Finite Group Databasebeta

← #26038949

Subgroups — #26038949

Order 11456.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#15668258C211yesmaximal, characteristic, complemented
2C22×C4×D179C211yesmaximal, characteristic, complemented
3#19153095C212yesmaximal, complemented
4#19153093C211yesmaximal, characteristic, complemented
5#19153092C211yesmaximal, characteristic, complemented
6#19153084C211yesmaximal, characteristic, complemented
7#19153081C212yesmaximal, complemented
8D4×Dic179C212yesmaximal, complemented
9(C22⋊D179)⋊C4C212yesmaximal, complemented
10#19153071C211yesmaximal, characteristic, complemented
11#121×C179C211yesmaximal, characteristic, complemented
12C23×D179C2211yescharacteristic
13#15668258—24no—
14C2×Dic358C2211yescharacteristic, complemented
15C2×C22⋊D179C2212yes—
16C2×C22⋊D179C2212yes—
17C22×Dic179C2211yescharacteristic
18C22×Dic179C2212yescomplemented
19C2×C4×D179—24no—
20C2×C4×D179C2211yescharacteristic
21C2×C4×D179C2211yescharacteristic
22C2×C4×D179C2211yescharacteristic
23C2×C4×D179C2211yescharacteristic
24C4⋊Dic179C2211yescharacteristic, complemented
25C4⋊Dic179C2212yescomplemented
26Dic179⋊C4C2212yescomplemented
27C22⋊Dic179C2212yes—
28C22⋊Dic179C2212yescomplemented
29(C2×D179)⋊C4C2212yes—
30(C2×D179)⋊C4C2212yes—
31C4×Dic179C2212yescomplemented
32C2×D4×C179C2212yes—
33C2×D4×C179C2211yescharacteristic, complemented
34C22×C4×C179C2211yescharacteristic
35C4⋊C4×C179C2211yescharacteristic, complemented
36C22⋊C4×C179C2212yescomplemented
37C22×D179—22no—
38C22×D179—22no—
39C22×D179—22no—
40C22×D179C2311yescharacteristic
41C22×D179C2311yescharacteristic
42Dic358—22no—
43C22⋊D179—24no—
44C22⋊D179—24no—
45C2×Dic179—22no—
46C2×Dic179—24no—
47C2×Dic179C2312yes—
48C2×Dic179C2312yes—
49C2×Dic179C2311yescharacteristic
50C2×Dic179C2311yescharacteristic
51C4×D179—22no—
52C4×D179—22no—
53C4×D179—22no—
54C4×D179D414yescomplemented
55C23×C179C2312yes—
56C23×C179C2311yescharacteristic
57D4×C179—24no—
58D4×C179—22no—
59C2×C4×C179—22no—
60C2×C4×C179C2311yescharacteristic
61C2×C4×C179C2312yes—
62C2×C4×C179C2311yescharacteristic
63C22×C179—21no—
64C22×C179—21no—
65C22×C179—22no—
66C22×C179—22no—
67C22×C179—22no—
68C22×C179#4112yescomplemented
69C22×C179C2411yescharacteristic
70C2×D179—28no—
71C2×D179—21no—
72C2×D179—21no—
73C2×D179—21no—
74C2×D179—21no—
75C2×D179C2×D412yes—
76C2×D179#4112yescomplemented
77C4×C179—21no—
78C4×C179—22no—
79C4×C179C2×D412yes—
80Dic179—21no—
81Dic179—22no—
82Dic179—22no—
83Dic179C2×D412yes—
84C2×C179—22no—
85C2×C179—42no—
86C2×C179C22×D411yescharacteristic
87C2×C179C2×#4111yescharacteristic
88C2×C179C2×#4111yescharacteristic
89D179—22no—
90D179—22no—
91C179#52511yescharacteristic, complemented, Sylow
92#525—1791nomaximal, Sylow
93C2×#41—1791no—
94C23×C4—1791no—
95#123—1792no—
96C22⋊Q8—1791no—
97C22⋊Q8—1791no—
