Finite Group Databasebeta

← D17⋊#356 #14545073

Subgroups — D17⋊#356 #14545073

Order 2176.

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

normal subgroups only

This group has 240 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
2#11763720C212yesmaximal, complemented
3#11763719C212yesmaximal, complemented
4C4⋊Q8×D17C211yesmaximal, characteristic
5C2×Q8×D17C411yescharacteristic
6C2×Q8×D17C2211yescharacteristic
7D17⋊M16—22no—
8C17×C4⋊Q8C411yescharacteristic
9#10587630C411yescharacteristic
10#10587630C2211yescharacteristic
11C4⋊C4×D17—22no—
12#10587574—22no—
13C42×D17C411yescharacteristic
14#10587555C2211yescharacteristic
15#10587545—22no—
16#10587545C2214yes—
17C2×Q8×C17C2×C411yescharacteristic
18C2×Q8×C17C2×C411yescharacteristic
19Q8×D17—41no—
20Q8×D17—41no—
21Q8×D17—41no—
22Q8×D17—41no—
23C2×C4×D17—21no—
24C2×C4×D17—21no—
25C2×C4×D17C2×C411yescharacteristic
26C2×C4×D17D412yes—
27C2×Dic34—21no—
28C2×Dic34C2×C411yescharacteristic
29C2×Dic34D412yes—
30C2×Dic34C2311yescharacteristic
31#63247—24no—
32#63243—42no—
33#63242—42no—
34C4⋊C4×C17—22no—
35C42×C17C2×C411yescharacteristic
36C4⋊Dic17—22no—
37#63225—22no—
38#63225—22no—
39C4×Dic17—21no—
40C4×Dic17C2×C411yescharacteristic
41C22×D17C22⋊C411yescharacteristic
42Q8×C17—41no—
43Q8×C17—41no—
44C2×C4×C17—21no—
45C2×C4×C17—21no—
46C2×C4×C17C22⋊C412yes—
47C2×C4×C17C22×C411yescharacteristic
48C2×Dic17—21no—
49C2×Dic17—21no—
50C2×Dic17C22⋊C411yescharacteristic
51C2×Dic17C2×D412yes—
52C4×D17—24no—
53C4×D17—22no—
54C4×D17—22no—
55C4×D17—21no—
56C4×D17—21no—
57C4×D17—41no—
58C4×D17—41no—
59Dic34—42no—
60Dic34—41no—
61Dic34—42no—
62Dic34—41no—
63#3624—44no—
64C2×#356—171nomaximal, Sylow
65C22×C17C2×C22⋊C411yescharacteristic
66C2×D17—22no—
67C2×D17#9911yescharacteristic
68C2×D17#9911yescharacteristic
69C4×C17—22no—
70C4×C17—21no—
71C4×C17—22no—
72C4×C17—41no—
73Dic17—22no—
74Dic17—22no—
75Dic17—21no—
76Dic17—41no—
77C2×C4⋊Q8—171no—
78C2×#101—172no—
79#356—172no—
80#356—172no—
81C2×C17—21no—
82C2×C17C2×#9911yescharacteristic
83D17—21no—
84D17#35612yescomplemented
85C22×Q8—171no—
86C22×Q8—171no—
87C2×M16—342no—
88C4⋊Q8—171no—
89C4⋊Q8—171no—
90C4⋊Q8—171no—
91C4⋊Q8—342no—
92C4⋊Q8#59511yescharacteristic
93C2×C4⋊C4—342no—
94C2×C42—171no—
95#101—174no—
96#101—342no—
97C17C2×#35611yescharacteristic, complemented, Sylow
98C2×Q8—172no—
99C2×Q8—171no—
100C2×Q8—171no—
101C2×Q8—341no—
102C2×Q8—681no—
103C2×Q8—681no—
104C2×Q8—681no—
105C2×Q8—681no—
106C2×Q8C2×#59511yescharacteristic
107C2×Q8C2×#59511yescharacteristic
108C22×C4—172no—
109C22×C4—171no—
110C22×C4—341no—
111C22×C4—341no—
112M16—344no—
113M16—682no—
114C2×C8—682no—
115C4⋊C4—22no—
116C4⋊C4—342no—
117C4⋊C4—342no—
118C4⋊C4—342no—
119C42—171no—
120C42—341no—
121C42C2×#59511yescharacteristic
122C23—171no—
123Q8—41no—
124Q8—41no—
125Q8—682no—
126Q8—681no—
127Q8—681no—
128Q8—682no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—171no—
132C2×C4—172no—
133C2×C4—342no—
134C2×C4—342no—
135C2×C4—341no—
136C2×C4—341no—
137C2×C4—344no—
138C2×C4—341no—
139C2×C4—341no—
140C2×C4—681no—
141C2×C4—681no—
142C2×C4C22⋊#59512yes—
143C2×C4C22×#59511yescharacteristic
144C8—684no—
145C22—171no—
146C22—171no—
147C22—342no—
148C22C2×C22⋊#59511yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—342no—
154C4—342no—
155C4—341no—
156C4—681no—
157C2—21no—
158C2—172no—
159C2—341no—
160C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C4⋊Q8C4
socleC2×C17C2×#99
prime radicalC22×D17C22⋊C4
cosocleC2×Dic34C23
hypercenterC4⋊Q8#595
hypocenterC17C2×#356
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11763720C2semidirect
3. #11763719C2semidirect
84. D17#356semidirect
97. C17C2×#356semidirect

Maximal subgroups

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

subgroupindex
D17⋊#1012
#117637202
#117637192
C4⋊Q8×D172
C2×#35617