Finite Group Databasebeta

← #31902427

Subgroups — #31902427

Order 18176.

372 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 93 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#22589137C211yesmaximal, characteristic
2#22588514C211yesmaximal, characteristic, complemented
3#22587114C211yesmaximal, characteristic, complemented
4#22586766C211yesmaximal, characteristic
5#22586764C211yesmaximal, characteristic
6#22586170C211yesmaximal, characteristic, complemented
7C71×#1571C211yesmaximal, characteristic, complemented
8C2×C4×D284C2211yescharacteristic
9#17986052C2211yescharacteristic
10C2×#14556063—22no—
11#17986023C2211yescharacteristic
12C2×(C2×D71)⋊C8—22no—
13C2×#14556039C2211yescharacteristic
14#17985751—22no—
15#17985728—22no—
16#17985726—22no—
17#431×C71C2211yescharacteristic
18#423×C71C2211yescharacteristic
19C2×C4⋊C8×C71C2211yescharacteristic
20#335×C71—22no—
21C22×D284Q811yescharacteristic
22C22×C4×D71—21no—
23C2×#11765849—21no—
24C22×C71⋊C8D412yes—
25C4×D284—21no—
26C4×D284—21no—
27C4×D284—21no—
28C4×D284—21no—
29C2×C4⋊Dic71Q811yescharacteristic
30C2×(C2×D71)⋊C4—21no—
31#14556063—44no—
32C4⋊(C71⋊C8)—21no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)—21no—
35C22⋊(C71⋊C8)—21no—
36(C2×D71)⋊C8—44no—
37#14556039—21no—
38#14556039—21no—
39C4×C71⋊C8D412yes—
40C2×M16×C71—21no—
41C22×C8×C71—21no—
42C2×C42×C71C2311yescharacteristic
43C4⋊C8×C71—21no—
44C4⋊C8×C71—21no—
45C4⋊C8×C71—21no—
46C4⋊C8×C71—21no—
47C4⋊C8×C71—21no—
48C22⋊C8×C71—21no—
49C4×C8×C71D412yes—
50C23×D71—21no—
51C2×D284—21no—
52C2×D284—21no—
53C2×D284—42no—
54C22×Dic71—21no—
55C2×C4×D71—41no—
56C2×C4×D71—41no—
57C2×C4×D71—42no—
58C2×C4×D71—42no—
59#11765849—42no—
60C2×C71⋊C8—22no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—22no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8D814yes—
65C4⋊Dic71—21no—
66C4⋊Dic71—21no—
67(C2×D71)⋊C4—41no—
68(C2×D71)⋊C4—41no—
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×Q811yescharacteristic
71C22×C4×C71#4111yescharacteristic
72M16×C71—42no—
73C2×C8×C71—22no—
74C2×C8×C71—41no—
75C2×C8×C71—41no—
76C2×C8×C71—42no—
77C42×C71C2×D411yescharacteristic
78C42×C71C2×D411yescharacteristic
79C42×C71#4111yescharacteristic
80C42×C71#4111yescharacteristic
81C23×C71C22⋊Q811yescharacteristic
82C22×D71—22no—
83C22×D71—41no—
84C22×D71—42no—
85C22×D71—41no—
86C22×D71—42no—
87C2×C4×C71—41no—
88C2×C4×C71—41no—
89C2×C4×C71C22⋊Q811yescharacteristic
90C2×C4×C71C22⋊Q811yescharacteristic
91C2×C4×C71#12311yescharacteristic
92C2×C4×C71#12311yescharacteristic
93C2×C4×C71#12311yescharacteristic
94C2×C4×C71#12711yescharacteristic
95C2×C4×C71C2×D812yes—
96C2×C4×C71#13712yes—
97C2×Dic71—41no—
98C2×Dic71—42no—
99D284—22no—
100D284—22no—
101D284—41no—
102D284—41no—
103C4×D71—82no—
104C4×D71—81no—
105C4×D71—81no—
106C8×C71—42no—
107C8×C71—81no—
108C8×C71—81no—
109C71⋊C8—24no—
110C71⋊C8—42no—
111C71⋊C8—81no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71#39911yescharacteristic
115C22×C71#47412yes—
116C2×D71—44no—
117C2×D71—42no—
118C2×D71—82no—
119C2×D71—81no—
120C2×D71—81no—
121C4×C71—21no—
122C4×C71—21no—
123C4×C71—21no—
124C4×C71—21no—
125C4×C71#48012yes—
126C4×C71#49311yescharacteristic
127C4×C71#49711yescharacteristic
128Dic71—81no—
129#10660—711nomaximal, Sylow
130C2×C71—22no—
131C2×C71#209111yescharacteristic
