Finite Group Databasebeta

← #37511031

Subgroups — #37511031

Order 22912.

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

normal subgroups only

This group has 224 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#26039169C211yesmaximal, characteristic, complemented
2#26038624C212yesmaximal
3#26038621C211yesmaximal, characteristic
4#26038237C212yesmaximal, complemented
5#356×C179C211yesmaximal, characteristic, complemented
6#19153161C2211yescharacteristic
7#19153146—22no—
8#19153144—21no—
9#19153112—22no—
10#19153111—21no—
11#19153111C2212yes—
12C4×Dic358—21no—
13#19153078—22no—
14#19153041—21no—
15#19153039—22no—
16#19153032C2211yescharacteristic
17C4⋊Q8×C179C2211yescharacteristic
18#101×C179C2212yes—
19#15668260—41no—
20#15668259—22no—
21#15668259—21no—
22C2×Dic358—21no—
23C2×Dic358—21no—
24C2×Dic358D412yes—
25Q8×D179—41no—
26Dic716—42no—
27#15668249—41no—
28#15668248—42no—
29#15668246—41no—
30#15668243—21no—
31C4⋊Dic179—21no—
32Dic179⋊C4—22no—
33Dic179⋊C4—22no—
34Dic179⋊C4—22no—
35C4×Dic179—21no—
36C4×Dic179—22no—
37C4×Dic179D412yes—
38C2×Q8×C179D411yescharacteristic
39C2×Q8×C179C2311yescharacteristic
40M16×C179—22no—
41C4⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43Dic358—24no—
44Dic358—21no—
45Dic358—22no—
46Dic358—42no—
47Dic358—81no—
48D716—21no—
49D716—22no—
50C22⋊D179—41no—
51C2×Dic179—22no—
52C2×Dic179—22no—
53C4×D179—42no—
54C4×D179—41no—
55C179⋊C8—41no—
56Q8×C179—41no—
57Q8×C179—41no—
58C2×C4×C179—21no—
59C2×C4×C179—21no—
60C2×C4×C179C2×D412yes—
61C2×C4×C179C2×D411yescharacteristic
62C8×C179—42no—
63C22×C179#12011yescharacteristic
64C2×D179—41no—
65C4×C179—22no—
66C4×C179—22no—
67C4×C179—21no—
68C4×C179—41no—
69Dic179—24no—
70Dic179—41no—
71Dic179—42no—
72C2×C179—21no—
73C2×C179B(4,𝔽2)11yescharacteristic
74D179—81no—
75C179#220511yescharacteristic, complemented, Sylow
76#2205—1791nomaximal, Sylow
77Q82—1791no—
78#456—1791no—
79#456—1792no—
80#356—1792no—
81#356D17911yescharacteristic, complemented
822−1+4—1791no—
83#137—3582no—
84#137—3581no—
85C4⋊Q8—3582no—
86C4⋊Q8—3581no—
87C4⋊Q8—1792no—
88C4⋊Q8C2×D17911yescharacteristic
89C4×Q8—3582no—
90C4×Q8—3581no—
91#104—3582no—
92#104—3581no—
93#101—1791no—
94#101C2×D17912yes—
95#41—3582no—
96#41—3581no—
97#41—7161no—
98C2×Q8—3581no—
99C2×Q8—3581no—
100C2×Q8—1792no—
101C2×Q8—7161no—
102C2×Q8D71611yescharacteristic
103C2×Q8C22×D17911yescharacteristic
104Q16—7162no—
105Q16—7161no—
106SD16—7162no—
107SD16—7161no—
108M16—22no—
109M16—3581no—
110C4⋊C4—22no—
111C4⋊C4—3582no—
112C4⋊C4—3581no—
113C4⋊C4—3582no—
114C4⋊C4—3582no—
115C42—3582no—
116C42—3581no—
117C42—1792no—
118C42D71611yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—3584no—
122Q8—3582no—
123Q8—3581no—
124Q8—14321no—
125Q8—7162no—
126D4—3582no—
127D4—3581no—
128D4—7161no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—3582no—
132C2×C4—3582no—
133C2×C4—7162no—
134C2×C4—7161no—
135C2×C4D4×D17912yes—
136C2×C4C2×D71611yescharacteristic
137C8—42no—
138C8—7161no—
139C22—7161no—
140C22#1915308311yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—3584no—
146C4—7162no—
147C4—7161no—
148C2—21no—
149C2—14321no—
150C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#356×C179C2
socleC2×C179B(4,𝔽2)
prime radical#19153161C22
hypercenter#356D179
hypocenterC179#2205
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #26039169C2semidirect
4. #26038237C2semidirect
5. #356×C179C2semidirect
75. C179#2205semidirect
81. #356D179semidirect

Maximal subgroups

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

subgroupindex
#260391692
#260386242
#260386212
#260382372
#356×C1792
#2205179