Finite Group Databasebeta

← #14528019

Subgroups — #14528019

Order 2176.

240 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 61 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#11763212C211yesmaximal, characteristic, complemented
2#11763089C211yesmaximal, characteristic, complemented
3#11763014C211yesmaximal, characteristic, complemented
4#11762555C211yesmaximal, characteristic, complemented
5D4⋊#3622C211yesmaximal, characteristic, complemented
6D4⋊#3622C211yesmaximal, characteristic, complemented
7C17×#547C211yesmaximal, characteristic
8C2×#41×C17—21no—
9C17×C4⋊Q8C2211yescharacteristic
10C17×#123—22no—
11C17×C22⋊Q8—21no—
12C17×#121—21no—
13C4×D4×C17C2211yescharacteristic
14C4×D4×C17C2211yescharacteristic
15C2×C4⋊C4×C17—21no—
16C2×#63229—21no—
17#10587555C2211yescharacteristic
18#10587503—21no—
19#10587481—21no—
20#10587478—22no—
21#10587478—21no—
22C4⋊#3622C2211yescharacteristic
23C4⋊#3622C2211yescharacteristic
24C4×#3622C2211yescharacteristic
25#41×C17—42no—
26#41×C17—42no—
27C2×Q8×C17—21no—
28C2×D4×C17—22no—
29C2×D4×C17D411yescharacteristic
30C2×D4×C17D411yescharacteristic
31C22×C4×C17—22no—
32C22×C4×C17—21no—
33C22×C4×C17—21no—
34C2×Dic34—21no—
35C4⋊C4×C17—22no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17D411yescharacteristic
38C4⋊C4×C17D412yes—
39C4⋊C4×C17D411yescharacteristic
40C22⋊C4×C17—22no—
41C22⋊C4×C17—21no—
42C22⋊C4×C17—21no—
43C42×C17C2311yescharacteristic
44#63229—42no—
45C4⋊Dic17—21no—
46C4⋊Dic17—21no—
47C2×#3622—21no—
48C2×#3622—21no—
49C2×#3622—21no—
50C23×C17—21no—
51C23×C17—21no—
52Q8×C17—41no—
53Q8×C17—41no—
54D4×C17—21no—
55D4×C17—21no—
56D4×C17—21no—
57D4×C17—44no—
58D4×C17D812yes—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—22no—
62C2×C4×C17—44no—
63C2×C4×C17—41no—
64C2×C4×C17—41no—
65C2×C4×C17—41no—
66C2×C4×C17—41no—
67C2×C4×C17C2×D411yescharacteristic
68C2×C4×C17C2×D411yescharacteristic
69C2×C4×C17C2×D411yescharacteristic
70C2×Dic17—41no—
71Dic34—41no—
72Dic34—41no—
73#3622—22no—
74#3622—41no—
75#3622—41no—
76#1647—171nomaximal, Sylow
77C22×C17—22no—
78C22×C17—41no—
79C22×C17—41no—
80C22×C17—41no—
81C22×C17—41no—
82C22×C17—41no—
83C22×C17#12011yescharacteristic
84C4×C17—21no—
85C4×C17—41no—
86C4×C17—42no—
87C4×C17—41no—
88C4×C17C2×D811yescharacteristic
89C4×C17#13511yescharacteristic
90C4×C17AGL(1,ℤ/8)11yescharacteristic
91C4×C17#13711yescharacteristic
92Dic17—81no—
93#547D1711yescharacteristic
94#500—171no—
95#474—171no—
96#461—171no—
97#332—171no—
98D4⋊C8—171no—
99D4⋊C8—171no—
100C2×C17—42no—
101C2×C17—81no—
102C2×C17#44711yescharacteristic
103C2×C17#45211yescharacteristic
104C2×C17#45611yescharacteristic
105C2×#41—21no—
106C2×SD16—341no—
107C4⋊Q8—171no—
108C4⋊Q8C2×D1711yescharacteristic
109#123—22no—
110C22⋊Q8—21no—
111#121—21no—
112C4×D4C2×D1711yescharacteristic
113C4×D4C2×D1711yescharacteristic
114C2×C4⋊C4—21no—
115#107—341no—
116#107—342no—
117C4⋊C8—171no—
118C4⋊C8—171no—
119#103—341no—
120#102—341no—
121C4×C8—171no—
122C17#164711yescharacteristic, complemented, Sylow
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—341no—
127C2×D4—22no—
128C2×D4C22⋊D1711yescharacteristic
129C2×D4C22⋊D1711yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—682no—
134C2×C8—341no—
135C2×C8—341no—
136C2×C8—341no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—341no—
140C4⋊C4—341no—
141C4⋊C4C22⋊D1711yescharacteristic
142C4⋊C4C22⋊D1711yescharacteristic
143C4⋊C4C22⋊D1712yes—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D1711yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—681no—
153Q8—681no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#6322812yes—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—681no—
168C2×C4C2×C22⋊D1711yescharacteristic
169C2×C4C2×C22⋊D1711yescharacteristic
170C2×C4C2×C22⋊D1711yescharacteristic
171C8—342no—
172C8—681no—
173C8—681no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#1058763911yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—42no—
185C4—1361no—
186C4C2×#6322811yescharacteristic
187C4#1058761811yescharacteristic
188C4#1058763611yescharacteristic
189C4#1058763711yescharacteristic
190C2—42no—
191C2—81no—
192C2#1176294311yescharacteristic
193C2#1176297111yescharacteristic
194C2#1176299211yescharacteristic
subgroupG/H
centerC22#10587639
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
Fitting, prime radicalC17×#547C2
socleC22×C17#120
hypercenter#547D17
hypocenterC17#1647
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #11763212C2semidirect
2. #11763089C2semidirect
3. #11763014C2semidirect
4. #11762555C2semidirect
5. D4⋊#3622C2semidirect
6. D4⋊#3622C2semidirect
122. C17#1647semidirect

Maximal subgroups

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

subgroupindex
#117632122
#117630892
#117630142
#117625552
D4⋊#36222
C17×#5472
#164717