Finite Group Databasebeta

← #36352788

Subgroups — #36352788

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#26012207C211yesmaximal, characteristic
2#26011768C211yesmaximal, characteristic, complemented
3#26011693C211yesmaximal, characteristic, complemented
4#26011240C211yesmaximal, characteristic, complemented
5#26011220C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic, complemented
7#487×C173C211yesmaximal, characteristic, complemented
8C2×#14583813—21no—
9#19149668—22no—
10#19149654—21no—
11#19149654—21no—
12C4×D692C2211yescharacteristic
13C4×D692C2211yescharacteristic
14#19149637—21no—
15#19149637—21no—
16#19149612—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149603—21no—
20#19149599—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#103×C173—22no—
24C4×C8×C173C2211yescharacteristic
25C2×D692—22no—
26C2×D692D411yescharacteristic
27C2×D692D411yescharacteristic
28C2×C4×D173—21no—
29C2×C4×D173—21no—
30#14583813—42no—
31C2×#13063436—21no—
32C2×#13063436—21no—
33C4⋊Dic173D411yescharacteristic
34C4⋊Dic173D411yescharacteristic
35#14583803—22no—
36#14583803—22no—
37#14583803—21no—
38#14583803—21no—
39C4×Dic173—22no—
40C2×Q8×C173—21no—
41C2×Q8×C173D412yes—
42C2×C8×C173—21no—
43C4⋊C4×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C42×C173C2311yescharacteristic
48C22×D173—21no—
49C22×D173—21no—
50D692—21no—
51D692—21no—
52D692—21no—
53D692—21no—
54D692—82no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C4×D173—41no—
58C4×D173—41no—
59C4×D173—41no—
60C4×D173—41no—
61#13063436—41no—
62#13063436—41no—
63Q8×C173—42no—
64Q8×C173—41no—
65Q8×C173—41no—
66C2×C4×C173—22no—
67C2×C4×C173—41no—
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C8×C173—41no—
72C22×C173#12011yescharacteristic
73C2×D173—41no—
74C2×D173—41no—
75C2×D173—41no—
76C2×D173—41no—
77C2×D173—41no—
78C2×D173—41no—
79C4×C173—21no—
80C4×C173—42no—
81C4×C173—81no—
82C4×C173#13511yescharacteristic
83C4×C173#13511yescharacteristic
84C4×C173AGL(1,ℤ/8)11yescharacteristic
85C4×C173#13711yescharacteristic
86Dic173—41no—
87Dic173—41no—
88C2×C173#44911yescharacteristic
89C2×C173#45211yescharacteristic
90C2×C173#45511yescharacteristic
91D173—81no—
92D173—81no—
93C173#164611yescharacteristic, complemented, Sylow
94#1646—1731nomaximal, Sylow
95#555—1731no—
96#487D17311yescharacteristic, complemented
97#474—1731no—
98#461—1731no—
99#333—1731no—
100D4⋊C8—1731no—
101D4⋊C8—1731no—
102C2×SD16—3461no—
103C4⋊Q8C2×D17311yescharacteristic
104C4⋊Q8C2×D17311yescharacteristic
105#124—3462no—
106#121—3461no—
107#121—3461no—
108C4×D4—1731no—
109C4×D4—1731no—
110#117—3461no—
111#117—3461no—
112#107—3461no—
113C4⋊C8—1731no—
114C4⋊C8—1731no—
115#103—22no—
116#103—3461no—
117#102—3461no—
118C4×C8C2×D17311yescharacteristic
119C2×Q8—21no—
120C2×Q8C22⋊D17312yes—
121C2×D4—3462no—
122C2×D4—1731no—
123C2×D4—1731no—
124C22×C4—3461no—
125C22×C4—3461no—
126SD16—6922no—
127C2×C8—21no—
128C2×C8—3461no—
129C2×C8—3461no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1731no—
135C4⋊C4—1731no—
136C22⋊C4—3462no—
137C22⋊C4—3462no—
138C22⋊C4—3461no—
139C22⋊C4—3461no—
140C42—3462no—
141C42C22×D17311yescharacteristic
142C23—3461no—
143C23—3461no—
144Q8—42no—
145Q8—41no—
146Q8—41no—
147D4—3461no—
148D4—3461no—
149D4—3461no—
150D4—3461no—
151D4—13842no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—6921no—
157C2×C4—6921no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4D4×D17311yescharacteristic
161C2×C4D4×D17311yescharacteristic
162C2×C4C2×C22⋊D17311yescharacteristic
163C8—41no—
164C8—6921no—
165C8—6921no—
166C22—6921no—
167C22—6921no—
168C22—6921no—
169C22—6921no—
170C22—6921no—
171C22—6921no—
172C22#1914964711yescharacteristic
173C4—21no—
174C4—42no—
175C4—81no—
176C4—6921no—
177C4—6921no—
178C4#1914970011yescharacteristic
179C4#1914970211yescharacteristic
180C4#1914970611yescharacteristic
181C4#1914971111yescharacteristic
182C2—13841no—
183C2—13841no—
184C2#2601163511yescharacteristic
185C2#2601165211yescharacteristic
186C2#2601166811yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#487×C173C2
socleC22×C173#120
prime radical#26012207C2
hypercenter#487D173
hypocenterC173#1646
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #26011768C2semidirect
3. #26011693C2semidirect
4. #26011240C2semidirect
5. #26011220C2semidirect
6. #26011220C2semidirect
7. #487×C173C2semidirect
93. C173#1646semidirect
96. #487D173semidirect

Maximal subgroups

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

subgroupindex
#260122072
#260117682
#260116932
#260112402
#260112202
#487×C1732
#1646173