Finite Group Databasebeta

← #31894882

Subgroups — #31894882

Order 18176.

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

normal subgroups only

This group has 554 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#22593339C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic, complemented
3#22587007C212yesmaximal
4#22586242C212yesmaximal, complemented
5C71×#2983C211yesmaximal, characteristic, complemented
6C2×#14556125C2211yescharacteristic
7C2×#14556117—21no—
8#17986023C2212yes—
9C2×#14556047—21no—
10#17985916C414yes—
11#17985901—22no—
12#17985900—21no—
13C4×C4⋊Dic71C2211yescharacteristic
14#17985768—21no—
15#17985742—24no—
16C2×C4×Q8×C71C2211yescharacteristic
17#446×C71—24no—
18#432×C71—22no—
19#431×C71C2212yes—
20C22×D284—21no—
21C22×D284—21no—
22C22×D284C2×C412yes—
23C2×#11765849—22no—
24#14556125—22no—
25#14556125—22no—
26#14556117—44no—
27C2×C4⋊Dic71—21no—
28C2×(C2×D71)⋊C4—24no—
29C2×(C2×D71)⋊C4—24no—
30C2×C4×Dic71—21no—
31C4⋊(C71⋊C8)—22no—
32C4⋊(C71⋊C8)—22no—
33#14556047—42no—
34C71⋊#95C2×C412yes—
35C22×Q8×C71D411yescharacteristic
36C4×Q8×C71—22no—
37C4×Q8×C71—22no—
38C2×C4⋊C4×C71C2×C412yes—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C42×C71D412yes—
41C2×C42×C71C2311yescharacteristic
42C4⋊C8×C71—24no—
43C4⋊C8×C71D414yes—
44C22⋊C8×C71—22no—
45#97×C71—24no—
46C4×C8×C71—22no—
47C23×D71—22no—
48C2×D284—24no—
49C2×D284—44no—
50C2×D284—41no—
51C2×D284—42no—
52C2×D284—41no—
53C2×D284—41no—
54C2×D284—42no—
55C2×D284—41no—
56C2×D284—82no—
57C22×Dic71—22no—
58#11765849—42no—
59#11765849—42no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—42no—
62(C2×D71)⋊C4—48no—
63(C2×D71)⋊C4—84no—
64C4×Dic71—42no—
65C2×Q8×C71—22no—
66C2×Q8×C71—21no—
67C2×Q8×C71—22no—
68C2×Q8×C71—21no—
69C22×C4×C71C22⋊C412yes—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—24no—
75C4⋊C4×C71—24no—
76C4⋊C4×C71—22no—
77C42×C71—22no—
78C42×C71—22no—
79C42×C71C2×D412yes—
80C42×C71#4112yes—
81C23×C71C2×C22⋊C411yescharacteristic
82C22×D71—44no—
83C22×D71—44no—
84C22×D71—42no—
85C22×D71—42no—
86C22×D71—42no—
87Q8×C71—42no—
88Q8×C71—42no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—41no—
95C2×C4×C71—42no—
96C2×C4×C71C4×D412yes—
97C2×C4×C71#12011yescharacteristic
98C2×C4×C71#12112yes—
99C2×C4×C71#12311yescharacteristic
100C2×C4×C71AGL(1,ℤ/8)12yes—
101C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
102C2×C4×C71#13711yescharacteristic
103C2×Dic71—42no—
104C2×Dic71—42no—
105C2×Dic71—42no—
106D284—84no—
107D284—82no—
108D284—82no—
109D284—82no—
110D284—82no—
111C8×C71—44no—
112C71⋊C8—82no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71#38611yescharacteristic
116C22×C71#41711yescharacteristic
117C22×C71#41811yescharacteristic
118C2×D71—84no—
119C2×D71—84no—
120C2×D71—82no—
121C2×D71—82no—
122C2×D71—82no—
123C4×C71—21no—
124C4×C71—22no—
125C4×C71—42no—
126C4×C71—42no—
127C4×C71#45412yes—
128C4×C71#46812yes—
129C4×C71#46912yes—
130Dic71—82no—
131#9406—711nomaximal, Sylow
132C2×C71—21no—
133C2×C71—21no—
134C2×C71#188811yescharacteristic
135C2×C71#189511yescharacteristic
136C2×C71#191411yescharacteristic
137D71—162no—
