Finite Group Databasebeta

← #37507979

Subgroups — #37507979

Order 22912.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#26039029C211yesmaximal, characteristic
2#26038578C211yesmaximal, characteristic, complemented
3#26038350C211yesmaximal, characteristic, complemented
4#26038176C211yesmaximal, characteristic, complemented
5#26038174C211yesmaximal, characteristic, complemented
6#26038162C211yesmaximal, characteristic, complemented
7#451×C179C211yesmaximal, characteristic, complemented
8C2×#15668246—21no—
9#19153108—21no—
10#19153102—21no—
11#19153098—21no—
12#19153088—21no—
13#19153087C2211yescharacteristic
14#19153083—21no—
15(C22⋊D179)⋊C4—21no—
16C2×(C2×D179)⋊C4—21no—
17C2×(C2×D179)⋊C4C2211yescharacteristic
18#19153037—21no—
19C22⋊(C179⋊C8)C2211yescharacteristic
20C179⋊#95—21no—
21C179⋊#95C2211yescharacteristic
22C22×Q8×C179C2211yescharacteristic
23C2×Q16×C179—21no—
24C22⋊Q8×C179C2211yescharacteristic
25#103×C179—21no—
26C22⋊C8×C179C2211yescharacteristic
27C23×D179D411yescharacteristic
28C2×D716—21no—
29C2×D716—21no—
30C2×D716D411yescharacteristic
31C2×C22⋊D179—21no—
32C22×Dic179—21no—
33C22×Dic179D411yescharacteristic
34C2×C4×D179—21no—
35#15668246—42no—
36C2×C179⋊C8—21no—
37C4⋊Dic179D411yescharacteristic
38Dic179⋊C4—21no—
39(C2×D179)⋊C4—21no—
40(C2×D179)⋊C4—21no—
41(C2×D179)⋊C4—21no—
42(C2×D179)⋊C4—21no—
43(C2×D179)⋊C4—21no—
44(C2×D179)⋊C4—21no—
45(C2×D179)⋊C4—42no—
46C4×Dic179—21no—
47C2×Q8×C179—21no—
48C2×Q8×C179—21no—
49C2×Q8×C179—42no—
50C2×Q8×C179D411yescharacteristic
51C22×C4×C179—21no—
52C22×C4×C179C2311yescharacteristic
53Q16×C179—42no—
54C2×C8×C179—21no—
55C4⋊C4×C179—21no—
56C4⋊C4×C179D411yescharacteristic
57C22⋊C4×C179—21no—
58C22×D179—21no—
59C22×D179—21no—
60C22×D179—21no—
61C22×D179—21no—
62C22×D179—41no—
63C22×D179—41no—
64D716—41no—
65D716—81no—
66D716—81no—
67C22⋊D179—41no—
68C22⋊D179—41no—
69C2×Dic179—21no—
70C2×Dic179—21no—
71C2×Dic179—41no—
72C2×Dic179—41no—
73C2×Dic179—41no—
74C2×Dic179—41no—
75C4×D179—81no—
76C179⋊C8—41no—
77C23×C179C2×D411yescharacteristic
78Q8×C179—22no—
79Q8×C179—22no—
80Q8×C179—41no—
81Q8×C179—41no—
82Q8×C179—41no—
83C2×C4×C179—22no—
84C2×C4×C179—21no—
85C2×C4×C179—21no—
86C2×C4×C179—41no—
87C2×C4×C179—41no—
88C2×C4×C179—41no—
89C2×C4×C179C2×D411yescharacteristic
90C2×C4×C179C2×D411yescharacteristic
91C8×C179—41no—
92C22×C179—21no—
93C22×C179—21no—
94C22×C179#12011yescharacteristic
95C22×C179#13511yescharacteristic
96C22×C179#13711yescharacteristic
97C2×D179—42no—
98C2×D179—41no—
99C2×D179—41no—
100C2×D179—41no—
101C2×D179—41no—
102C2×D179—41no—
103C2×D179—41no—
104C2×D179—81no—
105C4×C179—21no—
106C4×C179—21no—
107C4×C179—41no—
108C4×C179—42no—
109C4×C179—41no—
110Dic179—41no—
111Dic179—41no—
112Dic179—81no—
113C2×C179—21no—
114C2×C179—21no—
115C2×C179#45211yescharacteristic
116C2×C179#45511yescharacteristic
