Finite Group Databasebeta

← #37510993

Subgroups — #37510993

Order 22912.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#26039119C211yesmaximal, characteristic, complemented
2#26038630C211yesmaximal, characteristic, complemented
3#26038627C211yesmaximal, characteristic
4#26038616C211yesmaximal, characteristic, complemented
5#26038228C211yesmaximal, characteristic, complemented
6#26038225C211yesmaximal, characteristic, complemented
7#354×C179C211yesmaximal, characteristic, complemented
8#19153158C2211yescharacteristic
9#19153144—21no—
10#19153106—21no—
11#19153097—21no—
12#19153095—21no—
13#19153094C2211yescharacteristic
14#19153089—21no—
15#19153083—21no—
16#19153081C2211yescharacteristic
17(C22⋊D179)⋊C4—21no—
18#19153070—21no—
19C22⋊C4×D179—21no—
20#19153041—21no—
21#19153030C2211yescharacteristic
22#19153027—21no—
23#19153027—21no—
24#124×C179C2211yescharacteristic
25#99×C179C2211yescharacteristic
26#99×C179C2211yescharacteristic
27C23×D179D411yescharacteristic
28#15668259—21no—
29#15668258—41no—
30C2×D716—41no—
31C2×C22⋊D179—21no—
32C2×C22⋊D179—21no—
33C2×C22⋊D179—21no—
34D4×D179—41no—
35C22×Dic179D411yescharacteristic, complemented
36C2×C4×D179—21no—
37C2×C4×D179—21no—
38#15668249—41no—
39#15668246—41no—
40#15668243—21no—
41C4⋊Dic179—21no—
42C4⋊Dic179—41no—
43Dic179⋊C4—21no—
44Dic179⋊C4—21no—
45Dic179⋊C4—21no—
46C22⋊Dic179—21no—
47C22⋊Dic179D411yescharacteristic
48C22⋊Dic179D411yescharacteristic, complemented
49(C2×D179)⋊C4—21no—
50(C2×D179)⋊C4—21no—
51(C2×D179)⋊C4—21no—
52(C2×D179)⋊C4—41no—
53(C2×D179)⋊C4—41no—
54C4×Dic179—21no—
55C2×Q8×C179D411yescharacteristic, complemented
56C2×D4×C179C2311yescharacteristic
57C22⋊C4×C179—21no—
58C22⋊C4×C179—21no—
59C22⋊C4×C179—21no—
60C22⋊C4×C179—21no—
61C42×C179D411yescharacteristic, complemented
62C22×D179—21no—
63C22×D179—21no—
64C22×D179—21no—
65C22×D179—42no—
66C22×D179—41no—
67Dic358—21no—
68D716—21no—
69D716—82no—
70C22⋊D179—42no—
71C22⋊D179—41no—
72C22⋊D179—41no—
73C22⋊D179—41no—
74C2×Dic179—21no—
75C2×Dic179—21no—
76C2×Dic179—21no—
77C2×Dic179—21no—
78C2×Dic179—21no—
79C2×Dic179—41no—
80C4×D179—41no—
81C4×D179—42no—
82C4×D179—81no—
83C179⋊C8—41no—
84C23×C179C2×D411yescharacteristic
85C23×C179C2×D411yescharacteristic
86Q8×C179—41no—
87D4×C179—41no—
88C2×C4×C179—21no—
89C2×C4×C179—21no—
90C2×C4×C179—41no—
91C2×C4×C179—41no—
92C2×C4×C179C2×D411yescharacteristic
93C22×C179—21no—
94C22×C179—21no—
95C22×C179—41no—
96C22×C179—41no—
97C22×C179#12011yescharacteristic
98C2×D179—22no—
99C2×D179—41no—
100C2×D179—41no—
101C2×D179—42no—
102C2×D179—41no—
103C2×D179—81no—
104C2×D179—81no—
105C2×D179—81no—
106C4×C179—21no—
107C4×C179—41no—
108C4×C179—41no—
109C4×C179—81no—
110C4×C179—81no—
111Dic179—41no—
112Dic179—41no—
113Dic179—41no—
114Dic179—41no—
115C2×C179—21no—
116C2×C179—41no—
117C2×C179—41no—
118C2×C179B(4,𝔽2)11yescharacteristic
119D179—42no—
120D179—81no—
