Finite Group Databasebeta

← #17949477

Subgroups — #17949477

Order 4352.

276 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 50 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
1C2×#11763719C211yesmaximal, characteristic
2C2×#11763669C211yesmaximal, characteristic
3#14529177C211yesmaximal, characteristic
4C2×#10587544—21no—
5C2×#10587545—21no—
6#11763719—21no—
7#11763719C412yescomplemented
8#11763669—21no—
9#11763669C412yescomplemented
10C2×#10587630C2211yescharacteristic
11#11762817—21no—
12C2×#10587507C411yescharacteristic
13#11762568—21no—
14C17×#519C411yescharacteristic
15C22×Q8×C17C2×C411yescharacteristic
16C22×Dic34D411yescharacteristic
17C2×#63247—41no—
18C22×#3624—41no—
19C4×Q8×C17—42no—
20C17×#117—21no—
21C17×#117—41no—
22#10587630—41no—
23#10587630C2×C411yescharacteristic
24#10587630C2×C411yescharacteristic
25#10587630D411yescharacteristic
26#10587630Q811yescharacteristic
27C2×#63225—41no—
28C2×C4×Dic17D411yescharacteristic
29C2×#63223—21no—
30#10587545—22no—
31#10587545—41no—
32#10587544—22no—
33#10587544—41no—
34#10587507—41no—
35#10587481—42no—
36C2×Q8×C17—21no—
37C2×Q8×C17—41no—
38C2×Q8×C17—41no—
39C2×Q8×C17C4211yescharacteristic
40C2×Q8×C17C4⋊C411yescharacteristic
41C22×C4×C17—21no—
42C22×C4×C17C22⋊C411yescharacteristic
43C22×Dic17—21no—
44C22×Dic17—21no—
45C2×Dic34—21no—
46C2×Dic34—41no—
47C2×Dic34—41no—
48C2×Dic34C22⋊C411yescharacteristic
49C2×Dic34C4⋊C411yescharacteristic
50#63247—42no—
51#63247—81no—
52C2×#3624—42no—
53C2×#3624—82no—
54C2×#3624—81no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—42no—
57C22⋊C4×C17—41no—
58C42×C17—41no—
59C42×C17—42no—
60#63225—41no—
61#63225—41no—
62#63225—41no—
63#63225—41no—
64C4×Dic17—21no—
65C4×Dic17C22⋊C411yescharacteristic
66C4×Dic17C4⋊C411yescharacteristic
67#63223—41no—
68#63223—41no—
69C2×#3622—41no—
70#7491—171nomaximal, Sylow
71C23×C17#9911yescharacteristic
72Q8×C17—41no—
73Q8×C17—41no—
74Q8×C17—42no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—42no—
82C2×C4×C17#9511yescharacteristic
83C2×C4×C17#9911yescharacteristic
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—21no—
90C2×Dic17—41no—
91C2×Dic17—41no—
92Dic34—41no—
93Dic34—41no—
94Dic34—81no—
95#3624—82no—
96#3624—82no—
97#3622—81no—
98C2×#356—171no—
99#1886—171no—
100C2×#333—171no—
101C22×C17—22no—
102C22×C17#34211yescharacteristic
103C22×C17AΣL(1,𝔽16)11yescharacteristic
104C22×C17#35211yescharacteristic
105C4×C17—21no—
106C4×C17—21no—
107C4×C17—41no—
108C4×C17—41no—
109C4×C17—82no—
110Dic17—21no—
111Dic17—22no—
112Dic17—21no—
113Dic17—41no—
114Dic17—41no—
115C2×C4⋊Q8—171no—
116#519#59511yescharacteristic
117C2×C4⋊C8—341no—
118#419—341no—
119C2×#101—171no—
120C2×#101—341no—
121#356—172no—
122#356—341no—
123#339—341no—
124#333—172no—
125#333—341no—
126C2×C17—21no—
127C2×C17—21no—
128C2×C17#139011yescharacteristic
129C2×C17#140512yes—
130C22×Q8—171no—
131C22×Q8C2×#59511yescharacteristic
132C2×M16—341no—
133C2×M16—681no—
134C22×C8—681no—
135C4⋊Q8—171no—
136C4⋊Q8—171no—
137C4⋊Q8—171no—
138C4⋊Q8—171no—
139C4⋊Q8—681no—
140C4×Q8—42no—
141#117—21no—
142#117—41no—
143C2×C4⋊C4—681no—
144C2×C42—171no—
145C4⋊C8—342no—
146C4⋊C8—681no—
147#103—682no—
148#101—342no—
149#101—681no—
150#101—681no—
151C17#749111yescharacteristic, complemented, Sylow
152C2×Q8—21no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—171no—
156C2×Q8—171no—
157C2×Q8—341no—
158C2×Q8—681no—
159C2×Q8—681no—
160C2×Q8C4×#59511yescharacteristic
161C2×Q8C4⋊#59511yescharacteristic
162C22×C4—21no—
163C22×C4—341no—
164C22×C4—341no—
165C22×C4C22⋊#59511yescharacteristic
166M16—681no—
167M16—682no—
168M16—681no—
169M16—1361no—
170C2×C8—681no—
171C2×C8—682no—
172C2×C8—1362no—
173C2×C8—1361no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—681no—
177C4⋊C4—681no—
178C4⋊C4—681no—
179C4⋊C4—681no—
180C22⋊C4—41no—
181C42—41no—
182C42—42no—
183C42—171no—
184C42—171no—
185C42—341no—
186C23#1058755111yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—42no—
190Q8—681no—
191Q8—681no—
192Q8—1361no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—341no—
203C2×C4—341no—
204C2×C4—341no—
205C2×C4—341no—
206C2×C4—681no—
207C2×C4—681no—
208C2×C4#1058753911yescharacteristic
209C2×C4(C2×#595)⋊C411yescharacteristic
210C8—1361no—
211C8—1362no—
212C8—1362no—
213C22—22no—
214C22#1176368711yescharacteristic
215C22#1176370911yescharacteristic
216C22#1176371111yescharacteristic
217C4—21no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—341no—
223C4—342no—
224C4—341no—
225C4—681no—
226C4—681no—
227C2—21no—
228C2—21no—
229C2#1454386811yescharacteristic
230C2#1454390912yes—
subgroupG/H
centerC22#11763709
commutatorC2×Q8×C17C42
FrattiniC22×Q8C2×#595
FittingC17×#519C4
socleC22×C17AΣL(1,𝔽16)
prime radicalC23×C17#99
cosocleC2×#10587630C22
hypercenter#519#595
hypocenterC17#7491
LCSG ▷ C2×Q8×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C22×Q8 ◁ #519

Complements

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

NH ≅ G/Ntype
7. #11763719C4semidirect
9. #11763669C4semidirect
151. C17#7491semidirect

Maximal subgroups

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

subgroupindex
C2×#117637192
C2×#117636692
#145291772
#749117