Finite Group Databasebeta

← #31886706

Subgroups — #31886706

Order 18176.

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

normal subgroups only

This group has 422 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#22589748C212yesmaximal, complemented
3#22586166C212yesmaximal, complemented
4#22586161C211yesmaximal, characteristic
5C71×#2804C211yesmaximal, characteristic
6C2×#14556061—21no—
7C2×#14556052—22no—
8C2×#14556049—21no—
9#17985869C2211yescharacteristic
10#17985761—22no—
11#17985754C2212yes—
12#17985754C2211yescharacteristic
13#400×C71—21no—
14#393×C71—22no—
15#391×C71C2212yes—
16#390×C71C2211yescharacteristic
17#386×C71—21no—
18#385×C71—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556061—42no—
24#14556052—44no—
25#14556049—42no—
26C22×Q8×C71D412yes—
27C22×D4×C71D411yescharacteristic
28C23×C4×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71D412yes—
31C2×C4⋊C4×C71D411yescharacteristic
32C2×C22⋊C4×C71—22no—
33C2×C22⋊C4×C71—22no—
34C2×C22⋊C4×C71—21no—
35C2×C42×C71C2311yescharacteristic
36#95×C71—22no—
37#95×C71—21no—
38#95×C71—22no—
39#95×C71—22no—
40C22×Dic71—41no—
41C2×C71⋊C8—22no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—42no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—42no—
47C4⋊Dic71—41no—
48C4⋊Dic71—41no—
49C4⋊Dic71—42no—
50C4⋊Dic71—42no—
51C24×C71—21no—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×Q8×C71—42no—
55C2×Q8×C71—42no—
56C2×D4×C71—21no—
57C2×D4×C71—41no—
58C2×D4×C71—41no—
59C2×D4×C71Q1612yes—
60C22×C4×C71—22no—
61C22×C4×C71—21no—
62C22×C4×C71—22no—
63C22×C4×C71—81no—
64C22×C4×C71—82no—
65C22×C4×C71C2×D411yescharacteristic
66C22×C4×C71C2×D412yes—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—81no—
70C22⋊C4×C71—42no—
71C22⋊C4×C71—42no—
72C22⋊C4×C71—82no—
73C22⋊C4×C71—81no—
74C42×C71—41no—
75C23×C71—22no—
76C23×C71—42no—
77C23×C71—41no—
78C23×C71—42no—
79C23×C71—41no—
80C23×C71#12011yescharacteristic
81Q8×C71—44no—
82Q8×C71—82no—
83D4×C71—42no—
84D4×C71—81no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—41no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—84no—
98C2×C4×C71C2×Q1611yescharacteristic
99C2×C4×C71#13512yes—
100C2×C4×C71AGL(1,ℤ/8)12yes—
101C2×C4×C71#13711yescharacteristic
102C2×Dic71—82no—
103C2×Dic71—81no—
104C71⋊C8—42no—
105C71⋊C8—82no—
106C22×C71—42no—
107C22×C71—44no—
108C22×C71—82no—
109C22×C71—81no—
110C22×C71—81no—
111C22×C71#44912yes—
112C22×C71#45111yescharacteristic
113C22×C71#45311yescharacteristic
114C22×C71#45512yes—
115C22×C71#45811yescharacteristic
116C4×C71—22no—
117C4×C71—22no—
118C4×C71—21no—
119C4×C71—21no—
120C4×C71—82no—
121C4×C71—81no—
122C4×C71—82no—
123Dic71—161no—
124#7821—711nomaximal, Sylow
125C2×C71—82no—
126C2×C71#161212yes—
127C2×C71#161411yescharacteristic
128C2×C71#162011yescharacteristic
129C2×C71#164412yes—
130C2×C71#165911yescharacteristic
131#2804D7111yescharacteristic
132#2093—711no—
133#2072—712no—
134#1272—712no—
135#1270—711no—
136C71#782111yescharacteristic, complemented, Sylow
137C2×#107—1421no—
138C2×#103—1422no—
139C2×#102—1421no—
140#400—21no—
141#393—22no—
142#391C2×D7112yes—
143#390C2×D7111yescharacteristic
144#386—21no—
145#385—22no—
146#384—711no—
147#340—1422no—
148#336—712no—
149#336—711no—
150C22×Q8C22⋊D7112yes—
151C22×D4C22⋊D7111yescharacteristic
152C23×C4—21no—
153C22×C8—1422no—
154C22×C8—1421no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—1422no—
157C2×C4⋊C4—1421no—
158C2×C4⋊C4C22⋊D7112yes—
159C2×C4⋊C4C22⋊D7111yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42C22×D7111yescharacteristic
164#107—2842no—
165#103—2844no—
166#102—2842no—
167#95—22no—
168#95—21no—
169#95—22no—
170#95—22no—
171C24—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#1176585512yes—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—81no—
184C22×C4—82no—
185C22×C4—2841no—
186C22×C4C2×C22⋊D7111yescharacteristic
187C22×C4C2×C22⋊D7112yes—
188C2×C8—2841no—
189C2×C8—2841no—
190C2×C8—2842no—
191C2×C8—2842no—
192C2×C8—2842no—
193C2×C8—1422no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—2841no—
198C4⋊C4—2841no—
199C4⋊C4—2842no—
200C4⋊C4—2842no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1455609711yescharacteristic
212Q8—44no—
213Q8—82no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—5682no—
230C2×C4—5681no—
231C2×C4C2×#1176585511yescharacteristic
232C2×C4#1455614812yes—
233C2×C4#1455615312yes—
234C2×C4#1455615911yescharacteristic
235C8—2842no—
236C8—5682no—
237C22—44no—
238C22—42no—
239C22—81no—
240C22—82no—
241C22—81no—
242C22#1798614412yes—
243C22#1798615511yescharacteristic
244C22#1798616711yescharacteristic
245C22#1798617712yes—
246C22#1798618911yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—81no—
252C4—82no—
253C4—82no—
254C4—11361no—
255C2—82no—
256C2#2258735412yes—
257C2#2258736511yescharacteristic
258C2#2258739911yescharacteristic
259C2#2258755412yes—
260C2#2258764611yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2804C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2804D71
hypocenterC71#7821
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #22589748C2semidirect
3. #22586166C2semidirect
136. C71#7821semidirect

Maximal subgroups

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

subgroupindex
#225898682
#225897482
#225861662
#225861612
C71×#28042
#782171