Finite Group Databasebeta

← #37508239

Subgroups — #37508239

Order 22912.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039164C211yesmaximal, characteristic
2#26038721C211yesmaximal, characteristic
3#26038646C211yesmaximal, characteristic
4#26038185C211yesmaximal, characteristic
5#26038171C211yesmaximal, characteristic
6#26038171C211yesmaximal, characteristic
7#500×C179C211yesmaximal, characteristic
8C2×#15668249—21no—
9#19153111—21no—
10#19153104—21no—
11#19153103—22no—
12Q8×Dic179—21no—
13C4×Dic358C2211yescharacteristic
14C4×Dic358C2211yescharacteristic
15#19153078—21no—
16#19153044—21no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18C4⋊(C179⋊C8)C2211yescharacteristic
19#19153037—21no—
20#19153036—21no—
21C4⋊Q8×C179C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23#107×C179—22no—
24C4×C8×C179C2211yescharacteristic
25C2×Dic358D411yescharacteristic
26C2×Dic358D411yescharacteristic
27#15668249—42no—
28C2×C179⋊C8—21no—
29C2×C179⋊C8—21no—
30C4⋊Dic179—22no—
31C4⋊Dic179D411yescharacteristic
32C4⋊Dic179D411yescharacteristic
33Dic179⋊C4—22no—
34Dic179⋊C4—22no—
35Dic179⋊C4—21no—
36Dic179⋊C4—21no—
37C4×Dic179—22no—
38C4×Dic179—21no—
39C4×Dic179—21no—
40C2×Q8×C179—21no—
41C2×Q8×C179—21no—
42C2×C8×C179—21no—
43C4⋊C4×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179D412yes—
47C42×C179C2311yescharacteristic
48Dic358—21no—
49Dic358—21no—
50Dic358—21no—
51Dic358Q1612yescomplemented
52C2×Dic179—21no—
53C2×Dic179—21no—
54C2×Dic179—21no—
55C2×Dic179—21no—
56C179⋊C8—41no—
57C179⋊C8—41no—
58Q8×C179—41no—
59Q8×C179—41no—
60Q8×C179—41no—
61Q8×C179—41no—
62C2×C4×C179—22no—
63C2×C4×C179—41no—
64C2×C4×C179C2×D411yescharacteristic
65C2×C4×C179C2×D411yescharacteristic
66C2×C4×C179C2×D411yescharacteristic
67C8×C179—22no—
68C22×C179#12011yescharacteristic
69C4×C179—21no—
70C4×C179—42no—
71C4×C179—81no—
72C4×C179C2×Q1611yescharacteristic
73C4×C179#13511yescharacteristic
74C4×C179AGL(1,ℤ/8)11yescharacteristic
75C4×C179#13711yescharacteristic
76Dic179—22no—
77Dic179—41no—
78Dic179—41no—
79Dic179—41no—
80C2×C179#44911yescharacteristic
81C2×C179#45111yescharacteristic
82C2×C179#45611yescharacteristic
83C179#164811yescharacteristic, complemented, Sylow
84#1648—1791nomaximal, Sylow
85#557—1791no—
86#500D17911yescharacteristic
87#475—1791no—
88#462—1791no—
89#332—1791no—
90Q8⋊C8—1791no—
91Q8⋊C8—1791no—
92C2×Q16—3581no—
93C4⋊Q8—3581no—
94C4⋊Q8C2×D17911yescharacteristic
95C4⋊Q8C2×D17911yescharacteristic
96#125—3581no—
97#125—3582no—
98C4×Q8—3581no—
99C4×Q8—3581no—
100C4×Q8—1791no—
101C4×Q8—1791no—
102#107—22no—
103#106—3581no—
104C4⋊C8—1791no—
105C4⋊C8—1791no—
106#103—3581no—
107#103—3581no—
108C4×C8C2×D17911yescharacteristic
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—1791no—
112C2×Q8—1791no—
113Q16—7162no—
114C2×C8—21no—
115C2×C8—3581no—
116C2×C8—3581no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—3582no—
121C4⋊C4—3582no—
122C4⋊C4—3582no—
123C4⋊C4—3581no—
124C4⋊C4—3581no—
125C4⋊C4—1791no—
126C4⋊C4—1791no—
127C4⋊C4C22⋊D17912yes—
128C42—3582no—
129C42—3581no—
130C42—3581no—
131C42C22×D17911yescharacteristic
132Q8—41no—
133Q8—41no—
134Q8—41no—
135Q8—41no—
136Q8—3581no—
137Q8—3581no—
138Q8—3581no—
139Q8—1792no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—3581no—
143C2×C4—3581no—
144C2×C4—3581no—
145C2×C4—3581no—
146C2×C4D4×D17911yescharacteristic
147C2×C4D4×D17911yescharacteristic
148C2×C4C2×C22⋊D17911yescharacteristic
149C8—22no—
150C8—7161no—
151C8—7161no—
152C22#1915308111yescharacteristic
153C4—21no—
154C4—42no—
155C4—81no—
156C4—3582no—
157C4—7161no—
158C4—7161no—
159C4—7161no—
160C4Q16×D17911yescharacteristic
161C4#1915313211yescharacteristic
162C4#1915313711yescharacteristic
163C4#1915314511yescharacteristic
164C2#2603858111yescharacteristic
165C2#2603859311yescharacteristic
166C2#2603862111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#500×C179C2
socle, prime radicalC22×C179#120
hypercenter#500D179
hypocenterC179#1648
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
51. Dic358Q16semidirect
83. C179#1648semidirect

Maximal subgroups

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

subgroupindex
#260391642
#260387212
#260386462
#260381852
#260381712
#500×C1792
#1648179