Finite Group Databasebeta

← #31894817

Subgroups — #31894817

Order 18176.

558 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 135 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#22593332C211yesmaximal, characteristic
2#22588332C211yesmaximal, characteristic
3#22587011C212yesmaximal, complemented
4#22586242C212yesmaximal, complemented
5C71×#2972C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7C2×#14556115—21no—
8#17986023C2212yes—
9C2×#14556045—21no—
10#17985924C414yes—
11#17985904—21no—
12#17985859—22no—
13C4×C4⋊Dic71C2211yescharacteristic
14#17985768—21no—
15#17985744—24no—
16C2×C4×D4×C71C2211yescharacteristic
17C8×D4×C71—24no—
18#431×C71C2212yes—
19C2×C22⋊C8×C71—22no—
20C22×Dic142—21no—
21C2×#11765849—22no—
22#14556128—22no—
23#14556128—22no—
24#14556115—42no—
25#14556115—42no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71C2×C412yes—
29C2×C22⋊Dic71—24no—
30C2×C4×Dic71—21no—
31C4⋊(C71⋊C8)—22no—
32C4⋊(C71⋊C8)—22no—
33#14556045—42no—
34C71⋊#95—24no—
35C71⋊#95C2×C412yes—
36C22×D4×C71D411yescharacteristic
37C23×C4×C71D412yes—
38C22×C8×C71—24no—
39C4×D4×C71—22no—
40C4×D4×C71—22no—
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C22⋊C4×C71C2×C412yes—
43C2×C42×C71C2311yescharacteristic
44C4⋊C8×C71D414yes—
45C22⋊C8×C71—24no—
46C22⋊C8×C71—22no—
47C4×C8×C71—22no—
48C2×Dic142—42no—
49C2×Dic142—41no—
50C2×Dic142—41no—
51C2×Dic142—42no—
52C22×Dic71—22no—
53C22×Dic71—22no—
54#11765849—42no—
55#11765849—42no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—24no—
58C4⋊Dic71—42no—
59C4⋊Dic71—41no—
60C4⋊Dic71—41no—
61C22⋊Dic71—48no—
62C4×Dic71—41no—
63C4×Dic71—41no—
64C24×C71C22⋊C412yes—
65C2×D4×C71—22no—
66C2×D4×C71—22no—
67C2×D4×C71D812yes—
68C2×D4×C71SD1612yes—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—44no—
72C22×C4×C71C22⋊C412yes—
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C2×C8×C71—24no—
77C2×C8×C71—84no—
78C4⋊C4×C71—22no—
79C22⋊C4×C71—24no—
80C42×C71C2×D412yes—
81C42×C71#4112yes—
82C23×C71—22no—
83C23×C71—22no—
84C23×C71—42no—
85C23×C71—42no—
86C23×C71#10214yes—
87C23×C71C2×C22⋊C411yescharacteristic
88D4×C71—24no—
89D4×C71—24no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—41no—
93C2×C4×C71—44no—
94C2×C4×C71—42no—
95C2×C4×C71—82no—
96C2×C4×C71C4×D412yes—
97C2×C4×C71#12011yescharacteristic
98C2×C4×C71#12112yes—
99C2×C4×C71#12311yescharacteristic
100C2×C4×C71C2×D811yescharacteristic
101C2×C4×C71C2×SD1611yescharacteristic
102C2×C4×C71#13512yes—
103C2×Dic71—42no—
104C2×Dic71—42no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109Dic142—82no—
110Dic142—82no—
111C8×C71—44no—
112C71⋊C8—82no—
113C22×C71—21no—
114C22×C71—21no—
115C22×C71—24no—
116C22×C71—44no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71—82no—
120C22×C71#38611yescharacteristic
121C22×C71C2×#10211yescharacteristic
122C22×C71#41611yescharacteristic
123C4×C71—21no—
124C4×C71—22no—
125C4×C71—42no—
126C4×C71#45512yes—
127C4×C71#46512yes—
128C4×C71#46612yes—
129Dic71—82no—
130Dic71—82no—
131#9393—711nomaximal, Sylow
132C2×C71—21no—
133C2×C71—21no—
134C2×C71—44no—
135C2×C71#188511yescharacteristic
136C2×C71#189311yescharacteristic
