Finite Group Databasebeta

← #37510953

Subgroups — #37510953

Order 22912.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#26039239C211yesmaximal, characteristic
2#26038882C212yesmaximal, complemented
3#26038697C211yesmaximal, characteristic, complemented
4#26038250C212yesmaximal, complemented
5#349×C179C211yesmaximal, characteristic, complemented
6C2×#15668259—21no—
7#19153146—22no—
8#19153141—22no—
9#19153135—21no—
10#19153135C2212yes—
11C2×Dic716C2211yescharacteristic, complemented
12#19153092—21no—
13Dic1432—22no—
14#19153058—22no—
15#19153045C2211yescharacteristic, complemented
16#19153038—21no—
17C2×M16×C179C2211yescharacteristic
18M32×C179C2212yescomplemented
19#15668259—22no—
20#15668259—22no—
21#15668259—41no—
22#15668259—41no—
23C2×Dic358—21no—
24C2×Dic358—21no—
25C2×D716—21no—
26C2×C22⋊D179—41no—
27C2×C4×D179—41no—
28Dic716—21no—
29Dic716—22no—
30Dic716D412yes—
31#15668248—22no—
32#15668248—22no—
33D1432—21no—
34D1432D412yes—
35C4⋊Dic179—21no—
36Dic179⋊C4—41no—
37C22⋊Dic179—41no—
38C22×C4×C179D411yescharacteristic
39M16×C179—22no—
40C2×C8×C179D411yescharacteristic
41C2×C8×C179C2311yescharacteristic
42C16×C179—22no—
43C22×D179—41no—
44Dic358—22no—
45Dic358—41no—
46Dic358—41no—
47Dic358—41no—
48D716—22no—
49D716—41no—
50C22⋊D179—42no—
51C22⋊D179—42no—
52C2×Dic179—41no—
53C2×Dic179—41no—
54C2×Dic179—41no—
55C4×D179—42no—
56C4×D179—42no—
57C23×C179D811yescharacteristic
58C2×C4×C179—21no—
59C2×C4×C179—21no—
60C2×C4×C179D811yescharacteristic
61C2×C4×C179C2×D411yescharacteristic
62C8×C179—21no—
63C8×C179C2×D412yes—
64C22×C179—21no—
65C22×C179—41no—
66C22×C179C2×D811yescharacteristic
67C2×D179—42no—
68C2×D179—82no—
69C4×C179—21no—
70C4×C179#12011yescharacteristic
71C4×C179AGL(1,ℤ/8)11yescharacteristic
72Dic179—42no—
73Dic179—81no—
74Dic179—81no—
75C2×C179—21no—
76C2×C179—41no—
77C2×C179#44711yescharacteristic
78D179—82no—
79C179#219111yescharacteristic, complemented, Sylow
80#2191—1791nomaximal, Sylow
81#575—1791no—
82#510—1792no—
83#470—1791no—
84#360—1792no—
85#349D17911yescharacteristic, complemented
86C2×#41—3581no—
87#137—3582no—
88AGL(1,ℤ/8)—3582no—
89#135—3581no—
90#135—1792no—
91C2×Q16—1791no—
92C2×M16C2×D17911yescharacteristic
93C22⋊Q8—3581no—
94Q32—3582no—
95SD32—3582no—
96M32C2×D17912yescomplemented
97#106—1791no—
98#103—3581no—
99#41—3582no—
100#41—3582no—
101#41—7161no—
102#41—7161no—
103C2×Q8—3581no—
104C2×Q8—3581no—
105C2×D4—3581no—
106C2×D4—7161no—
107C22×C4—7161no—
108C22×C4D71611yescharacteristic
109Q16—3581no—
110Q16—3582no—
111Q16—1792no—
112SD16—3582no—
113SD16—3582no—
114D8—3581no—
115D8—1792no—
116M16—22no—
117C2×C8D71611yescharacteristic
118C2×C8C22×D17911yescharacteristic
119C4⋊C4—3581no—
120C4⋊C4—7161no—
121C22⋊C4—7161no—
122C16—22no—
123C23—7161no—
124C23D143211yescharacteristic
125Q8—3582no—
126Q8—7161no—
127Q8—7161no—
128Q8—7161no—
129D4—3582no—
130D4—7161no—
131D4—7162no—
132D4—7162no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—7162no—
136C2×C4—7161no—
137C2×C4—7161no—
138C2×C4—7161no—
139C2×C4—7162no—
140C2×C4D143211yescharacteristic
141C2×C4C2×D71611yescharacteristic
142C8—21no—
143C8D4×D17912yes—
144C22—21no—
145C22—41no—
146C22—14322no—
147C22—7162no—
148C22C2×D143211yescharacteristic
149C4—21no—
150C4—14321no—
151C4—14321no—
152C4—7162no—
153C4#1915308311yescharacteristic
154C4#1915314111yescharacteristic
155C2—21no—
156C2—41no—
157C2—14322no—
158C2#2603857511yescharacteristic
subgroupG/H
centerC2#26038575
commutator, cosocleC2×C8×C179C23
FrattiniC2×C8C22×D179
Fitting#349×C179C2
socleC2×C179#447
prime radical#26039239C2
hypercenter#349D179
hypocenterC179#2191
LCSG ▷ C2×C8×C179 ▷ C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #26038882C2semidirect
3. #26038697C2semidirect
4. #26038250C2semidirect
5. #349×C179C2semidirect
11. C2×Dic716C22semidirect
15. #19153045C22semidirect
18. M32×C179C22semidirect
79. C179#2191semidirect
85. #349D179semidirect
96. M32C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260392392
#260388822
#260386972
#260382502
#349×C1792
#2191179