Finite Group Databasebeta

← #31913712

Subgroups — #31913712

Order 18176.

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

normal subgroups only

This group has 786 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#22588255C212yesmaximal
2#22586465C214yesmaximal, complemented
3C71×#2309C211yesmaximal, characteristic
4C2×#14556043—22no—
5C2×#14556043C2214yes—
6#17985855—22no—
7C23×D4×C71C2211yescharacteristic
8C2×#117×C71C411yescharacteristic, complemented
9C2×#117×C71C411yescharacteristic, complemented
10#390×C71—22no—
11#390×C71C412yescomplemented
12#390×C71C2212yes—
13#379×C71—22no—
14C2×C22⋊Dic71—24no—
15C2×C22⋊Dic71—44no—
16#14556043—28no—
17#14556043—44no—
18C25×C71D412yes—
19C22×D4×C71—24no—
20C22×D4×C71—42no—
21C22×D4×C71—42no—
22C22×D4×C71C2×C411yescharacteristic
23C22×D4×C71D412yes—
24C22×D4×C71C2311yescharacteristic
25C23×C4×C71C2×C411yescharacteristic
26#117×C71—42no—
27#117×C71—42no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71C2×C412yes—
30C2×C22⋊C4×C71—21no—
31C2×C22⋊C4×C71—21no—
32C2×C22⋊C4×C71—24no—
33C2×C22⋊C4×C71—24no—
34C2×C42×C71—21no—
35C2×C42×C71C2×C412yes—
36C22×Dic71—44no—
37C22⋊Dic71—44no—
38C22⋊Dic71—44no—
39C22⋊Dic71—88no—
40C24×C71—24no—
41C24×C71—24no—
42C24×C71—42no—
43C24×C71—84no—
44C24×C71C22⋊C412yes—
45C24×C71C22⋊C411yescharacteristic
46C24×C71C2×D412yes—
47C2×D4×C71—24no—
48C2×D4×C71—24no—
49C2×D4×C71—22no—
50C2×D4×C71—41no—
51C2×D4×C71—41no—
52C2×D4×C71—42no—
53C2×D4×C71—41no—
54C2×D4×C71—41no—
55C2×D4×C71—416no—
56C2×D4×C71#4114yes—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71C22⋊C411yescharacteristic
63C22×C4×C71C22×C411yescharacteristic
64C4⋊C4×C71—42no—
65C4⋊C4×C71—42no—
66C22⋊C4×C71—42no—
67C22⋊C4×C71—42no—
68C22⋊C4×C71—84no—
69C22⋊C4×C71—84no—
70C42×C71—42no—
71C42×C71—42no—
72C23×C71—21no—
73C23×C71—21no—
74C23×C71—22no—
75C23×C71—22no—
76C23×C71—22no—
77C23×C71—22no—
78C23×C71—22no—
79C23×C71—42no—
80C23×C71—48no—
81C23×C71—44no—
82C23×C71—44no—
83C23×C71—48no—
84C23×C71—84no—
85C23×C71—88no—
86C23×C71—88no—
87C23×C71#9912yes—
88C23×C71C2×C22⋊C411yescharacteristic
89C23×C71#12314yes—
90D4×C71—44no—
91D4×C71—48no—
92D4×C71—44no—
93C2×C4×C71—28no—
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—44no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71#9912yes—
104C2×C4×C71#11712yes—
105C2×Dic71—84no—
106C2×Dic71—88no—
107C22×C71—22no—
108C22×C71—24no—
109C22×C71—22no—
110C22×C71—24no—
111C22×C71—44no—
112C22×C71—42no—
113C22×C71—42no—
114C22×C71—44no—
115C22×C71—44no—
116C22×C71—42no—
117C22×C71—44no—
118C22×C71—82no—
119C22×C71—82no—
120C22×C71—816no—
121C22×C71#38111yescharacteristic
122C22×C71C2×#9911yescharacteristic
123C22×C71C2×#9911yescharacteristic
124C4×C71—22no—
125C4×C71—22no—
126C4×C71—82no—
127C4×C71—82no—
128Dic71—164no—
129#12858—711nomaximal, Sylow
130C2×C71—21no—
131C2×C71—21no—
132C2×C71—44no—
133C2×C71—42no—
134C2×C71—82no—
135C2×C71#181911yescharacteristic
136C2×C71#212712yes—
137#2309D7111yescharacteristic
138#1781—712no—
139#1393—714no—
140C71#1285811yescharacteristic, complemented, Sylow
141C23×D4C2×D7111yescharacteristic
142C2×#117Dic7111yescharacteristic, complemented
143C2×#117Dic7111yescharacteristic, complemented
144C2×#99—714no—
145C2×#99—1422no—
146#390—22no—
147#390Dic7112yescomplemented
148#390C2×D7112yes—
149#379—22no—
150#379—1422no—
151C25C22⋊D7112yes—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C2×Dic7111yescharacteristic
156C22×D4C22⋊D7112yes—
157C22×D4C22×D7111yescharacteristic
158C23×C4C2×Dic7111yescharacteristic
159#117—42no—
160#117—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic7112yes—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—2844no—
168C2×C22⋊C4—1424no—
169C2×C42—21no—
170C2×C42C2×Dic7112yes—
171#99—2844no—
172#99—1428no—
173C24—24no—
174C24—24no—
175C24—42no—
176C24—84no—
177C24C22⋊Dic7112yes—
178C24C22⋊Dic7111yescharacteristic
179C24C2×C22⋊D7112yes—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—416no—
189C2×D4#1176586514yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—2844no—
196C22×C4C22⋊Dic7111yescharacteristic
197C22×C4C22×Dic7111yescharacteristic
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—84no—
203C22⋊C4—84no—
204C22⋊C4—2844no—
205C22⋊C4—2844no—
206C22⋊C4—5688no—
207C42—42no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—42no—
217C23—48no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—84no—
222C23—88no—
223C23—88no—
224C23#1455604312yes—
225C23C2×C22⋊Dic7111yescharacteristic
226C23#1455611114yes—
227D4—44no—
228D4—48no—
229D4—44no—
230C2×C4—28no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—5684no—
241C2×C4—5688no—
242C2×C4#1455604312yes—
243C2×C4#1455608712yes—
244C22—22no—
245C22—24no—
246C22—22no—
247C22—24no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22—44no—
252C22—44no—
253C22—44no—
254C22—42no—
255C22—82no—
256C22—82no—
257C22—816no—
258C22#1798586211yescharacteristic
259C22C2×#1455604311yescharacteristic
260C22C2×#1455604311yescharacteristic
261C4—22no—
262C4—22no—
263C4—82no—
264C4—82no—
265C4—11364no—
266C2—21no—
267C2—21no—
268C2—44no—
269C2—42no—
270C2—82no—
271C2#2258841411yescharacteristic
272C2#2259000912yes—
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×D4C22×D71
FittingC71×#2309C2
socleC22×C71C2×#99
prime radicalC23×D4×C71C22
cosocleC22×D4×C71C23
hypercenter#2309D71
hypocenterC71#12858
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #22586465C2semidirect
8. C2×#117×C71C4semidirect
9. C2×#117×C71C4semidirect
11. #390×C71C4semidirect
140. C71#12858semidirect
142. C2×#117Dic71semidirect
143. C2×#117Dic71semidirect
147. #390Dic71semidirect

Maximal subgroups

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

subgroupindex
#225882552
#225864652
C71×#23092
#1285871