Finite Group Databasebeta

← #16929513

Subgroups — #16929513

Order 4352.

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

normal subgroups only

This group has 424 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#14538823C211yesmaximal, characteristic
2#14537724C211yesmaximal, characteristic
3#14527442C211yesmaximal, characteristic
4#14527213C211yesmaximal, characteristic
5#14526681C211yesmaximal, characteristic
6#14526629C211yesmaximal, characteristic
7C17×#1985C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763144C412yes—
10#11763104—21no—
11#11763019—21no—
12#11762965C412yes—
13#11762937—22no—
14#11762937—22no—
15C4×#63231—22no—
16#11762862C2211yescharacteristic
17#11762862C2211yescharacteristic
18#11762835—21no—
19#11762833C2211yescharacteristic
20#11762764—21no—
21#11762557—21no—
22#11762554—21no—
23Q8⋊#3622—22no—
24C2×C4×Q8×C17C2211yescharacteristic
25C2×C17×C4⋊C8C2211yescharacteristic
26C2×C17×#97C2211yescharacteristic
27C17×#336—21no—
28C17×#336—21no—
29C22×Q8×C17C2×C411yescharacteristic
30C22×C8×C17—21no—
31C22×C8×C17D411yescharacteristic
32C22×C8×C17D411yescharacteristic
33C4×Q8×C17—21no—
34C4×Q8×C17—21no—
35C4×Q8×C17—21no—
36C4×Q8×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17C2×C411yescharacteristic
39C2×C42×C17—21no—
40C2×C42×C17C2311yescharacteristic
41C2×#63231—22no—
42#10587612—21no—
43#10587610—21no—
44#10587610C2×C412yes—
45C2×#63223—21no—
46#10587556—21no—
47#10587555D411yescharacteristic
48#10587555D411yescharacteristic
49C4×Dic34—21no—
50C4×Dic34D412yes—
51C17×C4⋊C8—21no—
52C17×C4⋊C8—21no—
53C17×#97—21no—
54C17×#97—21no—
55#10587506—22no—
56C22⋊#3622—21no—
57C22⋊#3622C2×C412yes—
58#10587486—22no—
59#10587485—24no—
60C8×Dic17—22no—
61#10587481—22no—
62#10587478—22no—
63C4⋊#3622—21no—
64C4⋊#3622—21no—
65C4⋊#3622—21no—
66C4⋊#3622D412yes—
67C4×#3622—21no—
68C4×#3622—21no—
69C2×Q8×C17—21no—
70C2×Q8×C17—21no—
71C2×Q8×C17—42no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17C22×C411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17#4111yescharacteristic
77C2×Dic34—21no—
78C2×Dic34C22⋊C412yes—
79C2×C8×C17—21no—
80C2×C8×C17—21no—
81C2×C8×C17—21no—
82C2×C8×C17—21no—
83C2×C8×C17—21no—
84C2×C8×C17—41no—
85C2×C8×C17—41no—
86C2×C8×C17—41no—
87C2×C8×C17Q1612yescomplemented
88C4⋊C4×C17—21no—
89C4⋊C4×C17—21no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C42×C17—41no—
93C42×C17—41no—
94C42×C17C2×D411yescharacteristic
95C42×C17C2×D411yescharacteristic
96C42×C17C2×D411yescharacteristic
97C42×C17#4111yescharacteristic
98C22⋊Dic17—22no—
99#63231—44no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17C22⋊C412yes—
104#63225—22no—
105#63225—22no—
106C4×Dic17—22no—
107#63223—41no—
108#63223—41no—
109C2×#3622—22no—
110C2×#3622—22no—
111C2×#3622—41no—
112#9048—171nomaximal, Sylow
113C23×C17C4×D411yescharacteristic
114Q8×C17—22no—
115Q8×C17—22no—
116Q8×C17—41no—
117Q8×C17—41no—
118C2×C4×C17—22no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17C2×C22⋊C411yescharacteristic
126C2×C4×C17C4×D411yescharacteristic
127C2×C4×C17#12011yescharacteristic
128C2×C4×C17#12111yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17#12311yescharacteristic
131C2×C4×C17C2×Q1611yescharacteristic
132C2×C4×C17#13511yescharacteristic
133C2×C4×C17#13711yescharacteristic
134C2×C4×C17#13711yescharacteristic
135C2×Dic17—22no—
136C2×Dic17—22no—
137Dic34—42no—
138Dic34—81no—
139C8×C17—22no—
140C8×C17—41no—
141C8×C17—81no—
142#3622—42no—
143#3622—42no—
144#3622—81no—
145#3104—171no—
146#2961—171no—
147#1985D1711yescharacteristic
148#1542—171no—
149#1487—171no—
150#1301—171no—
151#1281—171no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17#38611yescharacteristic
155C22×C17C4×Q1611yescharacteristic
156C22×C17#44111yescharacteristic
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—42no—
