Finite Group Databasebeta

← #37511028

Subgroups — #37511028

Order 22912.

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

normal subgroups only

This group has 236 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#26039155C211yesmaximal, characteristic, complemented
2#26038621C211yesmaximal, characteristic, complemented
3#26038614C212yesmaximal, complemented
4#26038235C212yesmaximal, complemented
5#356×C179C211yesmaximal, characteristic, complemented
6#19153161C2211yescharacteristic
7#19153144—21no—
8#19153141—22no—
9#19153102—21no—
10#19153102C2212yes—
11#19153088—22no—
12C4×D716—21no—
13#19153071—22no—
14#19153041—21no—
15#19153039—22no—
16#19153032C2211yescharacteristic
17C4⋊Q8×C179C2211yescharacteristic
18#101×C179C2212yes—
19#15668260—41no—
20#15668259—22no—
21#15668259—21no—
22Q8×D179—41no—
23C2×D716—21no—
24C2×D716—21no—
25C2×D716D412yes—
26C2×C4×D179—22no—
27#15668249—41no—
28#15668248—42no—
29#15668246—41no—
30D1432—42no—
31#15668243—21no—
32C4⋊Dic179—21no—
33(C2×D179)⋊C4—22no—
34(C2×D179)⋊C4—22no—
35(C2×D179)⋊C4—22no—
36C4×Dic179—21no—
37C4×Dic179D412yes—
38C2×Q8×C179D411yescharacteristic
39C2×Q8×C179C2311yescharacteristic
40M16×C179—22no—
41C4⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43C22×D179—22no—
44Dic358—22no—
45Dic358—21no—
46D716—22no—
47D716—21no—
48D716—42no—
49D716—42no—
50D716—81no—
51C22⋊D179—41no—
52C2×Dic179—22no—
53C4×D179—44no—
54C4×D179—42no—
55C4×D179—41no—
56C179⋊C8—41no—
57Q8×C179—41no—
58Q8×C179—41no—
59C2×C4×C179—21no—
60C2×C4×C179—21no—
61C2×C4×C179C2×D412yes—
62C2×C4×C179C2×D411yescharacteristic
63C8×C179—42no—
64C22×C179#12011yescharacteristic
65C2×D179—42no—
66C2×D179—42no—
67C2×D179—41no—
68C2×D179—42no—
69C4×C179—22no—
70C4×C179—22no—
71C4×C179—21no—
72C4×C179—41no—
73Dic179—42no—
74Dic179—41no—
75C2×C179—21no—
76C2×C179B(4,𝔽2)11yescharacteristic
77D179—81no—
78D179—82no—
79C179#220411yescharacteristic, complemented, Sylow
80#2204—1791nomaximal, Sylow
81#555—1791no—
82#456—1791no—
83#454—1792no—
84#356D17911yescharacteristic, complemented
85#355—1792no—
862−1+4—1791no—
87#137—3581no—
88AGL(1,ℤ/8)—3582no—
89C4⋊Q8C2×D17911yescharacteristic
90#124—3581no—
91#124—1792no—
92#121—3582no—
93C4×D4—3581no—
94#117—3582no—
95#104—3582no—
96#104—3581no—
97#101—1791no—
98#101C2×D17912yes—
99#41—3582no—
100#41—3581no—
101#41—7161no—
102C2×Q8—7161no—
103C2×Q8D71611yescharacteristic
104C2×Q8C22×D17911yescharacteristic
105C2×D4—3581no—
106C2×D4—3581no—
107C2×D4—1792no—
108C22×C4—3582no—
109Q16—7161no—
110SD16—7162no—
111SD16—7161no—
112D8—7162no—
113M16—22no—
114M16—3581no—
115C4⋊C4—22no—
116C4⋊C4—3581no—
117C22⋊C4—3582no—
118C22⋊C4—3582no—
119C22⋊C4—3582no—
120C42—3581no—
121C42—1792no—
122C42D71611yescharacteristic
123C23—3582no—
124Q8—41no—
125Q8—41no—
126Q8—3582no—
127Q8—3581no—
128D4—3582no—
129D4—3581no—
130D4—14321no—
131D4—7162no—
132D4—7162no—
133D4—7161no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—3582no—
137C2×C4—7164no—
138C2×C4—7162no—
139C2×C4—7161no—
140C2×C4D4×D17912yes—
141C2×C4C2×D71611yescharacteristic
142C8—42no—
143C8—7161no—
144C22—7162no—
145C22—7162no—
146C22—7161no—
147C22—7162no—
148C22#1915308311yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—7162no—
154C4—7161no—
155C2—21no—
156C2—14322no—
157C2—14321no—
158C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#356×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#356D179
hypocenterC179#2204
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. #26039155C2semidirect
2. #26038621C2semidirect
3. #26038614C2semidirect
4. #26038235C2semidirect
5. #356×C179C2semidirect
79. C179#2204semidirect
84. #356D179semidirect

Maximal subgroups

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

subgroupindex
#260391552
#260386212
#260386142
#260382352
#356×C1792
#2204179