Finite Group Databasebeta

← #31908048

Subgroups — #31908048

Order 18176.

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

normal subgroups only

This group has 414 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#22593325C211yesmaximal, characteristic
2C4×#14556047C211yesmaximal, characteristic
3#22586347C214yesmaximal
4C71×#1467C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×#14556126—21no—
7C2×#14556047—21no—
8C4×#11765849—21no—
9#17985929C412yes—
10#17985929C2212yes—
11#17985904—21no—
12#17985897—21no—
13#17985897—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—
24#14556126—44no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71C2×C412yes—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4—22no—
30C2×C4×Dic71—21no—
31#14556047—24no—
32#14556039—41no—
33C4×C71⋊C8—41no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95C2×C412yes—
37C22×Q8×C71D411yescharacteristic
38C4×Q8×C71—41no—
39C4×Q8×C71—41no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C4⋊C4×C71C2311yescharacteristic
43C2×C42×C71C2×C411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C4⋊C8×C71—41no—
47C22⋊C8×C71—22no—
48#97×C71—41no—
49C2×Dic142—42no—
50C2×Dic142—42no—
51C2×Dic142—42no—
52C22×Dic71—22no—
53C22×Dic71—22no—
54#11765849—24no—
55#11765849—42no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—24no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60C4⋊Dic71—42no—
61Dic71⋊C4—44no—
62Dic71⋊C4—44no—
63C4×Dic71—42no—
64C2×Q8×C71—22no—
65C2×Q8×C71—22no—
66C2×Q8×C71—41no—
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—42no—
75C4⋊C4×C71—41no—
76C4⋊C4×C71—41no—
77C4⋊C4×C71—41no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C23×C71C2×C22⋊C411yescharacteristic
85Q8×C71—44no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71#9911yescharacteristic
101C2×C4×C71#9911yescharacteristic
102C2×C4×C71#10412yes—
103C2×C4×C71AGL(1,ℤ/8)12yes—
104C2×Dic71—24no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110Dic142—84no—
111C8×C71—82no—
112C71⋊C8—44no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71C2×#9911yescharacteristic
116C22×C71#41811yescharacteristic
117C22×C71C2×#10411yescharacteristic
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—41no—
122C4×C71—41no—
123C4×C71—41no—
124C4×C71—41no—
125C4×C71#35612yes—
126Dic71—44no—
127Dic71—82no—
128#11782—711nomaximal, Sylow
129C2×C71—21no—
130C2×C71—21no—
131C2×C71#150711yescharacteristic
132C2×C71#212311yescharacteristic
133C2×C71C2×#35611yescharacteristic
134#2546—711no—
135C4×#101—711no—
136#1467D7111yescharacteristic
137#1346—714no—
138C71#1178211yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—711no—
140C2×C4⋊Q8—1421no—
141C2×C4×Q8C2×D7111yescharacteristic
142#432—21no—
143C2×#101—1421no—
144C4×M16—1421no—
145#398—712no—
146#398—712no—
147#391—1421no—
148#389—1421no—
149#389—1421no—
150C4×C4⋊C4—711no—
151#343—712no—
152#324—21no—
153#324C2×D7112yes—
154C22×Q8—1421no—
155C22×Q8D28411yescharacteristic
156C2×M16—1422no—
157C4⋊Q8—2841no—
158C4⋊Q8—2841no—
159C4⋊Q8—2844no—
160C4×Q8—41no—
161C4×Q8—41no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4—1422no—
164C2×C4⋊C4—1421no—
165C2×C4⋊C4—1422no—
166C2×C4⋊C4—1421no—
167C2×C4⋊C4C4×D7111yescharacteristic
168C2×C4⋊C4C22⋊D7111yescharacteristic
169C2×C4⋊C4C22×D7111yescharacteristic
170C2×C42—1421no—
171C2×C42C4×D7111yescharacteristic
172C2×C42D28411yescharacteristic
173C2×C42C22⋊D7111yescharacteristic
174C4⋊C8—41no—
175#101—1424no—
176C22⋊C8—22no—
177#97—41no—
178#97—2841no—
179C4×C8—2841no—
180#95—712no—
181#95—1422no—
182#95—1422no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—2842no—
187C2×Q8—2842no—
188C2×Q8—2842no—
189C22×C4—1422no—
190C22×C4—1422no—
191C22×C4(C2×D71)⋊C411yescharacteristic
192C22×C4(C2×D71)⋊C411yescharacteristic
193C22×C4(C2×D71)⋊C411yescharacteristic
194C22×C4(C2×D71)⋊C411yescharacteristic
195C22×C4C2×C4×D7111yescharacteristic
196C22×C4C2×C22⋊D7111yescharacteristic
197C22×C4C2×D28411yescharacteristic
198M16—2842no—
199M16—1424no—
200C2×C8—42no—
201C2×C8—2842no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—2842no—
206C4⋊C4—2844no—
207C4⋊C4—2841no—
208C4⋊C4—2841no—
209C4⋊C4—2844no—
210C4⋊C4—1424no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—2842no—
218C23C2×(C2×D71)⋊C411yescharacteristic
219Q8—44no—
220Q8—5684no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—2842no—
236C2×C4—2842no—
237C2×C4—2842no—
238C2×C4—2842no—
239C2×C4—2842no—
240C2×C4—1424no—
241C2×C4#1455604211yescharacteristic
242C2×C4#1455604211yescharacteristic
243C2×C4#1455605412yes—
244C2×C4#1455615612yes—
245C8—82no—
246C8—2844no—
247C22—21no—
248C22—21no—
249C22C2×#1455604211yescharacteristic
250C22#1798600511yescharacteristic
251C22C2×#1455605411yescharacteristic
252C4—21no—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—2844no—
260C4—5682no—
261C4#1798579712yes—
262C2—21no—
263C2—21no—
264C2#2258685711yescharacteristic
265C2#2258999311yescharacteristic
266C2C2×#1798579711yescharacteristic
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#11782
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
138. C71#11782semidirect

Maximal subgroups

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

subgroupindex
#225933252
C4×#145560472
#225863472
C71×#14672
#1178271