Finite Group Databasebeta

← #16954687

Subgroups — #16954687

Order 4352.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#11762608C212yesmaximal
2#14526957C214yesmaximal, complemented
3C17×#3139C211yesmaximal, characteristic
4C2×#10587507—22no—
5#11762608—22no—
6#11762608C2214yes—
7C17×#533C411yescharacteristic
8C2×C17×C4⋊Q8C2211yescharacteristic
9C17×#491C412yes—
10C17×#491C2212yes—
11C17×#487—22no—
12C17×#419—22no—
13C4×M16×C17C411yescharacteristic
14C22×Q8×C17D412yes—
15C2×M16×C17—21no—
16C17×C4⋊Q8—21no—
17C17×C4⋊Q8—41no—
18C17×C4⋊Q8C2×C411yescharacteristic
19C17×C4⋊Q8D412yes—
20C17×C4⋊Q8C2311yescharacteristic
21C17×#125C2×C412yes—
22C17×C22⋊Q8—41no—
23C4×Q8×C17—41no—
24C17×#117—21no—
25C2×C4⋊C4×C17—41no—
26C2×C42×C17C2×C411yescharacteristic
27C2×#63223—42no—
28C17×#103—24no—
29C17×#103—24no—
30C17×#97C2×C412yes—
31C4×C8×C17—21no—
32#10587507—24no—
33#10587507—42no—
34C2×Q8×C17—22no—
35C2×Q8×C17—41no—
36C2×Q8×C17—42no—
37C2×Q8×C17—42no—
38C2×Q8×C17C22⋊C412yes—
39C2×Q8×C17C2×D412yes—
40C22×C4×C17—21no—
41C22×C4×C17—22no—
42C22×C4×C17C22⋊C411yescharacteristic
43M16×C17—42no—
44C2×C8×C17—22no—
45C4⋊C4×C17—22no—
46C4⋊C4×C17—41no—
47C4⋊C4×C17—42no—
48C4⋊C4×C17—41no—
49C4⋊C4×C17—42no—
50C4⋊C4×C17—41no—
51C22⋊C4×C17—41no—
52C42×C17—21no—
53C42×C17—41no—
54C42×C17C22⋊C411yescharacteristic
55C42×C17C22×C411yescharacteristic
56#63223—44no—
57#63223—82no—
58C2×#3622—82no—
59#13599—171nomaximal, Sylow
60C23×C17#9911yescharacteristic
61Q8×C17—42no—
62Q8×C17—42no—
63Q8×C17—82no—
64Q8×C17—82no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—42no—
71C2×C4×C17—42no—
72C2×C4×C17—42no—
73C2×C4×C17—42no—
74C2×C4×C17#9911yescharacteristic
75C2×C4×C17C2×C22⋊C411yescharacteristic
76C8×C17—42no—
77#3622—84no—
78#3139D1711yescharacteristic
79C2×#356—172no—
80#1413—174no—
81C22×C17—41no—
82C22×C17#35412yes—
83C22×C17C2×#9911yescharacteristic
84C4×C17—22no—
85C4×C17—21no—
86C4×C17—21no—
87C4×C17—41no—
88C4×C17—42no—
89C4×C17—42no—
90C4×C17—82no—
91#533Dic1711yescharacteristic
92C2×C4⋊Q8C2×D1711yescharacteristic
93#491Dic1712yes—
94#491C2×D1712yes—
95#487—22no—
96#419—22no—
97C2×#101—342no—
98C4×M16Dic1711yescharacteristic
99#356—174no—
100#356—342no—
101C2×C17—21no—
102C2×C17—22no—
103C2×C17C2×#35411yescharacteristic
104C22×Q8C22⋊D1712yes—
105C2×M16—21no—
106C2×M16—682no—
107C4⋊Q8—21no—
108C4⋊Q8—41no—
109C4⋊Q8C2×Dic1711yescharacteristic
110C4⋊Q8C22⋊D1712yes—
111C4⋊Q8C22×D1711yescharacteristic
112#125C2×Dic1712yes—
113C22⋊Q8—41no—
114C4×Q8—41no—
115#117—21no—
116C2×C4⋊C4—41no—
117C2×C42C2×Dic1711yescharacteristic
118#103—24no—
119#103—24no—
120#101—344no—
121#101—682no—
122#97C2×Dic1712yes—
123C4×C8—21no—
124C17#1359911yescharacteristic, complemented, Sylow
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8C22⋊Dic1712yes—
130C2×Q8C2×C22⋊D1712yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4C22⋊Dic1711yescharacteristic
134M16—42no—
135M16—684no—
136M16—1362no—
137C2×C8—22no—
138C2×C8—1362no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C22⋊C4—41no—
146C42—21no—
147C42—41no—
148C42C22⋊Dic1711yescharacteristic
149C42C22×Dic1711yescharacteristic
150C23#1058750511yescharacteristic
151Q8—42no—
152Q8—42no—
153Q8—82no—
154Q8—82no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4#1058750511yescharacteristic
165C2×C4C2×C22⋊Dic1711yescharacteristic
166C8—42no—
167C8—1364no—
168C22—41no—
169C22#1176260412yes—
170C22C2×#1058750511yescharacteristic
171C4—22no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—42no—
176C4—42no—
177C4—82no—
178C2—21no—
179C2—22no—
180C2C2×#1176260411yescharacteristic
subgroupG/H
centerC2C2×#11762604
commutatorC42×C17C22×C4
FrattiniC4⋊Q8C22×D17
FittingC17×#3139C2
socleC2×C17C2×#354
prime radicalC23×C17#99
cosocleC17×C4⋊Q8C23
hypercenter#3139D17
hypocenterC17#13599
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #14526957C2semidirect
124. C17#13599semidirect

Maximal subgroups

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

subgroupindex
C2×#117626082
#145269572
C17×#31392
#1359917