71—24no—
69C8×D71—42no—
70C8×D71—42no—
71C8×D71—42no—
72C4×Dic71—21no—
73C4×Dic71—21no—
74C4×Dic71—21no—
75C4×Dic71—21no—
76C4×Dic71—21no—
77C4×Dic71—21no—
78C71⋊C16—42no—
79C22×C4×C71C2×C811yescharacteristic
80C22×C4×C71C2×C811yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C2×C8×C71—21no—
83C2×C8×C71—41no—
84C2×C8×C71—41no—
85C2×C8×C71C2×D412yes—
86C42×C71—21no—
87C42×C71—21no—
88C16×C71—42no—
89C23×C71C22×C811yescharacteristic
90C22×D71—21no—
91C22×D71—21no—
92C22×D71—22no—
93C22×D71—22no—
94C22×D71C22⋊C811yescharacteristic
95C22×D71C22⋊C811yescharacteristic
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—41no—
102C2×C4×C71M3212yescomplemented
103C2×C4×C71C2×C22⋊C411yescharacteristic
104C2×C4×C71C2×M1611yescharacteristic
105C2×Dic71—21no—
106C2×Dic71—21no—
107C2×Dic71—22no—
108C2×Dic71—21no—
109C2×Dic71—41no—
110C2×Dic71C22⋊C811yescharacteristic
111C2×Dic71C22⋊C811yescharacteristic
112C2×Dic71M3212yescomplemented
113C4×D71—24no—
114C4×D71—24no—
115C4×D71—22no—
116C4×D71—21no—
117C4×D71—21no—
118C4×D71—22no—
119C4×D71—42no—
120C4×D71—42no—
121C4×D71#9912yes—
122C4×D71#10112yes—
123C8×C71—22no—
124C8×C71—41no—
125C71⋊C8—22no—
126C71⋊C8—41no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71C2×C22⋊C811yescharacteristic
130C22×C71C2×M3211yescharacteristic
131C22×C71C2×M3211yescharacteristic
132C2×D71—24no—
133C2×D71—24no—
134C2×D71—21no—
135C2×D71—21no—
136C2×D71—21no—
137C2×D71—21no—
138C2×D71#32412yes—
139C2×D71#34912yes—
140C4×C71—22no—
141C4×C71—21no—
142C4×C71—41no—
143C4×C71C2×#9911yescharacteristic
144C4×C71C2×#10111yescharacteristic
145Dic71—22no—
146Dic71—21no—
147Dic71—41no—
148Dic71#32412yes—
149#11440—711nomaximal, Sylow
150C2×C71—21no—
151C2×C71—21no—
152C2×C71C2×#32411yescharacteristic
153C2×C71#211211yescharacteristic
154C2×C71C2×#34911yescharacteristic
155D71—22no—
156D71—22no—
157C22×C4⋊C8—711no—
158#2112—712no—
159#1322—711no—
160#1322—712no—
161#1322D7111yescharacteristic, complemented
162C71#1144011yescharacteristic, complemented, Sylow
163C23×C8—711no—
164C23×C8—711no—
165C22×C42—711no—
166C2×M32—1422no—
167C2×C4⋊C8—711no—
168C2×C4⋊C8—711no—
169C2×C4⋊C8—711no—
170C2×C4⋊C8—1422no—
171C2×C4⋊C8—1422no—
172C2×C4⋊C8C2×D7111yescharacteristic
173C22⋊C16—712no—
174C22⋊C16—1422no—
175C22⋊C16C2×D7112yes—
176C23×C4—711no—
177C23×C4—711no—
178C23×C4—711no—
179C22×C8—2842no—
180C22×C8—2842no—
181C22×C8—711no—
182C22×C8—712no—
183C22×C8—711no—
184C22×C8—712no—
185C22×C8—1421no—
186C22×C8—1421no—
187C22×C8C4×D7111yescharacteristic
188C22×C8C22×D7111yescharacteristic
189C2×C42—711no—
190C2×C42—711no—
191C2×C42—711no—
192C2×C42—1421no—
193C2×C42—1421no—
194C2×C42—1421no—
195C2×C42—1421no—
