Finite Group Databasebeta

← #32025589

Subgroups — #32025589

Order 18176.

1334 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 291 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#22590952C213yesmaximal
2#22589692C218yesmaximal, complemented
3C2×#17985766C213yesmaximal
4C2×#543×C71C211yesmaximal, characteristic
5C22×C22⋊Dic71C2213yes—
6C22×C22⋊Dic71C2211yescharacteristic
7#17985872—26no—
8#17985871C22112yes—
9#17985857—26no—
10#17985766C22112yes—
11#543×C71—26no—
12#543×C71C2214yes—
13C2×C22⋊Q8×C71—26no—
14C22×C22⋊C4×C71C2213yes—
15C23×Dic71—23no—
16C23×Dic71C2312yes—
17C2×Dic71⋊C4—212no—
18C2×C22⋊Dic71—23no—
19C2×C22⋊Dic71—23no—
20C2×C22⋊Dic71—23no—
21C2×C22⋊Dic71—26no—
22C2×C22⋊Dic71Q8112yescomplemented
23C2×C22⋊Dic71C2316yes—
24C71⋊#95—212no—
25C25×C71C2311yescharacteristic
26C22×Q8×C71—22no—
27C23×C4×C71D416yes—
28C22⋊Q8×C71—224no—
29C22⋊Q8×C71—224no—
30C2×C4⋊C4×C71—26no—
31C2×C22⋊C4×C71—26no—
32C2×C22⋊C4×C71—26no—
33C2×C22⋊C4×C71D416yes—
34C2×C22⋊C4×C71C2316yes—
35C22×Dic71—26no—
36C22×Dic71—212no—
37C22×Dic71—26no—
38C22×Dic71—82no—
39C22×Dic71—86no—
40Dic71⋊C4—812no—
41C22⋊Dic71—212no—
42C22⋊Dic71—26no—
43C22⋊Dic71—43no—
44C22⋊Dic71—41no—
45C22⋊Dic71C2×Q8112yescomplemented
46C24×C71—82no—
47C24×C71C2×D413yes—
48C24×C71C2×D413yes—
49C24×C71C2×Q816yes—
50C24×C71#4112yes—
51C24×C71C2411yescharacteristic
52C2×Q8×C71—28no—
53C2×Q8×C71—84no—
54C22×C4×C71—26no—
55C22×C4×C71—26no—
56C22×C4×C71—86no—
57C22×C4×C71C2×D4112yes—
58C4⋊C4×C71—224no—
59C22⋊C4×C71—212no—
60C22⋊C4×C71—212no—
61C23×C71—46no—
62C23×C71—46no—
63C23×C71—42no—
64C23×C71—86no—
65C23×C71—82no—
66C23×C71#12013yes—
67C23×C71#12011yescharacteristic
68C23×C71C22⋊Q816yes—
69C23×C71C22⋊Q816yes—
70C23×C71C4⋊Q816yes—
71C23×C71C4⋊Q816yes—
72C23×C71C22×D413yes—
73C23×C71C22×Q813yes—
74C23×C71C2×#4111yescharacteristic
75Q8×C71—88no—
76C2×C4×C71—212no—
77C2×C4×C71—412no—
78C2×C4×C71—412no—
79C2×C4×C71—86no—
80C2×Dic71—212no—
81C2×Dic71—412no—
82C2×Dic71—412no—
83C2×Dic71—42no—
84C2×Dic71—812no—
85C22×C71—26no—
86C22×C71—22no—
87C22×C71—26no—
88C22×C71—46no—
89C22×C71—46no—
90C22×C71—46no—
91C22×C71—46no—
92C22×C71—43no—
93C22×C71—41no—
94C22×C71#39516yes—
95C22×C71#39512yes—
96C22×C71C2×#12011yescharacteristic
97C22×C71C2×C22⋊Q813yes—
98C22×C71C2×C4⋊Q813yes—
99C4×C71—412no—
100Dic71—412no—
101Dic71—82no—
102#23565—711nomaximal, Sylow
103C2×C71—23no—
104C2×C71—23no—
105C2×C71—22no—
106C2×C71—26no—
107C2×C71C2×#39511yescharacteristic
108C2×C71#302612yes—
109C2×#543D7111yescharacteristic
110#2315—713no—
111#2062—718no—
112C2×#342—713no—
