Finite Group Databasebeta

← #37510639

Subgroups — #37510639

Order 22912.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#26038946C211yesmaximal, characteristic
2#26038423C211yesmaximal, characteristic, complemented
3#26038228C212yesmaximal, complemented
4#26038225C212yesmaximal, complemented
5#410×C179C211yesmaximal, characteristic, complemented
6C22×C4×D179C411yescharacteristic, complemented
7C2×#15668243—21no—
8#19153094C411yescharacteristic, complemented
9#19153094C2211yescharacteristic
10#19153089—21no—
11#19153084—21no—
12#19153081C411yescharacteristic, complemented
13#19153081C2211yescharacteristic
14C4×C22⋊D179—21no—
15C4×C22⋊D179—21no—
16#19153072C411yescharacteristic, complemented
17#19153032—21no—
18#19153030C2212yes—
19C2×#41×C179C2211yescharacteristic
20#117×C179—21no—
21#99×C179—21no—
22#99×C179C2212yes—
23C23×D179C2×C411yescharacteristic
24C2×C22⋊D179—21no—
25C2×C22⋊D179—21no—
26C22×Dic179C2×C411yescharacteristic
27C2×C4×D179—21no—
28C2×C4×D179—21no—
29C2×C4×D179—41no—
30C2×C4×D179—41no—
31#15668243—22no—
32#15668243—41no—
33C2×C179⋊C8—41no—
34C4⋊Dic179—21no—
35Dic179⋊C4—21no—
36Dic179⋊C4—21no—
37C22⋊Dic179—21no—
38C22⋊Dic179—21no—
39C22⋊Dic179C2×C411yescharacteristic
40C22⋊Dic179C2×C411yescharacteristic
41(C2×D179)⋊C4—21no—
42(C2×D179)⋊C4—21no—
43C4×Dic179—21no—
44#41×C179—41no—
45#41×C179—41no—
46C2×Q8×C179D411yescharacteristic
47C2×D4×C179C2×C411yescharacteristic
48C2×D4×C179D411yescharacteristic
49C2×D4×C179C2311yescharacteristic
50C22×C4×C179C2×C411yescharacteristic
51C22×C4×C179D411yescharacteristic
52C22×C4×C179D411yescharacteristic
53C4⋊C4×C179—41no—
54C22⋊C4×C179—22no—
55C42×C179—41no—
56C22×D179—21no—
57C22×D179—42no—
58C22×D179—41no—
59C22⋊D179—42no—
60C22⋊D179—42no—
61C2×Dic179—21no—
62C2×Dic179—21no—
63C2×Dic179—21no—
64C2×Dic179—41no—
65C4×D179—42no—
66C4×D179—41no—
67C4×D179—41no—
68C179⋊C8—42no—
69C23×C179C22⋊C411yescharacteristic
70C23×C179C22×C411yescharacteristic
71C23×C179C2×D411yescharacteristic
72Q8×C179—41no—
73D4×C179—41no—
74D4×C179—41no—
75D4×C179—41no—
76C2×C4×C179—21no—
77C2×C4×C179—21no—
78C2×C4×C179—21no—
79C2×C4×C179—21no—
80C2×C4×C179—21no—
81C2×C4×C179—21no—
82C2×C4×C179—42no—
83C2×C4×C179C22⋊C411yescharacteristic
84C2×C4×C179C22⋊C411yescharacteristic
85C2×C4×C179C22⋊C411yescharacteristic
86C2×C4×C179C2×D411yescharacteristic
87C22×C179—21no—
88C22×C179—21no—
89C22×C179—21no—
90C22×C179—41no—
91C22×C179—41no—
92C22×C179—41no—
93C22×C179C2×C22⋊C411yescharacteristic
94C2×D179—22no—
95C2×D179—41no—
96C2×D179—42no—
97C2×D179—41no—
98C2×D179—42no—
99C4×C179—21no—
100C4×C179—21no—
101C4×C179—21no—
102C4×C179—82no—
103C4×C179#9911yescharacteristic
104C4×C179#9911yescharacteristic
105Dic179—41no—
106Dic179—41no—
107Dic179—41no—
108C2×C179—21no—
109C2×C179—41no—
110C2×C179—41no—
111C2×C179—41no—
