Finite Group Databasebeta

← #31897093

Subgroups — #31897093

Order 18176.

402 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 127 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#22589330C211yesmaximal, characteristic
2#22589330C211yesmaximal, characteristic
3#22587096C211yesmaximal, characteristic
4#22587025C211yesmaximal, characteristic
5#22587021C211yesmaximal, characteristic
6#22586629C211yesmaximal, characteristic
7C2×#487×C71C211yesmaximal, characteristic
8#17986052C2211yescharacteristic
9#17986023C2211yescharacteristic
10#17986023C2211yescharacteristic
11C4×#11765849C2211yescharacteristic
12#17985768—22no—
13#17985768—22no—
14#17985750—22no—
15#17985748—22no—
16#17985745—22no—
17#17985721—21no—
18#17985720—21no—
19C2×C4⋊Q8×C71C2211yescharacteristic
20C2×C4⋊Q8×C71C2211yescharacteristic
21#487×C71—21no—
22#487×C71—21no—
23#487×C71—22no—
24C2×#103×C71C414yes—
25C2×C4×C8×C71C2211yescharacteristic
26C2×#11765849—21no—
27C2×#11765849—21no—
28C2×#11765849D412yes—
29C4⋊(C71⋊C8)—21no—
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)—21no—
32C4⋊(C71⋊C8)—21no—
33C4⋊(C71⋊C8)D412yes—
34C22⋊(C71⋊C8)—21no—
35#14556039—21no—
36C4×C71⋊C8—21no—
37C4×C71⋊C8—21no—
38C22×Q8×C71—21no—
39C22×Q8×C71C2×C412yes—
40C22×C8×C71C2×C412yes—
41C4⋊Q8×C71—21no—
42C4⋊Q8×C71—21no—
43C4⋊Q8×C71—21no—
44C4⋊Q8×C71—21no—
45C4⋊Q8×C71—21no—
46C4⋊Q8×C71—21no—
47C4⋊Q8×C71—21no—
48C4⋊Q8×C71—21no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71—21no—
51C2×C4⋊C4×C71—21no—
52C2×C4⋊C4×C71C2×C412yes—
53C2×C42×C71C2311yescharacteristic
54#103×C71—24no—
55#103×C71—24no—
56C4×C8×C71D411yescharacteristic
57C4×C8×C71D411yescharacteristic
58C4×C8×C71D412yes—
59#11765849—22no—
60#11765849—22no—
61#11765849—42no—
62#11765849—42no—
63C2×C71⋊C8—22no—
64C2×C71⋊C8—22no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×Q8×C71—22no—
68C2×Q8×C71—22no—
69C2×Q8×C71—44no—
70C2×Q8×C71—41no—
71C2×Q8×C71—41no—
72C2×Q8×C71—82no—
73C22×C4×C71—22no—
74C22×C4×C71—22no—
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71#4111yescharacteristic
78C2×C8×C71—42no—
79C2×C8×C71C22⋊C412yes—
80C2×C8×C71C22⋊C412yes—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—22no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C42×C71C2×D411yescharacteristic
90C42×C71C2×D411yescharacteristic
91C42×C71C2×D411yescharacteristic
92C42×C71#4111yescharacteristic
93C23×C71C4×D411yescharacteristic
94Q8×C71—44no—
95Q8×C71—82no—
96Q8×C71—82no—
97Q8×C71—82no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—82no—
103C2×C4×C71—82no—
104C2×C4×C71C2×C22⋊C411yescharacteristic
105C2×C4×C71C4×D411yescharacteristic
106C2×C4×C71#12111yescharacteristic
107C2×C4×C71#12111yescharacteristic
108C2×C4×C71#12411yescharacteristic
109C2×C4×C71#12711yescharacteristic
110C2×C4×C71#13712yes—
111C2×C4×C71#13712yes—
112C8×C71—22no—
113C8×C71—22no—
114C71⋊C8—42no—
115C71⋊C8—42no—
116C71⋊C8—81no—
117C71⋊C8—81no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71#39011yescharacteristic
121C22×C71#44112yes—
122C4×C71—21no—
123C4×C71—21no—
124C4×C71—82no—
125C4×C71—82no—
126C4×C71#41312yes—
127C4×C71#47012yes—
128C4×C71#47112yes—
129C4×C71#49711yescharacteristic
