Finite Group Databasebeta

← #31906856

Subgroups — #31906856

Order 18176.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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#22586674C212yesmaximal
2#22586286C212yesmaximal, complemented
3#22586284C212yesmaximal, complemented
4C71×#2285C211yesmaximal, characteristic
5#17986056—22no—
6#17985725C2214yes—
7#17985723—22no—
8C2×#117×C71C411yescharacteristic
9C2×#117×C71C2211yescharacteristic
10C22×C42×C71C411yescharacteristic
11#384×C71C411yescharacteristic
12#384×C71C2211yescharacteristic
13#382×C71C411yescharacteristic
14#382×C71C2211yescharacteristic
15#380×C71—22no—
16C4×C4⋊C4×C71—22no—
17C4⋊(C71⋊C8)—42no—
18C22⋊(C71⋊C8)—24no—
19#14556039—42no—
20C23×C4×C71—21no—
21C23×C4×C71C2×C411yescharacteristic
22#117×C71—41no—
23#117×C71—41no—
24#117×C71—41no—
25#117×C71—41no—
26C2×C4⋊C4×C71—22no—
27C2×C4⋊C4×C71—22no—
28C2×C4⋊C4×C71C2×C411yescharacteristic
29C2×C4⋊C4×C71C2×C411yescharacteristic
30C2×C4⋊C4×C71C2×C411yescharacteristic
31C2×C4⋊C4×C71C2311yescharacteristic
32C2×C22⋊C4×C71—21no—
33C2×C22⋊C4×C71D412yes—
34C2×C42×C71—21no—
35C2×C42×C71—22no—
36C2×C42×C71—22no—
37C2×C42×C71—21no—
38C2×C42×C71C2×C411yescharacteristic
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71D412yes—
41#95×C71—22no—
42#95×C71—22no—
43C2×C71⋊C8—44no—
44C24×C71C22⋊C411yescharacteristic
45C22×C4×C71—22no—
46C22×C4×C71—21no—
47C22×C4×C71—21no—
48C22×C4×C71—21no—
49C22×C4×C71—21no—
50C22×C4×C71—21no—
51C22×C4×C71—24no—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71—42no—
56C22×C4×C71—42no—
57C22×C4×C71C22⋊C412yes—
58C22×C4×C71C22⋊C411yescharacteristic
59C22×C4×C71C22×C411yescharacteristic
60C22×C4×C71C2×D412yes—
61C4⋊C4×C71—41no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—44no—
66C4⋊C4×C71—44no—
67C22⋊C4×C71—42no—
68C22⋊C4×C71—42no—
69C42×C71—24no—
70C42×C71—22no—
71C42×C71—22no—
72C42×C71—21no—
73C42×C71—21no—
74C42×C71—41no—
75C42×C71—41no—
76C42×C71M1612yes—
77C42×C71M1612yes—
78C23×C71—21no—
79C23×C71—21no—
80C23×C71—41no—
81C23×C71—41no—
82C23×C71#10112yes—
83C23×C71C2×C22⋊C411yescharacteristic
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—48no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71#9911yescharacteristic
108C2×C4×C71#9911yescharacteristic
109C2×C4×C71C2×M1611yescharacteristic
110C2×C4×C71C2×M1611yescharacteristic
111C71⋊C8—84no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71—22no—
115C22×C71—42no—
116C22×C71—42no—
117C22×C71—41no—
118C22×C71—41no—
119C22×C71#40711yescharacteristic
120C22×C71C2×#9911yescharacteristic
121C22×C71C2×#10111yescharacteristic
122C4×C71—24no—
123C4×C71—21no—
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—21no—
127C4×C71—42no—
128C4×C71—42no—
129C4×C71—42no—
130C4×C71—81no—
131C4×C71—81no—
132#11513—711nomaximal, Sylow
133C2×C71—21no—
134C2×C71—21no—
135C2×C71—42no—
136C2×C71#146311yescharacteristic
137C2×C71#212611yescharacteristic
138C2×C71#213211yescharacteristic
139#2285D7111yescharacteristic
140#1464—712no—
141#1325—712no—
142#1324—712no—
143C71#1151311yescharacteristic, complemented, Sylow
144C2×#117Dic7111yescharacteristic
145C2×#117C2×D7111yescharacteristic
146C22×C42Dic7111yescharacteristic
147#432—1422no—
148#384Dic7111yescharacteristic
149#384C2×D7111yescharacteristic
150#382Dic7111yescharacteristic
151#382C2×D7111yescharacteristic
152#380—22no—
153C4×C4⋊C4—22no—
154#324—714no—
155#323—1422no—
156C23×C4—21no—
157C23×C4C2×Dic7111yescharacteristic
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×Dic7111yescharacteristic
165C2×C4⋊C4C2×Dic7111yescharacteristic
166C2×C4⋊C4C2×Dic7111yescharacteristic
167C2×C4⋊C4C22×D7111yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4C22⋊D7112yes—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42C2×Dic7111yescharacteristic
175C2×C42C2×Dic7111yescharacteristic
176C2×C42C22⋊D7112yes—
177C4⋊C8—2842no—
178C22⋊C8—1424no—
179#97—2842no—
180#95—22no—
181#95—22no—
182C24C22⋊Dic7111yescharacteristic
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C22⋊Dic7112yes—
196C22×C4C22⋊Dic7111yescharacteristic
197C22×C4C22×Dic7111yescharacteristic
198C22×C4C2×C22⋊D7112yes—
199C2×C8—2844no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—24no—
209C42—22no—
210C42—22no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C42#1176584912yes—
216C42#1176584912yes—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23#1455604712yes—
222C23C2×C22⋊Dic7111yescharacteristic
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4#1455604311yescharacteristic
247C2×C4#1455604311yescharacteristic
248C2×C4C2×#1176584911yescharacteristic
249C2×C4C2×#1176584911yescharacteristic
250C8—5684no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22#1798596311yescharacteristic
259C22C2×#1455604311yescharacteristic
260C22C2×#1455604711yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—81no—
270C4—81no—
271C2—21no—
272C2—21no—
273C2—42no—
274C2#2258667011yescharacteristic
275C2#2259000511yescharacteristic
276C2#2259002511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2285C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2285D71
hypocenterC71#11513
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #22586286C2semidirect
3. #22586284C2semidirect
143. C71#11513semidirect

Maximal subgroups

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

subgroupindex
#225866742
#225862862
#225862842
C71×#22852
#1151371