Finite Group Databasebeta

← #17951920

Subgroups — #17951920

Order 4352.

412 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 118 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#14544077C212yesmaximal
2#14543997C211yesmaximal, characteristic, complemented
3C2×#11763650C211yesmaximal, characteristic, complemented
4#14543617C212yesmaximal, complemented
5#14537427C211yesmaximal, characteristic
6#11763825C2212yes—
7C2×C4×#3624C2212yes—
8#11763659C414yes—
9#11763657C414yes—
10#11763651—22no—
11#11763650—22no—
12#11763277C2211yescharacteristic
13#11762869—22no—
14#11762861—22no—
15#11762836C411yescharacteristic
16#11762777C411yescharacteristic
17#11762775C411yescharacteristic
18M16×Dic17C2211yescharacteristic
19C2×C8×Dic17C2211yescharacteristic
20C17×#424C411yescharacteristic
21C2×#63252C2×C411yescharacteristic
22C22×#3624—22no—
23C2×M16×C17C2×C411yescharacteristic
24C22×C8×C17C2×C411yescharacteristic
25C17×#117C2×C411yescharacteristic
26C2×C4×Dic17C2311yescharacteristic
27C2×#63223C2×C411yescharacteristic
28C22×#3622C2×C411yescharacteristic
29#10587579C2×C412yes—
30#10587573C2×C412yes—
31#10587567—22no—
32#10587558—21no—
33C42×D17—21no—
34C22⋊#3624—22no—
35C4⋊#3624C2×C414yes—
36C4×#3624—22no—
37C4×#3624—22no—
38C4×#3624C2×C414yes—
39C17×C4⋊C8—22no—
40#10587491—22no—
41#10587491—22no—
42#10587486—21no—
43C8×Dic17—21no—
44C8×Dic17D412yes—
45C8×Dic17D412yes—
46C4⋊#3622—22no—
47C22×C4×C17C22×C411yescharacteristic
48C2×C22⋊D17—21no—
49C22×Dic17#4112yes—
50#63252—22no—
51#63252—22no—
52C2×D68C4211yescharacteristic
53C2×C4×D17—21no—
54C2×C4×D17—22no—
55C2×Dic34C4211yescharacteristic
56C2×#3624—24no—
57C2×#3624—24no—
58C2×#3624—44no—
59M16×C17—22no—
60C2×C8×C17—22no—
61C2×C8×C17—21no—
62C2×C8×C17C22⋊C412yes—
63C4⋊C4×C17C4212yes—
64C22⋊C4×C17—21no—
65C42×C17—21no—
66#63227—22no—
67#63225—22no—
68C4×Dic17—22no—
69C4×Dic17C22×C412yes—
70C4×Dic17C2×D411yescharacteristic
71C4×Dic17C2×D411yescharacteristic
72#63223—22no—
73C2×#3622—22no—
74C2×#3622—21no—
75C2×#3622C22⋊C412yes—
76#8188—171nomaximal, Sylow
77C23×C17#11711yescharacteristic
78C22×D17—21no—
79C2×C4×C17—22no—
80C2×C4×C17C2×C4211yescharacteristic
81C2×C4×C17C2×C22⋊C411yescharacteristic
82C2×C4×C17#13112yes—
83C2×C4×C17#13112yes—
84C22⋊D17—42no—
85C2×Dic17—24no—
86C2×Dic17—21no—
87C2×Dic17C4×D412yes—
88C2×Dic17C4×D412yes—
89D68—22no—
90C4×D17—41no—
91C4×D17—41no—
92C4×D17—44no—
93Dic34—22no—
94#3624—44no—
95#3624—44no—
96C8×C17—22no—
97C8×C17—41no—
98#3622—22no—
99#3622—41no—
100#2922—171no—
101#1476—172no—
102#1452—171no—
103C2×#322—171no—
104#1278—172no—
105C22×C17—22no—
106C22×C17C4×C22⋊C411yescharacteristic
107C22×C17#43312yes—
108C2×D17—41no—
109C2×D17—42no—
110C4×C17—42no—
111C4×C17#40512yes—
