Finite Group Databasebeta

← #37511025

Subgroups — #37511025

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#26038612C212yesmaximal, complemented
3#26038238C211yesmaximal, characteristic, complemented
4#26038235C212yesmaximal, complemented
5#456×C179C211yesmaximal, characteristic, complemented
6#19153138—22no—
7#19153102—21no—
8#19153102C2212yes—
9#19153088—22no—
10C4×D716—21no—
11#19153071—22no—
12#19153040—22no—
13#19153032C2212yes—
142−1+4×C179C2211yescharacteristic
15#137×C179—21no—
16C4⋊Q8×C179C2211yescharacteristic
17#104×C179—21no—
18#101×C179C2211yescharacteristic
19C2×D716—21no—
20C2×D716—21no—
21C2×D716D412yes—
22C2×C4×D179—22no—
23#15668246—42no—
24#15668244—42no—
25#15668243—22no—
26C4⋊Dic179—21no—
27(C2×D179)⋊C4—22no—
28(C2×D179)⋊C4—22no—
29(C2×D179)⋊C4—22no—
30C4×Dic179—21no—
31C4×Dic179D412yes—
32#41×C179—21no—
33#41×C179—22no—
34#41×C179—41no—
35C2×Q8×C179—41no—
36C2×Q8×C179D411yescharacteristic
37C2×Q8×C179C2311yescharacteristic
38Q16×C179—41no—
39SD16×C179—41no—
40M16×C179—21no—
41C4⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43C22×D179—22no—
44D716—42no—
45D716—42no—
46D716—81no—
47C2×Dic179—22no—
48C4×D179—44no—
49C179⋊C8—42no—
50Q8×C179—21no—
51Q8×C179—22no—
52Q8×C179—41no—
53Q8×C179—41no—
54D4×C179—21no—
55D4×C179—22no—
56D4×C179—41no—
57C2×C4×C179—21no—
58C2×C4×C179—21no—
59C2×C4×C179—41no—
60C2×C4×C179—42no—
61C2×C4×C179C2×D412yes—
62C2×C4×C179C2×D411yescharacteristic
63C8×C179—41no—
64C22×C179—41no—
65C22×C179#12011yescharacteristic
66C2×D179—42no—
67C2×D179—42no—
68C2×D179—42no—
69C4×C179—22no—
70C4×C179—21no—
71C4×C179—22no—
72C4×C179—41no—
73C4×C179—41no—
74Dic179—42no—
75C2×C179—21no—
76C2×C179—81no—
77C2×C179B(4,𝔽2)11yescharacteristic
78D179—82no—
79C179#220411yescharacteristic, complemented, Sylow
80#2204—1791nomaximal, Sylow
81#555—1791no—
82#456D17911yescharacteristic, complemented
83#454—1792no—
84#356—1791no—
85#355—1792no—
862−1+4C2×D17911yescharacteristic
87#137—21no—
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—21no—
96#104—3582no—
97#101—1792no—
98#101C2×D17911yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D17911yescharacteristic
104C2×Q8C22×D17911yescharacteristic
105C2×D4—3581no—
106C2×D4—3581no—
107C2×D4—1792no—
108C22×C4—3582no—
109Q16—41no—
110SD16—41no—
111SD16—7162no—
112D8—7162no—
113M16—21no—
114M16—3582no—
115C4⋊C4—22no—
116C4⋊C4—3581no—
117C22⋊C4—3582no—
118C22⋊C4—3582no—
119C22⋊C4—3582no—
120C42—3581no—
121C42—1792no—
122C42C22⋊D17911yescharacteristic
123C23—3582no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—14321no—
132D4—7162no—
133D4—7162no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—3582no—
139C2×C4—7164no—
140C2×C4D4×D17912yes—
141C2×C4C2×C22⋊D17911yescharacteristic
142C8—41no—
143C8—7162no—
144C22—41no—
145C22—7162no—
146C22—7162no—
147C22—7162no—
148C22#1915308111yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—7162no—
155C2—21no—
156C2—81no—
157C2—14322no—
158C2#2603862511yescharacteristic
subgroupG/H
centerC2#26038625
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#456×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#456D179
hypocenterC179#2204
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26039155C2semidirect
2. #26038612C2semidirect
3. #26038238C2semidirect
4. #26038235C2semidirect
5. #456×C179C2semidirect
79. C179#2204semidirect
82. #456D179semidirect

Maximal subgroups

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

subgroupindex
#260391552
#260386122
#260382382
#260382352
#456×C1792
#2204179