Finite Group Databasebeta

← #17981544

Subgroups — #17981544

Order 4352.

566 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 97 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#14545925C211yesmaximal, characteristic
2#14545925C211yesmaximal, characteristic
3#14545784C214yesmaximal, complemented
4C2×#11763760C211yesmaximal, characteristic
5C2×C22⋊#3633—21no—
6C2×C22⋊#3633—21no—
7(C2×#3633)⋊C4—21no—
8(C2×#3633)⋊C4—21no—
9(C2×#3633)⋊C4C2212yes—
10(C2×#3633)⋊C4C2212yes—
11C22×C4⋊#595—21no—
12C22×C4⋊#595C411yescharacteristic
13C22×C4×#595C2211yescharacteristic
14#11763760—21no—
15#11763760—21no—
16#11763760C412yes—
17#11763760C2212yes—
18C2×(C2×#595)⋊C4—21no—
19C2×C4⋊C4×D17C411yescharacteristic
20C23×#595—21no—
21C23×#595D411yescharacteristic
22C23×#595D411yescharacteristic
23C22×#3633—22no—
24C22×#3633—22no—
25C22×C4×D17—21no—
26C22×C4×D17C2×C411yescharacteristic
27C2×C4⋊#595—22no—
28C2×C4⋊#595—22no—
29C2×C4⋊#595—21no—
30C2×C4⋊#595—42no—
31C2×C4⋊#595—44no—
32C2×C4⋊#595C2×C412yes—
33C2×C4×#595—42no—
34C2×C4×#595C2×C411yescharacteristic
35C2×C4×#595D411yescharacteristic
36C2×C4×#595D411yescharacteristic
37C2×C4×#595C2311yescharacteristic
38C22⋊#3633—42no—
39C22⋊#3633—42no—
40C2×C4⋊C4×C17C811yescharacteristic, complemented
41C2×C4⋊Dic17C811yescharacteristic, complemented
42C2×#63225—21no—
43C4⋊C4×D17—21no—
44C4⋊C4×D17—21no—
45C4⋊C4×D17C812yescomplemented
46C4⋊C4×D17C2×C412yes—
47(C2×#595)⋊C4—22no—
48(C2×#595)⋊C4—22no—
49C23×D17C22⋊C411yescharacteristic
50C22×#595—22no—
51C22×#595—22no—
52C22×#595—22no—
53C22×#595—22no—
54C22×#595—22no—
55C22×#595—41no—
56C22×#595—41no—
57C22×#595—42no—
58C22×#595—42no—
59C22×#595—42no—
60C22×#595C22⋊C411yescharacteristic
61C22×#595C22⋊C411yescharacteristic
62C22×#595C2×D411yescharacteristic
63C22×#595C2×D411yescharacteristic
64C2×#3633—42no—
65C2×#3633—42no—
66C2×#3633—42no—
67C2×#3633—42no—
68C2×#3633—42no—
69C2×#3633—42no—
70C22×C4×C17—21no—
71C22×C4×C17C2×C811yescharacteristic
72C22×Dic17—21no—
73C22×Dic17M1611yescharacteristic
74C2×C4×D17—22no—
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—42no—
79C2×C4×D17—42no—
80C2×C4×D17C22⋊C411yescharacteristic
81C2×C4×D17C2×C811yescharacteristic
82C2×C4×D17M1611yescharacteristic
83C2×C4×D17C22×C411yescharacteristic
84C4⋊#595—42no—
85C4⋊#595—42no—
86C4⋊#595—44no—
87C4⋊#595—44no—
88C4×#595—41no—
89C4×#595—41no—
90C4×#595—41no—
91C4×#595—41no—
92C4⋊C4×C17—21no—
93C4⋊C4×C17C2×C812yes—
94C4⋊Dic17—21no—
95C4⋊Dic17C2×C812yes—
96#63225—22no—
97#63225—22no—
98#7593—171nomaximal, Sylow
99C23×C17C22⋊C811yescharacteristic
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17C22⋊C811yescharacteristic
105C22×D17#9912yes—
106C22×D17#10112yes—
107C22×D17C2×C22⋊C411yescharacteristic
108C2×#595—21no—
109C2×#595—21no—
110C2×#595—21no—
111C2×#595—21no—
112C2×#595—21no—
113C2×#595—21no—
114C2×#595—21no—
115C2×#595—21no—
116C2×#595—21no—
117C2×#595—21no—
118C2×#595—44no—
119C2×#595—44no—
120C2×#595—42no—
121C2×#595—42no—
122C2×#595—42no—
123C2×#595—84no—
124C2×#595AGL(1,ℤ/8)12yes—
125C2×#595#13712yes—
126#3633—82no—
127#3633—82no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17—42no—
131C2×C4×C17C22⋊C811yescharacteristic
132C2×C4×C17C22×C811yescharacteristic
133C2×Dic17—22no—
134C2×Dic17—22no—
135C2×Dic17—42no—
136C2×Dic17C22⋊C811yescharacteristic
137C2×Dic17C2×M1611yescharacteristic
138C4×D17—22no—
139C4×D17—22no—
140C4×D17—21no—
141C4×D17—21no—
142C4×D17—21no—
143C4×D17—21no—
144C4×D17—44no—
145C4×D17—42no—
146C4×D17—42no—
147C2×#389—171no—
148#1817—171no—
149#1817—171no—
150#1271—174no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17#32412yes—
154C22×C17C2×C22⋊C811yescharacteristic
155C2×D17—21no—
156C2×D17—21no—
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—22no—
161C2×D17#32412yes—
162C2×D17C2×#9911yescharacteristic
163C2×D17C2×#10111yescharacteristic
164C2×D17#41811yescharacteristic
