Finite Group Databasebeta

← #31894867

Subgroups — #31894867

Order 18176.

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

normal subgroups only

This group has 570 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#22593293C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22587024C212yesmaximal, complemented
4#22586242C212yesmaximal, complemented
5C71×#2975C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7C2×#14556107—22no—
8#17986023C2212yes—
9C2×#14556045—21no—
10#17985935C414yes—
11#17985902—21no—
12#17985868—21no—
13C4×C4⋊Dic71C2211yescharacteristic
14#17985768—21no—
15#17985746—24no—
16C2×C4×D4×C71C2211yescharacteristic
17#436×C71—24no—
18#431×C71C2212yes—
19#407×C71—22no—
20C22×Dic142—21no—
21C2×#11765849—22no—
22#14556128—22no—
23#14556128—22no—
24#14556107—48no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71C2×C412yes—
28C2×Dic71⋊C4—24no—
29C2×C22⋊Dic71—24no—
30C2×C4×Dic71—21no—
31C4⋊(C71⋊C8)—22no—
32C4⋊(C71⋊C8)—22no—
33#14556045—42no—
34C71⋊#95C2×C412yes—
35C22×D4×C71D411yescharacteristic
36C23×C4×C71D412yes—
37C2×M16×C71—24no—
38C4×D4×C71—22no—
39C4×D4×C71—22no—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71C2×C412yes—
42C2×C42×C71C2311yescharacteristic
43C4⋊C8×C71D414yes—
44C22⋊C8×C71—24no—
45C22⋊C8×C71—22no—
46C4×C8×C71—22no—
47C2×Dic142—44no—
48C2×Dic142—41no—
49C2×Dic142—41no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52#11765849—42no—
53#11765849—42no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—24no—
56C4⋊Dic71—24no—
57C4⋊Dic71—42no—
58Dic71⋊C4—48no—
59C22⋊Dic71—48no—
60C4×Dic71—41no—
61C4×Dic71—41no—
62C24×C71C22⋊C412yes—
63C2×D4×C71—22no—
64C2×D4×C71—21no—
65C2×D4×C71—22no—
66C2×D4×C71—21no—
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—44no—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74M16×C71—84no—
75C2×C8×C71—24no—
76C4⋊C4×C71—22no—
77C22⋊C4×C71—24no—
78C42×C71C2×D412yes—
79C42×C71#4112yes—
80C23×C71—22no—
81C23×C71—22no—
82C23×C71—22no—
83C23×C71—42no—
84C23×C71—42no—
85C23×C71C2×C22⋊C411yescharacteristic
86D4×C71—42no—
87D4×C71—42no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—41no—
91C2×C4×C71—44no—
92C2×C4×C71—42no—
93C2×C4×C71—82no—
94C2×C4×C71C4×D412yes—
95C2×C4×C71#12011yescharacteristic
96C2×C4×C71#12112yes—
97C2×C4×C71#12311yescharacteristic
98C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
99C2×C4×C71#13712yes—
100C2×C4×C71#13711yescharacteristic
101C2×Dic71—24no—
102C2×Dic71—42no—
103C2×Dic71—42no—
104C2×Dic71—44no—
105C2×Dic71—42no—
106Dic142—82no—
107Dic142—82no—
108C8×C71—44no—
109C71⋊C8—82no—
110C22×C71—21no—
111C22×C71—21no—
112C22×C71—44no—
113C22×C71—42no—
114C22×C71—42no—
115C22×C71—42no—
116C22×C71—82no—
117C22×C71#38611yescharacteristic
118C22×C71#41711yescharacteristic
119C22×C71#41911yescharacteristic
120C4×C71—21no—
121C4×C71—22no—
122C4×C71—42no—
123C4×C71#45312yes—
124C4×C71#46812yes—
125C4×C71#47012yes—
126Dic71—44no—
127Dic71—82no—
128#9403—711nomaximal, Sylow
129C2×C71—21no—
