Finite Group Databasebeta

← #31886694

Subgroups — #31886694

Order 18176.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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#22589868C211yesmaximal, characteristic
2#22589816C212yesmaximal, complemented
3#22586163C212yesmaximal, complemented
4#22586161C211yesmaximal, characteristic
5C71×#2724C211yesmaximal, characteristic
6C2×#14556061—21no—
7C2×#14556059—22no—
8C2×#14556049—21no—
9#17985869C2211yescharacteristic
10#17985761—22no—
11#17985754C2212yes—
12#17985754C2211yescharacteristic
13C2×#123×C71—21no—
14#398×C71—21no—
15#397×C71—22no—
16#390×C71C2211yescharacteristic
17#389×C71C2212yes—
18#385×C71—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556061—42no—
24#14556059—44no—
25#14556049—42no—
26C22×D4×C71D411yescharacteristic
27C23×C4×C71—21no—
28#123×C71—42no—
29#123×C71—42no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—21no—
32C2×C4⋊C4×C71D412yes—
33C2×C4⋊C4×C71D412yes—
34C2×C4⋊C4×C71D411yescharacteristic
35C2×C22⋊C4×C71—22no—
36C2×C22⋊C4×C71—21no—
37C2×C22⋊C4×C71—22no—
38C2×C42×C71C2311yescharacteristic
39#95×C71—22no—
40#95×C71—21no—
41#95×C71—22no—
42C22×Dic71—41no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8—24no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—41no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—41no—
49C4⋊Dic71—41no—
50C4⋊Dic71—41no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C24×C71—21no—
54C2×D4×C71—21no—
55C2×D4×C71—41no—
56C2×D4×C71—41no—
57C2×D4×C71SD1612yes—
58C22×C4×C71—21no—
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71—42no—
62C22×C4×C71—81no—
63C22×C4×C71—81no—
64C22×C4×C71C2×D411yescharacteristic
65C22×C4×C71C2×D412yes—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—81no—
73C22⋊C4×C71—42no—
74C22⋊C4×C71—44no—
75C22⋊C4×C71—42no—
76C22⋊C4×C71—42no—
77C42×C71—41no—
78C23×C71—22no—
79C23×C71—42no—
80C23×C71—41no—
81C23×C71—42no—
82C23×C71—41no—
83C23×C71#12011yescharacteristic
84D4×C71—42no—
85D4×C71—81no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—41no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—84no—
99C2×C4×C71C2×SD1611yescharacteristic
100C2×C4×C71#13512yes—
101C2×C4×C71AGL(1,ℤ/8)12yes—
102C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
103C2×Dic71—82no—
104C2×Dic71—81no—
105C71⋊C8—44no—
106C71⋊C8—42no—
107C22×C71—42no—
108C22×C71—44no—
109C22×C71—82no—
110C22×C71—81no—
111C22×C71—81no—
112C22×C71#44912yes—
113C22×C71#45011yescharacteristic
114C22×C71#45311yescharacteristic
115C22×C71#45612yes—
116C22×C71#45811yescharacteristic
117C4×C71—22no—
118C4×C71—22no—
119C4×C71—21no—
120C4×C71—21no—
121C4×C71—82no—
122C4×C71—81no—
123C4×C71—82no—
124Dic71—161no—
125#7819—711nomaximal, Sylow
126C2×C71—82no—
127C2×C71#160912yes—
128C2×C71#161311yescharacteristic
129C2×C71#162011yescharacteristic
130C2×C71#165412yes—
131C2×C71#166011yescharacteristic
132#2724D7111yescharacteristic
133#2093—711no—
134#2084—712no—
135#1271—712no—
136#1270—711no—
137C71#781911yescharacteristic, complemented, Sylow
138C2×#123—21no—
139C2×#107—1421no—
140C2×#106—1422no—
141C2×#102—1421no—
142#398—21no—
143#397—22no—
144#390C2×D7111yescharacteristic
145#389C2×D7112yes—
146#385—22no—
147#384—711no—
148#340—1422no—
149#336—712no—
150#336—711no—
151C22×D4C22⋊D7111yescharacteristic
152C23×C4—21no—
153C22×C8—1422no—
154C22×C8—1421no—
155#123—42no—
156#123—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—1422no—
160C2×C4⋊C4—1421no—
161C2×C4⋊C4C22⋊D7112yes—
162C2×C4⋊C4C22⋊D7112yes—
163C2×C4⋊C4C22⋊D7111yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C42C22×D7111yescharacteristic
168#107—2842no—
169#106—2844no—
170#102—2842no—
171#95—22no—
172#95—21no—
173#95—22no—
174C24—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4#1176585312yes—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—2841no—
186C22×C4C2×C22⋊D7111yescharacteristic
187C22×C4C2×C22⋊D7112yes—
188C2×C8—2842no—
189C2×C8—2841no—
190C2×C8—2842no—
191C2×C8—2841no—
192C2×C8—1422no—
193C2×C8—1424no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4—2841no—
202C4⋊C4—2841no—
203C4⋊C4—2842no—
204C4⋊C4—2842no—
205C22⋊C4—42no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—41no—
210C23—22no—
211C23—42no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23#1455609711yescharacteristic
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—5682no—
232C2×C4—5681no—
233C2×C4C2×#1176585311yescharacteristic
234C2×C4#1455614812yes—
235C2×C4#1455615312yes—
236C2×C4#1455615411yescharacteristic
237C8—2844no—
238C8—2842no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—81no—
243C22—82no—
244C22#1798614412yes—
245C22#1798615011yescharacteristic
246C22#1798616711yescharacteristic
247C22#1798618212yes—
248C22#1798618911yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—82no—
255C4—82no—
256C4—11361no—
257C2—82no—
258C2#2258733712yes—
259C2#2258736011yescharacteristic
260C2#2258739911yescharacteristic
261C2#2258762412yes—
262C2#2258765111yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2724C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2724D71
hypocenterC71#7819
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #22589816C2semidirect
3. #22586163C2semidirect
137. C71#7819semidirect

Maximal subgroups

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

subgroupindex
#225898682
#225898162
#225861632
#225861612
C71×#27242
#781971