Finite Group Databasebeta

← #36352698

Subgroups — #36352698

Order 22144.

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

normal subgroups only

This group has 214 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#26012202C211yesmaximal, characteristic
2#26011870C211yesmaximal, characteristic, complemented
3#26011787C211yesmaximal, characteristic, complemented
4#26011782C211yesmaximal, characteristic
5#26011221C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic
7#333×C173C211yesmaximal, characteristic, complemented
8C2×#14583816—22no—
9#19149679C2211yescharacteristic
10#19149670—21no—
11#19149659—21no—
12#19149655—22no—
13C4×Dic346C2211yescharacteristic
14C4×D692C2211yescharacteristic
15#19149641—21no—
16C4⋊C4×D173—21no—
17#19149613—21no—
18#19149611—21no—
19#19149604—21no—
20#19149601—21no—
21C4×#13063436C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25C2×Dic346—21no—
26C2×Dic346D411yescharacteristic
27C2×D692D411yescharacteristic
28C2×C4×D173—22no—
29C2×C4×D173—21no—
30#14583816—44no—
31C2×#13063436—21no—
32C4⋊Dic173—22no—
33C4⋊Dic173—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173D411yescharacteristic
36C4⋊Dic173D411yescharacteristic
37#14583805—22no—
38#14583805—21no—
39#14583803—22no—
40#14583803—21no—
41C4×Dic173—21no—
42C2×Q8×C173D412yes—
43C2×C8×C173—21no—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C42×C173C2311yescharacteristic
48C22×D173—21no—
49Dic346—21no—
50Dic346—21no—
51Dic346—42no—
52D692—21no—
53D692Q1612yescomplemented
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—41no—
58C4×D173—41no—
59C4×D173—41no—
60C4×D173—82no—
61#13063436—22no—
62Q8×C173—42no—
63C2×C4×C173—22no—
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173C2×D411yescharacteristic
66C2×C4×C173C2×D411yescharacteristic
67C8×C173—41no—
68C8×C173—41no—
69C22×C173#12011yescharacteristic
70C2×D173—22no—
71C2×D173—41no—
72C2×D173—41no—
73C4×C173—21no—
74C4×C173—42no—
75C4×C173C2×Q1611yescharacteristic
76C4×C173#13511yescharacteristic
77C4×C173#13711yescharacteristic
78C4×C173#13711yescharacteristic
79Dic173—41no—
80Dic173—41no—
81Dic173—41no—
82Dic173—81no—
83C2×C173#45111yescharacteristic
84C2×C173#45211yescharacteristic
85C2×C173#45311yescharacteristic
86D173—42no—
87C173#163211yescharacteristic, complemented, Sylow
88#1632—1731nomaximal, Sylow
89#554—1731no—
90#494—1731no—
91#477—1731no—
92#476—1731no—
93#333D17311yescharacteristic, complemented
94Q8⋊C8—1731no—
95D4⋊C8—1731no—
96C2×Q16—3462no—
97C4⋊Q8—1731no—
98C4⋊Q8C2×D17311yescharacteristic
99#125—3461no—
100C22⋊Q8—3461no—
101C22⋊Q8—3462no—
102C4×Q8—1731no—
103C4×D4—3461no—
104C4×D4—1731no—
105C2×C4⋊C4—3461no—
106#107—3461no—
107#106—3461no—
108C4⋊C8C2×D17311yescharacteristic
109C4⋊C8C2×D17311yescharacteristic
110#103—3461no—
111#102—3461no—
112C4×C8—1731no—
113C2×Q8—3461no—
114C2×Q8—1731no—
115C2×Q8D69212yes—
116C2×D4—1731no—
117C22×C4—3462no—
118C22×C4—3461no—
119Q16—6924no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3461no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—3462no—
126C4⋊C4—3462no—
127C4⋊C4—3461no—
128C4⋊C4—3461no—
129C4⋊C4—3461no—
130C4⋊C4—1731no—
131C4⋊C4—1731no—
132C22⋊C4—3462no—
133C22⋊C4—3461no—
134C42—3461no—
135C42C22×D17311yescharacteristic
136C23—3461no—
137Q8—42no—
138Q8—3461no—
139Q8—3461no—
140Q8—6922no—
141D4—3461no—
142D4—1732no—
143C2×C4—22no—
144C2×C4—3461no—
145C2×C4—3461no—
146C2×C4—3461no—
147C2×C4—13842no—
148C2×C4—6921no—
149C2×C4—6921no—
150C2×C4—6921no—
151C2×C4D4×D17311yescharacteristic
152C2×C4D4×D17311yescharacteristic
153C2×C4C2×D69211yescharacteristic
154C8—41no—
155C8—41no—
156C8—3462no—
157C22—3462no—
158C22—6921no—
159C22—6921no—
160C22#1914964911yescharacteristic
161C4—21no—
162C4—42no—
163C4—13841no—
164C4—6921no—
165C4—6921no—
166C4—6921no—
167C4Q16×D17311yescharacteristic
168C4#1914970011yescharacteristic
169C4#1914970811yescharacteristic
170C4#1914971111yescharacteristic
171C2—6922no—
172C2#2601164811yescharacteristic
173C2#2601165611yescharacteristic
174C2#2601166111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#333×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#333D173
hypocenterC173#1632
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #26011870C2semidirect
3. #26011787C2semidirect
5. #26011221C2semidirect
7. #333×C173C2semidirect
53. D692Q16semidirect
87. C173#1632semidirect
93. #333D173semidirect

Maximal subgroups

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

subgroupindex
#260122022
#260118702
#260117872
#260117822
#260112212
#260112182
#333×C1732
#1632173