Finite Group Databasebeta

← #37511026

Subgroups — #37511026

Order 22912.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#26038621C211yesmaximal, characteristic, complemented
2#26038611C212yesmaximal, complemented
3#26038238C211yesmaximal, characteristic, complemented
4#26038236C212yesmaximal, complemented
5#555×C179C211yesmaximal, characteristic, complemented
6#19153161C2211yescharacteristic
7#19153144—21no—
8#19153139—22no—
9#19153041—22no—
10#19153041—21no—
11#19153032C2212yes—
12#19153032C2211yescharacteristic
13C4⋊Q8×C179C2211yescharacteristic
14#124×C179—21no—
15#124×C179C2212yes—
16#121×C179—22no—
17C4×D4×C179—21no—
18#117×C179—22no—
19#15668260—41no—
20#15668259—22no—
21#15668259—21no—
22Q8×D179—41no—
23#15668249—41no—
24#15668247—42no—
25#15668246—41no—
26#15668244—42no—
27#15668243—22no—
28#15668243—21no—
29C2×Q8×C179D411yescharacteristic
30C2×Q8×C179C2311yescharacteristic
31C2×D4×C179—21no—
32C2×D4×C179—21no—
33C2×D4×C179D412yes—
34C22×C4×C179—22no—
35C4⋊C4×C179—21no—
36C4⋊C4×C179—22no—
37C22⋊C4×C179—22no—
38C22⋊C4×C179—22no—
39C22⋊C4×C179—22no—
40C42×C179—21no—
41C42×C179D412yes—
42C42×C179D411yescharacteristic
43Dic358—22no—
44Dic358—21no—
45D716—22no—
46D716—21no—
47C22⋊D179—41no—
48C4×D179—42no—
49C4×D179—41no—
50C179⋊C8—42no—
51C179⋊C8—41no—
52C23×C179—22no—
53Q8×C179—41no—
54Q8×C179—41no—
55D4×C179—42no—
56D4×C179—42no—
57D4×C179—81no—
58C2×C4×C179—21no—
59C2×C4×C179—22no—
60C2×C4×C179—21no—
61C2×C4×C179—44no—
62C2×C4×C179C2×D412yes—
63C2×C4×C179C2×D411yescharacteristic
64C22×C179—42no—
65C22×C179—42no—
66C22×C179—42no—
67C22×C179#12011yescharacteristic
68C2×D179—41no—
69C4×C179—22no—
70C4×C179—22no—
71C4×C179—21no—
72C4×C179—42no—
73C4×C179—41no—
74Dic179—41no—
75C2×C179—21no—
76C2×C179—82no—
77C2×C179B(4,𝔽2)11yescharacteristic
78D179—81no—
79C179#220411yescharacteristic, complemented, Sylow
80#2204—1791nomaximal, Sylow
81#555D17911yescharacteristic, complemented
82#456—1791no—
83#454—1792no—
84#356—1791no—
85#355—1792no—
862−1+4—1791no—
87#137—3581no—
88AGL(1,ℤ/8)—3582no—
89C4⋊Q8C2×D17911yescharacteristic
90#124—21no—
91#124C2×D17912yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—3582no—
96#104—3581no—
97#101—1791no—
98#101—1792no—
99#41—3582no—
100#41—3581no—
101#41—7161no—
102C2×Q8—7161no—
103C2×Q8C22⋊D17911yescharacteristic
104C2×Q8C22×D17911yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D17912yes—
108C22×C4—22no—
109Q16—7161no—
110SD16—7162no—
111SD16—7161no—
112D8—7162no—
113M16—3582no—
114M16—3581no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D17912yes—
122C42C22⋊D17911yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—3582no—
127Q8—3581no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—3582no—
132D4—3581no—
133D4—7161no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—7162no—
139C2×C4—7161no—
140C2×C4C2×C22⋊D17911yescharacteristic
141C2×C4C2×C22⋊D17912yes—
142C8—7162no—
143C8—7161no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—7161no—
148C22#1915308211yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—7161no—
155C2—21no—
156C2—82no—
157C2—14321no—
158C2#2603862611yescharacteristic
subgroupG/H
centerC2#26038626
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#555×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#555D179
hypocenterC179#2204
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #26038621C2semidirect
2. #26038611C2semidirect
3. #26038238C2semidirect
4. #26038236C2semidirect
5. #555×C179C2semidirect
79. C179#2204semidirect
81. #555D179semidirect

Maximal subgroups

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

subgroupindex
#260386212
#260386112
#260382382
#260382362
#555×C1792
#2204179