Finite Group Databasebeta

← #37509538

Subgroups — #37509538

Order 22912.

314 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 93 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#26038917C211yesmaximal, characteristic, complemented
2#26038456C212yesmaximal
3C2×#19153032C211yesmaximal, characteristic, complemented
4#26038195C212yesmaximal, complemented
5#536×C179C211yesmaximal, characteristic
6C2×#15668243C2212yes—
7Q8×Dic179—24no—
8#19153072—22no—
9#19153072C2212yes—
10#19153040—24no—
11#19153032—22no—
12C2×#41×C179C2211yescharacteristic
13C22×Q8×C179C2211yescharacteristic
14C4⋊Q8×C179C412yescomplemented
15#124×C179C412yescomplemented
16#123×C179—22no—
17C22⋊Q8×C179—22no—
18#117×C179C2211yescharacteristic
19#15668243—22no—
20#15668243—22no—
21C2×C179⋊C8—22no—
22C4⋊Dic179—24no—
23C4⋊Dic179—22no—
24C22⋊Dic179—22no—
25C4×Dic179—24no—
26C4×Dic179D414yes—
27#41×C179—22no—
28#41×C179—22no—
29C2×Q8×C179—22no—
30C2×Q8×C179—22no—
31C2×Q8×C179C2×C411yescharacteristic
32C2×Q8×C179C2×C412yes—
33C2×Q8×C179D412yes—
34C2×D4×C179—21no—
35C2×D4×C179C2×C411yescharacteristic
36C22×C4×C179—21no—
37C22×C4×C179D412yes—
38C22×C4×C179C2311yescharacteristic
39C4⋊C4×C179—24no—
40C4⋊C4×C179—21no—
41C22⋊C4×C179—21no—
42C22⋊C4×C179—24no—
43C42×C179C2×C412yes—
44C2×Dic179—24no—
45C179⋊C8—42no—
46C23×C179—21no—
47C23×C179C2×D411yescharacteristic
48Q8×C179—24no—
49Q8×C179—24no—
50Q8×C179—22no—
51D4×C179—22no—
52D4×C179—41no—
53D4×C179—41no—
54C2×C4×C179—22no—
55C2×C4×C179—22no—
56C2×C4×C179—21no—
57C2×C4×C179—22no—
58C2×C4×C179—42no—
59C2×C4×C179C22⋊C414yes—
60C2×C4×C179C22×C411yescharacteristic
61C2×C4×C179C2×D412yes—
62C2×C4×C179C2×D411yescharacteristic
63C2×C4×C179#4112yes—
64C22×C179—41no—
65C22×C179—42no—
66C22×C179—41no—
67C22×C179C2×C22⋊C411yescharacteristic
68C22×C179#12011yescharacteristic
69C22×C179#12311yescharacteristic
70C4×C179—24no—
71C4×C179—42no—
72C4×C179—41no—
73C4×C179C4×D412yes—
74C4×C179#12112yes—
75Dic179—44no—
76C2×C179—21no—
77C2×C179—21no—
78C2×C179—21no—
79C2×C179—81no—
80C2×C179#38611yescharacteristic
81C179#191211yescharacteristic, complemented, Sylow
82#1912—1791nomaximal, Sylow
83#536D17911yescharacteristic
84#519—1791no—
85#421—1792no—
86C2×#101—1791no—
87H(3,ℤ/4)—1792no—
88C2×#41C2×D17911yescharacteristic
89C22×Q8C2×D17911yescharacteristic
90C2×M16—1792no—
91C4⋊Q8Dic17912yescomplemented
92#124Dic17912yescomplemented
93#123—22no—
94C22⋊Q8—22no—
95C4×Q8—3584no—
96#117—3582no—
97#117—1792no—
98#117C2×D17911yescharacteristic
99#104—3584no—
100#101—3582no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic17911yescharacteristic
106C2×Q8C2×Dic17912yes—
107C2×Q8C22⋊D17912yes—
108C2×D4—21no—
109C2×D4C2×Dic17911yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D17912yes—
112C22×C4C22×D17911yescharacteristic
113M16—3582no—
114M16—3582no—
115C2×C8—3582no—
116C4⋊C4—24no—
117C4⋊C4—21no—
118C4⋊C4—3584no—
119C4⋊C4—3582no—
120C22⋊C4—21no—
121C22⋊C4—24no—
122C22⋊C4—3582no—
123C42—3584no—
124C42—1794no—
125C42C2×Dic17912yes—
126C23—21no—
127C23C2×C22⋊D17911yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—22no—
131D4—22no—
132D4—41no—
133D4—41no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—3584no—
140C2×C4C22⋊Dic17914yes—
141C2×C4C22×Dic17911yescharacteristic
142C2×C4D4×D17912yes—
143C2×C4C2×C22⋊D17911yescharacteristic
144C2×C4#1566825812yes—
145C8—7162no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22C2×C22⋊Dic17911yescharacteristic
150C22#1915308111yescharacteristic
151C22#1915309411yescharacteristic
152C4—24no—
153C4—42no—
154C4—41no—
155C4—7164no—
156C4D4×Dic17912yes—
157C4#1915308512yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—81no—
162C2#2603831911yescharacteristic
subgroupG/H
centerC2#26038319
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#536×C179C2
socleC2×C179#386
prime radicalC2×#41×C179C22
cosocleC22×C4×C179C23
hypercenter#536D179
hypocenterC179#1912
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #26038917C2semidirect
3. C2×#19153032C2semidirect
4. #26038195C2semidirect
14. C4⋊Q8×C179C4semidirect
15. #124×C179C4semidirect
81. C179#1912semidirect
91. C4⋊Q8Dic179semidirect
92. #124Dic179semidirect

Maximal subgroups

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

subgroupindex
#260389172
#260384562
C2×#191530322
#260381952
#536×C1792
#1912179