Finite Group Databasebeta

← #16931936

Subgroups — #16931936

Order 4352.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#14539160C211yesmaximal, characteristic
2#14537886C211yesmaximal, characteristic, complemented
3#14527453C212yesmaximal
4#14526673C212yesmaximal, complemented
5C17×#1561C211yesmaximal, characteristic, complemented
6C2×#10587560C2211yescharacteristic
7#11763175C414yes—
8#11763157—22no—
9#11762940—22no—
10Q8×#3622—22no—
11#11762870—22no—
12#11762862C2212yes—
13#11762813—22no—
14C4×#63223C2211yescharacteristic
15#11762552—22no—
16#11762552—22no—
17C17×#533C2211yescharacteristic
18C17×#423C2212yes—
19C17×#332—22no—
20C22×D68—21no—
21C22×D68—22no—
22C2×M16×C17—22no—
23C17×C4⋊Q8D412yes—
24C17×#125C2×C412yes—
25C17×C22⋊Q8—22no—
26C4×Q8×C17—22no—
27C17×#117D412yes—
28C2×C42×C17C2311yescharacteristic
29C2×#63223—21no—
30#10587560—22no—
31#10587560—21no—
32#10587560C2×C412yes—
33C17×C4⋊C8—22no—
34C17×C4⋊C8—22no—
35C22⋊#3622—22no—
36#10587480—28no—
37C4⋊#3622—24no—
38C4⋊#3622D412yes—
39C4⋊#3622D412yes—
40#10587474—22no—
41#10587474C2×C412yes—
42C4×#3622—22no—
43C4×#3622—21no—
44C4×#3622—22no—
45C23×D17—41no—
46C2×Q8×C17—22no—
47C22×C4×C17C2×D411yescharacteristic
48C22×C4×C17#4112yes—
49C2×D68—22no—
50C2×D68—24no—
51C2×D68—44no—
52C2×D68—41no—
53C2×D68—42no—
54C2×D68—82no—
55C2×D68—82no—
56M16×C17—82no—
57C2×C8×C17—42no—
58C4⋊C4×C17—24no—
59C4⋊C4×C17—22no—
60C4⋊C4×C17C22⋊C414yes—
61C22⋊C4×C17—22no—
62C42×C17—22no—
63C42×C17C22×C411yescharacteristic
64C42×C17C2×D412yes—
65C42×C17C2×D411yescharacteristic
66#63223—42no—
67C2×#3622—22no—
68C2×#3622—22no—
69C2×#3622—22no—
70#9438—171nomaximal, Sylow
71C23×C17#12311yescharacteristic
72C22×D17—42no—
73C22×D17—82no—
74C22×D17—84no—
75Q8×C17—42no—
76Q8×C17—42no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—24no—
81C2×C4×C17C2×C22⋊C411yescharacteristic
82C2×C4×C17C4×D412yes—
83C2×C4×C17#12011yescharacteristic
84C2×C4×C17#12112yes—
85D68—44no—
86D68—84no—
87D68—82no—
88D68—82no—
89D68—82no—
90C8×C17—82no—
91#3622—42no—
92#3622—42no—
93#3622—42no—
94#3138—171no—
95#2986—171no—
96#1561D1711yescharacteristic, complemented
97#1545—172no—
98#1299—172no—
99C22×C17—21no—
100C22×C17—22no—
101C22×C17#38611yescharacteristic
102C2×D17—82no—
103C2×D17—84no—
104C2×D17—164no—
105C4×C17—21no—
106C4×C17—22no—
107C4×C17—44no—
108C4×C17#45412yes—
109C4×C17#47112yes—
110C4×C17#47212yes—
111#533C2×D1711yescharacteristic
112C2×#127—171no—
113#490—174no—
114#486—342no—
115#446—342no—
116C8×Q8—342no—
117#433—342no—
118#431—172no—
119#423C2×D1712yes—
120#418—342no—
121C4×M16—171no—
122#332—22no—
123#331—342no—
124#331—342no—
125C2×C17—41no—
126C2×C17#188811yescharacteristic
127C2×C17#190712yes—
128D17—162no—
129C22×D4—341no—
130C22×D4—342no—
131C2×M16—22no—
132C2×M16—341no—
133C4⋊Q8C22⋊D1712yes—
134#127—172no—
135#127—342no—
136#127—341no—
137#125C4×D1712yes—
138C22⋊Q8—22no—
139C4×Q8—22no—
140#117D6812yes—
141C2×C42C22×D1711yescharacteristic
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—172no—
145C4⋊C8—172no—
146C4⋊C8—344no—
147#102—348no—
148C22⋊C8—342no—
149#97—172no—
150#97—342no—
151C4×C8—342no—
152C4×C8—342no—
153C4×C8—341no—
154C17#943811yescharacteristic, complemented, Sylow
155C24—681no—
156C2×Q8—22no—
157C2×D4—344no—
158C2×D4—342no—
159C2×D4—684no—
160C2×D4—681no—
161C2×D4—682no—
162C2×D4—1362no—
163C2×D4—1362no—
164C22×C4C2×D6811yescharacteristic
165C22×C4#6325412yes—
166M16—82no—
167M16—682no—
168C2×C8—42no—
169C2×C8—342no—
170C2×C8—342no—
171C2×C8—342no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4#6322714yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D1711yescharacteristic
178C42D4×D1712yes—
179C42C2×C22⋊D1711yescharacteristic
180C23—682no—
181C23—1362no—
182C23—1364no—
183C23#1058757211yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—684no—
187D4—1364no—
188D4—1362no—
189D4—1362no—
190D4—1362no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4#1058756712yes—
196C2×C4#1058756811yescharacteristic
197C2×C4C2×#6322711yescharacteristic
198C2×C4#1058762512yes—
199C8—82no—
200C8—682no—
201C8—682no—
202C8—682no—
203C22—22no—
204C22—21no—
205C22—2724no—
206C22—1364no—
207C22—1362no—
208C22#1176268611yescharacteristic
209C4—22no—
210C4—21no—
211C4—44no—
212C4#1176298412yes—
213C4#1176307412yes—
214C4#1176307712yes—
215C2—41no—
216C2—2722no—
217C2#1452918511yescharacteristic
218C2#1452929012yes—
subgroupG/H
centerC22#11762686
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1561C2
socleC22×C17#386
prime radicalC2×#10587560C22
cosocleC2×C42×C17C23
hypercenter#1561D17
hypocenterC17#9438
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #14537886C2semidirect
4. #14526673C2semidirect
5. C17×#1561C2semidirect
96. #1561D17semidirect
154. C17#9438semidirect

Maximal subgroups

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

subgroupindex
#145391602
#145378862
#145274532
#145266732
C17×#15612
#943817