Finite Group Databasebeta

← #36369827

Subgroups — #36369827

Order 22144.

234 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 54 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#26012477C211yesmaximal, characteristic
2#26012470C211yesmaximal, characteristic
3#26012394C214yesmaximal, complemented
4#26012067C211yesmaximal, characteristic
5D173⋊M16—21no—
6#19149758—21no—
7#19149740—21no—
8#19149739C2212yes—
9#19149738—21no—
10#19149737C2212yes—
11C2×#14583826C2211yescharacteristic
12C22×Dic346C411yescharacteristic, complemented
13#19149679—21no—
14#19149667—21no—
15#19149663C412yescomplemented
16#19149663C2212yes—
17#19149659—21no—
18#19149659—21no—
19#117×C173C411yescharacteristic, complemented
20#14583836—41no—
21#14583835—22no—
22D173⋊C8—41no—
23C4⋊#10621924—41no—
24C22⋊#10621924—22no—
25C4×#10621924—41no—
26#14583826—41no—
27#14583826—41no—
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346—41no—
31C2×Dic346—41no—
32C2×Dic346D411yescharacteristic
33C2×D692D411yescharacteristic
34C2×C22⋊D173C2×C411yescharacteristic
35C2×C22⋊D173C2311yescharacteristic
36C22×Dic173C2×C412yes—
37C2×C4×D173D411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C4⋊Dic173—22no—
40C4⋊Dic173—22no—
41#14583803—22no—
42#14583803—22no—
43C22×C4×C173C2×C411yescharacteristic
44C4⋊C4×C173—21no—
45C22⋊C4×C173C2×C412yes—
46C42×C173—21no—
47C2×#10621924—42no—
48#13063443—42no—
49C22×D173C22⋊C411yescharacteristic
50C22×D173C2×D411yescharacteristic
51Dic346—41no—
52Dic346—42no—
53Dic346—41no—
54Dic346—41no—
55D692—41no—
56C22⋊D173—41no—
57C22⋊D173—41no—
58C2×Dic173—22no—
59C2×Dic173—42no—
60C2×Dic173C22⋊C411yescharacteristic
61C2×Dic173C2×D411yescharacteristic
62C4×D173—21no—
63C4×D173—21no—
64C4×D173—21no—
65C4×D173—21no—
66C23×C173C22×C411yescharacteristic
67C2×C4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173—22no—
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C22⋊C411yescharacteristic
72C22×C173—21no—
73C22×C173—41no—
74C22×C173C2×C22⋊C411yescharacteristic
75C2×D173—21no—
76C2×D173—21no—
77C2×D173—41no—
78C2×D173—41no—
79#10621924—82no—
80C4×C173—21no—
81C4×C173—42no—
82C4×C173#9911yescharacteristic
83C4×C173#9911yescharacteristic
84Dic173—21no—
85Dic173—21no—
86Dic173—42no—
87C2×C173—21no—
88C2×C173—41no—
89C2×C173C2×#9911yescharacteristic
90D173—41no—
91D173—41no—
92C173#212311yescharacteristic, complemented, Sylow
93#2123—1731nomaximal, Sylow
94#536—1731no—
95#413—1731no—
96#410—1731no—
97#352—1734no—
98C2×#41—1731no—
99C22×Q8—1731no—
100C2×M16—3461no—
101C4⋊Q8—3461no—
102#124—3461no—
103#123—1732no—
104#123—1732no—
105C22⋊Q8—3461no—
106C22⋊Q8—3461no—
107#117—3461no—
108#117#1062192411yescharacteristic, complemented
109#101—3461no—
110#100—1732no—
111#99—3461no—
112#99—1732no—
113#41—6921no—
114#41—6921no—
115C2×Q8—3461no—
116C2×Q8—3461no—
117C2×Q8—1731no—
118C2×Q8—6921no—
119C2×Q8—6921no—
120C2×D4—1731no—
121C2×D4—1731no—
122C2×D4—1731no—
123C22×C4—1731no—
124C22×C4—1731no—
125C22×C4—1732no—
126C22×C4C2×#1062192411yescharacteristic
127M16—3462no—
128M16—6921no—
129C2×C8—6921no—
130C4⋊C4—21no—
131C4⋊C4—3462no—
132C4⋊C4—3462no—
133C4⋊C4—6921no—
134C22⋊C4—3462no—
135C22⋊C4—3462no—
136C22⋊C4—3462no—
137C22⋊C4C2×#1062192412yes—
138C42—21no—
139C42—6921no—
140C23—1731no—
141C23—1731no—
142C23C22×#1062192411yescharacteristic
143Q8—6921no—
144Q8—6922no—
145Q8—6921no—
146Q8—6921no—
147D4—6921no—
148D4—6921no—
149D4—6921no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—3461no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4—3462no—
158C2×C4—1731no—
159C2×C4—1731no—
160C2×C4—6922no—
161C2×C4—6922no—
162C2×C4C22⋊#1062192411yescharacteristic
163C2×C4C22⋊#1062192411yescharacteristic
164C8—6922no—
165C22—21no—
166C22—41no—
167C22—3461no—
168C22—3461no—
169C22—6921no—
170C22—6921no—
171C22C2×C22⋊#1062192411yescharacteristic
172C4—21no—
173C4—42no—
174C4—3461no—
175C4—3461no—
176C4—13842no—
177C4—6922no—
178C4#1914973811yescharacteristic
179C4#1914973811yescharacteristic
180C2—21no—
181C2—41no—
182C2—6921no—
183C2—6921no—
184C2C2×#1914973811yescharacteristic
subgroupG/H
centerC2C2×#19149738
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#117×C173C4
socleC2×C173C2×#99
prime radicalC2×#14583826C22
cosocleC2×C22⋊D173C23
hypercenter#117#10621924
hypocenterC173#2123
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #26012394C2semidirect
12. C22×Dic346C4semidirect
15. #19149663C4semidirect
19. #117×C173C4semidirect
92. C173#2123semidirect
108. #117#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124772
#260124702
#260123942
#260120672
#2123173