130C4×C71#49711yescharacteristic
131#9803—711nomaximal, Sylow
132C2×C71—22no—
133C2×C71#196711yescharacteristic
134C2×C71#197111yescharacteristic
135C2×C71#197711yescharacteristic
136C2×#487D7111yescharacteristic
137#1997—711no—
138#1997—711no—
139#1566—711no—
140#1549—711no—
141#1548—711no—
142#1451—711no—
143C71#980311yescharacteristic, complemented, Sylow
144C2×C4⋊Q8C2×D7111yescharacteristic
145C2×C4⋊Q8C2×D7111yescharacteristic
146#487—21no—
147#487—21no—
148#487—22no—
149#431—711no—
150#423—711no—
151#423—711no—
152C2×#103Dic7114yes—
153C4×M16—711no—
154C2×C4×C8C2×D7111yescharacteristic
155#343—1422no—
156#343—1422no—
157#334—1422no—
158#333—1422no—
159#332—1422no—
160#322—1421no—
161#322—1421no—
162C22×Q8—21no—
163C22×Q8C2×Dic7112yes—
164C2×M16—712no—
165C2×M16—1421no—
166C2×M16—1421no—
167C22×C8C2×Dic7112yes—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C2×Dic7112yes—
180C2×C42C22×D7111yescharacteristic
181C4⋊C8—712no—
182C4⋊C8—1421no—
183C4⋊C8—1421no—
184C4⋊C8—1421no—
185C4⋊C8—1421no—
186#103—24no—
187#103—24no—
188C22⋊C8—1421no—
189#97—1421no—
190C4×C8—1421no—
191C4×C8—1421no—
192C4×C8C22⋊D7111yescharacteristic
193C4×C8C22⋊D7111yescharacteristic
194C4×C8C22⋊D7112yes—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—44no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—82no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4C22×Dic7111yescharacteristic
204C22×C4D4×D7111yescharacteristic
205C22×C4#1176586411yescharacteristic
206M16—2842no—
207M16—2842no—
208M16—1422no—
209M16—1422no—
210C2×C8—42no—
211C2×C8—2841no—
212C2×C8—2841no—
213C2×C8—1422no—
214C2×C8—1422no—
215C2×C8C22⋊Dic7112yes—
216C2×C8C22⋊Dic7112yes—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C42D4×D7111yescharacteristic
226C42C2×C22⋊D7111yescharacteristic
227C42C2×C22⋊D7111yescharacteristic
228C42#1176586411yescharacteristic
229C23D4×Dic7111yescharacteristic
230Q8—44no—
231Q8—82no—
232Q8—82no—
233Q8—82no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4C2×C22⋊Dic7111yescharacteristic
241C2×C4D4×Dic7111yescharacteristic
242C2×C4#1455609911yescharacteristic
243C2×C4#1455609911yescharacteristic
244C2×C4#1455611511yescharacteristic
245C2×C4#1455612411yescharacteristic
246C2×C4#1455616012yes—
247C2×C4#1455616012yes—
248C8—22no—
249C8—22no—
250C8—2842no—
251C8—2842no—
252C8—5681no—
253C8—5681no—
254C22—21no—
255C22—21no—
256C22#1798590211yescharacteristic
257C22#1798610612yes—
258C4—21no—
259C4—21no—
260C4—82no—
261C4—82no—
262C4#1798598312yes—
263C4#1798625412yes—
264C4#1798625912yes—
265C4#1798639211yescharacteristic
266C4#1798639211yescharacteristic
267C2—22no—
268C2#2258919011yescharacteristic
269C2#2258921211yescharacteristic
270C2#2258924511yescharacteristic
subgroupG/H
centerC22#17985902
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#487×C71C2
socleC22×C71#390
prime radicalC23×C71C4×D4
cosocleC2×C42×C71C23
hypercenterC2×#487D71
hypocenterC71#9803
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
143. C71#9803semidirect

Maximal subgroups

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

subgroupindex
#225893302
#225870962
#225870252
#225870212
#225866292
C2×#487×C712
#980371