Finite Group Databasebeta

← #31907121

Subgroups — #31907121

Order 18176.

374 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 95 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#22586684C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586286C214yesmaximal
4C71×#2567C211yesmaximal, characteristic
5#17986056—21no—
6#17986052—21no—
7#17985725—21no—
8#17985725—21no—
9#17985725C2212yes—
10#17985725C2212yes—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×C4×Q8×C71C2211yescharacteristic
13#395×C71—21no—
14#395×C71—21no—
15#391×C71C412yes—
16#389×C71—21no—
17#389×C71—21no—
18#384×C71C2212yes—
19C4×C4⋊C4×C71C411yescharacteristic
20C4⋊(C71⋊C8)—41no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24#14556039—41no—
25C4×C71⋊C8—41no—
26C22×Q8×C71—21no—
27C22×Q8×C71C2×C411yescharacteristic
28C4⋊Q8×C71—41no—
29C4⋊Q8×C71—41no—
30C4×Q8×C71—41no—
31C4×Q8×C71—41no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71C2×C412yes—
37C2×C4⋊C4×C71D411yescharacteristic
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71C2311yescharacteristic
40C2×C42×C71C2×C412yes—
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C2×C42×C71D411yescharacteristic
44#95×C71—21no—
45#95×C71—22no—
46#95×C71—22no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C2×Q8×C71—22no—
50C2×Q8×C71—22no—
51C2×Q8×C71—41no—
52C2×Q8×C71—42no—
53C2×Q8×C71—82no—
54C22×C4×C71—22no—
55C22×C4×C71—22no—
56C22×C4×C71C22⋊C411yescharacteristic
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C22⋊C411yescharacteristic
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22×C411yescharacteristic
61C22×C4×C71C2×D411yescharacteristic
62C22×C4×C71C2×D411yescharacteristic
63C4⋊C4×C71—24no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—82no—
71C4⋊C4×C71—82no—
72C42×C71—21no—
73C42×C71—21no—
74C42×C71—21no—
75C42×C71—21no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—24no—
79C23×C71C2×C22⋊C411yescharacteristic
80Q8×C71—44no—
81Q8×C71—84no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—24no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71#9911yescharacteristic
103C2×C4×C71#9911yescharacteristic
104C2×C4×C71#10112yes—
105C2×C4×C71#13112yes—
106C71⋊C8—82no—
107C71⋊C8—82no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71#40811yescharacteristic
111C22×C71C2×#9911yescharacteristic
112C22×C71C2×#10111yescharacteristic
113C4×C71—21no—
114C4×C71—22no—
115C4×C71—22no—
116C4×C71—44no—
117C4×C71—41no—
118C4×C71—41no—
119C4×C71—41no—
120C4×C71—41no—
121C4×C71—82no—
122C4×C71#35612yes—
123#11574—711nomaximal, Sylow
124C2×C71—21no—
125C2×C71—21no—
126C2×C71#147211yescharacteristic
127C2×C71#212811yescharacteristic
128C2×C71C2×#35611yescharacteristic
129#2567D7111yescharacteristic
130#1467—711no—
131#1461—711no—
132#1325—714no—
133C71#1157411yescharacteristic, complemented, Sylow
134C2×C4⋊Q8Dic7111yescharacteristic
135C2×C4×Q8C2×D7111yescharacteristic
136#432—1421no—
137#431—1421no—
138#395—21no—
139#395—21no—
140#391Dic7112yes—
141#389—21no—
142#389—21no—
143#384C2×D7112yes—
144C4×C4⋊C4Dic7111yescharacteristic
145#324—712no—
146#324—712no—
147#324—1421no—
148#324—1421no—
149C22×Q8—21no—
150C22×Q8C2×Dic7111yescharacteristic
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C2×Dic7112yes—
160C2×C4⋊C4C22⋊D7111yescharacteristic
161C2×C4⋊C4C22⋊D7111yescharacteristic
162C2×C4⋊C4C22×D7111yescharacteristic
163C2×C42C2×Dic7112yes—
164C2×C42C2×Dic7111yescharacteristic
165C2×C42C22⋊D7111yescharacteristic
166C2×C42C22⋊D7111yescharacteristic
167C4⋊C8—2841no—
168C4⋊C8—2841no—
169C22⋊C8—1422no—
170C22⋊C8—1422no—
171#97—2841no—
172C4×C8—2841no—
173#95—21no—
174#95—22no—
175#95—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—41no—
179C2×Q8—42no—
180C2×Q8—82no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4C22⋊Dic7111yescharacteristic
184C22×C4C22⋊Dic7111yescharacteristic
185C22×C4C22⋊Dic7111yescharacteristic
186C22×C4C22⋊Dic7111yescharacteristic
187C22×C4C22×Dic7111yescharacteristic
188C22×C4C2×C22⋊D7111yescharacteristic
189C22×C4C2×C22⋊D7111yescharacteristic
190C2×C8—2842no—
191C2×C8—2842no—
192C4⋊C4—24no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—24no—
208C23C2×C22⋊Dic7111yescharacteristic
209Q8—44no—
210Q8—84no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4#1455604311yescharacteristic
232C2×C4#1455604311yescharacteristic
233C2×C4#1455604712yes—
234C2×C4#1455613512yes—
235C8—5682no—
236C8—5682no—
237C22—21no—
238C22—21no—
239C22#1798596611yescharacteristic
240C22C2×#1455604311yescharacteristic
241C22C2×#1455604711yescharacteristic
242C4—21no—
243C4—22no—
244C4—22no—
245C4—44no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—82no—
251C4#1798579812yes—
252C2—21no—
253C2—21no—
254C2#2258670411yescharacteristic
255C2#2259001211yescharacteristic
256C2C2×#1798579811yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2567C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2567D71
hypocenterC71#11574
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
133. C71#11574semidirect

Maximal subgroups

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

subgroupindex
#225866842
#225866642
#225862862
C71×#25672
#1157471