Finite Group Databasebeta

← #31891807

Subgroups — #31891807

Order 18176.

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

normal subgroups only

This group has 500 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#22588509C212yesmaximal, complemented
3C2×#17985739C211yesmaximal, characteristic
4#22586221C212yesmaximal, complemented
5C71×#2379C211yesmaximal, characteristic
6C2×#14556061—22no—
7C2×#14556059—22no—
8C2×#14556052—21no—
9C2×#14556049—21no—
10C2×#14556039C2211yescharacteristic
11C2×#14556039C2212yes—
12#17985869C2211yescharacteristic
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—22no—
21C22×C71⋊C8—21no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—22no—
24#14556061—44no—
25#14556059—44no—
26#14556052—42no—
27#14556049—42no—
28#14556039—22no—
29#14556039—22no—
30#14556039C2×C414yescomplemented
31C22×Q8×C71D411yescharacteristic
32C22×D4×C71D411yescharacteristic
33C23×C4×C71—21no—
34#124×C71—22no—
35#124×C71C2×C412yes—
36#124×C71Q812yes—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71D412yes—
39C2×C4⋊C4×C71D412yes—
40C2×C22⋊C4×C71—22no—
41C2×C22⋊C4×C71—22no—
42C2×C22⋊C4×C71—21no—
43C2×C42×C71—21no—
44C2×C42×C71C2311yescharacteristic
45#95×C71—22no—
46#95×C71—22no—
47#95×C71—22no—
48C22×Dic71—41no—
49C2×C71⋊C8—24no—
50C2×C71⋊C8—24no—
51C2×C71⋊C8—24no—
52C2×C71⋊C8—22no—
53C2×C71⋊C8—41no—
54C2×C71⋊C8—41no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57C4⋊Dic71—42no—
58C4⋊Dic71—42no—
59C24×C71—21no—
60C2×Q8×C71—21no—
61C2×Q8×C71—41no—
62C2×Q8×C71—41no—
63C2×Q8×C71C4⋊C412yes—
64C2×D4×C71—21no—
65C2×D4×C71—41no—
66C2×D4×C71—41no—
67C2×D4×C71C4⋊C412yes—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—82no—
72C22×C4×C71—81no—
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71C2×D412yes—
75C4⋊C4×C71—42no—
76C4⋊C4×C71—42no—
77C4⋊C4×C71—81no—
78C22⋊C4×C71—24no—
79C22⋊C4×C71—24no—
80C22⋊C4×C71—82no—
81C22⋊C4×C71—81no—
82C42×C71—81no—
83C42×C71C22×C411yescharacteristic
84C42×C71C2×Q811yescharacteristic
85C42×C71#4112yes—
86C23×C71—22no—
87C23×C71—42no—
88C23×C71—41no—
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×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—84no—
110C2×C4×C71C2×C4⋊C411yescharacteristic
111C2×C4×C71C4×D412yes—
112C2×C4×C71C22⋊Q812yes—
113C2×C4×C71#12311yescharacteristic
114C2×Dic71—81no—
115C2×Dic71—82no—
116C71⋊C8—28no—
117C71⋊C8—42no—
118C22×C71—42no—
119C22×C71—44no—
120C22×C71—81no—
121C22×C71—81no—
122C22×C71—82no—
123C22×C71#38511yescharacteristic
124C22×C71#45311yescharacteristic
125C22×C71#45312yes—
126C22×C71#45511yescharacteristic
127C22×C71#45612yes—
128C4×C71—21no—
129C4×C71—21no—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—42no—
133C4×C71—82no—
134C4×C71—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#2379D7111yescharacteristic
144#1959—711no—
145#1847—712no—
146C2×#329—711no—
147#1295—712no—
148C71#891011yescharacteristic, complemented, Sylow
149C2×#124C2×D7111yescharacteristic
150C2×#107—1422no—
151C2×#106—1422no—
152C2×#103—1421no—
153C2×#102—1421no—
154C2×#97—712no—
155C2×#97—711no—
156#391—21no—
157#388—22no—
158#386—21no—
159#385—22no—
160#384—711no—
161#382C2×D7112yes—
162#329—714no—
163C22×Q8C22⋊D7111yescharacteristic
164C22×D4C22⋊D7111yescharacteristic
165C23×C4—21no—
166C22×C8—1421no—
167C22×C8—1422no—
168#124—22no—
169#124Dic14212yes—
170#124C4×D7112yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—1421no—
173C2×C4⋊C4—1422no—
174C2×C4⋊C4C22⋊D7112yes—
175C2×C4⋊C4C22⋊D7112yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C42—21no—
180C2×C42C22×D7111yescharacteristic
181#107—2844no—
182#106—2844no—
183#103—2842no—
184#102—2842no—
185#97—714no—
186#97—1422no—
187#97—1422no—
188#95—22no—
189#95—22no—
190#95—22no—
191C24—21no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8Dic71⋊C412yes—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4Dic71⋊C412yes—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—82no—
204C22×C4—81no—
205C22×C4—2841no—
206C22×C4C2×C22⋊D7111yescharacteristic
207C22×C4C2×C22⋊D7112yes—
208C2×C8—2841no—
209C2×C8—2841no—
210C2×C8—1422no—
211C2×C8—1424no—
212C2×C8—1424no—
213C2×C8—1424no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—2841no—
218C4⋊C4—2841no—
219C4⋊C4—2842no—
220C4⋊C4—2842no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—82no—
224C22⋊C4—81no—
225C42—81no—
226C42C2×C4×D7111yescharacteristic
227C42C2×Dic14211yescharacteristic
228C42#1176586512yes—
229C23—22no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23#1455609711yescharacteristic
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—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—5681no—
254C2×C4—5682no—
255C2×C4C2×Dic71⋊C411yescharacteristic
256C2×C4C4×C22⋊D7112yes—
257C2×C4#1455610712yes—
258C2×C4#1455611111yescharacteristic
259C8—2842no—
260C8—1428no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—82no—
265C22—81no—
266C22#1798587211yescharacteristic
267C22#1798616711yescharacteristic
268C22#1798616712yes—
269C22#1798617711yescharacteristic
270C22#1798618212yes—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—82no—
277C4—82no—
278C4—11361no—
279C2—82no—
280C2#2258762911yescharacteristic
281C2#2258765111yescharacteristic
282C2#2258765712yes—
283C2#2258855011yescharacteristic
284C2#2258856612yes—
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#8910
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. #22588509C2semidirect
4. #22586221C2semidirect
13. #17985739C4semidirect
30. #14556039C2×C4semidirect
148. C71#8910semidirect

Maximal subgroups

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

subgroupindex
#225891572
#225885092
C2×#179857392
#225862212
C71×#23792
#891071