Finite Group Databasebeta

← #31908619

Subgroups — #31908619

Order 18176.

388 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 107 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#22588336C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586370C214yesmaximal
4C71×#2567C211yesmaximal, characteristic
5#17986052—21no—
6#17986023—21no—
7C2×#14556047—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—
22C4⋊(C71⋊C8)—41no—
23C4⋊(C71⋊C8)—41no—
24#14556047—42no—
25C22⋊(C71⋊C8)—22no—
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—42no—
49#11765849—42no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—42no—
52C2×Q8×C71—41no—
53C2×Q8×C71—41no—
54C2×Q8×C71—42no—
55C2×Q8×C71—82no—
56C2×Q8×C71SD1612yes—
57C2×Q8×C71Q1612yes—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22⋊C411yescharacteristic
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22×C411yescharacteristic
65C22×C4×C71C2×D411yescharacteristic
66C22×C4×C71C2×D411yescharacteristic
67C4⋊C4×C71—24no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—82no—
75C4⋊C4×C71—82no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—42no—
83C23×C71C2×C22⋊C411yescharacteristic
84Q8×C71—42no—
85Q8×C71—42no—
86Q8×C71—84no—
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—22no—
94C2×C4×C71—24no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71#9911yescharacteristic
107C2×C4×C71#9911yescharacteristic
108C2×C4×C71#10312yes—
109C2×C4×C71#10312yes—
110C2×C4×C71C2×SD1611yescharacteristic
111C2×C4×C71C2×Q1611yescharacteristic
112C71⋊C8—82no—
113C71⋊C8—82no—
114C22×C71—21no—
115C22×C71—21no—
116C22×C71C2×#9911yescharacteristic
117C22×C71C2×#10311yescharacteristic
118C22×C71#42111yescharacteristic
119C4×C71—21no—
120C4×C71—21no—
121C4×C71—22no—
122C4×C71—22no—
123C4×C71—21no—
124C4×C71—44no—
125C4×C71—41no—
126C4×C71—41no—
127C4×C71—41no—
128C4×C71—82no—
129C4×C71#35612yes—
130#11908—711nomaximal, Sylow
131C2×C71—21no—
132C2×C71—21no—
133C2×C71#152011yescharacteristic
134C2×C71#212211yescharacteristic
135C2×C71C2×#35611yescharacteristic
136#2567D7111yescharacteristic
137#1801—711no—
138#1461—711no—
139#1354—714no—
140C71#1190811yescharacteristic, complemented, Sylow
141C2×C4⋊Q8Dic7111yescharacteristic
142C2×C4×Q8C2×D7111yescharacteristic
143#431—1421no—
144#423—1421no—
145C2×#101—1421no—
146#395Dic7112yes—
147#395C2×D7112yes—
148#391—21no—
149#389—21no—
150#389—21no—
151#384—21no—
152C4×C4⋊C4Dic7111yescharacteristic
153#343—712no—
154#324—712no—
155#324—1421no—
156C22×Q8C2×Dic7112yes—
157C22×Q8C22⋊D7111yescharacteristic
158C2×M16—1422no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C2×Dic7111yescharacteristic
169C2×C4⋊C4C22⋊D7111yescharacteristic
170C2×C4⋊C4C22×D7111yescharacteristic
171C2×C42—21no—
172C2×C42C2×Dic7111yescharacteristic
173C2×C42C22⋊D7111yescharacteristic
174C2×C42C22⋊D7111yescharacteristic
175C4⋊C8—2841no—
176C4⋊C8—2841no—
177C4⋊C8—2841no—
178#101—2842no—
179C22⋊C8—1422no—
180C4×C8—2841no—
181#95—22no—
182#95—22no—
183#95C2×Dic7112yes—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—82no—
188C2×Q8#1176585212yes—
189C2×Q8#1176585512yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4C22⋊Dic7111yescharacteristic
193C22×C4C22⋊Dic7111yescharacteristic
194C22×C4C22⋊Dic7111yescharacteristic
195C22×C4C22⋊Dic7111yescharacteristic
196C22×C4C22×Dic7111yescharacteristic
197C22×C4C2×C22⋊D7111yescharacteristic
198C22×C4C2×C22⋊D7111yescharacteristic
199M16—2842no—
200M16—2842no—
201C2×C8—2842no—
202C2×C8—2842no—
203C4⋊C4—24no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—42no—
219C23C2×C22⋊Dic7111yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—84no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4#1455604311yescharacteristic
243C2×C4#1455604311yescharacteristic
244C2×C4#1455605212yes—
245C2×C4#1455605212yes—
246C2×C4C2×#1176585211yescharacteristic
247C2×C4C2×#1176585511yescharacteristic
248C8—5682no—
249C8—5682no—
250C22—21no—
251C22—21no—
252C22C2×#1455604311yescharacteristic
253C22C2×#1455605211yescharacteristic
254C22#1798601611yescharacteristic
255C4—21no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—44no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4#1798579812yes—
266C2—21no—
267C2—21no—
268C2#2258691211yescharacteristic
269C2#2258999011yescharacteristic
270C2C2×#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#11908
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
140. C71#11908semidirect

Maximal subgroups

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

subgroupindex
#225883362
#225866642
#225863702
C71×#25672
#1190871