Finite Group Databasebeta

← #37507524

Subgroups — #37507524

Order 22912.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#26038975C211yesmaximal, characteristic
2#26038493C211yesmaximal, characteristic, complemented
3#26038290C211yesmaximal, characteristic, complemented
4#26038176C211yesmaximal, characteristic, complemented
5#26038174C211yesmaximal, characteristic
6#26038172C211yesmaximal, characteristic
7#328×C179C211yesmaximal, characteristic, complemented
8#19153108—21no—
9#19153104C411yescharacteristic, complemented
10#19153102C411yescharacteristic, complemented
11#19153097—21no—
12#19153087C2211yescharacteristic
13Q8×Dic179C411yescharacteristic, complemented
14#19153075C411yescharacteristic, complemented
15(C22⋊D179)⋊C4—21no—
16#19153071—21no—
17C2×Dic179⋊C4—21no—
18C2×C4×Dic179—21no—
19C2×C4×Dic179C2211yescharacteristic
20Dic179⋊C8—21no—
21C22⋊(C179⋊C8)C2211yescharacteristic
22#19153023—21no—
23C179⋊#95C2211yescharacteristic
24C22⋊Q8×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26#95×C179C2211yescharacteristic
27C2×D716C2×C411yescharacteristic
28C2×C22⋊D179—21no—
29C22×Dic179—21no—
30C22×Dic179D411yescharacteristic
31C22×Dic179D411yescharacteristic
32C2×C4×D179—21no—
33C2×C179⋊C8—21no—
34C4⋊Dic179—21no—
35C4⋊Dic179—21no—
36C4⋊Dic179C2×C411yescharacteristic
37Dic179⋊C4—24no—
38Dic179⋊C4—21no—
39(C2×D179)⋊C4—21no—
40(C2×D179)⋊C4—21no—
41(C2×D179)⋊C4—21no—
42(C2×D179)⋊C4—21no—
43C4×Dic179—21no—
44C4×Dic179—42no—
45C4×Dic179C2×C411yescharacteristic
46C4×Dic179C2×C411yescharacteristic
47C4×Dic179D411yescharacteristic
48C4×Dic179D411yescharacteristic
49C2×Q8×C179C2×C411yescharacteristic
50C22×C4×C179—21no—
51C22×C4×C179C2311yescharacteristic
52C2×C8×C179—21no—
53C4⋊C4×C179—21no—
54C4⋊C4×C179C2×C411yescharacteristic
55C22⋊C4×C179—21no—
56C22×D179—21no—
57D716—41no—
58C22⋊D179—41no—
59C22⋊D179—41no—
60C2×Dic179—21no—
61C2×Dic179—21no—
62C2×Dic179—22no—
63C2×Dic179—22no—
64C2×Dic179—21no—
65C2×Dic179—41no—
66C2×Dic179—41no—
67C2×Dic179C22⋊C411yescharacteristic
68C2×Dic179C22⋊C411yescharacteristic
69C2×Dic179C22⋊C411yescharacteristic
70C2×Dic179C22⋊C411yescharacteristic
71C4×D179—42no—
72C179⋊C8—41no—
73C23×C179C2×D411yescharacteristic
74Q8×C179—41no—
75C2×C4×C179—22no—
76C2×C4×C179—21no—
77C2×C4×C179—21no—
78C2×C4×C179—41no—
79C2×C4×C179—41no—
80C2×C4×C179—41no—
81C2×C4×C179C22×C411yescharacteristic
82C2×C4×C179C2×D411yescharacteristic
83C8×C179—41no—
84C22×C179—21no—
85C22×C179—21no—
86C22×C179C2×C22⋊C411yescharacteristic
87C22×C179AGL(1,ℤ/8)11yescharacteristic
88C22×C179#13711yescharacteristic
89C2×D179—41no—
90C2×D179—41no—
91C2×D179—41no—
92C4×C179—21no—
93C4×C179—21no—
94C4×C179—42no—
95C4×C179—41no—
96C4×C179—41no—
97Dic179—21no—
98Dic179—21no—
99Dic179—21no—
100Dic179—42no—
101Dic179—41no—
102Dic179#10412yes—
103C2×C179—21no—
104C2×C179—21no—
