Finite Group Databasebeta

← #16931731

Subgroups — #16931731

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#14533773C211yesmaximal, characteristic
3#14528777C211yesmaximal, characteristic
4#14527456C211yesmaximal, characteristic
5#14526686C211yesmaximal, characteristic
6#14526685C211yesmaximal, characteristic
7C17×#1546C211yesmaximal, characteristic
8C2×#10587630—21no—
9C2×C4×Dic34C2211yescharacteristic
10#11762939—22no—
11#11762936—22no—
12#11762863—22no—
13#11762862C2211yescharacteristic
14#11762833C2211yescharacteristic
15C2×#10587495—21no—
16#11762740C414yes—
17#11762716—21no—
18#11762705—22no—
19C4×C4⋊Dic17C2211yescharacteristic
20#11762577—21no—
21#11762555—22no—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C17×#431C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C17×#332—22no—
26C22×Q8×C17—21no—
27C22×Dic34D411yescharacteristic
28C2×M16×C17—21no—
29C17×C4⋊Q8—21no—
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8—21no—
32C17×C4⋊Q8—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C412yes—
35C2×C42×C17C2311yescharacteristic
36#10587630—42no—
37#10587630—42no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×#63225—24no—
41C2×#63225C2×C412yes—
42C2×C4×Dic17—21no—
43C2×C4×Dic17D412yes—
44C2×#63223—21no—
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—
54#10587502—24no—
55#10587502C2×C412yes—
56C22⋊#3622—21no—
57#10587495—42no—
58#10587486—24no—
59#10587485—24no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4⋊#3622D412yes—
63C4×#3622—21no—
64C2×Q8×C17—41no—
65C2×Q8×C17—41no—
66C2×Q8×C17—41no—
67C2×Q8×C17—41no—
68C2×Q8×C17—41no—
69C2×Q8×C17—41no—
70C22×C4×C17—22no—
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×Dic17—22no—
75C22×Dic17C22⋊C412yes—
76C22×Dic17C22⋊C412yes—
77C2×Dic34—21no—
78C2×Dic34—21no—
79C2×Dic34—21no—
80C2×Dic34—21no—
81C2×Dic34—21no—
82C2×Dic34—21no—
83M16×C17—41no—
84M16×C17—41no—
85C2×C8×C17—22no—
86C2×C8×C17—41no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C42×C17C2×D411yescharacteristic
92C42×C17C2×D411yescharacteristic
93C42×C17#4111yescharacteristic
94C42×C17#4111yescharacteristic
95C4⋊Dic17—24no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98#63225—22no—
99#63225—22no—
100#63225—48no—
101C4×Dic17—22no—
102C4×Dic17—22no—
103C4×Dic17—41no—
104C4×Dic17—41no—
105#63223—41no—
106#63223—41no—
107C2×#3622—22no—
108C2×#3622—41no—
109#9402—171nomaximal, Sylow
110C23×C17C2×C22⋊C411yescharacteristic
111Q8×C17—81no—
112Q8×C17—81no—
113Q8×C17—81no—
114Q8×C17—81no—
115C2×C4×C17—22no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17C4×D411yescharacteristic
121C2×C4×C17C4×D411yescharacteristic
122C2×C4×C17#12011yescharacteristic
123C2×C4×C17#12111yescharacteristic
124C2×C4×C17#12111yescharacteristic
125C2×C4×C17#12311yescharacteristic
126C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C17#13711yescharacteristic
130C2×Dic17—24no—
131C2×Dic17—22no—
132C2×Dic17—22no—
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138Dic34—41no—
139Dic34—41no—
140Dic34—41no—
141Dic34—41no—
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#1546D1711yescharacteristic
151#1303—171no—
152#1303—171no—
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—82no—
162C4×C17#45612yes—
163C4×C17#46811yescharacteristic
164C4×C17#46811yescharacteristic
165C4×C17#46911yescharacteristic
166C4×C17#46911yescharacteristic
167Dic17—44no—
168Dic17—42no—
169Dic17—42no—
170C2×C4⋊Q8—341no—
171C2×C4⋊Q8C2×D1711yescharacteristic
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⋊C4—171no—
185#343—341no—
186#332—22no—
187#332—342no—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17#188611yescharacteristic
191C2×C17#189511yescharacteristic
192C2×C17#191511yescharacteristic
193C22×Q8—21no—
194C22×Q8—171no—
195C2×M16—21no—
196C2×M16—341no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
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—341no—
211C2×C4⋊C4—344no—
212C2×C4⋊C4C4×D1712yes—
213C2×C42—172no—
214C2×C42—341no—
215C2×C42C22×D1711yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—172no—
219C4⋊C8—344no—
220C4⋊C8—341no—
221C4⋊C8—341no—
222C4⋊C8C22⋊D1712yes—
223#101—682no—
224C22⋊C8—21no—
225C22⋊C8—341no—
226#97—344no—
227C4×C8—21no—
228C4×C8—341no—
229#95—172no—
230#95—344no—
231C17#940211yescharacteristic, complemented, Sylow
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—341no—
239C2×Q8—341no—
240C2×Q8—341no—
241C2×Q8—341no—
242C2×Q8—341no—
243C2×Q8—341no—
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—22no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—344no—
264C4⋊C4—342no—
265C4⋊C4—341no—
266C4⋊C4—342no—
267C4⋊C4—341no—
268C4⋊C4—688no—
269C42—342no—
270C42—342no—
271C42—681no—
272C42—681no—
273C42#6325211yescharacteristic
274C42D4×D1711yescharacteristic
275C42#6325411yescharacteristic
276C42C2×C22⋊D1711yescharacteristic
277C23C22⋊C4×D1711yescharacteristic
278Q8—81no—
279Q8—81no—
280Q8—81no—
281Q8—81no—
282Q8—681no—
283Q8—681no—
284Q8—681no—
285Q8—681no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—344no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—342no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4#1058756711yescharacteristic
300C2×C4#1058756911yescharacteristic
301C2×C4#1058757011yescharacteristic
302C2×C4#1058759711yescharacteristic
303C2×C4#1058759711yescharacteristic
304C2×C4#1058760011yescharacteristic
305C2×C4#1058760111yescharacteristic
306C2×C4C4×C22⋊D1711yescharacteristic
307C2×C4#1058762311yescharacteristic
308C2×C4#1058762411yescharacteristic
309C8—42no—
310C8—81no—
311C8—682no—
312C8—1361no—
313C22—21no—
314C22—21no—
315C22#1176268511yescharacteristic
316C22#1176280711yescharacteristic
317C22#1176281211yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—82no—
322C4—684no—
323C4—682no—
324C4—682no—
325C4#1176299012yes—
326C4#1176305211yescharacteristic
327C4#1176305411yescharacteristic
328C4#1176305911yescharacteristic
329C4#1176306111yescharacteristic
330C2—21no—
331C2—21no—
332C2#1452917411yescharacteristic
333C2#1452921911yescharacteristic
334C2#1452933311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1546C2
socleC22×C17#386
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1546D17
hypocenterC17#9402
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

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
#145337732
#145287772
#145274562
#145266862
#145266852
C17×#15462
#940217