Finite Group Databasebeta

← #16953794

Subgroups — #16953794

Order 4352.

444 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 77 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×#11762608C212yesmaximal
2#14529178C212yesmaximal
3#14528998C211yesmaximal, characteristic
4#14526911C211yesmaximal, characteristic
5C17×#2734C211yesmaximal, characteristic
6#11763287C2211yescharacteristic
7#11762849—21no—
8#11762818—22no—
9C2×#10587507C2212yes—
10C2×#10587507C2211yescharacteristic
11#11762608—24no—
12#11762569—22no—
13#11762569—21no—
14C2×C17×C4⋊Q8C2212yes—
15C17×#398—21no—
16C17×#395—22no—
17C17×#391C2211yescharacteristic
18C17×#389—21no—
19C17×#387—22no—
20C22×Q8×C17D412yes—
21C22×Q8×C17C2311yescharacteristic
22C17×C4⋊Q8—22no—
23C17×C4⋊Q8—22no—
24C17×C4⋊Q8—22no—
25C17×C4⋊Q8—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C42×C17D412yes—
33C2×C42×C17D411yescharacteristic
34Q8×Dic17—41no—
35Q8×Dic17—41no—
36#10587612—22no—
37#10587612—21no—
38C2×#63223—21no—
39C2×#63223—22no—
40C17×#95—21no—
41C17×#95—22no—
42C17×#95—22no—
43C17×#95—22no—
44#10587507—22no—
45#10587507—21no—
46#10587507—22no—
47#10587507D412yes—
48#10587506—42no—
49#10587506—42no—
50#10587479—41no—
51#10587478—41no—
52C2×Q8×C17—22no—
53C2×Q8×C17—22no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17—42no—
57C2×Q8×C17—42no—
58C2×Q8×C17C2×D411yescharacteristic
59C2×Q8×C17#4111yescharacteristic
60C2×Q8×C17#4112yes—
61C22×C4×C17—21no—
62C22×C4×C17—22no—
63C22×C4×C17—22no—
64C22×C4×C17—21no—
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C4⋊C4×C17—24no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—42no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—81no—
74C4⋊C4×C17—81no—
75C42×C17—22no—
76C42×C17—22no—
77C42×C17—41no—
78C22⋊Dic17—41no—
79C4⋊Dic17—21no—
80C4⋊Dic17—21no—
81C4⋊Dic17—22no—
82C4⋊Dic17—22no—
83C4×Dic17—41no—
84C4×Dic17—42no—
85#63223—22no—
86#63223—41no—
87#63223—42no—
88#63223—42no—
89C2×#3622—41no—
90C2×#3622—42no—
91#13428—171nomaximal, Sylow
92C23×C17#12011yescharacteristic
93Q8×C17—41no—
94Q8×C17—41no—
95Q8×C17—42no—
96Q8×C17—44no—
97Q8×C17—82no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—24no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17#12111yescharacteristic
114C2×C4×C17#12112yes—
115C2×C4×C17#12312yes—
116C2×C4×C17#12411yescharacteristic
117C2×Dic17—41no—
118C2×Dic17—41no—
119#3622—42no—
120#3622—82no—
121#2734D1711yescharacteristic
122C2×#356—172no—
123#1886—172no—
124#1857—171no—
125#1394—171no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17#39411yescharacteristic
129C22×C17B(4,𝔽2)11yescharacteristic
130C22×C17#45811yescharacteristic
131C4×C17—21no—
132C4×C17—22no—
133C4×C17—22no—
134C4×C17—21no—
135C4×C17—44no—
136C4×C17—42no—
137C4×C17—81no—
138C4×C17—82no—
139Dic17—81no—
140Dic17—81no—
141C2×C4⋊Q8C2×D1712yes—
142#519—171no—
143#428—341no—
144#419—342no—
145C2×#101—171no—
146C2×#101—172no—
147#398—21no—
148#395—22no—
149#391C2×D1711yescharacteristic
150#389—21no—
151#387—22no—
152#356—344no—
153#339—341no—
154#339—342no—
155C2×C17—21no—
156C2×C17—21no—
157C2×C17#204411yescharacteristic
158C2×C17#220311yescharacteristic
159C2×C17#220511yescharacteristic
160C22×Q8C22⋊D1712yes—
161C22×Q8C22×D1711yescharacteristic
162C2×M16—341no—
163C2×M16—342no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4×Q8—681no—
169C4×Q8—681no—
170#117—342no—
171#117—341no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22⋊D1711yescharacteristic
178C2×C42C22⋊D1712yes—
179C2×C42C22⋊D1711yescharacteristic
180#107—681no—
181#106—681no—
182#103—682no—
183#103—682no—
184#101—172no—
185#101—341no—
186#101—342no—
187#101—342no—
188#95—21no—
189#95—22no—
190#95—22no—
191#95—22no—
192C17#1342811yescharacteristic, complemented, Sylow
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8D4×D1711yescharacteristic
200C2×Q8#6325411yescharacteristic
201C2×Q8#6325412yes—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4C2×C22⋊D1712yes—
207C22×C4C2×C22⋊D1711yescharacteristic
208M16—342no—
209M16—681no—
210M16—682no—
211M16—682no—
212C2×C8—681no—
213C2×C8—682no—
214C4⋊C4—24no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—342no—
223C4⋊C4—342no—
224C4⋊C4—341no—
225C4⋊C4—341no—
226C22⋊C4—681no—
227C42—22no—
228C42—22no—
229C42—41no—
230C42—681no—
231C42—682no—
232C23#1058763911yescharacteristic
233Q8—41no—
234Q8—41no—
235Q8—42no—
236Q8—44no—
237Q8—82no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—681no—
254C2×C4—681no—
255C2×C4#1058762112yes—
256C2×C4#1058762211yescharacteristic
257C2×C4#1058762411yescharacteristic
258C2×C4#1058762512yes—
259C8—682no—
260C8—1362no—
261C22—21no—
262C22—21no—
263C22#1176272311yescharacteristic
264C22#1176299611yescharacteristic
265C22#1176299911yescharacteristic
266C4—21no—
267C4—22no—
268C4—22no—
269C4—21no—
270C4—44no—
271C4—42no—
272C4—82no—
273C4—81no—
274C4—1361no—
275C4—1361no—
276C2—21no—
277C2—21no—
278C2#1453003211yescharacteristic
279C2#1453080511yescharacteristic
280C2#1453081411yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC22×Q8×C17C23
FrattiniC22×Q8C22×D17
FittingC17×#2734C2
socleC22×C17B(4,𝔽2)
prime radicalC23×C17#120
hypercenter#2734D17
hypocenterC17#13428
LCSG ▷ C22×Q8×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2734

Complements

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

NH ≅ G/Ntype
192. C17#13428semidirect

Maximal subgroups

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

subgroupindex
C2×#117626082
#145291782
#145289982
#145269112
C17×#27342
#1342817