Finite Group Databasebeta

← D17×#356 #14530463

Subgroups — D17×#356 #14530463

Order 2176.

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

normal subgroups only

This group has 240 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
1C4⋊Q8×D17C211yesmaximal, characteristic
2#101×D17C212yesmaximal
3#11762608C211yesmaximal, characteristic, complemented
4#11762607C212yesmaximal, complemented
5C17×#356C211yesmaximal, characteristic, complemented
6C2×Q8×D17C411yescharacteristic
7C2×Q8×D17C2211yescharacteristic
8C17×C4⋊Q8C2211yescharacteristic
9#10587630C411yescharacteristic
10#10587630C2211yescharacteristic
11M16×D17—22no—
12C4⋊C4×D17—22no—
13#10587574—22no—
14C42×D17C411yescharacteristic
15#10587555C411yescharacteristic
16C17×#101C2212yes—
17#10587507C2212yes—
18#10587495—22no—
19C2×Q8×C17C2×C411yescharacteristic
20C2×Q8×C17C2311yescharacteristic
21Q8×D17—41no—
22Q8×D17—41no—
23Q8×D17—41no—
24Q8×D17—41no—
25C2×C4×D17—21no—
26C2×C4×D17—21no—
27C2×C4×D17C2×C411yescharacteristic
28C2×C4×D17D412yes—
29C2×Dic34—21no—
30C2×Dic34C2×C411yescharacteristic
31C2×Dic34C2×C411yescharacteristic
32C2×Dic34D412yes—
33M16×C17—22no—
34C4⋊C4×C17—22no—
35C42×C17C2×C411yescharacteristic
36C4⋊Dic17—22no—
37#63225—22no—
38#63225—22no—
39C4×Dic17—21no—
40C4×Dic17C2×C411yescharacteristic
41#63223—22no—
42#63218—42no—
43C8×D17—42no—
44C22×D17C22⋊C411yescharacteristic
45Q8×C17—41no—
46Q8×C17—41no—
47C2×C4×C17—21no—
48C2×C4×C17—21no—
49C2×C4×C17C22×C411yescharacteristic
50C2×C4×C17C2×D412yes—
51C2×Dic17—21no—
52C2×Dic17—21no—
53C2×Dic17C22⋊C411yescharacteristic
54C2×Dic17C22⋊C412yes—
55C4×D17—24no—
56C4×D17—21no—
57C4×D17—21no—
58C4×D17—22no—
59C4×D17—22no—
60C4×D17—41no—
61C4×D17—41no—
62Dic34—41no—
63Dic34—42no—
64Dic34—42no—
65Dic34—41no—
66C8×C17—42no—
67#3622—42no—
68C2×#356—171nomaximal, Sylow
69C22×C17C2×C22⋊C411yescharacteristic
70C2×D17—22no—
71C2×D17#9911yescharacteristic
72C2×D17#9911yescharacteristic
73C4×C17—22no—
74C4×C17—22no—
75C4×C17—21no—
76C4×C17—41no—
77Dic17—22no—
78Dic17—21no—
79Dic17—22no—
80Dic17—41no—
81C2×C4⋊Q8—171no—
82C2×#101—172no—
83#356—171no—
84#356—172no—
85#356D1711yescharacteristic, direct product factor
86C2×C17—21no—
87C2×C17C2×#9911yescharacteristic
88D17—21no—
89D17#35612yesdirect product factor
90C22×Q8—171no—
91C22×Q8—171no—
92C2×M16—342no—
93C4⋊Q8—171no—
94C4⋊Q8—171no—
95C4⋊Q8—171no—
96C4⋊Q8—342no—
97C4⋊Q8C2×D1711yescharacteristic
98C2×C4⋊C4—342no—
99C2×C42—171no—
100#101—172no—
101#101—342no—
102#101C2×D1712yes—
103C17C2×#35611yescharacteristic, complemented, Sylow
104C2×Q8—171no—
105C2×Q8—172no—
106C2×Q8—171no—
107C2×Q8—341no—
108C2×Q8—681no—
109C2×Q8—681no—
110C2×Q8—681no—
111C2×Q8—681no—
112C2×Q8C4×D1711yescharacteristic
113C2×Q8C22×D1711yescharacteristic
114C22×C4—171no—
115C22×C4—172no—
116C22×C4—341no—
117C22×C4—341no—
118M16—22no—
119M16—342no—
120M16—682no—
121C2×C8—682no—
122C4⋊C4—22no—
123C4⋊C4—342no—
124C4⋊C4—342no—
125C4⋊C4—342no—
126C42—171no—
127C42—341no—
128C42C4×D1711yescharacteristic
129C23—171no—
130Q8—41no—
131Q8—41no—
132Q8—682no—
133Q8—681no—
134Q8—681no—
135Q8—682no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—171no—
139C2×C4—172no—
140C2×C4—342no—
141C2×C4—342no—
142C2×C4—341no—
143C2×C4—341no—
144C2×C4—344no—
145C2×C4—341no—
146C2×C4—341no—
147C2×C4—681no—
148C2×C4—681no—
149C2×C4C2×C4×D1711yescharacteristic
150C2×C4D4×D1712yes—
151C8—42no—
152C8—682no—
153C22—171no—
154C22—171no—
155C22—342no—
156C22C22⋊C4×D1711yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—342no—
162C4—341no—
163C4—342no—
164C4—681no—
165C2—21no—
166C2—172no—
167C2—341no—
168C2#99×D1711yescharacteristic
subgroupG/H
centerC2#99×D17
commutatorC2×C4×C17C22×C4
FrattiniC2×Q8C22×D17
FittingC17×#356C2
socleC2×C17C2×#99
prime radicalC22×D17C22⋊C4
cosocleC2×Q8×C17C23
hypercenter#356D17
hypocenterC17C2×#356
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #11762608C2semidirect
4. #11762607C2semidirect
5. C17×#356C2semidirect
85. #356D17direct (N × H)
89. D17#356direct (N × H)
103. C17C2×#356semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D172
#101×D172
#117626082
#117626072
C17×#3562
C2×#35617