Finite Group Databasebeta

← #31908104

Subgroups — #31908104

Order 18176.

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

normal subgroups only

This group has 394 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
1C4×#14556047C211yesmaximal, characteristic
2#22586684C211yesmaximal, characteristic
3#22586370C214yesmaximal
4C71×#2567C211yesmaximal, characteristic
5#17986056—21no—
6C2×#14556047—21no—
7C4×#11765849—21no—
8#17985768C2212yes—
9#17985725—21no—
10#17985725C2212yes—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×C4×Q8×C71C2211yescharacteristic
13#395×C71C412yes—
14#395×C71C2212yes—
15#391×C71—21no—
16#389×C71—21no—
17#389×C71—21no—
18#384×C71—21no—
19C4×C4⋊C4×C71C411yescharacteristic
20C2×#11765849—22no—
21C4⋊(C71⋊C8)—41no—
22#14556047—24no—
23C22⋊(C71⋊C8)—22no—
24#14556039—41no—
25#14556039—41no—
26C4×C71⋊C8—41no—
27C22×Q8×C71C2×C412yes—
28C22×Q8×C71D411yescharacteristic
29C4⋊Q8×C71—41no—
30C4⋊Q8×C71—41no—
31C4×Q8×C71—41no—
32C4×Q8×C71—41no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71C2311yescharacteristic
41C2×C42×C71—21no—
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45#95×C71—22no—
46#95×C71—22no—
47#95×C71C2×C412yes—
48#11765849—24no—
49#11765849—42no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—42no—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×Q8×C71—41no—
55C2×Q8×C71—42no—
56C2×Q8×C71—82no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22⋊C411yescharacteristic
63C22×C4×C71C22×C411yescharacteristic
64C22×C4×C71C2×D411yescharacteristic
65C22×C4×C71C2×D411yescharacteristic
66C4⋊C4×C71—24no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—82no—
74C4⋊C4×C71—82no—
75C42×C71—21no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—42no—
82C23×C71C2×C22⋊C411yescharacteristic
83Q8×C71—44no—
84Q8×C71—84no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—24no—
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—42no—
98C2×C4×C71—42no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71#9911yescharacteristic
106C2×C4×C71#9911yescharacteristic
107C2×C4×C71#10412yes—
108C2×C4×C71#13712yes—
109C71⋊C8—44no—
110C71⋊C8—82no—
111C22×C71—21no—
112C22×C71—21no—
113C22×C71C2×#9911yescharacteristic
114C22×C71#41911yescharacteristic
115C22×C71C2×#10411yescharacteristic
116C4×C71—21no—
117C4×C71—22no—
118C4×C71—22no—
119C4×C71—44no—
120C4×C71—41no—
121C4×C71—41no—
122C4×C71—41no—
123C4×C71—41no—
124C4×C71—82no—
125C4×C71#35612yes—
126#11795—711nomaximal, Sylow
127C2×C71—21no—
128C2×C71—21no—
129C2×C71#150811yescharacteristic
130C2×C71#212211yescharacteristic
131C2×C71C2×#35611yescharacteristic
132#2567D7111yescharacteristic
133C4×#101—711no—
134#1467—711no—
135#1354—714no—
136C71#1179511yescharacteristic, complemented, Sylow
137C2×C4⋊Q8Dic7111yescharacteristic
138C2×C4×Q8C2×D7111yescharacteristic
139#432—1421no—
140C2×#101—1421no—
141C4×M16—1421no—
142#395Dic7112yes—
143#395C2×D7112yes—
144#391—21no—
145#389—21no—
146#389—21no—
147#384—21no—
148C4×C4⋊C4Dic7111yescharacteristic
149#343—712no—
150#324—712no—
151#324—1421no—
152C22×Q8C2×Dic7112yes—
153C22×Q8C22⋊D7111yescharacteristic
154C2×M16—1422no—
155C4⋊Q8—41no—
156C4⋊Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×Dic7111yescharacteristic
165C2×C4⋊C4C22⋊D7111yescharacteristic
166C2×C4⋊C4C22×D7111yescharacteristic
167C2×C42—21no—
168C2×C42C2×Dic7111yescharacteristic
169C2×C42C22⋊D7111yescharacteristic
170C2×C42C22⋊D7111yescharacteristic
171C4⋊C8—2841no—
172#101—1424no—
173C22⋊C8—1422no—
174#97—2841no—
175#97—2841no—
176C4×C8—2841no—
177#95—22no—
178#95—22no—
179#95C2×Dic7112yes—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—82no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C22⋊Dic7111yescharacteristic
188C22×C4C22⋊Dic7111yescharacteristic
189C22×C4C22⋊Dic7111yescharacteristic
190C22×C4C22⋊Dic7111yescharacteristic
191C22×C4C22×Dic7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×C22⋊D7111yescharacteristic
194M16—2842no—
195M16—1424no—
196C2×C8—2842no—
197C2×C8—2842no—
198C4⋊C4—24no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C23C2×C22⋊Dic7111yescharacteristic
215Q8—44no—
216Q8—84no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4#1455604311yescharacteristic
238C2×C4#1455604311yescharacteristic
239C2×C4#1455605512yes—
240C2×C4#1455615912yes—
241C8—2844no—
242C8—5682no—
243C22—21no—
244C22—21no—
245C22C2×#1455604311yescharacteristic
246C22#1798600811yescharacteristic
247C22C2×#1455605511yescharacteristic
248C4—21no—
249C4—22no—
250C4—22no—
251C4—44no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—41no—
256C4—82no—
257C4#1798579812yes—
258C2—21no—
259C2—21no—
260C2#2258686011yescharacteristic
261C2#2258999011yescharacteristic
262C2C2×#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#11795
LCSG ▷ C22×C4×C71 ▷ C23×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
136. C71#11795semidirect

Maximal subgroups

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

subgroupindex
C4×#145560472
#225866842
#225863702
C71×#25672
#1179571