Finite Group Databasebeta

← #37510026

Subgroups — #37510026

Order 22912.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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#26039003C211yesmaximal, characteristic
2C2×#19153102C211yesmaximal, characteristic, complemented
3C2×#19153040C211yesmaximal, characteristic, complemented
4#26038448C211yesmaximal, characteristic, complemented
5C2×#19153029C211yesmaximal, characteristic, complemented
6#26038200C211yesmaximal, characteristic, complemented
7C2×#137×C179C211yesmaximal, characteristic, complemented
8C22×D716C2211yescharacteristic
9C2×#15668243C2211yescharacteristic
10#19153109—21no—
11#19153102—22no—
12#19153102—21no—
13#19153102—22no—
14#19153087—22no—
15#19153081—22no—
16#19153072C2211yescharacteristic
17C2×(C2×D179)⋊C4—22no—
18C2×C4×Dic179C2211yescharacteristic
19#19153040—22no—
20#19153037—22no—
21#19153029—22no—
22C2×#41×C179C2211yescharacteristic
23C22×Q8×C179C2211yescharacteristic
24#137×C179—22no—
25#137×C179—22no—
26C2×Q16×C179—21no—
27C2×SD16×C179—21no—
28C2×M16×C179C2211yescharacteristic
29C23×D179D412yes—
30C2×D716—21no—
31C2×D716—21no—
32C2×D716—21no—
33C2×D716—21no—
34C2×D716—21no—
35C2×D716—21no—
36C2×C22⋊D179—24no—
37C22×Dic179—21no—
38#15668243—21no—
39#15668243—21no—
40C2×C179⋊C8—21no—
41C4⋊Dic179D412yes—
42C22⋊Dic179—21no—
43(C2×D179)⋊C4—28no—
44(C2×D179)⋊C4—24no—
45C4×Dic179—21no—
46C4×Dic179—21no—
47C4×Dic179D412yescomplemented
48#41×C179—21no—
49#41×C179—21no—
50#41×C179—21no—
51#41×C179—21no—
52C2×Q8×C179—21no—
53C2×Q8×C179—21no—
54C2×Q8×C179—21no—
55C2×Q8×C179—21no—
56C2×Q8×C179—21no—
57C2×Q8×C179D411yescharacteristic
58C2×Q8×C179D412yescomplemented
59C2×D4×C179—21no—
60C2×D4×C179D411yescharacteristic
61C22×C4×C179—21no—
62C22×C4×C179—21no—
63C22×C4×C179C2311yescharacteristic
64Q16×C179—24no—
65SD16×C179—24no—
66M16×C179—21no—
67M16×C179D412yescomplemented
68C2×C8×C179—21no—
69C22×D179—22no—
70C22×D179—22no—
71C22×D179—22no—
72C22×D179—44no—
73D716—41no—
74D716—41no—
75D716—41no—
76D716—41no—
77C22⋊D179—84no—
78C2×Dic179—22no—
79C2×Dic179—22no—
80C2×Dic179—42no—
81C179⋊C8—41no—
82C23×C179—21no—
83C23×C179C2×D411yescharacteristic
84Q8×C179—22no—
85Q8×C179—21no—
86Q8×C179—22no—
87Q8×C179—21no—
88Q8×C179—41no—
89Q8×C179—41no—
90D4×C179—21no—
91D4×C179—21no—
92D4×C179—41no—
93D4×C179—41no—
94C2×C4×C179—21no—
95C2×C4×C179—22no—
96C2×C4×C179—41no—
97C2×C4×C179—41no—
98C2×C4×C179—41no—
99C2×C4×C179—41no—
100C2×C4×C179C2×D411yescharacteristic
101C2×C4×C179C2×D411yescharacteristic
102C2×C4×C179C2×D411yescharacteristic
103C2×C4×C179C2×D411yescharacteristic
104C2×C4×C179C2×D411yescharacteristic
105C2×C4×C179#4111yescharacteristic
106C8×C179—22no—
107C22×C179—22no—
108C22×C179—41no—
109C22×C179—42no—
110C22×C179#12011yescharacteristic
111C22×C179#12011yescharacteristic
112C22×C179#12111yescharacteristic
113C2×D179—44no—
114C2×D179—42no—
115C2×D179—44no—
116C2×D179—82no—
117C4×C179—41no—
118C4×C179—42no—
119C4×C179#12011yescharacteristic
120C4×C179#12111yescharacteristic
121C4×C179#12111yescharacteristic
122C4×C179#12711yescharacteristic
123Dic179—42no—
124Dic179—42no—
125C2×C179—21no—
126C2×C179—21no—
127C2×C179—81no—
