Finite Group Databasebeta

← #36352711

Subgroups — #36352711

Order 22144.

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

normal subgroups only

This group has 234 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#26012144C211yesmaximal, characteristic
2#26011778C211yesmaximal, characteristic, complemented
3#26011778C211yesmaximal, characteristic, complemented
4#26011237C211yesmaximal, characteristic, complemented
5#26011220C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic, complemented
7#487×C173C211yesmaximal, characteristic, complemented
8C2×#14583827—21no—
9#19149664—22no—
10#19149655—21no—
11#19149654—21no—
12C4×D692C2211yescharacteristic
13C4×D692C2211yescharacteristic
14C4⋊C4×D173—21no—
15#19149610—21no—
16#19149610—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149599—21no—
20#19149599—21no—
21C4⋊Q8×C173C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#103×C173—22no—
24C4×C8×C173C2211yescharacteristic
25#14583827—42no—
26#14583827—42no—
27C2×D692—22no—
28C2×D692D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C4×D173—22no—
31C2×C4×D173—21no—
32C2×C4×D173—21no—
33C2×#13063436—21no—
34C2×#13063436—21no—
35C4⋊Dic173D411yescharacteristic
36C4⋊Dic173D411yescharacteristic
37#14583805—22no—
38#14583803—22no—
39#14583803—21no—
40#14583803—21no—
41C2×Q8×C173—21no—
42C2×Q8×C173—21no—
43C2×C8×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D412yes—
48C42×C173C2311yescharacteristic
49C22×D173—21no—
50C22×D173—21no—
51D692—21no—
52D692—21no—
53D692—21no—
54D692—44no—
55D692SD1612yes—
56C2×Dic173—21no—
57C2×Dic173—21no—
58C4×D173—44no—
59C4×D173—41no—
60C4×D173—41no—
61C4×D173—41no—
62C4×D173—41no—
63#13063436—41no—
64#13063436—41no—
65Q8×C173—42no—
66Q8×C173—41no—
67Q8×C173—41no—
68C2×C4×C173—22no—
69C2×C4×C173—41no—
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C2×C4×C173C2×D411yescharacteristic
73C8×C173—41no—
74C22×C173#12011yescharacteristic
75C2×D173—22no—
76C2×D173—41no—
77C2×D173—41no—
78C2×D173—41no—
79C2×D173—41no—
80C2×D173—41no—
81C4×C173—21no—
82C4×C173—42no—
83C4×C173—81no—
84C4×C173C2×SD1611yescharacteristic
85C4×C173#13511yescharacteristic
86C4×C173AGL(1,ℤ/8)11yescharacteristic
87C4×C173#13711yescharacteristic
88Dic173—41no—
89Dic173—41no—
90C2×C173#45011yescharacteristic
91C2×C173#45211yescharacteristic
92C2×C173#45411yescharacteristic
93D173—42no—
94D173—81no—
95C173#163511yescharacteristic, complemented, Sylow
96#1635—1731nomaximal, Sylow
97#547—1731no—
98#487D17311yescharacteristic, complemented
99#476—1731no—
100#476—1731no—
101#332—1731no—
102D4⋊C8—1731no—
103D4⋊C8—1731no—
104C2×#41—3461no—
105C4⋊Q8C2×D17311yescharacteristic
106C4⋊Q8C2×D17311yescharacteristic
107#123—3462no—
108C22⋊Q8—3461no—
109#121—3461no—
110C4×D4—1731no—
111C4×D4—1731no—
112C2×C4⋊C4—3461no—
113#106—3461no—
114#106—3461no—
115C4⋊C8—1731no—
116C4⋊C8—1731no—
117#103—22no—
118#102—3461no—
119#102—3461no—
120C4×C8C2×D17311yescharacteristic
121#41—6922no—
122#41—6922no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×D4—3462no—
126C2×D4—1731no—
127C2×D4—1731no—
128C22×C4—3462no—
129C22×C4—3461no—
130C22×C4—3461no—
131C2×C8—21no—
132C2×C8—3461no—
133C2×C8—3461no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—3462no—
138C4⋊C4—1731no—
139C4⋊C4—1731no—
140C4⋊C4C22⋊D17312yes—
141C22⋊C4—3462no—
142C22⋊C4—3461no—
143C22⋊C4—3461no—
144C42C22×D17311yescharacteristic
145C23—3461no—
146C23—3461no—
147Q8—42no—
148Q8—41no—
149Q8—41no—
150D4—3461no—
151D4—3461no—
152D4—3461no—
153D4—1732no—
154D4—6924no—
155C2×C4—22no—
156C2×C4—41no—
157C2×C4—3461no—
158C2×C4—3461no—
159C2×C4—6924no—
160C2×C4—6921no—
161C2×C4—6921no—
162C2×C4—6921no—
163C2×C4—6921no—
164C2×C4D4×D17311yescharacteristic
165C2×C4D4×D17311yescharacteristic
166C2×C4C2×C22⋊D17311yescharacteristic
167C8—41no—
168C8—6921no—
169C8—6921no—
170C22—3462no—
171C22—6921no—
172C22—6921no—
173C22—6921no—
174C22—6921no—
175C22—6921no—
176C22#1914964711yescharacteristic
177C4—21no—
178C4—42no—
179C4—81no—
180C4—6921no—
181C4—6921no—
182C4SD16×D17311yescharacteristic
183C4#1914970211yescharacteristic
184C4#1914970611yescharacteristic
185C4#1914971111yescharacteristic
186C2—13841no—
187C2—6922no—
188C2#2601164011yescharacteristic
189C2#2601165211yescharacteristic
190C2#2601166311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#487×C173C2
socleC22×C173#120
prime radical#26012144C2
hypercenter#487D173
hypocenterC173#1635
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. #26011778C2semidirect
3. #26011778C2semidirect
4. #26011237C2semidirect
5. #26011220C2semidirect
6. #26011220C2semidirect
7. #487×C173C2semidirect
95. C173#1635semidirect
98. #487D173semidirect

Maximal subgroups

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

subgroupindex
#260121442
#260117782
#260112372
#260112202
#487×C1732
#1635173