121C179#219711yescharacteristic, complemented, Sylow
122#2197—1791nomaximal, Sylow
123#551—1791no—
124#458—1791no—
125B(4,𝔽2)—1791no—
126#455—1791no—
127#354D17911yescharacteristic, complemented
128#352—1791no—
129AΣL(1,𝔽16)—1791no—
130H(5,𝔽2)—1791no—
131#137—3581no—
132#126—3581no—
133#124C2×D17911yescharacteristic
134#123—3581no—
135#123—3581no—
136#123—1791no—
137C22⋊Q8—3581no—
138#120—3581no—
139#120—1791no—
140C4×D4—3581no—
141#117—3581no—
142C2×C22⋊C4—3581no—
143#104—3581no—
144#100—1791no—
145#99—3581no—
146#99—3581no—
147#99C2×D17911yescharacteristic
148#99C2×D17911yescharacteristic
149C24—1791no—
150#41—3581no—
151#41—7161no—
152C2×Q8D71611yescharacteristic, complemented
153C2×D4—3581no—
154C2×D4—3581no—
155C2×D4—3581no—
156C2×D4—7161no—
157C2×D4—7161no—
158C2×D4C22×D17911yescharacteristic
159C22×C4—3581no—
160C22×C4—3581no—
161C22×C4—1791no—
162Q16—7161no—
163SD16—7161no—
164M16—3581no—
165C4⋊C4—3581no—
166C4⋊C4—3581no—
167C4⋊C4—3581no—
168C4⋊C4—3581no—
169C4⋊C4—7161no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—3581no—
175C22⋊C4—3581no—
176C22⋊C4—3581no—
177C22⋊C4—3581no—
178C22⋊C4—1791no—
179C22⋊C4—1791no—
180C22⋊C4—7161no—
181C22⋊C4—7161no—
182C42—3581no—
183C42D71611yescharacteristic, complemented
184C23—3581no—
185C23—3581no—
186C23—3581no—
187C23—7161no—
188C23—7162no—
189C23D4×D17911yescharacteristic
190C23D4×D17911yescharacteristic
191Q8—41no—
192Q8—3581no—
193D4—41no—
194D4—3581no—
195D4—14322no—
196D4—7161no—
197D4—7161no—
198D4—7162no—
199D4—7161no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—3581no—
205C2×C4—3581no—
206C2×C4—3581no—
207C2×C4—3581no—
208C2×C4—3581no—
209C2×C4—14321no—
210C2×C4—7161no—
211C2×C4—7162no—
212C2×C4—7161no—
213C2×C4C2×D71611yescharacteristic
214C8—7161no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—3582no—
220C22—14321no—
221C22—14321no—
222C22—14321no—
223C22—7161no—
224C22—7162no—
225C22—7161no—
226C22—7161no—
227C22#1915308311yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—81no—
233C4—7161no—
234C4—7161no—
235C4—7161no—
236C4—7161no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—14321no—
241C2—7162no—
242C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×D4×C179C23
FrattiniC2×D4C22×D179
Fitting#354×C179C2
socleC2×C179B(4,𝔽2)
prime radical#26038627C2
hypercenter#354D179
hypocenterC179#2197
LCSG ▷ C2×D4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #26039119C2semidirect
2. #26038630C2semidirect
4. #26038616C2semidirect
5. #26038228C2semidirect
6. #26038225C2semidirect
7. #354×C179C2semidirect
35. C22×Dic179D4semidirect
48. C22⋊Dic179D4semidirect
55. C2×Q8×C179D4semidirect
61. C42×C179D4semidirect
121. C179#2197semidirect
127. #354D179semidirect
152. C2×Q8D716semidirect
183. C42D716semidirect

Maximal subgroups

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

subgroupindex
#260391192
#260386302
#260386272
#260386162
#260382282
#260382252
#354×C1792
#2197179