Finite Group Databasebeta

← #36352737

Subgroups — #36352737

Order 22144.

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

normal subgroups only

This group has 194 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#26012211C211yesmaximal, characteristic
2#26011842C211yesmaximal, characteristic, complemented
3#26011787C211yesmaximal, characteristic, complemented
4#26011781C211yesmaximal, characteristic
5#26011236C211yesmaximal, characteristic, complemented
6#26011223C211yesmaximal, characteristic, complemented
7D4⋊C8×C173C211yesmaximal, characteristic
8#19149679C2211yescharacteristic
9#19149679C2211yescharacteristic
10#19149672—22no—
11#19149666—21no—
12#19149658—21no—
13C4×Dic346C2211yescharacteristic
14#19149644—21no—
15#19149638—21no—
16#19149613—21no—
17#19149610—21no—
18C4⋊#13063436C2211yescharacteristic
19#19149604—22no—
20#19149604—21no—
21#19149600—21no—
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4×C8×C173C2211yescharacteristic
25C2×Dic346—21no—
26C2×Dic346—21no—
27C2×Dic346D411yescharacteristic
28C2×#13063436—21no—
29C4⋊Dic173—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173—21no—
32C4⋊Dic173D412yes—
33C4⋊Dic173D411yescharacteristic
34#14583805—22no—
35#14583805—21no—
36C22⋊Dic173—22no—
37C22⋊Dic173—22no—
38C4×Dic173—22no—
39C4×Dic173—21no—
40C2×D4×C173D411yescharacteristic
41C22×C4×C173—21no—
42C2×C8×C173—21no—
43C2×C8×C173—21no—
44C4⋊C4×C173D411yescharacteristic
45C22⋊C4×C173—21no—
46C42×C173C2311yescharacteristic
47Dic346—21no—
48Dic346—21no—
49Dic346—41no—
50Dic346—41no—
51Dic346—42no—
52C2×Dic173—22no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—41no—
56#13063436—41no—
57C23×C173—21no—
58D4×C173—21no—
59D4×C173—21no—
60C2×C4×C173—21no—
61C2×C4×C173—41no—
62C2×C4×C173—41no—
63C2×C4×C173C2×D411yescharacteristic
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173C2×D411yescharacteristic
66C8×C173—41no—
67C8×C173—41no—
68C22×C173—41no—
69C22×C173—41no—
70C22×C173—41no—
71C22×C173#12011yescharacteristic
72C4×C173—21no—
73C4×C173—41no—
74C4×C173#13511yescharacteristic
75C4×C173#13511yescharacteristic
76C4×C173#13711yescharacteristic
77C4×C173#13711yescharacteristic
78Dic173—41no—
79Dic173—42no—
80Dic173—41no—
81Dic173—81no—
82C2×C173—81no—
83C2×C173#45211yescharacteristic
84C2×C173#45211yescharacteristic
85C2×C173#45411yescharacteristic
86C173#163811yescharacteristic, complemented, Sylow
87#1638—1731nomaximal, Sylow
88#556—1731no—
89#487—1731no—
90#477—1731no—
91#476—1731no—
92#332—1731no—
93Q8⋊C8—1731no—
94D4⋊C8D17311yescharacteristic
95C4⋊Q8—1731no—
96C4⋊Q8—1731no—
97#126—3462no—
98#124—3461no—
99C22⋊Q8—3461no—
100C4×Q8—3461no—
101C4×Q8—1731no—
102C4×D4C2×D17311yescharacteristic
103#117—3461no—
104#107—3461no—
105#106—3461no—
106C4⋊C8—1731no—
107C4⋊C8C2×D17311yescharacteristic
108#103—3461no—
109#103—3462no—
110#102—3461no—
111C4×C8C2×D17311yescharacteristic
112C2×Q8—3461no—
113C2×Q8—3461no—
114C2×Q8—1731no—
115C2×D4D69211yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3461no—
120C4⋊C4—3462no—
121C4⋊C4—3461no—
122C4⋊C4—3461no—
123C4⋊C4—3461no—
124C4⋊C4—3461no—
125C4⋊C4—1732no—
126C4⋊C4—1731no—
127C4⋊C4D69211yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3462no—
130C22⋊C4—3462no—
131C42—3462no—
132C42—3461no—
133C42C22×D17311yescharacteristic
134C23—21no—
135Q8—3461no—
136Q8—3461no—
137Q8—6922no—
138Q8—6921no—
139Q8—6921no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—3461no—
146C2×C4—3461no—
147C2×C4—3462no—
148C2×C4—6921no—
149C2×C4D4×D17311yescharacteristic
150C2×C4D4×D17311yescharacteristic
151C2×C4C2×D69211yescharacteristic
152C8—41no—
153C8—41no—
154C8—6921no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#1914964911yescharacteristic
159C4—21no—
160C4—41no—
161C4—13841no—
162C4—6921no—
163C4—6922no—
164C4—6921no—
165C4#1914969811yescharacteristic
166C4#1914970111yescharacteristic
167C4#1914970811yescharacteristic
168C4#1914971211yescharacteristic
169C2—81no—
170C2#2601165411yescharacteristic
171C2#2601165511yescharacteristic
172C2#2601166511yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
FittingD4⋊C8×C173C2
socleC22×C173#120
prime radicalC2×D4×C173D4
hypercenterD4⋊C8D173
hypocenterC173#1638
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #26011842C2semidirect
3. #26011787C2semidirect
5. #26011236C2semidirect
6. #26011223C2semidirect
86. C173#1638semidirect

Maximal subgroups

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

subgroupindex
#260122112
#260118422
#260117872
#260117812
#260112362
#260112232
D4⋊C8×C1732
#1638173