Finite Group Databasebeta

← #14530477

Subgroups — #14530477

Order 2176.

230 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 55 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
1#11763384C211yesmaximal, characteristic
2#11762792C212yesmaximal
3#11762608C211yesmaximal, characteristic, complemented
4#11762607C212yesmaximal, complemented
5C17×#356C211yesmaximal, characteristic, complemented
6C2×#63256C2211yescharacteristic
7C2×Q8×D17C411yescharacteristic
8C17×C4⋊Q8C2211yescharacteristic
9#10587633C411yescharacteristic
10#10587630C2211yescharacteristic
11M16×D17—22no—
12#10587582—22no—
13#10587580—22no—
14#10587561C411yescharacteristic
15#10587558C411yescharacteristic
16C17×#101C2212yes—
17#10587507C2212yes—
18#10587494—22no—
19C2×Q8×C17C2×C411yescharacteristic
20C2×Q8×C17C2311yescharacteristic
21#63256—42no—
22Q8×D17—42no—
23C2×D68C2×C411yescharacteristic
24C2×D68D412yes—
25C2×C4×D17—21no—
26C2×C4×D17C2×C411yescharacteristic
27C2×C4×D17D412yes—
28C2×Dic34—21no—
29C2×Dic34C2×C411yescharacteristic
30M16×C17—22no—
31C4⋊C4×C17—22no—
32C42×C17C2×C411yescharacteristic
33#63227—21no—
34#63227—22no—
35#63227—22no—
36#63225—22no—
37#63225—21no—
38C4×Dic17C2×C411yescharacteristic
39#63223—22no—
40#63218—42no—
41C8×D17—42no—
42C22×D17C22⋊C411yescharacteristic
43C22×D17C22⋊C412yes—
44Q8×C17—41no—
45Q8×C17—41no—
46C2×C4×C17—21no—
47C2×C4×C17—21no—
48C2×C4×C17C22×C411yescharacteristic
49C2×C4×C17C2×D412yes—
50C2×Dic17—21no—
51C2×Dic17—21no—
52C2×Dic17C22⋊C411yescharacteristic
53D68—41no—
54D68—42no—
55C4×D17—22no—
56C4×D17—22no—
57C4×D17—22no—
58C4×D17—42no—
59Dic34—42no—
60Dic34—41no—
61C8×C17—42no—
62#3622—42no—
63#2135—171nomaximal, Sylow
64C22×C17C2×C22⋊C411yescharacteristic
65C2×D17—21no—
66C2×D17—22no—
67C2×D17—41no—
68C2×D17—42no—
69C4×C17—22no—
70C4×C17—21no—
71C4×C17—41no—
72C4×C17—41no—
73Dic17—41no—
74Dic17—41no—
75Dic17#9912yes—
76#536—171no—
77#413—172no—
78#356—171no—
79#356—172no—
80#356D1711yescharacteristic, complemented
81C2×C17—21no—
82C2×C17C2×#9911yescharacteristic
83D17—41no—
84D17—42no—
85C2×#41—171no—
86C22×Q8—171no—
87C2×M16—342no—
88C4⋊Q8—171no—
89C4⋊Q8C2×D1711yescharacteristic
90#124—171no—
91#124—171no—
92#123—342no—
93C22⋊Q8—342no—
94#117—171no—
95#101—172no—
96#101C2×D1712yes—
97#100—342no—
98C17#213511yescharacteristic, complemented, Sylow
99#41—682no—
100C2×Q8—171no—
101C2×Q8—341no—
102C2×Q8—682no—
103C2×Q8C4×D1711yescharacteristic
104C2×Q8C22×D1711yescharacteristic
105C2×D4—171no—
106C2×D4—172no—
107C22×C4—171no—
108C22×C4—172no—
109C22×C4—341no—
110M16—22no—
111M16—342no—
112M16—682no—
113C2×C8—682no—
114C4⋊C4—22no—
115C4⋊C4—341no—
116C4⋊C4—342no—
117C22⋊C4—341no—
118C22⋊C4—342no—
119C22⋊C4—342no—
120C42—171no—
121C42C4×D1711yescharacteristic
122C23—172no—
123C23—171no—
124Q8—41no—
125Q8—41no—
126Q8—681no—
127Q8—682no—
128D4—682no—
129D4—681no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—171no—
133C2×C4—342no—
134C2×C4—342no—
135C2×C4—342no—
136C2×C4—341no—
137C2×C4—341no—
138C2×C4—682no—
139C2×C4C2×C4×D1711yescharacteristic
140C2×C4D4×D1712yes—
141C8—42no—
142C8—682no—
143C22—341no—
144C22—342no—
145C22—682no—
146C22—681no—
147C22C22⋊C4×D1711yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—172no—
153C4—681no—
154C4—681no—
155C2—21no—
156C2—681no—
157C2—682no—
158C2#99×D1711yescharacteristic
subgroupG/H
centerC2#99×D17
commutatorC2×C4×C17C22×C4
FrattiniC2×Q8C22×D17
FittingC17×#356C2
socleC2×C17C2×#99
prime radicalC2×#63256C22
cosocleC2×Q8×C17C23
hypercenter#356D17
hypocenterC17#2135
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
80. #356D17semidirect
98. C17#2135semidirect

Maximal subgroups

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

subgroupindex
#117633842
#117627922
#117626082
#117626072
C17×#3562
#213517