105C2×C179#41011yescharacteristic
106C2×C179#41711yescharacteristic
107C2×C179C2×#10411yescharacteristic
108D179—81no—
109C179#150911yescharacteristic, complemented, Sylow
110#1509—1791nomaximal, Sylow
111#529—1791no—
112#432—1791no—
113C4×C4⋊C4—1791no—
114#328—1791no—
115#328D17911yescharacteristic, complemented
116#327—1791no—
117#327—1791no—
118#126—3581no—
119#125—1791no—
120#124—1791no—
121#123—3581no—
122C22⋊Q8C2×D17911yescharacteristic
123#121—1791no—
124C4×Q8—1791no—
125C4×D4—3581no—
126C4×D4—1791no—
127#117—3581no—
128C2×C4⋊C4—3581no—
129C2×C42—3581no—
130C2×C42—1791no—
131C4⋊C8—3581no—
132C22⋊C8—1791no—
133C22⋊C8C2×D17911yescharacteristic
134#97—3581no—
135#95—1791no—
136#95C2×D17911yescharacteristic
137C2×Q8C4×D17911yescharacteristic
138C2×D4—3581no—
139C2×D4—1791no—
140C22×C4—21no—
141C22×C4—3581no—
142C22×C4—3581no—
143C22×C4—1791no—
144C22×C4—1791no—
145C22×C4C22×D17911yescharacteristic
146C2×C8—21no—
147C2×C8—3581no—
148C4⋊C4—21no—
149C4⋊C4—3584no—
150C4⋊C4—3581no—
151C4⋊C4—3581no—
152C4⋊C4—3581no—
153C4⋊C4—1791no—
154C4⋊C4C4×D17911yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3581no—
157C22⋊C4—3581no—
158C22⋊C4—3581no—
159C22⋊C4—3581no—
160C42—3581no—
161C42—1791no—
162C42—1791no—
163C42—1791no—
164C42—1791no—
165C42—7162no—
166C23—3581no—
167C23D4×D17911yescharacteristic
168Q8—41no—
169D4—7161no—
170D4—7161no—
171D4—7161no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—3581no—
179C2×C4—3581no—
180C2×C4—3582no—
181C2×C4—3582no—
182C2×C4—3581no—
183C2×C4—1791no—
184C2×C4—1791no—
185C2×C4—1791no—
186C2×C4—1791no—
187C2×C4—7162no—
188C2×C4—7161no—
189C2×C4—7161no—
190C2×C4C2×C4×D17911yescharacteristic
191C2×C4D4×D17911yescharacteristic
192C8—41no—
193C8—7161no—
194C22—21no—
195C22—21no—
196C22—7161no—
197C22—7161no—
198C22—7161no—
199C22C22⋊C4×D17911yescharacteristic
200C22#1915314011yescharacteristic
201C22#1915314511yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—3581no—
208C4—3581no—
209C4—3581no—
210C4—1792no—
211C4—7162no—
212C4—7161no—
213C2—21no—
214C2—21no—
215C2—14321no—
216C2#2603841111yescharacteristic
217C2#2603843911yescharacteristic
218C2#104×D17911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D179
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#328×C179C2
socleC22×C179C2×C22⋊C4
prime radical#19153087C22
cosocleC22×C4×C179C23
hypercenter#328D179
hypocenterC179#1509
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #26038493C2semidirect
3. #26038290C2semidirect
4. #26038176C2semidirect
7. #328×C179C2semidirect
9. #19153104C4semidirect
10. #19153102C4semidirect
13. Q8×Dic179C4semidirect
14. #19153075C4semidirect
109. C179#1509semidirect
115. #328D179semidirect

Maximal subgroups

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

subgroupindex
#260389752
#260384932
#260382902
#260381762
#260381742
#260381722
#328×C1792
#1509179