Finite Group Databasebeta

← #16931733

Subgroups — #16931733

Order 4352.

486 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 109 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#14537867C211yesmaximal, characteristic
2#14533772C211yesmaximal, characteristic
3#14528777C211yesmaximal, characteristic
4#14527457C211yesmaximal, characteristic
5#14527455C211yesmaximal, characteristic
6#14526687C211yesmaximal, characteristic
7C17×#1303C211yesmaximal, characteristic
8C2×#10587555—21no—
9C2×#10587555C2211yescharacteristic
10C2×C4×Dic34C2211yescharacteristic
11#11762939—22no—
12#11762936—22no—
13#11762863—22no—
14#11762862C2211yescharacteristic
15#11762833C2211yescharacteristic
16C2×#10587495—21no—
17#11762741C414yes—
18#11762715—21no—
19#11762708—22no—
20#11762577—21no—
21#11762556—22no—
22#11762554—22no—
23C17×#431C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C4×C4⋊C4×C17C2211yescharacteristic
26C22×Dic34—21no—
27C22×Dic34D411yescharacteristic
28C2×M16×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C42×C17—21no—
31C2×C42×C17C2311yescharacteristic
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—24no—
34C2×C4⋊Dic17C2×C412yes—
35C2×C4⋊Dic17D411yescharacteristic
36C2×#63225C2×C412yes—
37C2×C4×Dic17D412yes—
38C2×#63223—21no—
39#10587555—21no—
40#10587555—21no—
41#10587555—21no—
42#10587555—21no—
43#10587555—42no—
44#10587555—42no—
45C4×Dic34—21no—
46C4×Dic34—21no—
47C4×Dic34—21no—
48C4×Dic34—21no—
49C17×C4⋊C8—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8D412yes—
52C17×C22⋊C8—21no—
53C4×C8×C17—21no—
54C17×#95C2×C412yes—
55#10587502—24no—
56C22⋊#3622—21no—
57#10587495—42no—
58#10587486—24no—
59#10587485—24no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4⋊#3622D412yes—
63C4×#3622—21no—
64C22×C4×C17—22no—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—22no—
69C22×Dic17C22⋊C412yes—
70C22×Dic17C22⋊C412yes—
71C2×Dic34—21no—
72C2×Dic34—21no—
73C2×Dic34—21no—
74C2×Dic34—21no—
75C2×Dic34—21no—
76C2×Dic34—21no—
77C2×Dic34—41no—
78C2×Dic34—41no—
79C2×Dic34—41no—
80C2×Dic34—41no—
81C2×Dic34—41no—
82C2×Dic34—41no—
83M16×C17—41no—
84M16×C17—41no—
85C2×C8×C17—22no—
86C2×C8×C17—41no—
87C4⋊C4×C17—24no—
88C42×C17—41no—
89C42×C17—41no—
90C42×C17C2×D411yescharacteristic
91C42×C17C2×D411yescharacteristic
92C42×C17#4111yescharacteristic
93C42×C17#4111yescharacteristic
94C4⋊Dic17—21no—
95C4⋊Dic17—21no—
96C4⋊Dic17—22no—
97C4⋊Dic17—22no—
98C4⋊Dic17—48no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101#63225—22no—
102#63225—22no—
103C4×Dic17—22no—
104C4×Dic17—22no—
105#63223—41no—
106#63223—41no—
107C2×#3622—22no—
108C2×#3622—41no—
109#9402—171nomaximal, Sylow
110C23×C17C2×C22⋊C411yescharacteristic
111C2×C4×C17—22no—
112C2×C4×C17—24no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17C4×D411yescharacteristic
117C2×C4×C17C4×D411yescharacteristic
118C2×C4×C17#12011yescharacteristic
119C2×C4×C17#12111yescharacteristic
120C2×C4×C17#12111yescharacteristic
121C2×C4×C17#12311yescharacteristic
122C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C17#13711yescharacteristic
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—22no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134Dic34—41no—
135Dic34—41no—
136Dic34—41no—
137Dic34—41no—
138Dic34—81no—
139Dic34—81no—
140Dic34—81no—
141Dic34—81no—
142C8×C17—42no—
143C8×C17—81no—
144#3622—42no—
145#3622—81no—
146#2983—171no—
147#2546—171no—
148#1801—171no—
149#1546—171no—
150#1546—171no—
151#1303—171no—
152#1303D1711yescharacteristic
153C22×C17—21no—
154C22×C17—21no—
155C22×C17#38611yescharacteristic
156C22×C17#41711yescharacteristic
157C22×C17#41811yescharacteristic
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—44no—
162C4×C17#45612yes—
163C4×C17#46811yescharacteristic