128C2×C179#39211yescharacteristic
129C2×C179#45512yes—
130D179—82no—
131C179#200411yescharacteristic, complemented, Sylow
132#2004—1791nomaximal, Sylow
133C2×#137D17911yescharacteristic, complemented
134#535—1791no—
135C2×#124—1791no—
136C2×#104—1791no—
137#419—1791no—
138C2×#100—1791no—
139#339—1791no—
140C2×#41C2×D17911yescharacteristic
141C22×Q8C2×D17911yescharacteristic
142C22×D4—1791no—
143#137—22no—
144#137—22no—
145C2×Q16—21no—
146C2×SD16—21no—
147C2×M16—1791no—
148C2×M16C2×D17911yescharacteristic
149#127—3581no—
150#124—3581no—
151#124—3582no—
152#124—3582no—
153#121—3582no—
154#120—3582no—
155#117—1791no—
156C2×C22⋊C4—3582no—
157C2×C42—1791no—
158#104—3582no—
159#103—3582no—
160#100—3582no—
161C24—1792no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D17911yescharacteristic
172C2×Q8C22⋊D17912yescomplemented
173C2×D4—21no—
174C2×D4—3584no—
175C2×D4—3581no—
176C2×D4—3581no—
177C2×D4—3581no—
178C2×D4—3581no—
179C2×D4—3581no—
180C2×D4—3581no—
181C2×D4C22⋊D17911yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—3581no—
185C22×C4C22×D17911yescharacteristic
186Q16—24no—
187SD16—24no—
188M16—21no—
189M16—3581no—
190M16—3581no—
191M16C22⋊D17912yescomplemented
192C2×C8—21no—
193C2×C8—3581no—
194C4⋊C4—1792no—
195C22⋊C4—3588no—
196C22⋊C4—3584no—
197C22⋊C4—3581no—
198C42—3581no—
199C42—3581no—
200C42—1792no—
201C23—21no—
202C23—3582no—
203C23—3582no—
204C23—3582no—
205C23—7164no—
206C23D4×D17911yescharacteristic
207Q8—22no—
208Q8—21no—
209Q8—22no—
210Q8—21no—
211Q8—41no—
212Q8—41no—
213D4—21no—
214D4—21no—
215D4—41no—
216D4—41no—
217D4—14324no—
218D4—7161no—
219D4—7161no—
220D4—7161no—
221D4—7161no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3582no—
229C2×C4—3582no—
230C2×C4—7162no—
231C2×C4D4×D17911yescharacteristic
232C2×C4D4×D17911yescharacteristic
233C2×C4C2×C22⋊D17911yescharacteristic
234C2×C4C2×C22⋊D17911yescharacteristic
235C2×C4C2×C22⋊D17911yescharacteristic
236C2×C4#1566825811yescharacteristic
237C8—22no—
238C8—7161no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—14322no—
243C22—7164no—
244C22—7162no—
245C22—7164no—
246C22#1915308111yescharacteristic
247C22#1915308111yescharacteristic
248C22#1915308411yescharacteristic
249C4—41no—
250C4—42no—
251C4—7162no—
252C4—7162no—
253C4#1915308211yescharacteristic
254C4#1915308511yescharacteristic
255C4#1915308511yescharacteristic
256C4#1915310911yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—14322no—
261C2#2603834711yescharacteristic
262C2#2603861512yes—
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
FittingC2×#137×C179C2
socleC22×C179#120
prime radical#26039003C2
hypercenterC2×#137D179
hypocenterC179#2004
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#19153102C2semidirect
3. C2×#19153040C2semidirect
4. #26038448C2semidirect
5. C2×#19153029C2semidirect
6. #26038200C2semidirect
7. C2×#137×C179C2semidirect
47. C4×Dic179D4semidirect
58. C2×Q8×C179D4semidirect
67. M16×C179D4semidirect
131. C179#2004semidirect
133. C2×#137D179semidirect
172. C2×Q8C22⋊D179semidirect
191. M16C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
#260390032
C2×#191531022
C2×#191530402
#260384482
C2×#191530292
#260382002
C2×#137×C1792
#2004179