98#121—1791no—
99#121D17911yescharacteristic, complemented
100#120—1792no—
101C4×D4—1792no—
102C4×D4—1792no—
103#117—1791no—
104C24—1791no—
105#41—3584no—
106C2×Q8—1791no—
107C2×D4—1792no—
108C2×D4—1792no—
109C2×D4C2×D17912yes—
110C2×D4C2×D17911yescharacteristic, complemented
111C22×C4—3584no—
112C22×C4—1791no—
113C22×C4—1792no—
114C22×C4—1791no—
115C22×C4—1791no—
116C22×C4—1791no—
117C22×C4—1791no—
118C22×C4C2×D17911yescharacteristic
119C4⋊C4—1792no—
120C4⋊C4—1791no—
121C4⋊C4—1792no—
122C4⋊C4C2×D17911yescharacteristic, complemented
123C22⋊C4—1792no—
124C22⋊C4—1792no—
125C22⋊C4—1792no—
126C22⋊C4—1792no—
127C22⋊C4C2×D17912yescomplemented
128C42—1792no—
129C23—3582no—
130C23—3582no—
131C23—3582no—
132C23—1791no—
133C23—1791no—
134C23C22×D17912yes—
135C23C22×D17911yescharacteristic
136Q8—3582no—
137D4—24no—
138D4—22no—
139D4—3584no—
140D4—3584no—
141C2×C4—22no—
142C2×C4—3582no—
143C2×C4—3582no—
144C2×C4—3582no—
145C2×C4—3584no—
146C2×C4—3582no—
147C2×C4—1792no—
148C2×C4—1791no—
149C2×C4—1792no—
150C2×C4—1791no—
151C2×C4—1794no—
152C2×C4C22×D17912yes—
153C2×C4C22×D17911yescharacteristic
154C2×C4C22×D17911yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—3588no—
161C22—3581no—
162C22—3581no—
163C22—3581no—
164C22—3581no—
165C22—1792no—
166C22—1792no—
167C22#1566825812yescomplemented
168C22C23×D17911yescharacteristic
169C4—22no—
170C4—21no—
171C4—3582no—
172C4—3581no—
173C4—3582no—
174C4—1792no—
175C4D4×D17912yes—
176C2—22no—
177C2—42no—
178C2—3582no—
179C2—3582no—
180C2C2×D4×D17911yescharacteristic
181C2#41×D17911yescharacteristic
182C2C2×#1566825811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D179
commutator, socle, cosocleC22×C179C24
Fitting#121×C179C2
hypercenter#121D179
hypocenterC179#525
LCSG ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15668258C2semidirect
2. C22×C4×D179C2semidirect
3. #19153095C2semidirect
4. #19153093C2semidirect
5. #19153092C2semidirect
6. #19153084C2semidirect
7. #19153081C2semidirect
8. D4×Dic179C2semidirect
9. (C22⋊D179)⋊C4C2semidirect
10. #19153071C2semidirect
11. #121×C179C2semidirect
14. C2×Dic358C22semidirect
18. C22×Dic179C22semidirect
24. C4⋊Dic179C22semidirect
25. C4⋊Dic179C22semidirect
26. Dic179⋊C4C22semidirect
28. C22⋊Dic179C22semidirect
31. C4×Dic179C22semidirect
33. C2×D4×C179C22semidirect
35. C4⋊C4×C179C22semidirect
36. C22⋊C4×C179C22semidirect
54. C4×D179D4semidirect
68. C22×C179#41semidirect
76. C2×D179#41semidirect
91. C179#525semidirect
99. #121D179semidirect
110. C2×D4C2×D179semidirect
122. C4⋊C4C2×D179semidirect
127. C22⋊C4C2×D179semidirect
167. C22#15668258semidirect

Maximal subgroups

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

subgroupindex
C2×#156682582
C22×C4×D1792
#191530952
#191530932
#191530922
#191530842
#191530812
D4×Dic1792
(C22⋊D179)⋊C42
#191530712
#121×C1792
#525179