Finite Group Databasebeta

← #31891809

Subgroups — #31891809

Order 18176.

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

normal subgroups only

This group has 500 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#22588508C212yesmaximal, complemented
4#22586222C212yesmaximal, complemented
5C2×#329×C71C211yesmaximal, characteristic
6C2×#14556062—22no—
7C2×#14556060—22no—
8C2×#14556052—21no—
9C2×#14556049—21no—
10C2×#14556039C2211yescharacteristic
11#17985906—21no—
12#17985892—22no—
13#17985879—21no—
14#17985874—22no—
15#17985869C2211yescharacteristic
16#17985865C2212yes—
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#14556062—44no—
31#14556060—44no—
32#14556052—42no—
33#14556049—42no—
34#14556039—22no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—22no—
38C22×Q8×C71D411yescharacteristic
39C22×D4×C71D411yescharacteristic
40C22×C8×C71—22no—
41#124×C71—22no—
42#124×C71C2×C412yes—
43#124×C71Q812yes—
44C2×C22⋊C4×C71—22no—
45C2×C42×C71C2311yescharacteristic
46#97×C71—22no—
47#97×C71C2×C414yescomplemented
48C22×Dic71—22no—
49C22×Dic71—22no—
50C22×Dic71—41no—
51C22×Dic71—82no—
52C22×Dic71—81no—
53C2×C71⋊C8—22no—
54C2×C71⋊C8—41no—
55C2×C71⋊C8—41no—
56C4⋊Dic71—42no—
57C4⋊Dic71—42no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60C4⋊Dic71—42no—
61C4⋊Dic71—42no—
62C4⋊Dic71—81no—
63C22⋊Dic71—82no—
64C22⋊Dic71—81no—
65C4×Dic71—81no—
66C24×C71—21no—
67C2×Q8×C71—21no—
68C2×Q8×C71—41no—
69C2×Q8×C71—41no—
70C2×Q8×C71C4⋊C412yes—
71C2×D4×C71—21no—
72C2×D4×C71—41no—
73C2×D4×C71—41no—
74C2×D4×C71C4⋊C412yes—
75C22×C4×C71—21no—
76C22×C4×C71C2×D412yes—
77C22×C4×C71C2×D411yescharacteristic
78C2×C8×C71—24no—
79C2×C8×C71—24no—
80C2×C8×C71—24no—
81C22⋊C4×C71—24no—
82C22⋊C4×C71—24no—
83C42×C71C22×C411yescharacteristic
84C42×C71C2×Q811yescharacteristic
85C42×C71#4112yes—
86C23×C71—22no—
87C23×C71—41no—
88C23×C71—42no—
89C23×C71—41no—
90C23×C71—42no—
91C23×C71#12011yescharacteristic
92Q8×C71—42no—
93Q8×C71—81no—
94D4×C71—42no—
95D4×C71—81no—
96C2×C4×C71—22no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71C2×C4⋊C411yescharacteristic
104C2×C4×C71C4×D412yes—
105C2×C4×C71C22⋊Q812yes—
106C2×C4×C71#12311yescharacteristic
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—84no—
114C2×Dic71—82no—
115C2×Dic71—81no—
116C8×C71—28no—
117C71⋊C8—42no—
118C22×C71—42no—
119C22×C71—44no—
120C22×C71—81no—
121C22×C71—81no—
122C22×C71—82no—
123C22×C71#38511yescharacteristic
124C22×C71#45312yes—
125C22×C71#45311yescharacteristic
126C22×C71#45511yescharacteristic
127C22×C71#45612yes—
128C4×C71—21no—
129C4×C71—21no—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—42no—
133Dic71—82no—
134Dic71—82no—
135Dic71—161no—
136#8910—711nomaximal, Sylow
137C2×C71—82no—
138C2×C71#165511yescharacteristic
139C2×C71#166011yescharacteristic
140C2×C71#166112yes—
141C2×C71#185611yescharacteristic
142C2×C71#185912yes—
143#2379—711no—
144#1959—711no—
145#1847—712no—
146C2×#329D7111yescharacteristic
147#1295—712no—
148C71#891011yescharacteristic, complemented, Sylow
149C2×#124C2×D7111yescharacteristic
150C2×#107—1422no—
151C2×#106—1422no—
152C2×#103—1421no—
153C2×#102—1421no—
154C2×#97—711no—
155C2×#97C2×D7112yes—
156#391—1421no—
157#388—1422no—
158#386—1421no—
159#385—1422no—
160#384—711no—
161#382—712no—
162#329Dic7114yescomplemented
163C22×Q8D28411yescharacteristic
164C22×D4D28411yescharacteristic
165C23×C4—1421no—
166C22×C8—22no—
167C22×C8—1421no—
168#124—22no—
169#124Dic14212yes—
170#124C2×Dic7112yes—
171C2×C4⋊C4—712no—
172C2×C4⋊C4—712no—
173C2×C4⋊C4—1422no—
174C2×C4⋊C4—1421no—
175C2×C4⋊C4—1421no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—1422no—
178C2×C22⋊C4—1421no—
179C2×C42—1421no—
180C2×C42C22×D7111yescharacteristic
181#107—2844no—
182#106—2844no—
183#103—2842no—
184#102—2842no—
185#97—22no—
186#97—1422no—
187#97C2×Dic7114yescomplemented
188#95—1422no—
189#95—1422no—
190#95—1422no—
191C24—21no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊Dic7112yes—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4C4⋊Dic7112yes—
200C22×C4—21no—
201C22×C4—2841no—
202C22×C4—5682no—
203C22×C4—5681no—
204C22×C4—1422no—
205C22×C4—1422no—
206C22×C4D4×D7112yes—
207C22×C4C2×D28411yescharacteristic
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—2841no—
212C2×C8—2841no—
213C2×C8—1422no—
214C4⋊C4—2842no—
215C4⋊C4—2842no—
216C4⋊C4—2841no—
217C4⋊C4—2841no—
218C4⋊C4—2842no—
219C4⋊C4—2842no—
220C4⋊C4—5681no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—5682no—
224C22⋊C4—5681no—
225C42—5681no—
226C42C22×Dic7111yescharacteristic
227C42C2×Dic14211yescharacteristic
228C42#1176586412yes—
229C23—22no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23#1455609811yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—42no—
238D4—81no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—2842no—
247C2×C4—2842no—
248C2×C4—2842no—
249C2×C4—2842no—
250C2×C4—2842no—
251C2×C4—2842no—
252C2×C4—5684no—
253C2×C4—5682no—
254C2×C4—5681no—
255C2×C4C2×C4⋊Dic7111yescharacteristic
256C2×C4D4×Dic7112yes—
257C2×C4#1455610512yes—
258C2×C4#1455611211yescharacteristic
259C8—28no—
260C8—2842no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—82no—
265C22—81no—
266C22#1798587411yescharacteristic
267C22#1798616812yes—
268C22#1798616911yescharacteristic
269C22#1798617911yescharacteristic
270C22#1798618312yes—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—5682no—
277C4—5682no—
278C4—11361no—
279C2—82no—
280C2#2258763011yescharacteristic
281C2#2258765211yescharacteristic
282C2#2258766012yes—
283C2#2258855211yescharacteristic
284C2#2258856812yes—
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#8910
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. #22588508C2semidirect
4. #22586222C2semidirect
19. #329×C71C4semidirect
47. #97×C71C2×C4semidirect
148. C71#8910semidirect
162. #329Dic71semidirect
187. #97C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915142
#225891572
#225885082
#225862222
C2×#329×C712
#891071