113C71#2356511yescharacteristic, complemented, Sylow
114#543—26no—
115#543C2×D7114yes—
116C2×C22⋊Q8—26no—
117C22×C22⋊C4—713no—
118C22×C22⋊C4—711no—
119C22×C22⋊C4C2×D7113yes—
120#385—7112no—
121#385—1426no—
122#380—1426no—
123#342—7112no—
124C25C22×D7111yescharacteristic
125C22×Q8—22no—
126C23×C4—712no—
127C23×C4—1423no—
128C23×C4C22⋊D7116yes—
129C22⋊Q8—224no—
130C22⋊Q8—224no—
131C2×C4⋊C4—26no—
132C2×C4⋊C4—14212no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—26no—
135C2×C22⋊C4—7112no—
136C2×C22⋊C4—716no—
137C2×C22⋊C4—1423no—
138C2×C22⋊C4—1423no—
139C2×C22⋊C4—1423no—
140C2×C22⋊C4—1426no—
141C2×C22⋊C4C22⋊D7116yes—
142C2×C22⋊C4C22×D7116yes—
143#95—14212no—
144C24—82no—
145C24C2×C22⋊D7113yes—
146C24C2×C22⋊D7113yes—
147C24Q8×D7116yes—
148C24#1176586612yes—
149C24C23×D7111yescharacteristic
150C2×Q8—28no—
151C2×Q8—84no—
152C22×C4—26no—
153C22×C4—26no—
154C22×C4—86no—
155C22×C4—5682no—
156C22×C4—5686no—
157C22×C4—1426no—
158C22×C4—1426no—
159C22×C4—14212no—
160C22×C4C2×C22⋊D71112yes—
161C4⋊C4—224no—
162C4⋊C4—56812no—
163C22⋊C4—212no—
164C22⋊C4—212no—
165C22⋊C4—2843no—
166C22⋊C4—2841no—
167C22⋊C4—7112no—
168C22⋊C4—1426no—
169C22⋊C4—14212no—
170C23—46no—
171C23—46no—
172C23—42no—
173C23—86no—
174C23—82no—
175C23#1455609713yes—
176C23#1455609711yescharacteristic
177C23#1455610416yes—
178C23#1455610416yes—
179C23#1455612616yes—
180C23#1455612616yes—
181C23C22×C22⋊D7113yes—
182C23C2×Q8×D7113yes—
183C23C2×#1176586611yescharacteristic
184Q8—88no—
185C2×C4—212no—
186C2×C4—412no—
187C2×C4—412no—
188C2×C4—86no—
189C2×C4—28412no—
190C2×C4—28412no—
191C2×C4—2842no—
192C2×C4—56812no—
193C2×C4—14212no—
194C22—22no—
195C22—26no—
196C22—26no—
197C22—46no—
198C22—43no—
199C22—41no—
200C22—46no—
201C22—46no—
202C22—46no—
203C22#1798591816yes—
204C22#1798591812yes—
205C22C2×#1455609711yescharacteristic
206C22C2×#1455610413yes—
207C22C2×#1455612613yes—
208C4—412no—
209C4—28412no—
210C4—5682no—
211C2—23no—
212C2—22no—
213C2—26no—
214C2—23no—
215C2C2×#1798591811yescharacteristic
216C2#2259773712yes—
subgroupG/H
centerC22C2×#14556097
commutator, cosocleC24×C71C24
FrattiniC24C23×D71
FittingC2×#543×C71C2
socleC22×C71C2×#120
prime radicalC25×C71C23
hypercenterC2×#543D71
hypocenterC71#23565
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C25 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #22589692C2semidirect
22. C2×C22⋊Dic71Q8semidirect
45. C22⋊Dic71C2×Q8semidirect
113. C71#23565semidirect

Maximal subgroups

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

subgroupindex
#225909522
#225896922
C2×#179857662
C2×#543×C712
#2356571