Finite Group Databasebeta

← #31891833

Subgroups — #31891833

Order 18176.

468 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 119 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#22591514C211yesmaximal, characteristic
2#22589157C211yesmaximal, characteristic
3#22588474C212yesmaximal, complemented
4#22586222C212yesmaximal, complemented
5C2×#329×C71C211yesmaximal, characteristic
6C2×#14556052—21no—
7C2×#14556049—21no—
8C2×#14556039C2211yescharacteristic
9#17985906—21no—
10#17985892—22no—
11#17985879—21no—
12#17985874—22no—
13#17985869C2211yescharacteristic
14#17985865C2212yes—
15#17985762—22no—
16#17985762—22no—
17C2×#124×C71C2211yescharacteristic
18C2×#97×C71C2212yes—
19#329×C71C414yescomplemented
20C23×Dic71—21no—
21C22×C71⋊C8—21no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71D412yes—
26C2×C4⋊Dic71D412yes—
27C2×C22⋊Dic71—22no—
28C2×C22⋊Dic71—21no—
29C2×C4×Dic71—21no—
30#14556052—42no—
31#14556049—42no—
32#14556039—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C22×Q8×C71D411yescharacteristic
37C22×D4×C71D411yescharacteristic
38C22×C8×C71—22no—
39#124×C71—22no—
40#124×C71C2×C412yes—
41#124×C71Q812yes—
42C2×C22⋊C4×C71—22no—
43C2×C42×C71C2311yescharacteristic
44#97×C71—22no—
45#97×C71C2×C414yescomplemented
46C22×Dic71—22no—
47C22×Dic71—22no—
48C22×Dic71—41no—
49C22×Dic71—81no—
50C22×Dic71—82no—
51C2×C71⋊C8—22no—
52C2×C71⋊C8—41no—
53C2×C71⋊C8—41no—
54C4⋊Dic71—42no—
55C4⋊Dic71—42no—
56C4⋊Dic71—41no—
57C4⋊Dic71—41no—
58C4⋊Dic71—44no—
59C4⋊Dic71—81no—
60C22⋊Dic71—81no—
61C22⋊Dic71—82no—
62C4×Dic71—81no—
63C24×C71—21no—
64C2×Q8×C71—21no—
65C2×Q8×C71—41no—
66C2×Q8×C71—41no—
67C2×Q8×C71C4⋊C412yes—
68C2×D4×C71—21no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71C4⋊C412yes—
72C22×C4×C71—21no—
73C22×C4×C71C2×D412yes—
74C22×C4×C71C2×D411yescharacteristic
75C2×C8×C71—24no—
76C2×C8×C71—42no—
77C2×C8×C71—42no—
78C22⋊C4×C71—24no—
79C22⋊C4×C71—24no—
80C42×C71C22×C411yescharacteristic
81C42×C71C2×Q811yescharacteristic
82C42×C71#4112yes—
83C23×C71—22no—
84C23×C71—41no—
85C23×C71—42no—
86C23×C71—41no—
87C23×C71—42no—
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×C71C2×C4⋊C411yescharacteristic
101C2×C4×C71C4×D412yes—
102C2×C4×C71C22⋊Q812yes—
103C2×C4×C71#12311yescharacteristic
104C2×Dic71—42no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—84no—
111C2×Dic71—81no—
112C2×Dic71—82no—
113C8×C71—44no—
114C71⋊C8—42no—
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#45512yes—
124C22×C71#45511yescharacteristic
125C4×C71—21no—
126C4×C71—21no—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—42no—
130Dic71—82no—
131Dic71—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#2379—711no—
141#1959—711no—
142#1836—712no—
143C2×#329D7111yescharacteristic
144#1295—712no—
145C71#891411yescharacteristic, complemented, Sylow
146C2×#124C2×D7111yescharacteristic
147C2×#103—1421no—
148C2×#102—1421no—
149C2×#97—711no—
150C2×#97C2×D7112yes—
151#391—1421no—
152#388—1422no—
153#386—1421no—
154#385—1422no—
155#384—711no—
156#382—712no—
157#340—1422no—
158#340—1422no—
159#329Dic7114yescomplemented
160C22×Q8D28411yescharacteristic
161C22×D4D28411yescharacteristic
162C23×C4—1421no—
163C22×C8—22no—
164C22×C8—1421no—
165#124—22no—
166#124Dic14212yes—
167#124C2×Dic7112yes—
168C2×C4⋊C4—712no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4—1422no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4—1421no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1422no—
175C2×C22⋊C4—1421no—
176C2×C42—1421no—
177C2×C42C22×D7111yescharacteristic
178#103—2842no—
179#102—2842no—
180#97—22no—
181#97—1422no—
182#97C2×Dic7114yescomplemented
183#95—1422no—
184#95—1422no—
185#95—1422no—
186C24—21no—
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8C4⋊Dic7112yes—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C4⋊Dic7112yes—
195C22×C4—21no—
196C22×C4—2841no—
197C22×C4—5681no—
198C22×C4—5682no—
199C22×C4—1422no—
200C22×C4—1422no—
201C22×C4D4×D7112yes—
202C22×C4C2×D28411yescharacteristic
203C2×C8—24no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—2841no—
207C2×C8—2841no—
208C2×C8—1422no—
209C4⋊C4—2842no—
210C4⋊C4—2842no—
211C4⋊C4—2841no—
212C4⋊C4—2841no—
213C4⋊C4—2844no—
214C4⋊C4—5681no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—5682no—
218C22⋊C4—5681no—
219C42—5681no—
220C42C22×Dic7111yescharacteristic
221C42C2×Dic14211yescharacteristic
222C42#1176586412yes—
223C23—22no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23#1455609811yescharacteristic
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—2842no—
241C2×C4—2842no—
242C2×C4—2842no—
243C2×C4—2842no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—5684no—
247C2×C4—5681no—
248C2×C4—5682no—
249C2×C4C2×C4⋊Dic7111yescharacteristic
250C2×C4D4×Dic7112yes—
251C2×C4#1455610512yes—
252C2×C4#1455611211yescharacteristic
253C8—44no—
254C8—2842no—
255C22—42no—
256C22—44no—
257C22—82no—
258C22—81no—
259C22—81no—
260C22#1798587411yescharacteristic
261C22#1798616911yescharacteristic
262C22#1798617312yes—
263C22#1798617812yes—
264C22#1798617911yescharacteristic
265C4—21no—
266C4—21no—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—5682no—
271C4—5682no—
272C4—11361no—
273C2—82no—
274C2#2258763911yescharacteristic
275C2#2258764711yescharacteristic
276C2#2258766512yes—
277C2#2258855211yescharacteristic
278C2#2258857512yes—
subgroupG/H
centerC23#14556098
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#329×C71C2
socleC23×C71#120
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenterC2×#329D71
hypocenterC71#8914
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
3. #22588474C2semidirect
4. #22586222C2semidirect
19. #329×C71C4semidirect
45. #97×C71C2×C4semidirect
145. C71#8914semidirect
159. #329Dic71semidirect
182. #97C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915142
#225891572
#225884742
#225862222
C2×#329×C712
#891471