Finite Group Databasebeta

← #37510690

Subgroups — #37510690

Order 22912.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039007C211yesmaximal, characteristic
2#26038422C212yesmaximal, complemented
3#26038236C211yesmaximal, characteristic, complemented
4#26038235C212yesmaximal, complemented
5#355×C179C211yesmaximal, characteristic, complemented
6C2×#15668260C2211yescharacteristic
7C2×D4×D179C411yescharacteristic
8M16×D179—22no—
9#19153110C411yescharacteristic
10#19153109C411yescharacteristic
11#19153102C2211yescharacteristic
12#19153086—22no—
13#19153083—22no—
14#19153070C411yescharacteristic
15#19153032C2212yescomplemented
16#19153029—22no—
17#124×C179C2211yescharacteristic
18#101×C179C2212yescomplemented
19C23×D179—21no—
20#15668260—42no—
21C2×D716—22no—
22C2×D716C2×C411yescharacteristic
23C2×D716C2×C411yescharacteristic
24C2×D716D412yes—
25C2×C22⋊D179—22no—
26C2×C22⋊D179—21no—
27D4×D179—42no—
28C2×C4×D179C2×C411yescharacteristic
29C2×C4×D179D412yes—
30#15668243—22no—
31#15668242—42no—
32C8×D179—42no—
33Dic179⋊C4—21no—
34(C2×D179)⋊C4—21no—
35(C2×D179)⋊C4—22no—
36C4×Dic179C2×C411yescharacteristic
37C2×Q8×C179C2311yescharacteristic
38C2×D4×C179C2×C411yescharacteristic
39M16×C179—22no—
40C22⋊C4×C179—22no—
41C42×C179C2×C411yescharacteristic
42C22×D179—21no—
43C22×D179—42no—
44C22×D179—82no—
45C22×D179C22⋊C411yescharacteristic
46C22×D179C22⋊C412yes—
47D716—41no—
48D716—42no—
49D716—41no—
50D716—82no—
51C22⋊D179—42no—
52C22⋊D179—82no—
53C2×Dic179—21no—
54C2×Dic179C22⋊C411yescharacteristic
55C4×D179—22no—
56C4×D179—22no—
57C4×D179—22no—
58C179⋊C8—42no—
59C23×C179—21no—
60Q8×C179—41no—
61D4×C179—41no—
62C2×C4×C179—21no—
63C2×C4×C179C22×C411yescharacteristic
64C2×C4×C179C2×D412yes—
65C8×C179—42no—
66C22×C179—42no—
67C22×C179—41no—
68C22×C179C2×C22⋊C411yescharacteristic
69C2×D179—21no—
70C2×D179—22no—
71C2×D179—42no—
72C2×D179—41no—
73C2×D179—42no—
74C2×D179—41no—
75C2×D179—84no—
76C4×C179—22no—
77C4×C179—21no—
78C4×C179—41no—
79Dic179—41no—
80Dic179#9912yes—
81C2×C179—21no—
82C2×C179—81no—
83C2×C179C2×#9911yescharacteristic
84D179—42no—
85D179—41no—
86D179—81no—
87C179#213411yescharacteristic, complemented, Sylow
88#2134—1791nomaximal, Sylow
89#535—1791no—
90#413—1792no—
91#355—1791no—
92#355—1792no—
93#355D17911yescharacteristic, complemented
94C2×#41—1791no—
95C22×D4—1791no—
96C2×M16—3582no—
97#127—1791no—
98#127—1791no—
99#124—1791no—
100#124C2×D17911yescharacteristic
101#121—3582no—
102#120—3582no—
103#117—1791no—
104#101—1792no—
105#101C2×D17912yescomplemented
106#100—3582no—
107C24—3581no—
108#41—7162no—
109C2×Q8C22×D17911yescharacteristic
110C2×D4—3582no—
111C2×D4—3582no—
112C2×D4—3581no—
113C2×D4—1791no—
114C2×D4—1791no—
115C2×D4—1792no—
116C2×D4—7162no—
117C2×D4C4×D17911yescharacteristic
118C22×C4—1791no—
119C22×C4—1792no—
120M16—22no—
121M16—3582no—
122M16—7162no—
123C2×C8—7162no—
124C4⋊C4—3581no—
125C22⋊C4—22no—
126C22⋊C4—3581no—
127C22⋊C4—3582no—
128C42—1791no—
129C42C4×D17911yescharacteristic
130C23—21no—
131C23—3581no—
132C23—14322no—
133C23—1791no—
134C23—1792no—
135C23—7162no—
136Q8—41no—
137D4—41no—
138D4—14322no—
139D4—14322no—
140D4—7162no—
141D4—7161no—
142D4—7161no—
143D4—7162no—
144C2×C4—21no—
145C2×C4—3582no—
146C2×C4—3582no—
147C2×C4—3582no—
148C2×C4—3581no—
149C2×C4—1791no—
150C2×C4C2×C4×D17911yescharacteristic
151C2×C4D4×D17912yes—
152C8—42no—
153C8—7162no—
154C22—42no—
155C22—41no—
156C22—3581no—
157C22—3582no—
158C22—14324no—
159C22—7162no—
160C22—7162no—
161C22—7161no—
162C22—7161no—
163C22C22⋊C4×D17911yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—1792no—
168C4—7161no—
169C2—21no—
170C2—81no—
171C2—14321no—
172C2—7161no—
173C2—7162no—
174C2#99×D17911yescharacteristic
subgroupG/H
centerC2#99×D179
commutatorC2×C4×C179C22×C4
FrattiniC2×Q8C22×D179
Fitting#355×C179C2
socleC2×C179C2×#99
prime radical#26039007C2
cosocleC2×Q8×C179C23
hypercenter#355D179
hypocenterC179#2134
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #26038422C2semidirect
3. #26038236C2semidirect
4. #26038235C2semidirect
5. #355×C179C2semidirect
15. #19153032C22semidirect
18. #101×C179C22semidirect
87. C179#2134semidirect
93. #355D179semidirect
105. #101C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260390072
#260384222
#260382362
#260382352
#355×C1792
#2134179