Finite Group Databasebeta

← #37508162

Subgroups — #37508162

Order 22912.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#26039104C211yesmaximal, characteristic
2#26038731C211yesmaximal, characteristic, complemented
3#26038726C211yesmaximal, characteristic
4#26038185C211yesmaximal, characteristic, complemented
5#26038171C211yesmaximal, characteristic, complemented
6#26038168C211yesmaximal, characteristic
7#487×C179C211yesmaximal, characteristic, complemented
8C2×Q8×D179—21no—
9#19153112—21no—
10#19153091—22no—
11#19153089—21no—
12C4×Dic358C2211yescharacteristic
13C4×D716C2211yescharacteristic
14#19153075—21no—
15#19153046—21no—
16#19153044—21no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18C4⋊(C179⋊C8)C2211yescharacteristic
19#19153036—21no—
20#19153033—21no—
21C4⋊Q8×C179C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23#103×C179—22no—
24C4×C8×C179C2211yescharacteristic
25C2×Dic358—22no—
26C2×Dic358D411yescharacteristic
27Q8×D179—42no—
28Q8×D179—42no—
29C2×D716D411yescharacteristic
30C2×C4×D179—22no—
31C2×C4×D179—21no—
32C2×C179⋊C8—21no—
33C2×C179⋊C8—21no—
34C4⋊Dic179D411yescharacteristic
35C4⋊Dic179D411yescharacteristic
36Dic179⋊C4—22no—
37Dic179⋊C4—21no—
38(C2×D179)⋊C4—22no—
39(C2×D179)⋊C4—21no—
40C4×Dic179—21no—
41C2×Q8×C179—21no—
42C2×Q8×C179—21no—
43C2×C8×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179—21no—
47C4⋊C4×C179D412yes—
48C42×C179C2311yescharacteristic
49C22×D179—21no—
50Dic358—21no—
51Dic358—44no—
52Dic358SD1612yes—
53D716—21no—
54D716Q1612yes—
55C2×Dic179—21no—
56C2×Dic179—21no—
57C2×Dic179—21no—
58C4×D179—44no—
59C4×D179—41no—
60C4×D179—41no—
61C179⋊C8—41no—
62C179⋊C8—41no—
63Q8×C179—42no—
64Q8×C179—41no—
65Q8×C179—41no—
66C2×C4×C179—22no—
67C2×C4×C179—41no—
68C2×C4×C179C2×D411yescharacteristic
69C2×C4×C179C2×D411yescharacteristic
70C2×C4×C179C2×D411yescharacteristic
71C8×C179—41no—
72C22×C179#12011yescharacteristic
73C2×D179—22no—
74C2×D179—41no—
75C2×D179—41no—
76C4×C179—21no—
77C4×C179—42no—
78C4×C179—81no—
79C4×C179C2×SD1611yescharacteristic
80C4×C179C2×Q1611yescharacteristic
81C4×C179AGL(1,ℤ/8)11yescharacteristic
82C4×C179#13711yescharacteristic
83Dic179—22no—
84Dic179—41no—
85Dic179—41no—
86C2×C179#45011yescharacteristic
87C2×C179#45111yescharacteristic
88C2×C179#45411yescharacteristic
89D179—42no—
90C179#163711yescharacteristic, complemented, Sylow
91#1637—1791nomaximal, Sylow
92D4×Q8—1791no—
93#487D17911yescharacteristic, complemented
94#477—1791no—
95#476—1791no—
96#332—1791no—
97Q8⋊C8—1791no—
98D4⋊C8—1791no—
99C22×Q8—3581no—
100C4⋊Q8—3581no—
101C4⋊Q8C2×D17911yescharacteristic
102C4⋊Q8C2×D17911yescharacteristic
103C22⋊Q8—3581no—
104C22⋊Q8—3582no—
105C4×Q8—1791no—
106C4×D4—3581no—
107C4×D4—1791no—
108#107—3581no—
109#106—3581no—
110C4⋊C8—1791no—
111C4⋊C8—1791no—
112#103—22no—
113#103—3581no—
114#102—3581no—
115C4×C8C2×D17911yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—3582no—
119C2×Q8—1791no—
120C2×Q8—7162no—
121C2×Q8—7162no—
122C2×D4—1791no—
123C22×C4—3582no—
124C22×C4—3581no—
125C2×C8—21no—
126C2×C8—3581no—
127C2×C8—3581no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3582no—
132C4⋊C4—3581no—
133C4⋊C4—1791no—
134C4⋊C4—1791no—
135C4⋊C4C22⋊D17912yes—
136C22⋊C4—3582no—
137C22⋊C4—3581no—
138C42—3581no—
139C42C22×D17911yescharacteristic
140C23—3581no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—3581no—
145Q8—1792no—
146Q8—7164no—
147D4—3581no—
148D4—1792no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—3581no—
152C2×C4—3581no—
153C2×C4—3581no—
154C2×C4—7164no—
155C2×C4—7161no—
156C2×C4—7161no—
157C2×C4D4×D17911yescharacteristic
158C2×C4D4×D17911yescharacteristic
159C2×C4C2×C22⋊D17911yescharacteristic
160C8—41no—
161C8—7161no—
162C8—7161no—
163C22—3582no—
164C22—7161no—
165C22—7161no—
166C22#1915308111yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—3582no—
171C4—7161no—
172C4—7161no—
173C4SD16×D17911yescharacteristic
174C4Q16×D17911yescharacteristic
175C4#1915314011yescharacteristic
176C4#1915314511yescharacteristic
177C2—7162no—
178C2#2603858811yescharacteristic
179C2#2603859311yescharacteristic
180C2#2603861111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#487×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#487D179
hypocenterC179#1637
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #26038731C2semidirect
4. #26038185C2semidirect
5. #26038171C2semidirect
7. #487×C179C2semidirect
90. C179#1637semidirect
93. #487D179semidirect

Maximal subgroups

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

subgroupindex
#260391042
#260387312
#260387262
#260381852
#260381712
#260381682
#487×C1792
#1637179