117C2×C179B(4,𝔽2)11yescharacteristic
118D179—81no—
119D179—81no—
120C179#160611yescharacteristic, complemented, Sylow
121#1606—1791nomaximal, Sylow
122#538—1791no—
123#451D17911yescharacteristic, complemented
124#448—1791no—
125#393—1791no—
126#328—1791no—
127#327—1791no—
128#323—1791no—
129C22×Q8C2×D17911yescharacteristic
130C2×Q16—21no—
131C2×SD16—3581no—
132#126—3581no—
133#124—3581no—
134#123—3581no—
135C22⋊Q8C2×D17911yescharacteristic
136#121—3581no—
137#121—1791no—
138#120—3581no—
139C4×D4—3581no—
140C2×C22⋊C4—3581no—
141C2×C22⋊C4—1791no—
142#103—21no—
143#103—3581no—
144C22⋊C8—1791no—
145C22⋊C8C2×D17911yescharacteristic
146#95—3581no—
147#95—1791no—
148C24—1791no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8C22⋊D17911yescharacteristic
153C2×D4—3581no—
154C2×D4—3581no—
155C2×D4—3581no—
156C2×D4—1791no—
157C22×C4—21no—
158C22×C4—3581no—
159C22×C4—3581no—
160C22×C4—1791no—
161C22×C4C22×D17911yescharacteristic
162Q16—42no—
163SD16—7162no—
164C2×C8—21no—
165C2×C8—3581no—
166C4⋊C4—21no—
167C4⋊C4—3581no—
168C4⋊C4—1791no—
169C4⋊C4C22⋊D17911yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—3581no—
172C22⋊C4—3581no—
173C22⋊C4—3581no—
174C22⋊C4—3581no—
175C22⋊C4—3581no—
176C22⋊C4—3581no—
177C22⋊C4—7162no—
178C42—3581no—
179C23—3581no—
180C23—3581no—
181C23—3581no—
182C23—3581no—
183C23—7161no—
184C23—7161no—
185C23D4×D17911yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191D4—14321no—
192D4—14321no—
193D4—7161no—
194D4—7161no—
195D4—7161no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3581no—
203C2×C4—3581no—
204C2×C4—14321no—
205C2×C4—7161no—
206C2×C4—7161no—
207C2×C4—7161no—
208C2×C4—7161no—
209C2×C4D4×D17911yescharacteristic
210C2×C4C2×C22⋊D17911yescharacteristic
211C8—41no—
212C8—7161no—
213C22—21no—
214C22—21no—
215C22—14321no—
216C22—7162no—
217C22—7161no—
218C22—7161no—
219C22—7161no—
220C22—7161no—
221C22—7161no—
222C22—7161no—
223C22#1915308111yescharacteristic
224C22#1915313611yescharacteristic
225C22#1915314511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—41no—
231C4—14321no—
232C4—7161no—
233C4—7161no—
234C2—21no—
235C2—21no—
236C2—14321no—
237C2—14321no—
238C2#2603860011yescharacteristic
239C2#2603861511yescharacteristic
240C2#2603862511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#451×C179C2
socleC22×C179#120
prime radical#26039029C2
hypercenter#451D179
hypocenterC179#1606
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #26038578C2semidirect
3. #26038350C2semidirect
4. #26038176C2semidirect
5. #26038174C2semidirect
6. #26038162C2semidirect
7. #451×C179C2semidirect
120. C179#1606semidirect
123. #451D179semidirect

Maximal subgroups

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

subgroupindex
#260390292
#260385782
#260383502
#260381762
#260381742
#260381622
#451×C1792
#1606179