Finite Group Databasebeta

← #31902420

Subgroups — #31902420

Order 18176.

356 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 89 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#22588480C211yesmaximal, characteristic, complemented
3#22587110C211yesmaximal, characteristic, complemented
4#22586766C211yesmaximal, characteristic
5#22586764C211yesmaximal, characteristic
6#22586170C211yesmaximal, characteristic, complemented
7C71×#1571C211yesmaximal, characteristic, complemented
8C2×C4×D284C2211yescharacteristic
9#17986052C2211yescharacteristic
10#17986023C2211yescharacteristic
11C2×(C2×D71)⋊C8—22no—
12C2×#14556039C2211yescharacteristic
13#17985763—22no—
14#17985749—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—
31C4⋊(C71⋊C8)—21no—
32C4⋊(C71⋊C8)—21no—
33C4⋊(C71⋊C8)—21no—
34C22⋊(C71⋊C8)—21no—
35(C2×D71)⋊C8—44no—
36#14556039—21no—
37#14556039—21no—
38C4×C71⋊C8D412yes—
39C2×M16×C71—21no—
40C22×C8×C71—21no—
41C2×C42×C71C2311yescharacteristic
42C4⋊C8×C71—21no—
43C4⋊C8×C71—21no—
44C4⋊C8×C71—21no—
45C4⋊C8×C71—21no—
46C4⋊C8×C71—21no—
47C22⋊C8×C71—21no—
48C4×C8×C71D412yes—
49C23×D71—21no—
50C2×D284—21no—
51C2×D284—21no—
52C2×D284—42no—
53C22×Dic71—21no—
54C2×C4×D71—41no—
55C2×C4×D71—41no—
56C2×C4×D71—42no—
57C2×C4×D71—42no—
58#11765849—42no—
59C2×C71⋊C8—22no—
60C2×C71⋊C8—22no—
61C2×C71⋊C8—22no—
62C2×C71⋊C8—22no—
63C2×C71⋊C8—41no—
64C4⋊Dic71—21no—
65C4⋊Dic71—21no—
66(C2×D71)⋊C4—41no—
67(C2×D71)⋊C4—41no—
68C22×C4×C71C2×D411yescharacteristic
69C22×C4×C71C2×Q811yescharacteristic
70C22×C4×C71#4111yescharacteristic
71M16×C71—42no—
72C2×C8×C71—22no—
73C2×C8×C71—41no—
74C2×C8×C71—41no—
75C2×C8×C71—42no—
76C42×C71C2×D411yescharacteristic
77C42×C71C2×D411yescharacteristic
78C42×C71#4111yescharacteristic
79C42×C71#4111yescharacteristic
80C23×C71C22⋊Q811yescharacteristic
81C22×D71—22no—
82C22×D71—41no—
83C22×D71—42no—
84C22×D71—41no—
85C22×D71—42no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71C22⋊Q811yescharacteristic
89C2×C4×C71C22⋊Q811yescharacteristic
90C2×C4×C71#12311yescharacteristic
91C2×C4×C71#12311yescharacteristic
92C2×C4×C71#12311yescharacteristic
93C2×C4×C71#12711yescharacteristic
94C2×C4×C71#13512yes—
95C2×C4×C71AGL(1,ℤ/8)12yes—
96C2×Dic71—41no—
97C2×Dic71—42no—
98D284—22no—
99D284—22no—
100D284—41no—
101D284—41no—
102C4×D71—82no—
103C4×D71—81no—
104C4×D71—81no—
105C8×C71—42no—
106C8×C71—81no—
107C8×C71—81no—
108C71⋊C8—42no—
109C71⋊C8—42no—
110C71⋊C8—81no—
111C22×C71—21no—
112C22×C71—21no—
113C22×C71#39911yescharacteristic
114C22×C71#47812yes—
115C2×D71—44no—
116C2×D71—42no—
117C2×D71—82no—
118C2×D71—81no—
119C2×D71—81no—
120C4×C71—21no—
121C4×C71—21no—
122C4×C71—21no—
123C4×C71—21no—
124C4×C71#48212yes—
125C4×C71#49511yescharacteristic
126C4×C71#49611yescharacteristic
127Dic71—81no—
128#10659—711nomaximal, Sylow
129C2×C71—22no—
130C2×C71#209111yescharacteristic
131C2×C71#209211yescharacteristic
132C2×C71#209811yescharacteristic
133D71—82no—
134#1955—711no—
135#1838—711no—
136#1571D7111yescharacteristic, complemented
137#1570—711no—
138#1486—711no—
139#1486—711no—
140#1275—711no—
