Finite Group Databasebeta

← #36355216

Subgroups — #36355216

Order 22144.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#26012062C211yesmaximal, characteristic
2#26011475C211yesmaximal, characteristic
3#26011280C214yesmaximal, complemented
4#410×C173C211yesmaximal, characteristic
5C22×Dic346C411yescharacteristic, complemented
6C2×#14583810—21no—
7#19149677—21no—
8#19149666—21no—
9#19149660C412yescomplemented
10#19149660C2212yes—
11#19149658—21no—
12#19149658—21no—
13#19149638C411yescharacteristic, complemented
14#19149598—21no—
15#19149596C2212yes—
16C2×#41×C173C2211yescharacteristic
17#117×C173—21no—
18#99×C173—21no—
19#99×C173C2212yes—
20C2×Dic346—21no—
21C2×Dic346—21no—
22C2×Dic346—41no—
23C2×Dic346—41no—
24C22×Dic173C2×C412yes—
25#14583810—22no—
26#14583810—41no—
27C2×#13063436—41no—
28C4⋊Dic173—21no—
29#14583805—22no—
30#14583805—22no—
31C22⋊Dic173—22no—
32C22⋊Dic173—22no—
33C22⋊Dic173C2×C412yes—
34C4×Dic173—21no—
35#41×C173—41no—
36#41×C173—41no—
37C2×Q8×C173D411yescharacteristic
38C2×D4×C173C2×C411yescharacteristic
39C2×D4×C173D411yescharacteristic
40C2×D4×C173C2311yescharacteristic
41C22×C4×C173C2×C411yescharacteristic
42C22×C4×C173D411yescharacteristic
43C22×C4×C173D411yescharacteristic
44C4⋊C4×C173—41no—
45C22⋊C4×C173—22no—
46C42×C173—41no—
47Dic346—42no—
48Dic346—41no—
49Dic346—41no—
50C2×Dic173—22no—
51C2×Dic173—22no—
52C2×Dic173—42no—
53#13063436—42no—
54C23×C173C22⋊C411yescharacteristic
55C23×C173C22×C411yescharacteristic
56C23×C173C2×D411yescharacteristic
57Q8×C173—41no—
58D4×C173—41no—
59D4×C173—41no—
60D4×C173—41no—
61C2×C4×C173—21no—
62C2×C4×C173—21no—
63C2×C4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—42no—
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22⋊C411yescharacteristic
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C22×C173—21no—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173—41no—
76C22×C173—41no—
77C22×C173—41no—
78C22×C173C2×C22⋊C411yescharacteristic
79C4×C173—21no—
80C4×C173—21no—
81C4×C173—21no—
82C4×C173—82no—
83C4×C173#9911yescharacteristic
84C4×C173#9911yescharacteristic
85Dic173—42no—
86Dic173—42no—
87C2×C173—21no—
88C2×C173—41no—
89C2×C173—41no—
90C2×C173—41no—
91C2×C173C2×#9911yescharacteristic
92C173#212311yescharacteristic, complemented, Sylow
93#2123—1731nomaximal, Sylow
94#536—1731no—
95#413—1731no—
96#410D17311yescharacteristic
97#352—1734no—
98C2×#41C2×D17311yescharacteristic
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—21no—
108#117—1731no—
109#101—3461no—
110#100—1732no—
111#99—21no—
112#99C2×D17312yes—
113#41—41no—
114#41—41no—
115C2×Q8—3461no—
116C2×Q8—3461no—
117C2×Q8—6921no—
118C2×Q8—6921no—
119C2×Q8D69211yescharacteristic
120C2×D4C4×D17311yescharacteristic
121C2×D4C22⋊D17311yescharacteristic
122C2×D4C22×D17311yescharacteristic
123C22×C4—1732no—
124C22×C4C4×D17311yescharacteristic
125C22×C4C22⋊D17311yescharacteristic
126C22×C4D69211yescharacteristic
127M16—3462no—
128M16—6921no—
129C2×C8—6921no—
130C4⋊C4—41no—
131C4⋊C4—3462no—
132C4⋊C4—3462no—
133C4⋊C4—3461no—
134C22⋊C4—22no—
135C22⋊C4—3462no—
136C22⋊C4—3462no—
137C22⋊C4—1732no—
138C42—41no—
139C42—3461no—
140C23#1458380311yescharacteristic
141C23C2×C4×D17311yescharacteristic
142C23C2×C22⋊D17311yescharacteristic
143Q8—41no—
144Q8—6922no—
145Q8—6921no—
146Q8—6921no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—3462no—
158C2×C4—3462no—
159C2×C4—6922no—
160C2×C4#1458380311yescharacteristic
161C2×C4#1458380311yescharacteristic
162C2×C4#1458380311yescharacteristic
163C2×C4C2×D69211yescharacteristic
164C8—6922no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×#1458380311yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—82no—
176C4—6922no—
177C4—6922no—
178C4#1914959311yescharacteristic
179C4#1914959311yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1914959311yescharacteristic
subgroupG/H
centerC2C2×#19149593
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#410×C173C2
socleC2×C173C2×#99
prime radicalC2×#41×C173C22
cosocleC2×D4×C173C23
hypercenter#410D173
hypocenterC173#2123
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #26011280C2semidirect
5. C22×Dic346C4semidirect
9. #19149660C4semidirect
13. #19149638C4semidirect
92. C173#2123semidirect

Maximal subgroups

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

subgroupindex
#260120622
#260114752
#260112802
#410×C1732
#2123173