Finite Group Databasebeta

← #32006720

Subgroups — #32006720

Order 18176.

1334 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 315 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#22590943C212yesmaximal, complemented
2#22589177C218yesmaximal, complemented
3#22588258C214yesmaximal, complemented
4C2×#537×C71C211yesmaximal, characteristic
5C2×#14556043C22116yes—
6#17985885—28no—
7#17985862C2214yes—
8#17985859C2214yes—
9C4×C22⋊Dic71C2214yes—
10C22×#41×C71C2212yes—
11#537×C71—28no—
12C2×C22⋊Q8×C71C414yescomplemented
13C2×#121×C71C414yescomplemented
14C2×#121×C71C2214yes—
15C22×C4⋊C4×C71C2211yescharacteristic
16C2×C4⋊Dic71—24no—
17C2×C22⋊Dic71C2318yes—
18C2×C4×Dic71—24no—
19#14556043—416no—
20C71⋊#95—24no—
21C71⋊#95—24no—
22C71⋊#95—28no—
23C2×#41×C71—28no—
24C2×#41×C71—48no—
25C22×Q8×C71C2×C412yes—
26C22×D4×C71C2×C412yes—
27C22×D4×C71C2×C414yes—
28C22×D4×C71C2314yes—
29C23×C4×C71C2×C414yes—
30C23×C4×C71C2311yescharacteristic
31C23×C4×C71C2312yes—
32C22⋊Q8×C71—216no—
33#121×C71—28no—
34#121×C71—28no—
35#121×C71—216no—
36C2×C4⋊C4×C71C2×C414yes—
37C2×C4⋊C4×C71C2×C414yes—
38C2×C4⋊C4×C71D418yes—
39C2×C22⋊C4×C71C2×C418yes—
40C22×Dic71—28no—
41C4⋊Dic71—84no—
42C22⋊Dic71—216no—
43C4×Dic71—84no—
44C24×C71C22×C414yes—
45C24×C71C2411yescharacteristic
46#41×C71—432no—
47C2×Q8×C71—24no—
48C2×Q8×C71—44no—
49C2×D4×C71—28no—
50C2×D4×C71—216no—
51C2×D4×C71—24no—
52C2×D4×C71—28no—
53C2×D4×C71—44no—
54C2×D4×C71—84no—
55C22×C4×C71—24no—
56C22×C4×C71—28no—
57C22×C4×C71—48no—
58C22×C4×C71—82no—
59C22×C4×C71C22×C414yes—
60C22×C4×C71C22×C412yes—
61C22×C4×C71C22×C414yes—
62C22×C4×C71C2×D414yes—
63C22×C4×C71C2×D414yes—
64C22×C4×C71C2×D414yes—
65C4⋊C4×C71—28no—
66C4⋊C4×C71C22⋊C4116yes—
67C22⋊C4×C71—216no—
68C23×C71—24no—
69C23×C71—28no—
70C23×C71—41no—
71C23×C71—41no—
72C23×C71—44no—
73C23×C71—44no—
74C23×C71C23×C411yescharacteristic
75C23×C71C22×D412yes—
76C23×C71H(5,𝔽2)14yes—
77Q8×C71—48no—
78D4×C71—48no—
79D4×C71—416no—
80D4×C71—88no—
81C2×C4×C71—24no—
82C2×C4×C71—28no—
83C2×C4×C71—24no—
84C2×C4×C71—28no—
85C2×C4×C71—44no—
86C2×C4×C71—44no—
87C2×C4×C71—44no—
88C2×C4×C71—416no—
89C2×C4×C71C2×C22⋊C418yes—
90C2×C4×C71C2×C22⋊C414yes—
91C2×Dic71—48no—
92C2×Dic71—48no—
93C2×Dic71—48no—
94C22×C71—22no—
95C22×C71—21no—
96C22×C71—22no—
97C22×C71—21no—
98C22×C71—44no—
99C22×C71—44no—
100C22×C71—44no—
101C22×C71—44no—
102C22×C71—88no—
103C22×C71C22×C22⋊C411yescharacteristic
104C22×C71#51812yes—
105C22×C71#53514yes—
106C4×C71—24no—
107C4×C71—28no—
108C4×C71—44no—
109Dic71—88no—
110#21796—711nomaximal, Sylow
111C2×C71—24no—
112C2×C71—22no—
113C2×C71—84no—
114C2×C71#230911yescharacteristic
115C2×C71#288311yescharacteristic
116C2×C71#288411yescharacteristic
117C2×#537D7111yescharacteristic
118#2313—712no—
119#1964—718no—
