Finite Group Databasebeta

← #37511013

Subgroups — #37511013

Order 22912.

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

normal subgroups only

This group has 226 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#26038628C212yesmaximal
2#26038616C211yesmaximal, characteristic
3#26038234C211yesmaximal, characteristic, complemented
4#26038229C212yesmaximal, complemented
5#563×C179C211yesmaximal, characteristic, complemented
6#19153158C2211yescharacteristic
7#19153144—21no—
8#19153096—22no—
9#19153041—21no—
10#19153030C2211yescharacteristic
11#19153028C2212yes—
12#19153027—22no—
13C4⋊Q8×C179—21no—
14#126×C179—22no—
15#125×C179—21no—
16#124×C179C2211yescharacteristic
17#123×C179C2212yes—
18C22⋊Q8×C179—22no—
19#15668259—21no—
20#15668258—41no—
21C2×C22⋊D179—22no—
22D4×D179—41no—
23#15668249—41no—
24#15668246—41no—
25#15668243—21no—
26Dic179⋊C4—42no—
27C22⋊Dic179—22no—
28(C2×D179)⋊C4—42no—
29C2×Q8×C179—21no—
30C2×Q8×C179D411yescharacteristic
31C2×D4×C179C2311yescharacteristic
32C22×C4×C179D412yes—
33C4⋊C4×C179—21no—
34C4⋊C4×C179—22no—
35C4⋊C4×C179—22no—
36C4⋊C4×C179—21no—
37C4⋊C4×C179—22no—
38C22⋊C4×C179—22no—
39C22⋊C4×C179—22no—
40C22⋊C4×C179D412yes—
41C42×C179—21no—
42C42×C179D411yescharacteristic
43C22×D179—22no—
44Dic358—21no—
45D716—21no—
46C22⋊D179—42no—
47C22⋊D179—41no—
48C2×Dic179—22no—
49C2×Dic179—42no—
50C4×D179—41no—
51C179⋊C8—41no—
52C23×C179C2×D412yes—
53Q8×C179—41no—
54Q8×C179—81no—
55D4×C179—41no—
56C2×C4×C179—22no—
57C2×C4×C179—21no—
58C2×C4×C179—22no—
59C2×C4×C179—21no—
60C2×C4×C179—42no—
61C2×C4×C179C2×D411yescharacteristic
62C22×C179—22no—
63C22×C179—42no—
64C22×C179#12011yescharacteristic
65C2×D179—41no—
66C2×D179—82no—
67C4×C179—21no—
68C4×C179—42no—
69C4×C179—41no—
70C4×C179—42no—
71C4×C179—41no—
72Dic179—41no—
73Dic179—82no—
74C2×C179—21no—
75C2×C179—42no—
76C2×C179B(4,𝔽2)11yescharacteristic
77D179—81no—
78C179#220111yescharacteristic, complemented, Sylow
79#2201—1791nomaximal, Sylow
80#563D17911yescharacteristic, complemented
81#458—1792no—
82#455—1791no—
83#354—1791no—
84#352—1792no—
85H(5,𝔽2)—1791no—
86#137—3581no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D17911yescharacteristic
91#123—3582no—
92#123C2×D17912yes—
93C22⋊Q8—22no—
94#104—3581no—
95#100—1791no—
96#99—3582no—
97#99—1792no—
98#41—3581no—
99#41—7161no—
100C2×Q8—21no—
101C2×Q8C22⋊D17911yescharacteristic
102C2×D4—3582no—
103C2×D4—7161no—
104C2×D4C22×D17911yescharacteristic
105C22×C4C22⋊D17912yes—
106Q16—7161no—
107SD16—7161no—
108M16—3581no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—7162no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3582no—
118C22⋊C4—7162no—
119C22⋊C4C22⋊D17912yes—
120C42—21no—
121C42C22⋊D17911yescharacteristic
122C23—3582no—
123C23C2×C22⋊D17912yes—
124Q8—41no—
125Q8—81no—
126Q8—3581no—
127D4—41no—
128D4—3581no—
129D4—7162no—
130D4—7161no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3582no—
137C2×C4—7161no—
138C2×C4—7162no—
139C2×C4C2×C22⋊D17911yescharacteristic
140C8—7161no—
141C22—22no—
142C22—42no—
143C22—14322no—
144C22—7161no—
145C22#1915308211yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—14322no—
152C4—7161no—
153C2—21no—
154C2—42no—
155C2—14321no—
156C2#2603862611yescharacteristic
subgroupG/H
centerC2#26038626
commutator, cosocleC2×D4×C179C23
FrattiniC2×D4C22×D179
Fitting#563×C179C2
socleC2×C179B(4,𝔽2)
prime radical#19153158C22
hypercenter#563D179
hypocenterC179#2201
LCSG ▷ C2×D4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #26038234C2semidirect
4. #26038229C2semidirect
5. #563×C179C2semidirect
78. C179#2201semidirect
80. #563D179semidirect

Maximal subgroups

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

subgroupindex
#260386282
#260386162
#260382342
#260382292
#563×C1792
#2201179