Finite Group Databasebeta

← #37508111

Subgroups — #37508111

Order 22912.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#26039163C211yesmaximal, characteristic, complemented
2#26038787C211yesmaximal, characteristic, complemented
3#26038640C211yesmaximal, characteristic, complemented
4#26038635C211yesmaximal, characteristic
5#26038169C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic, complemented
7#332×C179C211yesmaximal, characteristic, complemented
8C2×#15668248—21no—
9C2×D1432—21no—
10#19153110C2211yescharacteristic
11#19153108—22no—
12#19153102—21no—
13#19153093—21no—
14Q8×Dic179—21no—
15C4×Dic358C2211yescharacteristic
16C4×D716C2211yescharacteristic
17#19153071—21no—
18#19153035—21no—
19#19153035—21no—
20#19153033—22no—
21C4×C179⋊C8C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4⋊C8×C179C2211yescharacteristic
25C2×Dic358D411yescharacteristic
26C2×D716—21no—
27C2×D716—21no—
28C2×D716—21no—
29C2×D716D411yescharacteristic
30C2×C4×D179—21no—
31#15668248—42no—
32D1432—42no—
33C2×C179⋊C8—21no—
34C4⋊Dic179—22no—
35C4⋊Dic179D411yescharacteristic
36C4⋊Dic179D411yescharacteristic
37Dic179⋊C4—21no—
38(C2×D179)⋊C4—22no—
39(C2×D179)⋊C4—22no—
40(C2×D179)⋊C4—21no—
41C4×Dic179—22no—
42C4×Dic179—21no—
43C2×Q8×C179—21no—
44C2×C8×C179—21no—
45C2×C8×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179D412yes—
48C42×C179C2311yescharacteristic
49C22×D179—21no—
50C22×D179—41no—
51Dic358—21no—
52Dic358—21no—
53D716—21no—
54D716—21no—
55D716—42no—
56D716—41no—
57D716—41no—
58D716—41no—
59D716—41no—
60C2×Dic179—21no—
61C2×Dic179—21no—
62C2×Dic179—21no—
63C4×D179—41no—
64C4×D179—41no—
65C179⋊C8—41no—
66Q8×C179—41no—
67Q8×C179—41no—
68C2×C4×C179—22no—
69C2×C4×C179C2×D411yescharacteristic
70C2×C4×C179C2×D411yescharacteristic
71C2×C4×C179C2×D411yescharacteristic
72C8×C179—41no—
73C8×C179—41no—
74C22×C179#12011yescharacteristic
75C2×D179—41no—
76C2×D179—41no—
77C2×D179—41no—
78C2×D179—81no—
79C2×D179—81no—
80C2×D179—81no—
81C4×C179—21no—
82C4×C179—42no—
83C4×C179#13511yescharacteristic
84C4×C179#13511yescharacteristic
85C4×C179AGL(1,ℤ/8)11yescharacteristic
86C4×C179AGL(1,ℤ/8)11yescharacteristic
87Dic179—41no—
88Dic179—41no—
89Dic179—41no—
90C2×C179#44911yescharacteristic
91C2×C179#44911yescharacteristic
92C2×C179#45411yescharacteristic
93D179—81no—
94D179—161no—
95C179#162811yescharacteristic, complemented, Sylow
96#1628—1791nomaximal, Sylow
97#556—1791no—
98#486—1791no—
99#460—1791no—
100#459—1791no—
101#332D17911yescharacteristic, complemented
102Q8⋊C8—1791no—
103D4⋊C8—1791no—
104C2×SD16—3581no—
105C2×D8—3581no—
106C4⋊Q8C2×D17911yescharacteristic
107#127—1791no—
108#126—3582no—
109#124—3581no—
110C22⋊Q8—3581no—
111C4×Q8—3581no—
112C4×Q8—1791no—
113C4×D4—1791no—
114#117—3581no—
115C4⋊C8C2×D17911yescharacteristic
116C4⋊C8C2×D17911yescharacteristic
117#102—3581no—
118#102—3581no—
119#102—3582no—
120C4×C8—1791no—
121C2×Q8—21no—
122C2×Q8—1791no—
123C2×D4—3581no—
124C2×D4—3581no—
125C2×D4—3581no—
126C2×D4—1791no—
127C22×C4—3581no—
128SD16—7162no—
129D8—7162no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—3581no—
133C4⋊C4—21no—
134C4⋊C4—3582no—
135C4⋊C4—3581no—
136C4⋊C4—1791no—
137C4⋊C4—1791no—
138C4⋊C4D71612yes—
139C22⋊C4—3582no—
140C22⋊C4—3582no—
141C22⋊C4—3581no—
142C42—3582no—
143C42—3581no—
144C42C22×D17911yescharacteristic
145C23—3581no—
146C23—7161no—
147Q8—41no—
148Q8—41no—
149Q8—3581no—
150Q8—3581no—
151D4—3581no—
152D4—3581no—
153D4—7162no—
154D4—7161no—
155D4—7161no—
156D4—7161no—
157D4—7161no—
158C2×C4—22no—
159C2×C4—3581no—
160C2×C4—3581no—
161C2×C4—3581no—
162C2×C4—7161no—
163C2×C4—7161no—
164C2×C4D4×D17911yescharacteristic
165C2×C4D4×D17911yescharacteristic
166C2×C4C2×D71611yescharacteristic
167C8—41no—
168C8—41no—
169C8—7161no—
170C22—14321no—
171C22—14321no—
172C22—14321no—
173C22—7161no—
174C22—7161no—
175C22—7161no—
176C22#1915308311yescharacteristic
177C4—21no—
178C4—42no—
179C4—7161no—
180C4—7161no—
181C4—7161no—
182C4#1915313411yescharacteristic
183C4#1915313611yescharacteristic
184C4#1915313711yescharacteristic
185C4#1915314011yescharacteristic
186C2—28641no—
187C2—14321no—
188C2#2603858511yescharacteristic
189C2#2603858711yescharacteristic
190C2#2603861411yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#332×C179C2
socleC22×C179#120
prime radical#26038635C2
hypercenter#332D179
hypocenterC179#1628
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #26039163C2semidirect
2. #26038787C2semidirect
3. #26038640C2semidirect
5. #26038169C2semidirect
6. #26038166C2semidirect
7. #332×C179C2semidirect
95. C179#1628semidirect
101. #332D179semidirect

Maximal subgroups

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

subgroupindex
#260391632
#260387872
#260386402
#260386352
#260381692
#260381662
#332×C1792
#1628179