Finite Group Databasebeta

← #37511010

Subgroups — #37511010

Order 22912.

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

normal subgroups only

This group has 260 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#26039176C211yesmaximal, characteristic, complemented
2#26038630C211yesmaximal, characteristic, complemented
3#26038627C211yesmaximal, characteristic, complemented
4#26038606C211yesmaximal, characteristic, complemented
5#26038228C211yesmaximal, characteristic, complemented
6#26038225C211yesmaximal, characteristic, complemented
7#353×C179C211yesmaximal, characteristic, complemented
8#19153158C2211yescharacteristic
9#19153139—21no—
10#19153107—21no—
11#19153097—21no—
12#19153094—21no—
13#19153094C2211yescharacteristic
14#19153085—21no—
15#19153085—21no—
16#19153085—21no—
17#19153083—21no—
18#19153081—21no—
19#19153081C2211yescharacteristic
20#19153041—21no—
21#19153030C2211yescharacteristic
22#19153027—21no—
23#19153027—21no—
24#127×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—
34C2×C22⋊D179—21no—
35C2×C22⋊D179—21no—
36D4×D179—41no—
37C22×Dic179—21no—
38C22×Dic179D411yescharacteristic, complemented
39#15668247—41no—
40#15668244—41no—
41#15668243—21no—
42C4⋊Dic179—41no—
43Dic179⋊C4—21no—
44Dic179⋊C4—21no—
45Dic179⋊C4—21no—
46C22⋊Dic179—21no—
47C22⋊Dic179—21no—
48C22⋊Dic179D411yescharacteristic
49C22⋊Dic179D411yescharacteristic, complemented
50(C2×D179)⋊C4—21no—
51(C2×D179)⋊C4—21no—
52(C2×D179)⋊C4—21no—
53(C2×D179)⋊C4—41no—
54(C2×D179)⋊C4—41no—
55C2×D4×C179—21no—
56C2×D4×C179—21no—
57C2×D4×C179D411yescharacteristic, complemented
58C2×D4×C179C2311yescharacteristic
59C22⋊C4×C179—21no—
60C22⋊C4×C179—21no—
61C42×C179D411yescharacteristic, complemented
62C22×D179—21no—
63C22×D179—21no—
64C22×D179—21no—
65C22×D179—41no—
66C22×D179—81no—
67Dic358—21no—
68D716—21no—
69D716—82no—
70C22⋊D179—41no—
71C22⋊D179—41no—
72C22⋊D179—41no—
73C22⋊D179—81no—
74C22⋊D179—81no—
75C22⋊D179—81no—
76C2×Dic179—21no—
77C2×Dic179—21no—
78C2×Dic179—21no—
79C2×Dic179—21no—
80C2×Dic179—21no—
81C2×Dic179—41no—
82C2×Dic179—81no—
83C4×D179—41no—
84C179⋊C8—41no—
85C23×C179—21no—
86C23×C179C2×D411yescharacteristic
87C23×C179C2×D411yescharacteristic
88D4×C179—41no—
89D4×C179—41no—
90D4×C179—81no—
91D4×C179—81no—
92C2×C4×C179—21no—
93C2×C4×C179—41no—
94C2×C4×C179—41no—
95C2×C4×C179C2×D411yescharacteristic
96C22×C179—21no—
97C22×C179—21no—
98C22×C179—41no—
99C22×C179—41no—
100C22×C179—41no—
101C22×C179—41no—
102C22×C179—41no—
103C22×C179#12011yescharacteristic
104C2×D179—41no—
105C2×D179—41no—
106C2×D179—42no—
107C2×D179—41no—
108C2×D179—41no—
109C2×D179—81no—
110C2×D179—81no—
111C2×D179—81no—
112C4×C179—21no—
113C4×C179—41no—
114C4×C179—81no—
115C4×C179—81no—
116Dic179—41no—
117Dic179—41no—
118Dic179—41no—
119Dic179—41no—
120C2×C179—21no—
121C2×C179—41no—
122C2×C179—41no—
123C2×C179—81no—
124C2×C179B(4,𝔽2)11yescharacteristic
125D179—81no—
126D179—81no—
127C179#220011yescharacteristic, complemented, Sylow