165C2×D17#41911yescharacteristic
166C2×D17#42111yescharacteristic
167C2×D17#42111yescharacteristic
168#595—22no—
169#595—22no—
170#595—41no—
171#595—41no—
172#595—82no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—42no—
176Dic17—21no—
177Dic17—21no—
178Dic17—42no—
179C22×C4⋊C4—171no—
180C22×C4⋊C4—171no—
181C22×C4⋊C4—341no—
182C22×C42—171no—
183C2×C22⋊C8—341no—
184C2×C22⋊C8—341no—
185#389—172no—
186#389—172no—
187#389—341no—
188#389—341no—
189C2×#95—341no—
190#336—172no—
191#336—172no—
192#336—341no—
193#336—341no—
194C2×C17—22no—
195C2×C17C2×#32411yescharacteristic
196C2×C17#147811yescharacteristic
197C2×C17#147911yescharacteristic
198D17—22no—
199D17#151011yescharacteristic
200D17#151111yescharacteristic
201D17#153411yescharacteristic
202D17#155611yescharacteristic
203C23×C4—171no—
204C23×C4—171no—
205C23×C4—171no—
206C23×C4—341no—
207C23×C4—341no—
208C22×C8—342no—
209C22×C8—342no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—172no—
212C2×C4⋊C4—172no—
213C2×C4⋊C4—172no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—342no—
219C2×C4⋊C4—342no—
220C2×C4⋊C4—682no—
221C2×C4⋊C4—684no—
222C2×C4⋊C4#363311yescharacteristic, complemented
223C2×C42—171no—
224C2×C42—171no—
225C2×C42—171no—
226C2×C42—171no—
227C2×C42—682no—
228C22⋊C8—682no—
229C22⋊C8—682no—
230#95—342no—
231#95—342no—
232C17#759311yescharacteristic, complemented, Sylow
233C24—171no—
234C22×C4—21no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—342no—
251C22×C4—342no—
252C22×C4—341no—
253C22×C4—342no—
254C22×C4—682no—
255C22×C4—682no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4—682no—
259C22×C4—682no—
260C22×C4—682no—
261C22×C4C2×#363311yescharacteristic
262C2×C8—682no—
263C2×C8—682no—
264C2×C8—682no—
265C2×C8—682no—
266C2×C8—682no—
267C2×C8—682no—
268C4⋊C4—21no—
269C4⋊C4—172no—
270C4⋊C4—342no—
271C4⋊C4—342no—
272C4⋊C4—341no—
273C4⋊C4—684no—
274C4⋊C4—682no—
275C4⋊C4—682no—
276C4⋊C4—684no—
277C4⋊C4C2×#363312yes—
278C42—681no—
279C42—681no—
280C42—681no—
281C42—681no—
282C23—171no—
283C23—172no—
284C23—172no—
285C23—171no—
286C23—341no—
287C23—341no—
288C23—341no—
289C23—341no—
290C23C22⋊#363311yescharacteristic
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—172no—
295C2×C4—172no—
296C2×C4—171no—
297C2×C4—171no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—342no—
302C2×C4—341no—
303C2×C4—342no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—342no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—342no—
316C2×C4—682no—
317C2×C4—682no—
318C2×C4—684no—
319C2×C4—684no—
320C2×C4—684no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—1364no—
326C2×C4C22⋊#363311yescharacteristic
327C2×C4C22×#363311yescharacteristic
328C8—1362no—
329C8—1362no—
330C22—21no—
331C22—21no—
332C22—171no—
333C22—171no—
334C22—171no—
335C22—171no—
336C22—171no—
337C22—171no—
338C22—172no—
339C22—342no—
340C22—342no—
341C22—342no—
342C22—342no—
343C22—341no—
344C22—341no—
345C22#1176389512yes—
346C22C2×C22⋊#363311yescharacteristic
347C4—21no—
348C4—21no—
349C4—42no—
350C4—342no—
351C4—342no—
352C4—341no—
353C4—341no—
354C4—681no—
355C4—681no—
356C4—682no—
357C4—1362no—
358C2—22no—
359C2—171no—
360C2—171no—
361C2—171no—
362C2—171no—
363C2—342no—
364C2C2×#1176389511yescharacteristic
365C2#1454590411yescharacteristic
366C2#1454590511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#3633
commutatorC2×C4×C17C22×C8
FrattiniC2×C4C22×#3633
FittingC2×C4⋊C4×C17C8
socleC22×C17C2×C22⋊C8
prime radicalC23×D17C22⋊C4
cosocleC2×C4×#595C23
hypercenterC2×C4⋊C4#3633
hypocenterC17#7593
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14545784C2semidirect
40. C2×C4⋊C4×C17C8semidirect
41. C2×C4⋊Dic17C8semidirect
45. C4⋊C4×D17C8semidirect
222. C2×C4⋊C4#3633semidirect
232. C17#7593semidirect

Maximal subgroups

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

subgroupindex
#145459252
#145457842
C2×#117637602
#759317