Finite Group Databasebeta

← #31908101

Subgroups — #31908101

Order 18176.

378 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 87 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#22593552C211yesmaximal, characteristic
2C4×#14556047C211yesmaximal, characteristic
3#22586368C214yesmaximal
4C71×#1467C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×#14556047—21no—
7C4×#11765849—21no—
8#17985938C412yes—
9#17985938C2212yes—
10#17985906—21no—
11#17985905—21no—
12#17985905—21no—
13#17985864—21no—
14C4×C4⋊Dic71C411yescharacteristic
15#17985768C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17#432×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C22×Dic142—21no—
21C2×#11765849—22no—
22#14556128—41no—
23#14556128—41no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C412yes—
27C2×C4×Dic71—21no—
28#14556047—24no—
29#14556039—41no—
30C4×C71⋊C8—41no—
31C71⋊#95—22no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95C2×C412yes—
36C22×Q8×C71D411yescharacteristic
37C4×Q8×C71—41no—
38C4×Q8×C71—41no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C4⋊C8×C71—41no—
46C22⋊C8×C71—22no—
47#97×C71—41no—
48C2×Dic142—44no—
49C2×Dic142—41no—
50C2×Dic142—41no—
51C22×Dic71—22no—
52C22×Dic71—22no—
53#11765849—24no—
54#11765849—42no—
55C2×C71⋊C8—42no—
56C4⋊Dic71—42no—
57C4⋊Dic71—42no—
58C4⋊Dic71—42no—
59C4×Dic71—42no—
60C2×Q8×C71—22no—
61C2×Q8×C71—22no—
62C2×Q8×C71—41no—
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22×C411yescharacteristic
68C22×C4×C71C2×D411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C2×C8×C71—42no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—41no—
74C42×C71—21no—
75C42×C71—21no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C23×C71C2×C22⋊C411yescharacteristic
81Q8×C71—44no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—41no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71#9911yescharacteristic
97C2×C4×C71#9911yescharacteristic
98C2×C4×C71#10412yes—
99C2×C4×C71#13712yes—
100C2×Dic71—42no—
101C2×Dic71—42no—
102C2×Dic71—44no—
103C2×Dic71—42no—
104C2×Dic71—42no—
105Dic142—82no—
106Dic142—82no—
107C8×C71—82no—
108C71⋊C8—44no—
109C22×C71—21no—
110C22×C71—21no—
111C22×C71C2×#9911yescharacteristic
112C22×C71#41911yescharacteristic
113C22×C71C2×#10411yescharacteristic
114C4×C71—21no—
115C4×C71—22no—
116C4×C71—22no—
117C4×C71—41no—
118C4×C71—41no—
119C4×C71—41no—
120C4×C71—41no—
121C4×C71#35512yes—
122Dic71—82no—
123Dic71—82no—
124#11794—711nomaximal, Sylow
125C2×C71—21no—
126C2×C71—21no—
127C2×C71#150811yescharacteristic
128C2×C71#212311yescharacteristic
129C2×C71C2×#35511yescharacteristic
130#2568—711no—
131C4×#101—711no—
132#1467D7111yescharacteristic
133#1353—714no—
134C71#1179411yescharacteristic, complemented, Sylow
135C2×C4⋊Q8—711no—
136C2×C4×Q8C2×D7111yescharacteristic
137#432—21no—
138C2×#101—1421no—
139C4×M16—1421no—
140#400—712no—
141#400—712no—
142#391—1421no—
143#391—1421no—
144#391—1421no—
145#382—1421no—
146C4×C4⋊C4—711no—
147#343—712no—
148#324—21no—
149#324C2×D7112yes—
150C22×Q8—1421no—
151C22×Q8D28411yescharacteristic
152C2×M16—1422no—
153C4⋊Q8—2841no—
154C4⋊Q8—2841no—
155C4×Q8—41no—
156C4×Q8—41no—
157C2×C4⋊C4—712no—
158C2×C4⋊C4—1421no—
159C2×C4⋊C4—1421no—
160C2×C4⋊C4C4×D7111yescharacteristic
161C2×C4⋊C4C22⋊D7111yescharacteristic
162C2×C4⋊C4C22×D7111yescharacteristic
163C2×C42—1421no—
164C2×C42C4×D7111yescharacteristic
165C2×C42D28411yescharacteristic
166C2×C42C22⋊D7111yescharacteristic
167C4⋊C8—41no—
168#101—1424no—
169C22⋊C8—22no—
170#97—41no—
171#97—2841no—
172C4×C8—2841no—
173#95—712no—
174#95—1422no—
175#95—1422no—
176#95—1422no—
177#95—1422no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—2844no—
182C2×Q8—2841no—
183C2×Q8—2841no—
184C22×C4—1422no—
185C22×C4—1422no—
186C22×C4(C2×D71)⋊C411yescharacteristic
187C22×C4(C2×D71)⋊C411yescharacteristic
188C22×C4(C2×D71)⋊C411yescharacteristic
189C22×C4(C2×D71)⋊C411yescharacteristic
190C22×C4C2×C4×D7111yescharacteristic
191C22×C4C2×C22⋊D7111yescharacteristic
192C22×C4C2×D28411yescharacteristic
193M16—2842no—
194M16—1424no—
195C2×C8—42no—
196C2×C8—2842no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—2842no—
201C4⋊C4—2842no—
202C4⋊C4—2842no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—2842no—
210C23C2×(C2×D71)⋊C411yescharacteristic
211Q8—44no—
212Q8—5682no—
213Q8—5682no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—2842no—
229C2×C4—2844no—
230C2×C4—2842no—
231C2×C4—2842no—
232C2×C4—2842no—
233C2×C4#1455604211yescharacteristic
234C2×C4#1455604211yescharacteristic
235C2×C4#1455605412yes—
236C2×C4#1455616112yes—
237C8—82no—
238C8—2844no—
239C22—21no—
240C22—21no—
241C22C2×#1455604211yescharacteristic
242C22#1798600911yescharacteristic
243C22C2×#1455605411yescharacteristic
244C4—21no—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—5682no—
252C4—5682no—
253C4#1798579512yes—
254C2—21no—
255C2—21no—
256C2#2258686111yescharacteristic
257C2#2258999311yescharacteristic
258C2C2×#1798579511yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1467C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1467D71
hypocenterC71#11794
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
134. C71#11794semidirect

Maximal subgroups

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

subgroupindex
#225935522
C4×#145560472
#225863682
C71×#14672
#1179471