Finite Group Databasebeta

← #31891831

Subgroups — #31891831

Order 18176.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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#22589157C211yesmaximal, characteristic
2#22588473C212yesmaximal, complemented
3C2×#17985739C211yesmaximal, characteristic
4#22586221C212yesmaximal, complemented
5C71×#2379C211yesmaximal, characteristic
6C2×#14556052—21no—
7C2×#14556049—21no—
8C2×#14556039C2211yescharacteristic
9C2×#14556039C2212yes—
10#17985869C2211yescharacteristic
11#17985761—22no—
12#17985761—22no—
13#17985739C414yescomplemented
14C2×#124×C71C2211yescharacteristic
15#391×C71—21no—
16#388×C71—22no—
17#386×C71—21no—
18#385×C71—22no—
19#382×C71C2212yes—
20C22×C71⋊C8—21no—
21C22×C71⋊C8—22no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—22no—
24#14556052—42no—
25#14556049—42no—
26#14556039—22no—
27#14556039—22no—
28#14556039C2×C414yescomplemented
29C22×Q8×C71D411yescharacteristic
30C22×D4×C71D411yescharacteristic
31C23×C4×C71—21no—
32#124×C71—22no—
33#124×C71C2×C412yes—
34#124×C71Q812yes—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71D412yes—
37C2×C4⋊C4×C71D412yes—
38C2×C22⋊C4×C71—22no—
39C2×C22⋊C4×C71—22no—
40C2×C22⋊C4×C71—21no—
41C2×C42×C71—21no—
42C2×C42×C71C2311yescharacteristic
43#95×C71—22no—
44#95×C71—22no—
45#95×C71—22no—
46C22×Dic71—41no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8—24no—
49C2×C71⋊C8—42no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—41no—
52C2×C71⋊C8—41no—
53C4⋊Dic71—41no—
54C4⋊Dic71—41no—
55C4⋊Dic71—44no—
56C24×C71—21no—
57C2×Q8×C71—21no—
58C2×Q8×C71—41no—
59C2×Q8×C71—41no—
60C2×Q8×C71C4⋊C412yes—
61C2×D4×C71—21no—
62C2×D4×C71—41no—
63C2×D4×C71—41no—
64C2×D4×C71C4⋊C412yes—
65C22×C4×C71—21no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—81no—
69C22×C4×C71—82no—
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71C2×D412yes—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—42no—
74C4⋊C4×C71—81no—
75C22⋊C4×C71—24no—
76C22⋊C4×C71—24no—
77C22⋊C4×C71—81no—
78C22⋊C4×C71—82no—
79C42×C71—81no—
80C42×C71C22×C411yescharacteristic
81C42×C71C2×Q811yescharacteristic
82C42×C71#4112yes—
83C23×C71—22no—
84C23×C71—41no—
85C23×C71—42no—
86C23×C71—42no—
87C23×C71—41no—
88C23×C71#12011yescharacteristic
89Q8×C71—42no—
90Q8×C71—81no—
91D4×C71—42no—
92D4×C71—81no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—21no—
97C2×C4×C71—22no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—84no—
107C2×C4×C71C2×C4⋊C411yescharacteristic
108C2×C4×C71C4×D412yes—
109C2×C4×C71C22⋊Q812yes—
110C2×C4×C71#12311yescharacteristic
111C2×Dic71—81no—
112C2×Dic71—82no—
113C71⋊C8—42no—
114C71⋊C8—44no—
115C22×C71—42no—
116C22×C71—44no—
117C22×C71—81no—
118C22×C71—81no—
119C22×C71—82no—
120C22×C71#38511yescharacteristic
121C22×C71#45311yescharacteristic
122C22×C71#45412yes—
123C22×C71#45511yescharacteristic
124C22×C71#45512yes—
125C4×C71—21no—
126C4×C71—21no—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—42no—
130C4×C71—82no—
131C4×C71—82no—
132Dic71—161no—
133#8914—711nomaximal, Sylow
134C2×C71—82no—
135C2×C71#165711yescharacteristic
136C2×C71#165911yescharacteristic
137C2×C71#166212yes—
138C2×C71#185611yescharacteristic
139C2×C71#186012yes—
140#2379D7111yescharacteristic
141#1959—711no—
142#1836—712no—
143C2×#329—711no—
144#1295—712no—
145C71#891411yescharacteristic, complemented, Sylow
146C2×#124C2×D7111yescharacteristic
147C2×#103—1421no—
148C2×#102—1421no—
149C2×#97—712no—
150C2×#97—711no—
151#391—21no—
152#388—22no—
153#386—21no—
154#385—22no—
155#384—711no—
156#382C2×D7112yes—
157#340—1422no—
158#340—1422no—
159#329—714no—
160C22×Q8C22⋊D7111yescharacteristic
161C22×D4C22⋊D7111yescharacteristic
162C23×C4—21no—
163C22×C8—1422no—
164C22×C8—1421no—
165#124—22no—
166#124Dic14212yes—
167#124C4×D7112yes—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—1422no—
170C2×C4⋊C4—1421no—
171C2×C4⋊C4C22⋊D7112yes—
172C2×C4⋊C4C22⋊D7112yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C42—21no—
177C2×C42C22×D7111yescharacteristic
178#103—2842no—
179#102—2842no—
180#97—714no—
181#97—1422no—
182#97—1422no—
183#95—22no—
184#95—22no—
185#95—22no—
186C24—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8Dic71⋊C412yes—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4Dic71⋊C412yes—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—81no—
199C22×C4—82no—
200C22×C4—2841no—
201C22×C4C2×C22⋊D7111yescharacteristic
202C22×C4C2×C22⋊D7112yes—
203C2×C8—2842no—
204C2×C8—2842no—
205C2×C8—2841no—
206C2×C8—2841no—
207C2×C8—1422no—
208C2×C8—1424no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—2841no—
213C4⋊C4—2841no—
214C4⋊C4—2844no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—81no—
218C22⋊C4—82no—
219C42—81no—
220C42C2×C4×D7111yescharacteristic
221C42C2×Dic14211yescharacteristic
222C42#1176586512yes—
223C23—22no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23#1455609711yescharacteristic
229Q8—42no—
230Q8—81no—
231D4—42no—
232D4—81no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—5681no—
248C2×C4—5682no—
249C2×C4C2×Dic71⋊C411yescharacteristic
250C2×C4C4×C22⋊D7112yes—
251C2×C4#1455610712yes—
252C2×C4#1455611111yescharacteristic
253C8—2842no—
254C8—2844no—
255C22—42no—
256C22—44no—
257C22—82no—
258C22—81no—
259C22—81no—
260C22#1798587211yescharacteristic
261C22#1798616711yescharacteristic
262C22#1798617212yes—
263C22#1798617711yescharacteristic
264C22#1798617712yes—
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—82no—
271C4—82no—
272C4—11361no—
273C2—82no—
274C2#2258763811yescharacteristic
275C2#2258764611yescharacteristic
276C2#2258766312yes—
277C2#2258855011yescharacteristic
278C2#2258857312yes—
subgroupG/H
centerC23#14556097
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2379C2
socleC23×C71#120
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2379D71
hypocenterC71#8914
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
2. #22588473C2semidirect
4. #22586221C2semidirect
13. #17985739C4semidirect
28. #14556039C2×C4semidirect
145. C71#8914semidirect

Maximal subgroups

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

subgroupindex
#225891572
#225884732
C2×#179857392
#225862212
C71×#23792
#891471