141C71#1065911yescharacteristic, complemented, Sylow
142C2×C4×D4—711no—
143#431—711no—
144#431C2×D7111yescharacteristic
145#423—711no—
146#423C2×D7111yescharacteristic
147C2×C4⋊C8C2×D7111yescharacteristic
148C2×C22⋊C8—1422no—
149C2×#97—711no—
150#341—1422no—
151#335—22no—
152#334—1422no—
153D4⋊C8—1422no—
154D4⋊C8—1422no—
155C22×D4—711no—
156C23×C4—1421no—
157C2×M16—21no—
158C2×M16—1421no—
159C22×C8—21no—
160C22×C8—712no—
161C4×D4—1421no—
162C4×D4—1421no—
163C4×D4—1421no—
164C4×D4—1421no—
165C2×C4⋊C4—711no—
166C2×C22⋊C4—1421no—
167C2×C42C22×D7111yescharacteristic
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—1421no—
174C4⋊C8—1421no—
175C4⋊C8—1421no—
176C22⋊C8—21no—
177C22⋊C8—2844no—
178C22⋊C8—1421no—
179#97—1421no—
180#97—1421no—
181C4×C8—712no—
182C4×C8C22⋊D7112yes—
183C24—1421no—
184C2×D4—2842no—
185C2×D4—1421no—
186C2×D4—1421no—
187C22×C4—2841no—
188C22×C4—2841no—
189C22×C4—2842no—
190C22×C4—2842no—
191C22×C4—1421no—
192C22×C4D4×D7111yescharacteristic
193C22×C4Q8×D7111yescharacteristic
194C22×C4#1176586511yescharacteristic
195M16—42no—
196M16—2842no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—2841no—
202C2×C8—1422no—
203C2×C8—1422no—
204C2×C8—1422no—
205C2×C8—1422no—
206C4⋊C4—1421no—
207C4⋊C4—1421no—
208C22⋊C4—2841no—
209C22⋊C4—2841no—
210C42D4×D7111yescharacteristic
211C42C2×C22⋊D7111yescharacteristic
212C42#1176586511yescharacteristic
213C42#1176586611yescharacteristic
214C23—2841no—
215C23—2842no—
216C23—2841no—
217C23—2842no—
218C23—1422no—
219C23#1455610611yescharacteristic
220D4—2841no—
221D4—2841no—
222D4—1422no—
223D4—1422no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—2841no—
227C2×C4—2842no—
228C2×C4—5682no—
229C2×C4—5681no—
230C2×C4—5681no—
231C2×C4#1455610411yescharacteristic
232C2×C4#1455610611yescharacteristic
233C2×C4#1455611111yescharacteristic
234C2×C4#1455611311yescharacteristic
235C2×C4#1455611311yescharacteristic
236C2×C4#1455612411yescharacteristic
237C2×C4#1455615112yes—
238C2×C4#1455615212yes—
239C8—42no—
240C8—81no—
241C8—81no—
242C8—2842no—
243C8—2842no—
244C8—5681no—
245C22—21no—
246C22—21no—
247C22—2844no—
248C22—2842no—
249C22—5682no—
250C22—5681no—
251C22—5681no—
252C22#1798593411yescharacteristic
253C22#1798630212yes—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—5681no—
259C4#1798632612yes—
260C4#1798638411yescharacteristic
261C4#1798638711yescharacteristic
262C2—22no—
263C2—5682no—
264C2#2258985511yescharacteristic
265C2#2258985811yescharacteristic
266C2#2258989611yescharacteristic
subgroupG/H
centerC2×C4#14556124
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1571C2
socleC22×C71#399
prime radicalC22×D284Q8
hypercenter#1571D71
hypocenterC71#10659
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. #22588480C2semidirect
3. #22587110C2semidirect
6. #22586170C2semidirect
7. C71×#1571C2semidirect
136. #1571D71semidirect
141. C71#10659semidirect

Maximal subgroups

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

subgroupindex
#225891372
#225884802
#225871102
#225867662
#225867642
#225861702
C71×#15712
#1065971