Finite Group Databasebeta

← D17⋊#353 #14545050

Subgroups — D17⋊#353 #14545050

Order 2176.

368 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D17⋊#99C212yesmaximal, complemented
2#11763714C212yesmaximal, complemented
3#11763713C212yesmaximal, complemented
4#127×D17C211yesmaximal, characteristic
5C2×D4×D17—22no—
6C2×D4×D17C411yescharacteristic, complemented
7C2×D4×D17C2211yescharacteristic
8C2×C22⋊#595—22no—
9C17×#127C411yescharacteristic, complemented
10#10587626—22no—
11#10587626C411yescharacteristic, complemented
12#10587626C2211yescharacteristic
13#10587560C2211yescharacteristic
14C42×D17C411yescharacteristic, complemented
15#10587551—22no—
16#10587539C2214yescomplemented
17C23×D17—21no—
18C23×D17D412yes—
19C22×#595—42no—
20C2×D4×C17—22no—
21C2×D4×C17C2×C411yescharacteristic
22C2×D4×C17C2×C411yescharacteristic, complemented
23C2×C22⋊D17—22no—
24C2×C22⋊D17—22no—
25C2×C22⋊D17—21no—
26C2×C22⋊D17D412yes—
27D4×D17—41no—
28D4×D17—41no—
29D4×D17—41no—
30D4×D17—41no—
31D4×D17—48no—
32C2×D68—22no—
33C2×D68C2×C411yescharacteristic, complemented
34C2×D68C2311yescharacteristic
35C2×C4×D17—21no—
36C2×C4×D17C2×C411yescharacteristic
37C22⋊#595—24no—
38C22⋊#595—42no—
39C42×C17C2×C411yescharacteristic, complemented
40C4×Dic17—21no—
41C4×Dic17C2×C411yescharacteristic, complemented
42C23×C17—21no—
43C23×C17C22⋊C412yes—
44C22×D17—22no—
45C22×D17—22no—
46C22×D17—21no—
47C22×D17—44no—
48C22×D17—44no—
49C22×D17—41no—
50C22×D17—41no—
51C22×D17C22⋊C411yescharacteristic
52C22×D17C2×D412yes—
53C2×#595—44no—
54C2×#595—84no—
55D4×C17—41no—
56D4×C17—41no—
57D4×C17—44no—
58C2×C4×C17—21no—
59C2×C4×C17C22×C411yescharacteristic
60C22⋊D17—42no—
61C22⋊D17—44no—
62C22⋊D17—44no—
63C22⋊D17—42no—
64C2×Dic17—21no—
65C2×Dic17C22⋊C411yescharacteristic
66D68—41no—
67D68—41no—
68D68—44no—
69C4×D17—24no—
70C4×D17—21no—
71C4×D17—21no—
72C2×#353—171nomaximal, Sylow
73C22×C17—22no—
74C22×C17—42no—
75C22×C17—42no—
76C22×C17—41no—
77C22×C17C2×C22⋊C411yescharacteristic
78C2×D17—22no—
79C2×D17—22no—
80C2×D17—44no—
81C2×D17—42no—
82C2×D17—42no—
83C2×D17—41no—
84C2×D17—82no—
85C2×D17—82no—
86C2×D17—82no—
87C2×D17#9911yescharacteristic
88C2×D17#9911yescharacteristic
89#595—84no—
90C4×C17—22no—
91C4×C17—21no—
92Dic17—22no—
93Dic17—21no—
94C2×#127—171no—
95C2×#99—172no—
96#353—172no—
97#353—172no—
98C2×C17—21no—
99C2×C17—42no—
100C2×C17—81no—
101C2×C17C2×#9911yescharacteristic
102D17—21no—
103D17—42no—
104D17—81no—
105D17#35312yescomplemented
106C22×D4—171no—
107C22×D4—171no—
108C22×D4—342no—
109#127—171no—
110#127—171no—
111#127—171no—
112#127—342no—
113#127#59511yescharacteristic, complemented
114C2×C22⋊C4—342no—
115C2×C42—171no—
116#99—174no—
117#99—342no—
118C17C2×#35311yescharacteristic, complemented, Sylow
119C24—172no—
120C24—341no—
121C2×D4—22no—
122C2×D4—172no—
123C2×D4—171no—
124C2×D4—171no—
125C2×D4—342no—
126C2×D4—341no—
127C2×D4—342no—
128C2×D4—342no—
129C2×D4—681no—
130C2×D4—681no—
131C2×D4—681no—
132C2×D4—681no—
133C2×D4—688no—
134C2×D4C2×#59511yescharacteristic
135C2×D4C2×#59511yescharacteristic, complemented
136C22×C4—171no—
137C22×C4—341no—
138C22×C4—682no—
139C22⋊C4—344no—
140C22⋊C4—682no—
141C42—171no—
142C42—341no—
143C42C2×#59511yescharacteristic, complemented
144C23—21no—
145C23—171no—
146C23—172no—
147C23—342no—
148C23—342no—
149C23—341no—
150C23—681no—
151C23—681no—
152C23—684no—
153C23—684no—
154C23C22⋊#59512yes—
155D4—41no—
156D4—41no—
157D4—44no—
158D4—681no—
159D4—681no—
160D4—684no—
161D4—682no—
162D4—684no—
163D4—684no—
164D4—682no—
165C2×C4—21no—
166C2×C4—171no—
167C2×C4—341no—
168C2×C4—341no—
169C2×C4—344no—
170C2×C4—341no—
171C2×C4—684no—
172C2×C4—1364no—
173C2×C4C22×#59511yescharacteristic
174C22—22no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—171no—
179C22—171no—
180C22—342no—
181C22—342no—
182C22—682no—
183C22—682no—
184C22—684no—
185C22—681no—
186C22—1362no—
187C22—1362no—
188C22—1362no—
189C22C2×C22⋊#59511yescharacteristic
190C4—22no—
191C4—21no—
192C4—342no—
193C4—341no—
194C4—1364no—
195C2—21no—
196C2—42no—
197C2—81no—
198C2—172no—
199C2—341no—
200C2—682no—
201C2—1361no—
202C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#127C4
socleC2×C17C2×#99
prime radical#127×D17C2
cosocleC2×D68C23
hypercenter#127#595
hypocenterC17C2×#353
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. D17⋊#99C2semidirect
2. #11763714C2semidirect
3. #11763713C2semidirect
6. C2×D4×D17C4semidirect
9. C17×#127C4semidirect
11. #10587626C4semidirect
14. C42×D17C4semidirect
16. #10587539C22semidirect
22. C2×D4×C17C2×C4semidirect
33. C2×D68C2×C4semidirect
39. C42×C17C2×C4semidirect
41. C4×Dic17C2×C4semidirect
105. D17#353semidirect
113. #127#595semidirect
118. C17C2×#353semidirect
135. C2×D4C2×#595semidirect
143. C42C2×#595semidirect

Maximal subgroups

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

subgroupindex
D17⋊#992
#117637142
#117637132
#127×D172
C2×#35317