Finite Group Databasebeta

← #14527951

Subgroups — #14527951

Order 2176.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D68C211yesmaximal, characteristic
2#11763160C211yesmaximal, characteristic, complemented
3#11763104C211yesmaximal, characteristic, complemented
4#11763100C211yesmaximal, characteristic
5#11762554C211yesmaximal, characteristic, complemented
6#11762538C211yesmaximal, characteristic
7C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D17—21no—
9C4×Q8×C17C2211yescharacteristic
10#10587583—22no—
11#10587583—21no—
12C4×D68—21no—
13C4×D68C2211yescharacteristic
14#10587555—21no—
15#10587555C2211yescharacteristic
16#10587555C2211yescharacteristic
17C17×C4⋊C8C2211yescharacteristic
18C4×C8×C17C2211yescharacteristic
19#10587506—21no—
20#10587492—21no—
21#10587488—21no—
22#10587487—22no—
23#10587478—21no—
24C4⋊#3622C2211yescharacteristic
25C2×Q8×C17D411yescharacteristic
26Q8×D17—42no—
27Q8×D17—42no—
28C2×D68D411yescharacteristic
29C2×C4×D17—22no—
30C2×C4×D17—21no—
31C2×Dic34—22no—
32C2×Dic34—21no—
33C2×Dic34—21no—
34C2×C8×C17—21no—
35C2×C8×C17—21no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17D411yescharacteristic
38C42×C17—21no—
39C42×C17C2311yescharacteristic
40#63227—22no—
41#63227—21no—
42C4⋊Dic17—22no—
43C4⋊Dic17—21no—
44C4⋊Dic17—21no—
45C4⋊Dic17—21no—
46C4⋊Dic17D412yes—
47C4⋊Dic17D411yescharacteristic
48C2×#3622—21no—
49C22×D17—21no—
50Q8×C17—21no—
51Q8×C17SD1612yes—
52C2×C4×C17—21no—
53C2×C4×C17—21no—
54C2×C4×C17C2×D411yescharacteristic
55C2×C4×C17C2×D411yescharacteristic
56C2×C4×C17C2×D411yescharacteristic
57C2×Dic17—21no—
58C2×Dic17—22no—
59C2×Dic17—41no—
60D68—21no—
61D68Q1612yes—
62C4×D17—44no—
63C4×D17—41no—
64C4×D17—41no—
65Dic34—44no—
66Dic34—41no—
67Dic34—41no—
68Dic34—42no—
69C8×C17—41no—
70C8×C17—41no—
71#3622—41no—
72#1637—171nomaximal, Sylow
73C22×C17#12011yescharacteristic
74C2×D17—22no—
75C2×D17—41no—
76C2×D17—41no—
77C4×C17—22no—
78C4×C17—21no—
79C4×C17—41no—
80C4×C17C2×SD1611yescharacteristic
81C4×C17C2×Q1611yescharacteristic
82C4×C17AGL(1,ℤ/8)11yescharacteristic
83C4×C17#13711yescharacteristic
84Dic17—41no—
85Dic17—42no—
86Dic17—81no—
87D4×Q8—171no—
88#487—171no—
89#477—171no—
90#476—171no—
91#332—171no—
92Q8⋊C8D1711yescharacteristic, complemented
93D4⋊C8—171no—
94C2×C17#45011yescharacteristic
95C2×C17#45111yescharacteristic
96C2×C17#45411yescharacteristic
97D17—42no—
98C22×Q8—341no—
99C4⋊Q8—171no—
100C4⋊Q8—171no—
101C4⋊Q8—341no—
102C22⋊Q8—342no—
103C22⋊Q8—341no—
104C4×Q8C2×D1711yescharacteristic
105C4×D4—171no—
106C4×D4—341no—
107#107—341no—
108#106—341no—
109C4⋊C8—171no—
110C4⋊C8C2×D1711yescharacteristic
111#103—342no—
112#103—341no—
113#102—341no—
114C4×C8C2×D1711yescharacteristic
115C17#163711yescharacteristic, complemented, Sylow
116C2×Q8—342no—
117C2×Q8—341no—
118C2×Q8—341no—
119C2×Q8—682no—
120C2×Q8—682no—
121C2×Q8D6811yescharacteristic
122C2×D4—171no—
123C22×C4—342no—
124C22×C4—341no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—341no—
128C4⋊C4—21no—
129C4⋊C4—172no—
130C4⋊C4—171no—
131C4⋊C4—342no—
132C4⋊C4—341no—
133C4⋊C4—341no—
134C4⋊C4—341no—
135C4⋊C4D6811yescharacteristic
136C22⋊C4—342no—
137C22⋊C4—341no—
138C42—21no—
139C42C22×D1711yescharacteristic
140C23—341no—
141Q8—21no—
142Q8—684no—
143Q8—682no—
144Q8—681no—
145Q8—681no—
146Q8#6321912yes—
147D4—172no—
148D4—341no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—342no—
152C2×C4—341no—
153C2×C4—684no—
154C2×C4—681no—
155C2×C4—681no—
156C2×C4—681no—
157C2×C4C2×D6811yescharacteristic
158C2×C4D4×D1711yescharacteristic
159C2×C4D4×D1711yescharacteristic
160C8—41no—
161C8—41no—
162C8—681no—
163C22—342no—
164C22—681no—
165C22—681no—
166C22#1058756811yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—682no—
171C4—681no—
172C4—1361no—
173C4C2×#6321911yescharacteristic
174C4#1058759411yescharacteristic
175C4#1058760111yescharacteristic
176C4Q16×D1711yescharacteristic
177C2—682no—
178C2#1176296211yescharacteristic
179C2#1176296611yescharacteristic
180C2#1176298311yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×Q8⋊C8C2
socleC22×C17#120
prime radicalC2×D68D4
hypercenterQ8⋊C8D17
hypocenterC17#1637
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #11763160C2semidirect
3. #11763104C2semidirect
5. #11762554C2semidirect
7. C17×Q8⋊C8C2semidirect
92. Q8⋊C8D17semidirect
115. C17#1637semidirect

Maximal subgroups

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

subgroupindex
Q8×D682
#117631602
#117631042
#117631002
#117625542
#117625382
C17×Q8⋊C82
#163717