112C4×C17#40811yescharacteristic
113C4×C17#40811yescharacteristic
114Dic17—22no—
115Dic17—41no—
116Dic17#44414yes—
117C4×#41—171no—
118#433—342no—
119#431—172no—
120#431—342no—
121#424—171no—
122#424#59511yescharacteristic
123#408—171no—
124#408—171no—
125C4×M16—171no—
126C2×C4×C8—171no—
127C2×C4×C8—172no—
128Q8⋊C8—174no—
129D4⋊C8—174no—
130#322—342no—
131#322—342no—
132C2×C17—22no—
133C2×C17#175011yescharacteristic
134C2×C17#176512yes—
135D17—81no—
136C2×#41—171no—
137C2×M16—171no—
138C2×M16C2×#59511yescharacteristic
139C22×C8—171no—
140C22×C8—342no—
141C22×C8C2×#59511yescharacteristic
142C4×Q8—172no—
143C4×D4—172no—
144C4×D4—342no—
145#117—341no—
146#117C2×#59511yescharacteristic
147C2×C42—171no—
148C2×C42—341no—
149C4⋊C8—22no—
150C4⋊C8—174no—
151C4⋊C8—342no—
152C22⋊C8—342no—
153C22⋊C8—342no—
154C22⋊C8—342no—
155#97—341no—
156C4×C8—172no—
157C4×C8—172no—
158C4×C8—174no—
159C4×C8—341no—
160C4×C8—342no—
161C4×C8—342no—
162C17#818811yescharacteristic, complemented, Sylow
163#41—342no—
164#41—342no—
165C2×Q8—171no—
166C2×D4—171no—
167C2×D4—341no—
168C22×C4—172no—
169C22×C4—342no—
170C22×C4—341no—
171C22×C4C22×#59511yescharacteristic
172M16—22no—
173M16—342no—
174C2×C8—22no—
175C2×C8—21no—
176C2×C8—172no—
177C2×C8—342no—
178C2×C8—344no—
179C2×C8—344no—
180C2×C8—341no—
181C2×C8—684no—
182C2×C8C22⋊#59512yes—
183C4⋊C4—342no—
184C4⋊C4C4×#59512yes—
185C22⋊C4—21no—
186C22⋊C4—342no—
187C42—21no—
188C42—172no—
189C42—171no—
190C42—171no—
191C42—342no—
192C23—341no—
193C23#1058767711yescharacteristic
194Q8—342no—
195D4—342no—
196D4—682no—
197C2×C4—22no—
198C2×C4—172no—
199C2×C4—172no—
200C2×C4—344no—
201C2×C4—341no—
202C2×C4—684no—
203C2×C4—681no—
204C2×C4—681no—
205C2×C4C2×C4×#59511yescharacteristic
206C2×C4#1058767812yes—
207C2×C4#1058768012yes—
208C2×C4C2×C22⋊#59511yescharacteristic
209C8—22no—
210C8—41no—
211C8—342no—
212C8—684no—
213C8—684no—
214C8—681no—
215C22—22no—
216C22—681no—
217C22—682no—
218C22C4×C22⋊#59511yescharacteristic
219C22#1176383412yes—
220C4—42no—
221C4—174no—
222C4—342no—
223C4—681no—
224C4#1176377412yes—
225C4#1176378211yescharacteristic
226C4#1176378411yescharacteristic
227C2—22no—
228C2—1361no—
229C2#1454451311yescharacteristic
230C2#1454456212yes—
subgroupG/H
centerC2×C4C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#424C4
socleC22×C17C4×C22⋊C4
prime radicalC2×#63252C2×C4
cosocleC2×C4×Dic17C23
hypercenter#424#595
hypocenterC17#8188
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #14543997C2semidirect
3. C2×#11763650C2semidirect
4. #14543617C2semidirect
162. C17#8188semidirect

Maximal subgroups

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

subgroupindex
#145440772
#145439972
C2×#117636502
#145436172
#145374272
#818817