120#1782—714no—
121C71#2179611yescharacteristic, complemented, Sylow
122C22×#41C2×D7112yes—
123#537—28no—
124C2×C22⋊Q8Dic7114yescomplemented
125C2×#121Dic7114yescomplemented
126C2×#121C2×D7114yes—
127C22×C4⋊C4C2×D7111yescharacteristic
128C2×#99—7116no—
129#387—1428no—
130#381—714no—
131#380—714no—
132C4×C22⋊C4—714no—
133C2×#41—28no—
134C2×#41—48no—
135C22×Q8C2×Dic7112yes—
136C22×D4C2×Dic7112yes—
137C22×D4C2×Dic7114yes—
138C22×D4C22×D7114yes—
139C23×C4C2×Dic7114yes—
140C23×C4C22×D7111yescharacteristic
141C23×C4C22×D7112yes—
142C22⋊Q8—216no—
143#121—28no—
144#121—28no—
145#121—216no—
146C2×C4⋊C4—1424no—
147C2×C4⋊C4C2×Dic7114yes—
148C2×C4⋊C4C2×Dic7114yes—
149C2×C4⋊C4C22⋊D7118yes—
150C2×C22⋊C4—718no—
151C2×C22⋊C4C2×Dic7118yes—
152C2×C42—1424no—
153#99—28416no—
154#95—1424no—
155#95—1424no—
156#95—1428no—
157C24C22×Dic7114yes—
158C24C23×D7111yescharacteristic
159#41—432no—
160C2×Q8—24no—
161C2×Q8—44no—
162C2×D4—28no—
163C2×D4—216no—
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—44no—
167C2×D4—84no—
168C22×C4—24no—
169C22×C4—28no—
170C22×C4—48no—
171C22×C4—82no—
172C22×C4—1428no—
173C22×C4C22×Dic7114yes—
174C22×C4C22×Dic7112yes—
175C22×C4C22×Dic7114yes—
176C22×C4C2×C22⋊D7114yes—
177C22×C4C2×C22⋊D7114yes—
178C22×C4C2×C22⋊D7114yes—
179C4⋊C4—28no—
180C4⋊C4—5684no—
181C4⋊C4C22⋊Dic71116yes—
182C22⋊C4—216no—
183C22⋊C4—14216no—
184C42—5684no—
185C23—24no—
186C23—28no—
187C23—41no—
188C23—41no—
189C23—44no—
190C23—44no—
191C23C23×Dic7111yescharacteristic
192C23C22×C22⋊D7112yes—
193C23#1455617314yes—
194Q8—48no—
195D4—48no—
196D4—416no—
197D4—88no—
198C2×C4—28no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—28no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—416no—
206C2×C4—2848no—
207C2×C4—2848no—
208C2×C4—2848no—
209C2×C4C2×C22⋊Dic7118yes—
210C2×C4C2×C22⋊Dic7114yes—
211C22—22no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—44no—
216C22—44no—
217C22—44no—
218C22—44no—
219C22—88no—
220C22C22×C22⋊Dic7111yescharacteristic
221C22#1798647412yes—
222C22#1798656614yes—
223C4—28no—
224C4—24no—
225C4—44no—
226C4—5688no—
227C2—24no—
228C2—22no—
229C2—84no—
230C2#2259092211yescharacteristic
231C2#2259680111yescharacteristic
232C2#2259680411yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic71
commutatorC23×C71C23×C4
FrattiniC24C23×D71
Fitting, prime radicalC2×#537×C71C2
socleC22×C71C22×C22⋊C4
cosocleC24×C71C24
hypercenterC2×#537D71
hypocenterC71#21796
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #22590943C2semidirect
2. #22589177C2semidirect
3. #22588258C2semidirect
12. C2×C22⋊Q8×C71C4semidirect
13. C2×#121×C71C4semidirect
121. C71#21796semidirect
124. C2×C22⋊Q8Dic71semidirect
125. C2×#121Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225891772
#225882582
C2×#537×C712
#2179671