128#2200—1791nomaximal, Sylow
129#560—1791no—
130#458—1791no—
131B(4,𝔽2)—1791no—
132#453—1791no—
133#353D17911yescharacteristic, complemented
134#352—1791no—
135AΣL(1,𝔽16)—1791no—
136H(5,𝔽2)—1791no—
137AGL(1,ℤ/8)—3581no—
138#127C2×D17911yescharacteristic
139#126—3581no—
140#123—3581no—
141#123—3581no—
142#123—1791no—
143#121—3581no—
144#121—3581no—
145#121—3581no—
146#120—3581no—
147#120—3581no—
148#120—1791no—
149#104—3581no—
150#100—1791no—
151#99—3581no—
152#99—3581no—
153#99C2×D17911yescharacteristic
154#99C2×D17911yescharacteristic
155C24—1791no—
156#41—3581no—
157#41—7161no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—3581no—
161C2×D4—3581no—
162C2×D4—3581no—
163C2×D4—3581no—
164C2×D4—3581no—
165C2×D4—7161no—
166C2×D4—7161no—
167C2×D4D71611yescharacteristic, complemented
168C2×D4C22×D17911yescharacteristic
169C22×C4—3581no—
170C22×C4—1791no—
171SD16—7161no—
172D8—7161no—
173M16—3581no—
174C4⋊C4—3581no—
175C4⋊C4—3581no—
176C4⋊C4—3581no—
177C4⋊C4—7161no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—3581no—
181C22⋊C4—3581no—
182C22⋊C4—3581no—
183C22⋊C4—3581no—
184C22⋊C4—3581no—
185C22⋊C4—1791no—
186C22⋊C4—1791no—
187C22⋊C4—7161no—
188C22⋊C4—7161no—
189C42D71611yescharacteristic, complemented
190C23—21no—
191C23—3581no—
192C23—3581no—
193C23—3581no—
194C23—14321no—
195C23—7161no—
196C23D4×D17911yescharacteristic
197C23D4×D17911yescharacteristic
198Q8—3581no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203D4—3581no—
204D4—14322no—
205D4—14321no—
206D4—14321no—
207D4—14321no—
208D4—7161no—
209D4—7161no—
210D4—7161no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3581no—
215C2×C4—3581no—
216C2×C4—3581no—
217C2×C4—3581no—
218C2×C4—3581no—
219C2×C4—14321no—
220C2×C4—7161no—
221C2×C4—7161no—
222C2×C4C2×D71611yescharacteristic
223C8—7161no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—14321no—
232C22—14321no—
233C22—14321no—
234C22—7161no—
235C22—7161no—
236C22—7162no—
237C22—7161no—
238C22—7161no—
239C22#1915308311yescharacteristic
240C4—21no—
241C4—41no—
242C4—81no—
243C4—81no—
244C4—7161no—
245C4—7161no—
246C4—7161no—
247C4—7161no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—14321no—
253C2—14321no—
254C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×D4×C179C23
FrattiniC2×D4C22×D179
Fitting#353×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#353D179
hypocenterC179#2200
LCSG ▷ C2×D4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #26039176C2semidirect
2. #26038630C2semidirect
3. #26038627C2semidirect
4. #26038606C2semidirect
5. #26038228C2semidirect
6. #26038225C2semidirect
7. #353×C179C2semidirect
38. C22×Dic179D4semidirect
49. C22⋊Dic179D4semidirect
57. C2×D4×C179D4semidirect
61. C42×C179D4semidirect
127. C179#2200semidirect
133. #353D179semidirect
167. C2×D4D716semidirect
189. C42D716semidirect

Maximal subgroups

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

subgroupindex
#260391762
#260386302
#260386272
#260386062
#260382282
#260382252
#353×C1792
#2200179