Finite Group Databasebeta

← D17⋊#355 #14545070

Subgroups — D17⋊#355 #14545070

Order 2176.

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

normal subgroups only

This group has 260 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⋊#101C212yesmaximal, complemented
2#11763718C212yesmaximal, complemented
3#11763717C212yesmaximal, complemented
4#124×D17C211yesmaximal, characteristic
5C2×Q8×D17C2211yescharacteristic
6C2×D4×D17C411yescharacteristic
7D17⋊M16—22no—
8C17×#124C411yescharacteristic
9#10587633C411yescharacteristic
10#10587622C2211yescharacteristic
11#10587571—22no—
12C22⋊C4×D17—22no—
13#10587561C2211yescharacteristic
14C42×D17C411yescharacteristic
15#10587545—22no—
16#10587545C2214yescomplemented
17C23×D17—21no—
18C2×Q8×C17C2×C411yescharacteristic
19C2×D4×C17C2×C411yescharacteristic
20C2×C22⋊D17—21no—
21Q8×D17—41no—
22Q8×D17—41no—
23D4×D17—41no—
24D4×D17—41no—
25C2×D68C2×C411yescharacteristic
26C2×C4×D17—21no—
27C2×C4×D17C2×C411yescharacteristic
28C2×C4×D17D412yes—
29C2×Dic34D412yes—
30C2×Dic34C2311yescharacteristic
31#63247—24no—
32#63243—42no—
33#63242—42no—
34C22⋊C4×C17—22no—
35C42×C17C2×C411yescharacteristic
36C22⋊Dic17—22no—
37#63227—22no—
38#63227—22no—
39C4×Dic17—21no—
40C4×Dic17C2×C411yescharacteristic
41C23×C17—21no—
42C22×D17—21no—
43C22×D17—44no—
44C22×D17—41no—
45C22×D17—41no—
46C22×D17C22⋊C411yescharacteristic
47Q8×C17—41no—
48D4×C17—41no—
49C2×C4×C17—21no—
50C2×C4×C17C22⋊C412yes—
51C2×C4×C17C22×C411yescharacteristic
52C22⋊D17—42no—
53C2×Dic17—21no—
54C2×Dic17C22⋊C411yescharacteristic
55C2×Dic17C2×D412yes—
56D68—41no—
57C4×D17—22no—
58C4×D17—22no—
59C4×D17—21no—
60C4×D17—21no—
61C4×D17—42no—
62Dic34—42no—
63Dic34—41no—
64#3624—44no—
65C2×#355—171nomaximal, Sylow
66C22×C17—42no—
67C22×C17—41no—
68C22×C17C2×C22⋊C411yescharacteristic
69C2×D17—22no—
70C2×D17—42no—
71C2×D17—41no—
72C2×D17—82no—
73C2×D17—82no—
74C2×D17#9911yescharacteristic
75C2×D17#9911yescharacteristic
76C4×C17—21no—
77C4×C17—22no—
78C4×C17—41no—
79Dic17—22no—
80Dic17—21no—
81Dic17—41no—
82C2×#124—171no—
83C2×#101—172no—
84#355—172no—
85#355—172no—
86C2×C17—21no—
87C2×C17—81no—
88C2×C17C2×#9911yescharacteristic
89D17—21no—
90D17—81no—
91D17#35512yescomplemented
92C22×Q8—171no—
93C22×D4—171no—
94C2×M16—342no—
95#124—171no—
96#124—171no—
97#124—171no—
98#124—342no—
99#124#59511yescharacteristic
100C2×C22⋊C4—342no—
101C2×C42—171no—
102#101—174no—
103#101—342no—
104C17C2×#35511yescharacteristic, complemented, Sylow
105C24—341no—
106C2×Q8—172no—
107C2×Q8—171no—
108C2×Q8—681no—
109C2×Q8—681no—
110C2×Q8C2×#59511yescharacteristic
111C2×D4—171no—
112C2×D4—341no—
113C2×D4—681no—
114C2×D4—681no—
115C2×D4C2×#59511yescharacteristic
116C22×C4—172no—
117C22×C4—171no—
118C22×C4—341no—
119M16—344no—
120M16—682no—
121C2×C8—682no—
122C22⋊C4—22no—
123C22⋊C4—342no—
124C22⋊C4—342no—
125C22⋊C4—342no—
126C42—171no—
127C42—341no—
128C42C2×#59511yescharacteristic
129C23—21no—
130C23—171no—
131C23—341no—
132C23—681no—
133C23—681no—
134C23—684no—
135Q8—41no—
136Q8—682no—
137Q8—681no—
138D4—41no—
139D4—681no—
140D4—682no—
141C2×C4—21no—
142C2×C4—171no—
143C2×C4—172no—
144C2×C4—342no—
145C2×C4—342no—
146C2×C4—341no—
147C2×C4—341no—
148C2×C4—341no—
149C2×C4—682no—
150C2×C4C22⋊#59512yes—
151C2×C4C22×#59511yescharacteristic
152C8—684no—
153C22—42no—
154C22—41no—
155C22—171no—
156C22—171no—
157C22—342no—
158C22—682no—
159C22—681no—
160C22—1362no—
161C22—1362no—
162C22C2×C22⋊#59511yescharacteristic
163C4—22no—
164C4—21no—
165C4—41no—
166C4—342no—
167C4—341no—
168C4—681no—
169C2—21no—
170C2—81no—
171C2—172no—
172C2—341no—
173C2—1361no—
174C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#124C4
socleC2×C17C2×#99
prime radicalC2×D4×D17C4
cosocleC2×Dic34C23
hypercenter#124#595
hypocenterC17C2×#355
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. D17⋊#101C2semidirect
2. #11763718C2semidirect
3. #11763717C2semidirect
16. #10587545C22semidirect
91. D17#355semidirect
104. C17C2×#355semidirect

Maximal subgroups

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

subgroupindex
D17⋊#1012
#117637182
#117637172
#124×D172
C2×#35517