196C2×C42C4×D7111yescharacteristic
197M32—2844no—
198C2×C16—22no—
199C2×C16—1422no—
200C4⋊C8—22no—
201C4⋊C8—1422no—
202C4⋊C8—1422no—
203C4⋊C8—1422no—
204C24—711no—
205C22×C4—2842no—
206C22×C4—711no—
207C22×C4—711no—
208C22×C4—711no—
209C22×C4—711no—
210C22×C4—711no—
211C22×C4—711no—
212C22×C4—711no—
213C22×C4—1421no—
214C22×C4—1421no—
215C22×C4—1422no—
216C22×C4—1422no—
217C22×C4—1422no—
218C22×C4—1421no—
219C22×C4—1421no—
220C22×C4—1422no—
221C22×C4C8×D7111yescharacteristic
222C22×C4C8×D7111yescharacteristic
223C22×C4C2×C4×D7111yescharacteristic
224C2×C8—21no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—2842no—
228C2×C8—2842no—
229C2×C8—2842no—
230C2×C8—2841no—
231C2×C8—2841no—
232C2×C8—712no—
233C2×C8—1424no—
234C2×C8—1421no—
235C2×C8D4×D7112yes—
236C42—21no—
237C42—21no—
238C42—1421no—
239C42—1421no—
240C42—1421no—
241C42—1421no—
242C42—1421no—
243C42—1421no—
244C16—42no—
245C16—2842no—
246C23—711no—
247C23—711no—
248C23—1422no—
249C23—1422no—
250C23—1421no—
251C23—1421no—
252C23C2×C8×D7111yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—2842no—
260C2×C4—2842no—
261C2×C4—2841no—
262C2×C4—711no—
263C2×C4—712no—
264C2×C4—712no—
265C2×C4—712no—
266C2×C4—711no—
267C2×C4—1422no—
268C2×C4—1422no—
269C2×C4—1424no—
270C2×C4—1424no—
271C2×C4—1421no—
272C2×C4—1422no—
273C2×C4—1421no—
274C2×C4—1421no—
275C2×C4—1421no—
276C2×C4—1421no—
277C2×C4#1455606712yescomplemented
278C2×C4C22⋊C4×D7111yescharacteristic
279C2×C4M16×D7111yescharacteristic
280C8—22no—
281C8—41no—
282C8—2841no—
283C8—1422no—
284C22—21no—
285C22—21no—
286C22—712no—
287C22—712no—
288C22—1424no—
289C22—1424no—
290C22—1421no—
291C22—1421no—
292C22—1421no—
293C22—1421no—
294C22C22⋊C8×D7111yescharacteristic
295C22M32×D7111yescharacteristic
296C22C2×#1455606711yescharacteristic
297C4—21no—
298C4—22no—
299C4—41no—
300C4—2841no—
301C4—712no—
302C4—1421no—
303C4—1422no—
304C4#99×D7111yescharacteristic
305C4#101×D7111yescharacteristic
306C2—21no—
307C2—21no—
308C2—1422no—
309C2—1422no—
310C2#324×D7111yescharacteristic
311C2#2258994911yescharacteristic
312C2#349×D7111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC23×C71C22×C8
FrattiniC22×C8C22×D71
FittingC71×#1322C2
socleC22×C71C2×C22⋊C8
prime radicalC23×D71C2×C8
cosocleC22×C8×C71C23
hypercenter#1322D71
hypocenterC71#11440
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
3. #22586280C2semidirect
4. #22586279C2semidirect
5. C71×#1322C2semidirect
102. C2×C4×C71M32semidirect
112. C2×Dic71M32semidirect
161. #1322D71semidirect
162. C71#11440semidirect
277. C2×C4#14556067semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C8×D712
#225899492
#225862802
#225862792
C71×#13222
#1144071