164C4×C17#46811yescharacteristic
165C4×C17#46911yescharacteristic
166C4×C17#46911yescharacteristic
167Dic17—42no—
168Dic17—42no—
169Dic17—82no—
170C2×C4⋊Q8—171no—
171C2×C4⋊Q8—341no—
172C2×C4×Q8—171no—
173#446—342no—
174#446—342no—
175#432—342no—
176#431—171no—
177#431C2×D1711yescharacteristic
178#423—171no—
179#423C2×D1711yescharacteristic
180C2×#101—341no—
181#398—174no—
182#391—341no—
183#389—342no—
184C4×C4⋊C4C2×D1711yescharacteristic
185#343—341no—
186#332—342no—
187#332—342no—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17#188611yescharacteristic
191C2×C17#189511yescharacteristic
192C2×C17#191511yescharacteristic
193C22×Q8—171no—
194C22×Q8—341no—
195C2×M16—21no—
196C2×M16—341no—
197C4⋊Q8—341no—
198C4⋊Q8—341no—
199C4⋊Q8—341no—
200C4⋊Q8—341no—
201C4⋊Q8—682no—
202C4⋊Q8—682no—
203C4×Q8—341no—
204C4×Q8—341no—
205C4×Q8—341no—
206C4×Q8—341no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—172no—
209C2×C4⋊C4—171no—
210C2×C4⋊C4—172no—
211C2×C4⋊C4—344no—
212C2×C4⋊C4—341no—
213C2×C42—21no—
214C2×C42—172no—
215C2×C42C22×D1711yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—172no—
219C4⋊C8—344no—
220C4⋊C8—341no—
221C4⋊C8—341no—
222C4⋊C8D6812yes—
223#101—682no—
224C22⋊C8—21no—
225C22⋊C8—341no—
226#97—344no—
227C4×C8—21no—
228C4×C8—341no—
229#95—344no—
230#95C4×D1712yes—
231C17#940211yescharacteristic, complemented, Sylow
232C2×Q8—341no—
233C2×Q8—341no—
234C2×Q8—341no—
235C2×Q8—341no—
236C2×Q8—341no—
237C2×Q8—341no—
238C2×Q8—681no—
239C2×Q8—681no—
240C2×Q8—681no—
241C2×Q8—681no—
242C2×Q8—681no—
243C2×Q8—681no—
244C22×C4—22no—
245C22×C4—172no—
246C22×C4—172no—
247C22×C4—342no—
248C22×C4C2×C4×D1711yescharacteristic
249C22×C4D4×D1711yescharacteristic
250C22×C4D4×D1711yescharacteristic
251M16—41no—
252M16—41no—
253M16—681no—
254M16—681no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—342no—
258C2×C8—681no—
259C4⋊C4—24no—
260C4⋊C4—342no—
261C4⋊C4—341no—
262C4⋊C4—342no—
263C4⋊C4—341no—
264C4⋊C4—342no—
265C4⋊C4—342no—
266C4⋊C4—688no—
267C4⋊C4—681no—
268C4⋊C4—681no—
269C42—41no—
270C42—41no—
271C42—342no—
272C42—342no—
273C42C2×D6811yescharacteristic
274C42#6325211yescharacteristic
275C42D4×D1711yescharacteristic
276C42#6325611yescharacteristic
277C23C22⋊C4×D1711yescharacteristic
278Q8—681no—
279Q8—681no—
280Q8—681no—
281Q8—681no—
282Q8—1361no—
283Q8—1361no—
284Q8—1361no—
285Q8—1361no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—342no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4C4×D6811yescharacteristic
300C2×C4#1058756811yescharacteristic
301C2×C4#1058757011yescharacteristic
302C2×C4#1058757911yescharacteristic
303C2×C4#1058758011yescharacteristic
304C2×C4#1058758111yescharacteristic
305C2×C4#1058759711yescharacteristic
306C2×C4#1058760011yescharacteristic
307C2×C4#1058760011yescharacteristic
308C2×C4#1058760411yescharacteristic
309C8—42no—
310C8—81no—
311C8—682no—
312C8—1361no—
313C22—21no—
314C22—21no—
315C22#1176268711yescharacteristic
316C22#1176280911yescharacteristic
317C22#1176281211yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—44no—
322C4—682no—
323C4—682no—
324C4—1362no—
325C4#1176299312yes—
326C4#1176305611yescharacteristic
327C4#1176305811yescharacteristic
328C4#1176306111yescharacteristic
329C4#1176306311yescharacteristic
330C2—21no—
331C2—21no—
332C2#1452917611yescharacteristic
333C2#1452922111yescharacteristic
334C2#1452933511yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1303C2
socleC22×C17#386
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1303D17
hypocenterC17#9402
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
231. C17#9402semidirect

Maximal subgroups

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

subgroupindex
#145378672
#145337722
#145287772
#145274572
#145274552
#145266872
C17×#13032
#940217