132C2×C71#209311yescharacteristic
133C2×C71#209811yescharacteristic
134D71—82no—
135#1955—711no—
136#1849—711no—
137#1571—711no—
138#1571D7111yescharacteristic, complemented
139#1486—711no—
140#1486—711no—
141#1275—711no—
142C71#1066011yescharacteristic, complemented, Sylow
143C2×C4×D4—711no—
144#431—711no—
145#431C2×D7111yescharacteristic
146C2×#108—1422no—
147#423—711no—
148#423C2×D7111yescharacteristic
149C2×C4⋊C8C2×D7111yescharacteristic
150C2×C22⋊C8—1422no—
151C2×#97—711no—
152#335—22no—
153#335—1422no—
154D4⋊C8—1422no—
155D4⋊C8—1422no—
156C22×D4—711no—
157C23×C4—1421no—
158C2×M16—21no—
159C2×M16—1421no—
160C22×C8—21no—
161C22×C8—712no—
162C4×D4—1421no—
163C4×D4—1421no—
164C4×D4—1421no—
165C4×D4—1421no—
166C2×C4⋊C4—711no—
167C2×C22⋊C4—1421no—
168C2×C42C22×D7111yescharacteristic
169#108—2844no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—1421no—
176C4⋊C8—1421no—
177C4⋊C8—1421no—
178C22⋊C8—21no—
179C22⋊C8—2844no—
180C22⋊C8—1421no—
181#97—1421no—
182#97—1421no—
183C4×C8—712no—
184C4×C8C22⋊D7112yes—
185C24—1421no—
186C2×D4—2842no—
187C2×D4—1421no—
188C2×D4—1421no—
189C22×C4—2842no—
190C22×C4—2841no—
191C22×C4—2841no—
192C22×C4—2842no—
193C22×C4—1421no—
194C22×C4D4×D7111yescharacteristic
195C22×C4Q8×D7111yescharacteristic
196C22×C4#1176586511yescharacteristic
197M16—42no—
198M16—2842no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—2841no—
204C2×C8—714no—
205C2×C8—1422no—
206C2×C8—1422no—
207C2×C8—1422no—
208C4⋊C4—1421no—
209C4⋊C4—1421no—
210C22⋊C4—2841no—
211C22⋊C4—2841no—
212C42D4×D7111yescharacteristic
213C42C2×C22⋊D7111yescharacteristic
214C42#1176586511yescharacteristic
215C42#1176586611yescharacteristic
216C23—2841no—
217C23—2842no—
218C23—2841no—
219C23—2842no—
220C23—1422no—
221C23#1455610611yescharacteristic
222D4—2841no—
223D4—2841no—
224D4—1422no—
225D4—1422no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—2841no—
229C2×C4—2842no—
230C2×C4—5682no—
231C2×C4—5681no—
232C2×C4—5681no—
233C2×C4#1455610411yescharacteristic
234C2×C4#1455610611yescharacteristic
235C2×C4#1455611111yescharacteristic
236C2×C4#1455611311yescharacteristic
237C2×C4#1455611311yescharacteristic
238C2×C4#1455612411yescharacteristic
239C2×C4D8×D7112yes—
240C2×C4#1455616012yes—
241C8—42no—
242C8—81no—
243C8—81no—
244C8—2842no—
245C8—5681no—
246C8—1424no—
247C22—21no—
248C22—21no—
249C22—2844no—
250C22—2842no—
251C22—5682no—
252C22—5681no—
253C22—5681no—
254C22#1798593411yescharacteristic
255C22#1798627812yes—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—21no—
260C4—5681no—
261C4#1798631412yes—
262C4#1798637411yescharacteristic
263C4#1798639211yescharacteristic
264C2—22no—
265C2—5682no—
266C2#2258985511yescharacteristic
267C2#2258986511yescharacteristic
268C2#2258989611yescharacteristic
subgroupG/H
centerC2×C4#14556124
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1571C2
socleC22×C71#399
prime radicalC22×D284Q8
hypercenter#1571D71
hypocenterC71#10660
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
2. #22588514C2semidirect
3. #22587114C2semidirect
6. #22586170C2semidirect
7. C71×#1571C2semidirect
138. #1571D71semidirect
142. C71#10660semidirect

Maximal subgroups

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

subgroupindex
#225891372
#225885142
#225871142
#225867662
#225867642
#225861702
C71×#15712
#1066071