130C2×C71—21no—
131C2×C71—82no—
132C2×C71#188711yescharacteristic
133C2×C71#189411yescharacteristic
134C2×C71#191511yescharacteristic
135#2975D7111yescharacteristic
136#2542—711no—
137#1800—711no—
138#1549—712no—
139#1303—712no—
140C71#940311yescharacteristic, complemented, Sylow
141C2×C4⋊Q8—711no—
142C2×C22⋊Q8—1422no—
143C2×C4×D4C2×D7111yescharacteristic
144#436—24no—
145#431C2×D7112yes—
146#423—712no—
147C2×#100—1421no—
148#407—22no—
149#399—714no—
150#390—1421no—
151#384—1421no—
152C4×C4⋊C4—711no—
153#343—1421no—
154#332—1424no—
155C22×Q8—1421no—
156C22×D4D28411yescharacteristic
157C23×C4C22⋊D7112yes—
158C2×M16—24no—
159C2×M16—1422no—
160C4⋊Q8—1422no—
161C4⋊Q8—1422no—
162C22⋊Q8—2848no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—712no—
166C2×C4⋊C4—1424no—
167C2×C4⋊C4—1421no—
168C2×C4⋊C4—1421no—
169C2×C4⋊C4D28411yescharacteristic
170C2×C22⋊C4—1424no—
171C2×C22⋊C4C4×D7112yes—
172C2×C42—1421no—
173C2×C42C22×D7111yescharacteristic
174C4⋊C8—1422no—
175C4⋊C8—1422no—
176C4⋊C8C22⋊D7114yes—
177#100—2842no—
178C22⋊C8—24no—
179C22⋊C8—22no—
180C4×C8—22no—
181#95—712no—
182C24(C2×D71)⋊C412yes—
183C2×Q8—2844no—
184C2×Q8—2841no—
185C2×Q8—2841no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—1422no—
194C22×C4—1422no—
195C22×C4(C2×D71)⋊C412yes—
196C22×C4C2×C4×D7111yescharacteristic
197C22×C4C2×C22⋊D7111yescharacteristic
198C22×C4C2×D28411yescharacteristic
199M16—84no—
200M16—2842no—
201M16—2842no—
202C2×C8—24no—
203C2×C8—2842no—
204C4⋊C4—22no—
205C4⋊C4—2848no—
206C4⋊C4—2842no—
207C4⋊C4—1424no—
208C4⋊C4—1424no—
209C22⋊C4—24no—
210C22⋊C4—2848no—
211C42—2841no—
212C42—2841no—
213C42C2×C22⋊D7112yes—
214C42#1176586512yes—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23C2×(C2×D71)⋊C411yescharacteristic
221Q8—5682no—
222Q8—5682no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—2842no—
232C2×C4—2842no—
233C2×C4—2844no—
234C2×C4—2842no—
235C2×C4—1424no—
236C2×C4C4×C22⋊D7112yes—
237C2×C4#1455609711yescharacteristic
238C2×C4#1455610212yes—
239C2×C4#1455611111yescharacteristic
240C2×C4#1455615611yescharacteristic
241C2×C4#1455616112yes—
242C2×C4#1455616111yescharacteristic
243C8—44no—
244C8—5682no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#1798587811yescharacteristic
253C22#1798600111yescharacteristic
254C22#1798600911yescharacteristic
255C4—21no—
256C4—22no—
257C4—42no—
258C4—2844no—
259C4—5682no—
260C4#1798616712yes—
261C4#1798624512yes—
262C4#1798625712yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2258873811yescharacteristic
267C2#2258877311yescharacteristic
268C2#2258889211yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2975C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2975D71
hypocenterC71#9403
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #22587024C2semidirect
4. #22586242C2semidirect
140. C71#9403semidirect

Maximal subgroups

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

subgroupindex
#225932932
#225883322
#225870242
#225862422
C71×#29752
#940371