162C4×C17—81no—
163C4×C17#45111yescharacteristic
164C4×C17#45211yescharacteristic
165C4×C17#46211yescharacteristic
166C4×C17#46411yescharacteristic
167Dic17—42no—
168Dic17—42no—
169#533—171no—
170C2×C4×Q8C2×D1711yescharacteristic
171#484—172no—
172#477—341no—
173#462—341no—
174#451—172no—
175#446—342no—
176#446—342no—
177C4×Q16—342no—
178#431—171no—
179#431—171no—
180#424—341no—
181#423—171no—
182C2×C4⋊C8C2×D1711yescharacteristic
183#405—341no—
184C2×#97C2×D1711yescharacteristic
185#336—21no—
186#336—21no—
187#333—341no—
188#332—341no—
189Q8⋊C8—342no—
190C2×C17—21no—
191C2×C17—21no—
192C2×C17#187811yescharacteristic
193C2×C17#188111yescharacteristic
194C2×C17#189111yescharacteristic
195C22×Q8C4×D1711yescharacteristic
196C2×Q16—342no—
197C2×M16—341no—
198C22×C8—21no—
199C22×C8C22⋊D1711yescharacteristic
200C22×C8C22⋊D1711yescharacteristic
201C4⋊Q8—171no—
202C4⋊Q8—171no—
203#125—341no—
204C22⋊Q8—172no—
205C22⋊Q8—341no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—172no—
211C4×Q8—341no—
212#117—341no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C4×D1711yescharacteristic
215C2×C42—21no—
216C2×C42C22×D1711yescharacteristic
217#107—342no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—172no—
221C4⋊C8—344no—
222C4⋊C8—341no—
223C4⋊C8—341no—
224C4⋊C8—341no—
225#103—342no—
226#103—342no—
227C22⋊C8—172no—
228C22⋊C8—341no—
229#97—21no—
230#97—21no—
231#97—342no—
232C4×C8—342no—
233C4×C8—341no—
234C4×C8—341no—
235C17#904811yescharacteristic, complemented, Sylow
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—42no—
239C2×Q8—172no—
240C2×Q8—341no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4C2×C4×D1711yescharacteristic
244C22×C4#6325211yescharacteristic
245C22×C4C2×C22⋊D1711yescharacteristic
246Q16—684no—
247M16—681no—
248M16—681no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—342no—
258C2×C8—342no—
259C2×C8—681no—
260C2×C8#6323112yescomplemented
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—172no—
266C4⋊C4—342no—
267C4⋊C4—342no—
268C4⋊C4—341no—
269C4⋊C4—341no—
270C4⋊C4—341no—
271C22⋊C4—342no—
272C42—41no—
273C42—41no—
274C42—342no—
275C42D4×D1711yescharacteristic
276C42D4×D1711yescharacteristic
277C42D4×D1711yescharacteristic
278C42#6325411yescharacteristic
279C23C4×C22⋊D1711yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284Q8—682no—
285Q8—1361no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4C22⋊C4×D1711yescharacteristic
296C2×C4#1058756711yescharacteristic
297C2×C4#1058756911yescharacteristic
298C2×C4#1058757011yescharacteristic
299C2×C4#1058762311yescharacteristic
300C2×C4#1058762411yescharacteristic
301C2×C4#1058762811yescharacteristic
302C2×C4C2×#6323111yescharacteristic
303C2×C4#1058763611yescharacteristic
304C2×C4#1058763711yescharacteristic
305C8—22no—
306C8—41no—
307C8—81no—
308C8—682no—
309C8—682no—
310C8—1361no—
311C22—21no—
312C22—21no—
313C22#1176268511yescharacteristic
314C22C4×#6323111yescharacteristic
315C22#1176291411yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—42no—
321C4—81no—
322C4—682no—
323C4—682no—
324C4#1176296411yescharacteristic
325C4#1176296911yescharacteristic
326C4#1176301611yescharacteristic
327C4#1176302811yescharacteristic
328C2—21no—
329C2—21no—
330C2#1452912811yescharacteristic
331C2#1452914911yescharacteristic
332C2#1452919911yescharacteristic
subgroupG/H
centerC2×C4#10587623
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1985C2
socleC22×C17#386
prime radicalC23×C17C4×D4
cosocleC2×C42×C17C23
hypercenter#1985D17
hypocenterC17#9048
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
87. C2×C8×C17Q16semidirect
235. C17#9048semidirect
260. C2×C8#63231semidirect

Maximal subgroups

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

subgroupindex
#145388232
#145377242
#145274422
#145272132
#145266812
#145266292
C17×#19852
#904817