Finite Group Databasebeta

← #36352747

Subgroups — #36352747

Order 22144.

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

normal subgroups only

This group has 212 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#26012226C211yesmaximal, characteristic
2#26011839C211yesmaximal, characteristic
3#26011773C211yesmaximal, characteristic, complemented
4#26011234C211yesmaximal, characteristic
5Q8⋊#13063436C211yesmaximal, characteristic, complemented
6#26011221C211yesmaximal, characteristic, complemented
7#462×C173C211yesmaximal, characteristic, complemented
8#19149676C2211yescharacteristic
9#19149668—21no—
10#19149667—21no—
11#19149664—22no—
12C4×Dic346C2211yescharacteristic
13C4×D692—21no—
14#19149641—21no—
15#19149610—21no—
16C4⋊#13063436C2211yescharacteristic
17#19149602—21no—
18#19149599—22no—
19C4×#13063436C2211yescharacteristic
20C2×Q16×C173—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4×Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24#103×C173—21no—
25C2×Dic346D411yescharacteristic
26C2×D692—21no—
27C2×D692—21no—
28C2×D692D412yes—
29C2×C4×D173—22no—
30C2×#13063436—21no—
31C2×#13063436—21no—
32C4⋊Dic173D411yescharacteristic
33#14583805—21no—
34#14583803—22no—
35#14583803—22no—
36#14583803—22no—
37C4×Dic173—21no—
38C2×Q8×C173—21no—
39C2×Q8×C173D411yescharacteristic
40Q16×C173—42no—
41C2×C8×C173—21no—
42C4⋊C4×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173D411yescharacteristic
46C42×C173—21no—
47C42×C173C2311yescharacteristic
48C22×D173—22no—
49Dic346—21no—
50Dic346—21no—
51D692—42no—
52D692—41no—
53D692—41no—
54D692—41no—
55D692—41no—
56C2×Dic173—21no—
57C2×Dic173—21no—
58C4×D173—82no—
59#13063436—41no—
60#13063436—41no—
61Q8×C173—21no—
62Q8×C173—21no—
63Q8×C173—41no—
64Q8×C173—41no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—41no—
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C8×C173—41no—
72C22×C173#12011yescharacteristic
73C2×D173—42no—
74C2×D173—42no—
75C2×D173—42no—
76C4×C173—21no—
77C4×C173—41no—
78C4×C173—41no—
79C4×C173—81no—
80C4×C173#13511yescharacteristic
81C4×C173#13511yescharacteristic
82C4×C173AGL(1,ℤ/8)11yescharacteristic
83C4×C173#13711yescharacteristic
84Dic173—41no—
85Dic173—41no—
86C2×C173#44911yescharacteristic
87C2×C173#45211yescharacteristic
88C2×C173#45511yescharacteristic
89D173—82no—
90C173#164011yescharacteristic, complemented, Sylow
91#1640—1731nomaximal, Sylow
92#559—1731no—
93#486—1731no—
94#475—1731no—
95#462D17311yescharacteristic, complemented
96#331—1731no—
97Q8⋊C8—1731no—
98Q8⋊C8—1731no—
99C2×Q16—21no—
100C4⋊Q8C2×D17311yescharacteristic
101#127—1731no—
102#124—3461no—
103#124—3461no—
104#123—3462no—
105C4×Q8—1731no—
106C4×Q8C2×D17311yescharacteristic
107C4×D4—3461no—
108C4×D4—3461no—
109#106—3461no—
110C4⋊C8—1731no—
111C4⋊C8C2×D17311yescharacteristic
112#103—21no—
113#103—3461no—
114#102—3462no—
115C4×C8—1731no—
116C2×Q8—21no—
117C2×Q8—1731no—
118C2×Q8C22⋊D17311yescharacteristic
119C2×D4—3461no—
120C2×D4—3461no—
121C2×D4—1732no—
122C22×C4—3462no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—3461no—
126C2×C8—3461no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—3461no—
131C4⋊C4—1731no—
132C4⋊C4C22⋊D17311yescharacteristic
133C22⋊C4—3462no—
134C22⋊C4—3462no—
135C22⋊C4—3462no—
136C42—21no—
137C42—3461no—
138C42C22×D17311yescharacteristic
139C23—3462no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—3461no—
145Q8—3461no—
146D4—6922no—
147D4—6921no—
148D4—6921no—
149D4—6921no—
150D4—6921no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—13842no—
157C2×C4D4×D17311yescharacteristic
158C2×C4D4×D17311yescharacteristic
159C2×C4C2×C22⋊D17311yescharacteristic
160C8—41no—
161C8—6921no—
162C8—6921no—
163C22—6922no—
164C22—6922no—
165C22—6922no—
166C22#1914964711yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—6921no—
172C4—6921no—
173C4#1914969911yescharacteristic
174C4#1914970211yescharacteristic
175C4#1914970511yescharacteristic
176C4#1914971111yescharacteristic
177C2—13842no—
178C2#2601163411yescharacteristic
179C2#2601165211yescharacteristic
180C2#2601166711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#462×C173C2
socleC22×C173#120
prime radical#19149676C22
hypercenter#462D173
hypocenterC173#1640
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #26011773C2semidirect
5. Q8⋊#13063436C2semidirect
6. #26011221C2semidirect
7. #462×C173C2semidirect
90. C173#1640semidirect
95. #462D173semidirect

Maximal subgroups

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

subgroupindex
#260122262
#260118392
#260117732
#260112342
Q8⋊#130634362
#260112212
#462×C1732
#1640173