Finite Group Databasebeta

← #16955113

Subgroups — #16955113

Order 4352.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×#11762648C211yesmaximal, characteristic
2C2×#11762646C211yesmaximal, characteristic
3#14531035C211yesmaximal, characteristic, complemented
4#14530630C211yesmaximal, characteristic
5#14526979C211yesmaximal, characteristic, complemented
6#14526964C211yesmaximal, characteristic, complemented
7C17×#1416C211yesmaximal, characteristic, complemented
8C2×Dic136C2211yescharacteristic
9C2×#10587500—21no—
10C2×D272C2211yescharacteristic
11#11763187—21no—
12#11762891—21no—
13#11762648—22no—
14#11762646—22no—
15C2×#10587465C2211yescharacteristic
16#11762609—21no—
17C16×Dic17C2211yescharacteristic
18C2×C17×Q32C2211yescharacteristic
19C2×C17×C32C2211yescharacteristic
20C17×#366C2211yescharacteristic
21C2×Q16×C17—21no—
22#10587630—41no—
23C2×#63231—41no—
24C2×Dic68—21no—
25C2×D136—21no—
26#10587579—41no—
27C17×Q32—21no—
28C17×Q32—21no—
29C2×C16×C17C2311yescharacteristic
30C17×#107—21no—
31#10587500—42no—
32C8×Dic17D411yescharacteristic
33C2×#63214D411yescharacteristic
34#10587480—41no—
35Dic136—21no—
36Dic136—21no—
37D272—21no—
38D272D412yes—
39C17×C32—21no—
40#10587465—21no—
41C2×Q8×C17—41no—
42C2×D68—41no—
43C2×C4×D17—81no—
44C2×Dic34—41no—
45Q16×C17—41no—
46Q16×C17—41no—
47C2×C8×C17C2×D411yescharacteristic
48C4⋊C4×C17—41no—
49#63231—82no—
50#63227—81no—
51#63225—81no—
52#63225—81no—
53C4×Dic17D811yescharacteristic
54C2×#3622D811yescharacteristic
55Dic68—41no—
56Dic68—41no—
57D136—22no—
58D136—41no—
59C16×C17C2×D411yescharacteristic
60C16×C17#4111yescharacteristic
61#63214—21no—
62#13687—171nomaximal, Sylow
63C22×D17—81no—
64Q8×C17—81no—
65Q8×C17—81no—
66C2×C4×C17—81no—
67C2×C4×C17—81no—
68C2×C4×C17C2×D811yescharacteristic
69C2×Dic17—81no—
70C2×Dic17D1611yescharacteristic
71C2×Dic17D1611yescharacteristic
72D68—42no—
73D68—81no—
74C4×D17—162no—
75Dic34—81no—
76Dic34—81no—
77C8×C17#12111yescharacteristic
78C8×C17AGL(1,ℤ/8)11yescharacteristic
79#3622—21no—
80C2×SD64—171no—
81C2×SD64—171no—
82#2247—171no—
83C4×D16—171no—
84C4⋊C32—171no—
85#1416—171no—
86#1416D1711yescharacteristic, complemented
87C22×C17C2×D1611yescharacteristic
88C2×D17—82no—
89C2×D17—161no—
90C2×D17—161no—
91C4×C17—161no—
92C4×C17—161no—
93C4×C17#45911yescharacteristic
94C4×C17#50911yescharacteristic
95Dic17—21no—
96Dic17—161no—
97Dic17SD6412yescomplemented
98C2×Q32—171no—
99C2×Q32—341no—
100C2×Q32C2×D1711yescharacteristic
101C2×D16—171no—
102#494—341no—
103C4×D8—341no—
104SD64—342no—
105SD64—342no—
106C2×C32—171no—
107C2×C32C2×D1711yescharacteristic
108#366C2×D1711yescharacteristic
109#357—341no—
110C4×C16—171no—
111C2×C17#220611yescharacteristic
112C2×C17C2×SD6411yescharacteristic
113C2×C17#226411yescharacteristic
114D17—162no—
115C2×Q16—21no—
116C2×Q16—341no—
117C2×Q16—681no—
118C2×D8—341no—
119C4⋊Q8—681no—
120C4×D4—681no—
121Q32—21no—
122Q32—21no—
123Q32—341no—
124Q32—341no—
125Q32—682no—
126D16—172no—
127D16—341no—
128C2×C16—171no—
129C2×C16C22×D1711yescharacteristic
130#107—21no—
131#102—681no—
132C4×C8—171no—
133C32—21no—
134C32—341no—
135C17#1368711yescharacteristic, complemented, Sylow
136C2×Q8—41no—
137C2×Q8—681no—
138C2×D4—681no—
139C22×C4—1361no—
140Q16—41no—
141Q16—41no—
142Q16—681no—
143Q16—681no—
144Q16—1362no—
145D8—342no—
146D8—681no—
147C2×C8—171no—
148C2×C8D4×D1711yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—1361no—
151C4⋊C4—1361no—
152C22⋊C4—1361no—
153C42—171no—
154C16—341no—
155C16#6325211yescharacteristic
156C16D4×D1711yescharacteristic
157C23—1361no—
158Q8—81no—
159Q8—81no—
160Q8—1361no—
161Q8—1361no—
162D4—682no—
163D4—1361no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—2722no—
167C2×C4—171no—
168C2×C4—171no—
169C2×C4—1361no—
170C2×C4D8×D1711yescharacteristic
171C8—341no—
172C8#1058757011yescharacteristic
173C8#1058760011yescharacteristic
174C22—2721no—
175C22—2721no—
176C22—1362no—
177C22D16×D1711yescharacteristic
178C4—161no—
179C4—161no—
180C4—2721no—
181C4—172no—
182C4—341no—
183C4#1176300311yescharacteristic
184C4#1176324611yescharacteristic
185C2—2722no—
186C2#1453081811yescharacteristic
187C2D17×SD6411yescharacteristic
188C2#1453110311yescharacteristic
subgroupG/H
centerC22D16×D17
commutator, cosocleC2×C16×C17C23
FrattiniC2×C16C22×D17
FittingC17×#1416C2
socleC22×C17C2×D16
prime radicalC2×D272C22
hypercenter#1416D17
hypocenterC17#13687
LCSG ▷ C2×C16×C17 ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1416

Complements

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

NH ≅ G/Ntype
3. #14531035C2semidirect
5. #14526979C2semidirect
6. #14526964C2semidirect
7. C17×#1416C2semidirect
86. #1416D17semidirect
97. Dic17SD64semidirect
135. C17#13687semidirect

Maximal subgroups

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

subgroupindex
C2×#117626482
C2×#117626462
#145310352
#145306302
#145269792
#145269642
C17×#14162
#1368717