Finite Group Databasebeta

← #14530439

Subgroups — #14530439

Order 2176.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#11763380C211yesmaximal, characteristic
2#11762783C211yesmaximal, characteristic
3#11762600C214yesmaximal, complemented
4C17×#413C211yesmaximal, characteristic
5C2×#41×C17C2211yescharacteristic
6C22×Dic34C411yescharacteristic, complemented
7C2×M16×C17—21no—
8#10587630—21no—
9#10587622—21no—
10#10587621C412yescomplemented
11#10587621C2212yes—
12#10587612—21no—
13#10587612C411yescharacteristic, complemented
14#10587610—21no—
15#10587610—21no—
16C17×#101—21no—
17C17×#100C2212yes—
18#10587505—21no—
19#10587505C2212yes—
20#41×C17—41no—
21#41×C17—41no—
22C2×Q8×C17D411yescharacteristic
23C2×D4×C17C2×C411yescharacteristic
24C2×D4×C17D411yescharacteristic
25C2×D4×C17C2311yescharacteristic
26C22×C4×C17C2×C411yescharacteristic
27C22×C4×C17D411yescharacteristic
28C22×C4×C17D411yescharacteristic
29C22×Dic17C2×C412yes—
30C2×Dic34—21no—
31C2×Dic34—21no—
32C2×Dic34—41no—
33C2×Dic34—41no—
34M16×C17—22no—
35M16×C17—41no—
36C2×C8×C17—41no—
37C22⋊Dic17—22no—
38C22⋊Dic17—22no—
39C22⋊Dic17—22no—
40C22⋊Dic17C2×C412yes—
41C4⋊Dic17—21no—
42C4⋊Dic17—41no—
43#63225—22no—
44#63225—22no—
45C4×Dic17—21no—
46C4×Dic17—41no—
47C23×C17C22⋊C411yescharacteristic
48C23×C17C22×C411yescharacteristic
49C23×C17C2×D411yescharacteristic
50Q8×C17—41no—
51D4×C17—41no—
52D4×C17—41no—
53D4×C17—41no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17C22⋊C411yescharacteristic
61C2×C4×C17C22⋊C411yescharacteristic
62C2×C4×C17C22⋊C411yescharacteristic
63C2×C4×C17C2×D411yescharacteristic
64C2×Dic17—22no—
65C2×Dic17—22no—
66C2×Dic17—42no—
67C2×Dic17—42no—
68Dic34—42no—
69Dic34—41no—
70Dic34—41no—
71C8×C17—42no—
72#2123—171nomaximal, Sylow
73C22×C17—21no—
74C22×C17—21no—
75C22×C17—21no—
76C22×C17—41no—
77C22×C17—41no—
78C22×C17—41no—
79C22×C17C2×C22⋊C411yescharacteristic
80C4×C17—21no—
81C4×C17—21no—
82C4×C17—21no—
83C4×C17#9911yescharacteristic
84C4×C17#9911yescharacteristic
85Dic17—42no—
86Dic17—42no—
87Dic17—82no—
88#536—171no—
89#413D1711yescharacteristic
90#410—171no—
91#352—174no—
92C2×C17—21no—
93C2×C17—41no—
94C2×C17—41no—
95C2×C17—41no—
96C2×C17C2×#9911yescharacteristic
97C2×#41C2×D1711yescharacteristic
98C22×Q8—171no—
99C2×M16—21no—
100C4⋊Q8—341no—
101#124—341no—
102#123—172no—
103#123—172no—
104C22⋊Q8—341no—
105C22⋊Q8—341no—
106#117—171no—
107#117—341no—
108#101—21no—
109#100C2×D1712yes—
110#99—172no—
111#99—341no—
112C17#212311yescharacteristic, complemented, Sylow
113#41—41no—
114#41—41no—
115C2×Q8—341no—
116C2×Q8—341no—
117C2×Q8—681no—
118C2×Q8—681no—
119C2×Q8C22⋊D1711yescharacteristic
120C2×D4C4×D1711yescharacteristic
121C2×D4D6811yescharacteristic
122C2×D4C22×D1711yescharacteristic
123C22×C4—172no—
124C22×C4C4×D1711yescharacteristic
125C22×C4D6811yescharacteristic
126C22×C4C22⋊D1711yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—342no—
131C4⋊C4—342no—
132C4⋊C4—341no—
133C4⋊C4—681no—
134C22⋊C4—172no—
135C22⋊C4—342no—
136C22⋊C4—342no—
137C22⋊C4—342no—
138C42—341no—
139C42—681no—
140C23#6322711yescharacteristic
141C23C2×C4×D1711yescharacteristic
142C23C2×D6811yescharacteristic
143Q8—41no—
144Q8—682no—
145Q8—681no—
146Q8—681no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—342no—
157C2×C4—342no—
158C2×C4—682no—
159C2×C4—682no—
160C2×C4#6322711yescharacteristic
161C2×C4#6322711yescharacteristic
162C2×C4#6322711yescharacteristic
163C2×C4C2×C22⋊D1711yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×#6322711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—682no—
176C4—682no—
177C4—1362no—
178C4#1058747711yescharacteristic
179C4#1058747711yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1058747711yescharacteristic
subgroupG/H
centerC2C2×#10587477
commutatorC23×C17C22×C4
FrattiniC2×D4C22×D17
FittingC17×#413C2
socleC2×C17C2×#99
prime radicalC2×#41×C17C22
cosocleC2×D4×C17C23
hypercenter#413D17
hypocenterC17#2123
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #11762600C2semidirect
6. C22×Dic34C4semidirect
10. #10587621C4semidirect
13. #10587612C4semidirect
112. C17#2123semidirect

Maximal subgroups

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

subgroupindex
#117633802
#117627832
#117626002
C17×#4132
#212317