Finite Group Databasebeta

← #16945513

Subgroups — #16945513

Order 4352.

272 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 75 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#14539443C211yesmaximal, characteristic
2#14530600C211yesmaximal, characteristic
3#14526818C214yesmaximal, complemented
4C17×#2110C211yesmaximal, characteristic
5#11763421C412yes—
6#11763211C411yescharacteristic
7#11763189C411yescharacteristic
8#11763178C2212yes—
9#11763067—21no—
10#11763050—21no—
11#11762917—21no—
12C4×Dic68—21no—
13#11762630—21no—
14#11762591—21no—
15#11762587C2212yes—
16C2×C17×M32—21no—
17C17×#405C2211yescharacteristic
18C17×#347—21no—
19C17×#347C2212yes—
20C2×M16×C17D411yescharacteristic
21C22×C8×C17D411yescharacteristic
22C17×#117C2×C411yescharacteristic
23#10587610—22no—
24C2×Dic68—21no—
25C2×Dic68—21no—
26#10587575—42no—
27#10587565—42no—
28#10587556C2×C412yes—
29#10587555C2×C412yes—
30C4×Dic34—22no—
31C17×M32—22no—
32C17×M32—22no—
33C2×C16×C17—22no—
34C17×#97D411yescharacteristic
35C17×#97C2311yescharacteristic
36C4×C8×C17C2×C411yescharacteristic
37C4×C8×C17D411yescharacteristic
38#10587489—21no—
39#10587488—21no—
40#10587487—22no—
41#10587487—22no—
42#10587483—22no—
43C22×C4×C17C22⋊C411yescharacteristic
44C2×Dic34—22no—
45M16×C17—41no—
46C2×C8×C17—21no—
47C2×C8×C17—21no—
48C2×C8×C17C22⋊C411yescharacteristic
49C2×C8×C17C22⋊C411yescharacteristic
50C2×C8×C17C2×D411yescharacteristic
51C2×C8×C17C2×D411yescharacteristic
52C4⋊C4×C17—21no—
53C22⋊C4×C17—21no—
54C42×C17C22⋊C411yescharacteristic
55C42×C17C22×C411yescharacteristic
56C22⋊Dic17—42no—
57C4⋊Dic17—22no—
58C4⋊Dic17—42no—
59#63225—42no—
60#63225—42no—
61#63225—42no—
62C4×Dic17—42no—
63Dic68—42no—
64Dic68—41no—
65Dic68—41no—
66C16×C17—24no—
67#63214—42no—
68#11914—171nomaximal, Sylow
69C23×C17#10011yescharacteristic
70C2×C4×C17—21no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17#10011yescharacteristic
75C2×C4×C17C2×C22⋊C411yescharacteristic
76C2×Dic17—42no—
77C2×Dic17—42no—
78Dic34—42no—
79Dic34—42no—
80C8×C17—21no—
81C8×C17—21no—
82C8×C17—21no—
83C8×C17#10411yescharacteristic
84C8×C17#10411yescharacteristic
85#3177—171no—
86#2166—171no—
87#2110D1711yescharacteristic
88#1356—174no—
89C22×C17—21no—
90C22×C17—41no—
91C22×C17C2×#10011yescharacteristic
92C4×C17—21no—
93C4×C17—41no—
94C4×C17—41no—
95C4×C17C2×#10411yescharacteristic
96C4×C17#42111yescharacteristic
97Dic17—82no—
98Dic17—82no—
99#541—172no—
100C2×M32—21no—
101#499—171no—
102#494—171no—
103#491—172no—
104#470—341no—
105#467—341no—
106#441—341no—
107C4×Q16—341no—
108#405C2×D1711yescharacteristic
109#364—341no—
110#349—341no—
111#347—21no—
112#347—172no—
113#347C2×D1712yes—
114C2×C17—21no—
115C2×C17—41no—
116C2×C17#153011yescharacteristic
117C2×Q16—341no—
118C2×Q16—341no—
119C2×M16D6811yescharacteristic
120C22×C8C22⋊D1711yescharacteristic
121C4⋊Q8—172no—
122#126—682no—
123#125—172no—
124#125—682no—
125C22⋊Q8—342no—
126C4×Q8—342no—
127#117C4×D1711yescharacteristic
128M32—22no—
129M32—22no—
130M32—342no—
131C2×C16—22no—
132#107—341no—
133#106—341no—
134#103—342no—
135#103—342no—
136#97C22⋊D1711yescharacteristic
137#97C22×D1711yescharacteristic
138C4×C8C4×D1711yescharacteristic
139C4×C8D6811yescharacteristic
140C17#1191411yescharacteristic, complemented, Sylow
141C2×Q8—342no—
142C22×C4#6322711yescharacteristic
143Q16—682no—
144Q16—681no—
145Q16—681no—
146M16—41no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8#6322711yescharacteristic
150C2×C8#6322711yescharacteristic
151C2×C8C2×D6811yescharacteristic
152C2×C8C2×C22⋊D1711yescharacteristic
153C4⋊C4—21no—
154C4⋊C4—342no—
155C4⋊C4—682no—
156C4⋊C4—682no—
157C4⋊C4—682no—
158C4⋊C4—682no—
159C22⋊C4—21no—
160C22⋊C4—682no—
161C42—682no—
162C42#6322711yescharacteristic
163C42C2×C4×D1711yescharacteristic
164C16—24no—
165C16—682no—
166C23#1058749411yescharacteristic
167Q8—682no—
168Q8—682no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—682no—
174C2×C4—682no—
175C2×C4#1058749411yescharacteristic
176C2×C4C2×#6322711yescharacteristic
177C8—21no—
178C8—21no—
179C8—21no—
180C8#1058747611yescharacteristic
181C8#1058747611yescharacteristic
182C22—21no—
183C22—41no—
184C22C2×#1058749411yescharacteristic
185C4—21no—
186C4—41no—
187C4—41no—
188C4—1362no—
189C4—1362no—
190C4C2×#1058747611yescharacteristic
191C4#1176282511yescharacteristic
192C2—21no—
193C2—41no—
194C2#1452739411yescharacteristic
subgroupG/H
centerC2#14527394
commutatorC42×C17C22×C4
Frattini#97C22×D17
FittingC17×#2110C2
socleC2×C17#1530
prime radicalC23×C17#100
cosocleC17×#97C23
hypercenter#2110D17
hypocenterC17#11914
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
3. #14526818C2semidirect
140. C17#11914semidirect

Maximal subgroups

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

subgroupindex
#145394432
#145306002
#145268182
C17×#21102
#1191417