138#2983D7111yescharacteristic, complemented
139#2548—711no—
140#1801—711no—
141#1545—712no—
142#1303—712no—
143C71#940611yescharacteristic, complemented, Sylow
144C2×#127—711no—
145C2×#124—1421no—
146C2×C4×Q8C2×D7111yescharacteristic
147#446—24no—
148#432—22no—
149#431C2×D7112yes—
150#423—712no—
151C2×#101—1421no—
152#394—714no—
153#390—1421no—
154#390—1422no—
155C4×C4⋊C4—711no—
156#343—1421no—
157#331—1424no—
158C22×Q8D28411yescharacteristic
159C22×D4—712no—
160C22×D4—1421no—
161C22×D4—1421no—
162C2×M16—1422no—
163#127—1422no—
164#127—1422no—
165#124—2844no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—1421no—
169C2×C4⋊C4C4×D7112yes—
170C2×C4⋊C4D28411yescharacteristic
171C2×C22⋊C4—1424no—
172C2×C22⋊C4—1424no—
173C2×C42—1421no—
174C2×C42C22⋊D7112yes—
175C2×C42C22×D7111yescharacteristic
176C4⋊C8—24no—
177C4⋊C8—1422no—
178C4⋊C8—1422no—
179C4⋊C8C22⋊D7114yes—
180#101—2842no—
181C22⋊C8—22no—
182#97—24no—
183C4×C8—22no—
184#95—712no—
185C24—1422no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×D4—2844no—
191C2×D4—2841no—
192C2×D4—2841no—
193C2×D4—2842no—
194C2×D4—2841no—
195C2×D4—2842no—
196C2×D4—2841no—
197C2×D4—5682no—
198C2×D4—1424no—
199C22×C4—1422no—
200C22×C4(C2×D71)⋊C412yes—
201C22×C4(C2×D71)⋊C412yes—
202C22×C4C2×C4×D7111yescharacteristic
203C22×C4C2×C22⋊D7111yescharacteristic
204C22×C4C2×D28411yescharacteristic
205M16—2842no—
206M16—2842no—
207C2×C8—24no—
208C2×C8—2842no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—2842no—
212C22⋊C4—2848no—
213C22⋊C4—5684no—
214C42—22no—
215C42—22no—
216C42—2842no—
217C42C2×C22⋊D7112yes—
218C42#1176586512yes—
219C23—2844no—
220C23—2844no—
221C23—2842no—
222C23—2842no—
223C23—2842no—
224C23C2×(C2×D71)⋊C411yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—5684no—
228D4—5682no—
229D4—5682no—
230D4—5682no—
231D4—5682no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—2842no—
240C2×C4—2842no—
241C2×C4—2842no—
242C2×C4C4×C22⋊D7112yes—
243C2×C4#1455609711yescharacteristic
244C2×C4#1455610212yes—
245C2×C4#1455611111yescharacteristic
246C2×C4#1455615612yes—
247C2×C4#1455615611yescharacteristic
248C2×C4#1455616111yescharacteristic
249C8—44no—
250C8—5682no—
251C22—21no—
252C22—21no—
253C22—5684no—
254C22—5684no—
255C22—5682no—
256C22—5682no—
257C22—5682no—
258C22#1798587811yescharacteristic
259C22#1798600111yescharacteristic
260C22#1798600511yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—5682no—
266C4#1798617212yes—
267C4#1798624512yes—
268C4#1798625212yes—
269C2—21no—
270C2—21no—
271C2—11362no—
272C2#2258874311yescharacteristic
273C2#2258877811yescharacteristic
274C2#2258888711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2983C2
socleC22×C71#386
prime radicalC2×#14556125C22
cosocleC2×C42×C71C23
hypercenter#2983D71
hypocenterC71#9406
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. #22588336C2semidirect
4. #22586242C2semidirect
5. C71×#2983C2semidirect
138. #2983D71semidirect
143. C71#9406semidirect

Maximal subgroups

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

subgroupindex
#225933392
#225883362
#225870072
#225862422
C71×#29832
#940671