112C2×C179C2×#9911yescharacteristic
113D179—42no—
114C179#212011yescharacteristic, complemented, Sylow
115#2120—1791nomaximal, Sylow
116#525—1791no—
117#413—1791no—
118#410D17911yescharacteristic, complemented
119#352—1792no—
120AΣL(1,𝔽16)—1792no—
121C2×#41C2×D17911yescharacteristic
122C23×C4—1791no—
123C2×M16—3581no—
124#123—1791no—
125#123—1791no—
126C22⋊Q8—3581no—
127#121—3581no—
128#120—1791no—
129#120—1791no—
130C4×D4—3581no—
131C4×D4—3581no—
132#117—21no—
133#117—1791no—
134#101—3581no—
135#100—1792no—
136#99—21no—
137#99C2×D17912yes—
138C24—1791no—
139#41—41no—
140#41—41no—
141C2×Q8D71611yescharacteristic
142C2×D4—3581no—
143C2×D4—3581no—
144C2×D4C4×D17911yescharacteristic
145C2×D4C22⋊D17911yescharacteristic
146C2×D4C22×D17911yescharacteristic
147C22×C4—3581no—
148C22×C4—3581no—
149C22×C4—1791no—
150C22×C4—7161no—
151C22×C4—7161no—
152C22×C4C4×D17911yescharacteristic
153C22×C4C22⋊D17911yescharacteristic
154C22×C4D71611yescharacteristic
155M16—3582no—
156M16—7161no—
157C2×C8—7161no—
158C4⋊C4—41no—
159C4⋊C4—3581no—
160C4⋊C4—3581no—
161C4⋊C4—3581no—
162C22⋊C4—22no—
163C22⋊C4—3581no—
164C22⋊C4—3581no—
165C22⋊C4—3581no—
166C22⋊C4—3581no—
167C22⋊C4—1791no—
168C22⋊C4—1791no—
169C42—41no—
170C42—3581no—
171C23—3581no—
172C23—7161no—
173C23—7162no—
174C23(C2×D179)⋊C411yescharacteristic
175C23C2×C4×D17911yescharacteristic
176C23C2×C22⋊D17911yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—7162no—
182D4—7162no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—3581no—
191C2×C4—3581no—
192C2×C4—3581no—
193C2×C4—7162no—
194C2×C4—7161no—
195C2×C4—7161no—
196C2×C4—7161no—
197C2×C4(C2×D179)⋊C411yescharacteristic
198C2×C4(C2×D179)⋊C411yescharacteristic
199C2×C4(C2×D179)⋊C411yescharacteristic
200C2×C4C2×D71611yescharacteristic
201C8—7162no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—3582no—
209C22—7161no—
210C22—7162no—
211C22—7162no—
212C22—7161no—
213C22C2×(C2×D179)⋊C411yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—7161no—
219C4—7161no—
220C4—7161no—
221C4#1915302711yescharacteristic
222C4#1915302711yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—7162no—
228C2C2×#1915302711yescharacteristic
subgroupG/H
centerC4#19153027
commutatorC23×C179C22×C4
FrattiniC2×D4C22×D179
Fitting#410×C179C2
socleC2×C179C2×#99
prime radical#26038946C2
cosocleC2×D4×C179C23
hypercenter#410D179
hypocenterC179#2120
LCSG ▷ C23×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #26038423C2semidirect
3. #26038228C2semidirect
4. #26038225C2semidirect
5. #410×C179C2semidirect
6. C22×C4×D179C4semidirect
8. #19153094C4semidirect
12. #19153081C4semidirect
16. #19153072C4semidirect
114. C179#2120semidirect
118. #410D179semidirect

Maximal subgroups

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

subgroupindex
#260389462
#260384232
#260382282
#260382252
#410×C1792
#2120179