Finite Group Databasebeta

← #16950601

Subgroups — #16950601

Order 4352.

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

normal subgroups only

This group has 342 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
1#14537424C211yesmaximal, characteristic
2#14528885C212yesmaximal
3#14526912C212yesmaximal
4#14526907C211yesmaximal, characteristic
5C17×#1391C211yesmaximal, characteristic
6#41×Dic17C2211yescharacteristic
7#11762867—22no—
8#11762859—22no—
9#11762837C411yescharacteristic, complemented
10#11762835C411yescharacteristic, complemented
11#11762793C2212yes—
12#11762776C2211yescharacteristic
13M16×Dic17C411yescharacteristic, complemented
14C2×C8×Dic17C411yescharacteristic, complemented
15#11762568—22no—
16#11762565—22no—
17C17×#410C2212yes—
18C17×#408C2211yescharacteristic
19C2×#41×C17C2311yescharacteristic
20C2×M16×C17C2×C411yescharacteristic
21C22×C8×C17C2×C411yescharacteristic
22C17×#117—22no—
23Q8×Dic17—21no—
24D4×Dic17—22no—
25D4×Dic17—21no—
26#10587612C2×C411yescharacteristic
27#10587612D412yes—
28C2×C4×Dic17C2×C411yescharacteristic
29C2×C4×Dic17D412yes—
30C2×#63223—22no—
31C2×#63223C2×C411yescharacteristic
32C22×#3622C2×C411yescharacteristic
33C17×#99—22no—
34C17×#99—22no—
35C17×C22⋊C8—22no—
36#10587507—22no—
37#10587504—22no—
38C22⋊#3622—22no—
39#10587486—21no—
40#10587485—22no—
41#10587485—22no—
42C8×Dic17—21no—
43C8×Dic17—21no—
44C8×Dic17—21no—
45#41×C17—41no—
46#41×C17—41no—
47C2×Q8×C17#4111yescharacteristic
48C2×D4×C17#4112yes—
49C2×D4×C17#4111yescharacteristic
50C22×C4×C17C22×C411yescharacteristic
51C22×C4×C17C2×D412yes—
52C22×Dic17—21no—
53C22×Dic17—22no—
54M16×C17—41no—
55C2×C8×C17—21no—
56C2×C8×C17—21no—
57C2×C8×C17—41no—
58C4⋊C4×C17—42no—
59C22⋊C4×C17—42no—
60C42×C17—42no—
61C22⋊Dic17—21no—
62C22⋊Dic17—22no—
63C4⋊Dic17—21no—
64C4⋊Dic17—22no—
65C4×Dic17—21no—
66C4×Dic17—22no—
67C4×Dic17C22⋊C411yescharacteristic
68C4×Dic17C22⋊C411yescharacteristic
69C4×Dic17C22⋊C412yes—
70#63223—41no—
71#63223—42no—
72#63223—42no—
73C2×#3622—21no—
74C2×#3622—21no—
75C2×#3622—41no—
76C2×#3622—42no—
77#12864—171nomaximal, Sylow
78C23×C17#11711yescharacteristic
79C23×C17#12312yes—
80Q8×C17—41no—
81D4×C17—42no—
82D4×C17—41no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—82no—
88C2×C4×C17C2×C22⋊C411yescharacteristic
89C2×C4×C17#11711yescharacteristic
90C2×C4×C17#12312yes—
91C2×Dic17—21no—
92C2×Dic17—22no—
93C2×Dic17—21no—
94C2×Dic17—42no—
95C2×Dic17—44no—
96C8×C17—41no—
97C8×C17—41no—
98#3622—41no—
99#3622—41no—
100#3622—82no—
101#2922—171no—
102#1827—172no—
103#1395—172no—
104#1392—171no—
105#1391D1711yescharacteristic
106C22×C17—22no—
107C22×C17—21no—
108C22×C17—41no—
109C22×C17—42no—
110C22×C17#38111yescharacteristic
111C4×C17—22no—
112C4×C17—21no—
113C4×C17—162no—
114C4×C17#41011yescharacteristic
115C4×C17#41011yescharacteristic
116Dic17—22no—
117Dic17—42no—
118Dic17—41no—
119C4×#41—171no—
120#433—342no—
121#431—342no—
122#424—171no—
123#424—171no—
124#413—172no—
125#410C2×D1712yes—
126#408—171no—
127#408C2×D1711yescharacteristic
128C4×M16—171no—
129C2×C4×C8—171no—
130#339—342no—
131H(3,ℤ/4)—342no—
132C2×C17—21no—
133C2×C17—42no—
134C2×C17—41no—
135C2×C17#182011yescharacteristic
136C2×#41C22×D1711yescharacteristic
137C2×M16—171no—
138C2×M16—342no—
139C2×M16C4×D1711yescharacteristic
140C22×C8—171no—
141C22×C8C4×D1711yescharacteristic
142C4×Q8—341no—
143C4×D4—342no—
144C4×D4—341no—
145#117—22no—
146#117—171no—
147#117—172no—
148C2×C42—171no—
149C2×C42—172no—
150C4⋊C8—342no—
151C4⋊C8—342no—
152#101—342no—
153#100—342no—
154#99—22no—
155#99—22no—
156C22⋊C8—22no—
157C22⋊C8—342no—
158#97—341no—
159C4×C8—341no—
160C4×C8—341no—
161C4×C8—341no—
162C17#1286411yescharacteristic, complemented, Sylow
163#41—41no—
164#41—41no—
165C2×Q8#6325611yescharacteristic
166C2×D4#6325212yes—
167C2×D4#6325411yescharacteristic
168C22×C4—341no—
169C22×C4—342no—
170C22×C4C2×C4×D1711yescharacteristic
171C22×C4D4×D1712yes—
172M16—41no—
173M16—681no—
174M16—682no—
175M16—682no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—341no—
180C2×C8—341no—
181C2×C8—681no—
182C2×C8—682no—
183C4⋊C4—42no—
184C4⋊C4—341no—
185C4⋊C4—342no—
186C22⋊C4—42no—
187C22⋊C4—341no—
188C22⋊C4—342no—
189C42—42no—
190C42—171no—
191C42—171no—
192C42—172no—
193C42—341no—
194C42—342no—
195C23#1058755811yescharacteristic
196C23#1058756912yes—
197Q8—41no—
198D4—42no—
199D4—41no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—82no—
205C2×C4—341no—
206C2×C4—341no—
207C2×C4—342no—
208C2×C4—682no—
209C2×C4—684no—
210C2×C4C22⋊C4×D1711yescharacteristic
211C2×C4#1058757811yescharacteristic
212C2×C4#1058758012yes—
213C8—41no—
214C8—41no—
215C8—681no—
216C8—681no—
217C8—1362no—
218C22—22no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22#1176267111yescharacteristic
223C4—22no—
224C4—21no—
225C4—162no—
226C4—342no—
227C4—681no—
228C4—682no—
229C4#1176278111yescharacteristic
230C4#1176278111yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#1452886011yescharacteristic
subgroupG/H
centerC4#11762781
commutatorC22×C4×C17C22×C4
FrattiniC2×#41C22×D17
FittingC17×#1391C2
socleC2×C17#1820
prime radical, cosocleC2×#41×C17C23
hypercenter#1391D17
hypocenterC17#12864
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
9. #11762837C4semidirect
10. #11762835C4semidirect
13. M16×Dic17C4semidirect
14. C2×C8×Dic17C4semidirect
162. C17#12864semidirect

Maximal subgroups

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

subgroupindex
#145374242
#145288852
#145269122
#145269072
C17×#13912
#1286417