137C2×C71#191511yescharacteristic
138#2972D7111yescharacteristic
139#2547—711no—
140#1800—711no—
141#1546—712no—
142#1303—712no—
143C71#939311yescharacteristic, complemented, Sylow
144C2×C4⋊Q8—711no—
145C2×#124—1421no—
146C2×C4×D4C2×D7111yescharacteristic
147C8×D4—24no—
148#431C2×D7112yes—
149#423—712no—
150C2×#100—1421no—
151C2×C22⋊C8—22no—
152#397—714no—
153#391—1421no—
154#380—1422no—
155C4×C4⋊C4—711no—
156#343—1421no—
157#332—1424no—
158C22×Q8—1421no—
159C22×D4D28411yescharacteristic
160C23×C4C22⋊D7112yes—
161C2×M16—1422no—
162C22×C8—24no—
163C4⋊Q8—1422no—
164C4⋊Q8—1422no—
165#124—2842no—
166#124—2842no—
167C4×D4—22no—
168C4×D4—22no—
169C2×C4⋊C4—712no—
170C2×C4⋊C4—1421no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4D28411yescharacteristic
173C2×C22⋊C4—1424no—
174C2×C22⋊C4C4×D7112yes—
175C2×C42—1421no—
176C2×C42C22×D7111yescharacteristic
177C4⋊C8—1422no—
178C4⋊C8—1422no—
179C4⋊C8C22⋊D7114yes—
180#100—2842no—
181C22⋊C8—24no—
182C22⋊C8—22no—
183C4×C8—22no—
184#95—712no—
185#95—1424no—
186C24(C2×D71)⋊C412yes—
187C2×Q8—2842no—
188C2×Q8—2841no—
189C2×Q8—2841no—
190C2×Q8—2842no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4D56812yes—
194C2×D4#1176585412yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—44no—
198C22×C4—1422no—
199C22×C4—1422no—
200C22×C4(C2×D71)⋊C412yes—
201C22×C4C2×C4×D7111yescharacteristic
202C22×C4C2×C22⋊D7111yescharacteristic
203C22×C4C2×D28411yescharacteristic
204M16—2842no—
205M16—2842no—
206C2×C8—24no—
207C2×C8—84no—
208C2×C8—2842no—
209C4⋊C4—22no—
210C4⋊C4—2842no—
211C4⋊C4—2841no—
212C4⋊C4—2841no—
213C4⋊C4—1424no—
214C22⋊C4—24no—
215C22⋊C4—2848no—
216C42—2841no—
217C42—2841no—
218C42C2×C22⋊D7112yes—
219C42#1176586512yes—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23#1455605014yes—
225C23C2×(C2×D71)⋊C411yescharacteristic
226Q8—5682no—
227Q8—5682no—
228D4—24no—
229D4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—2842no—
237C2×C4—2842no—
238C2×C4—2842no—
239C2×C4—2842no—
240C2×C4—2842no—
241C2×C4—2842no—
242C2×C4C4×C22⋊D7112yes—
243C2×C4#1455609711yescharacteristic
244C2×C4#1455610212yes—
245C2×C4#1455611111yescharacteristic
246C2×C4C2×D56811yescharacteristic
247C2×C4C2×#1176585411yescharacteristic
248C2×C4#1455615012yes—
249C8—44no—
250C8—5682no—
251C22—21no—
252C22—21no—
253C22—24no—
254C22—44no—
255C22—42no—
256C22—42no—
257C22—82no—
258C22#1798587811yescharacteristic
259C22C2×#1455605011yescharacteristic
260C22#1798599611yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4—5682no—
265C4—5682no—
266C4#1798617712yes—
267C4#1798622812yes—
268C4#1798623512yes—
269C2—21no—
270C2—21no—
271C2—44no—
272C2#2258872811yescharacteristic
273C2#2258876811yescharacteristic
274C2#2258889211yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2972C2
socleC22×C71#386
prime radicalC22×D4×C71D4
cosocleC2×C42×C71C23
hypercenter#2972D71
hypocenterC71#9393
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
3. #22587011C2semidirect
4. #22586242C2semidirect
143. C71#9393semidirect

Maximal subgroups

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

subgroupindex
#225933322
#225883322
#225870112
#225862422
C71×#29722
#939371