Finite Group Databasebeta

← #17949693

Subgroups — #17949693

Order 4352.

520 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 118 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#14544700C211yesmaximal, characteristic, complemented
2C4×#10587544C211yesmaximal, characteristic, complemented
3#14543602C212yesmaximal, complemented
4#14543601C212yesmaximal
5#14531242C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×#10587544—21no—
8C2×C4×#3624—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587579C411yescharacteristic
13C2×#10587578—21no—
14#11762711C411yescharacteristic
15#11762710C2211yescharacteristic
16#11762708C2211yescharacteristic
17#11762701—21no—
18#11762699—21no—
19C4⋊C4×Dic17C411yescharacteristic
20#11762656—21no—
21C42×Dic17C2211yescharacteristic
22C17×#389C411yescharacteristic
23C22×D68C2×C411yescharacteristic
24C22×C4×D17—21no—
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×#63227—21no—
32C2×#63227—21no—
33C2×#63227—21no—
34C2×#63227—21no—
35C2×#63227—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4×Dic17—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45#10587579—22no—
46#10587579C814yescomplemented
47#10587578—42no—
48#10587578—42no—
49#10587544—24no—
50#10587544—42no—
51C4×#3624—42no—
52C17×#95—21no—
53#10587502—21no—
54#10587502—21no—
55C23×D17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—21no—
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C22⋊C411yescharacteristic
64C22×Dic17C2×D411yescharacteristic
65C22×Dic17C2×D411yescharacteristic
66C2×D68—21no—
67C2×D68—41no—
68C2×D68—41no—
69C2×D68C2×C812yes—
70C2×C4×D17—24no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C2×#3624—24no—
74C2×#3624—42no—
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17C2×C812yes—
83C42×C17—41no—
84#63227—24no—
85#63227—42no—
86#63227—42no—
87#63227—41no—
88#63227—41no—
89#63227—41no—
90#63227—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17—41no—
94C4×Dic17—22no—
95C4×Dic17—21no—
96C4×Dic17—21no—
97C4×Dic17—41no—
98C4×Dic17—41no—
99C4×Dic17—42no—
100C4×Dic17C2×C812yes—
101C4×Dic17M1612yes—
102C4×Dic17D812yes—
103C4×Dic17SD1612yes—
104#7574—171nomaximal, Sylow
105C23×C17C2×C22⋊C411yescharacteristic
106C22×D17—22no—
107C22×D17—41no—
108C22×D17—41no—
109C22×D17—41no—
110C22×D17—41no—
111C22×D17—41no—
112C22×D17—41no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—22no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17#10212yes—
127C2×C4×C17C22×C811yescharacteristic
128C2×C4×C17C2×M1611yescharacteristic
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17C22⋊C812yes—
139C2×Dic17C22⋊C812yes—
140C2×Dic17#10212yes—
141C2×Dic17#10412yes—
142C2×Dic17C2×D811yescharacteristic
143C2×Dic17C2×SD1611yescharacteristic
144C2×Dic17AGL(1,ℤ/8)11yescharacteristic
145C2×Dic17#13711yescharacteristic
146D68—42no—
147D68—42no—
148C4×D17—48no—
149#3624—44no—
150#3624—82no—
151#2293—171no—
152#1831—171no—
153C4×C4⋊C8—171no—
154#1271—172no—
155#1270—172no—
156C22×C17C2×C22⋊C811yescharacteristic
157C22×C17#41011yescharacteristic
158C22×C17#41311yescharacteristic
159C22×C17C2×#10211yescharacteristic
160C22×C17#41711yescharacteristic
161C22×C17C2×#10411yescharacteristic
162C22×C17#42111yescharacteristic
163C2×D17—42no—
164C2×D17—42no—
165C2×D17—42no—
166C2×D17—81no—
167C2×D17—81no—
168C2×D17—81no—
169C2×D17—81no—
170C4×C17—22no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—42no—
174C4×C17—41no—
175C4×C17—81no—
176Dic17—22no—
177Dic17—21no—
178Dic17—21no—
179Dic17—21no—
180Dic17—21no—
181Dic17—81no—
182Dic17D4⋊C814yes—
183C2×C4×D4—171no—
184C2×#117—341no—
185C2×C4⋊C8—341no—
186C2×C4⋊C8—341no—
187C2×C4×C8—341no—
188#390—171no—
189#390—171no—
190#389—171no—
191#389#59511yescharacteristic
192#388—341no—
193#388—341no—
194C4×C4⋊C4—171no—
195C4×C22⋊C4—341no—
196C43—171no—
197#336—172no—
198#336—172no—
199#336—341no—
200C2×C17#145811yescharacteristic
201C2×C17C2×D4⋊C811yescharacteristic
202C2×C17#147711yescharacteristic
203C2×C17#150911yescharacteristic
204C2×C17#151711yescharacteristic
205C2×C17#152911yescharacteristic
206C2×C17#154711yescharacteristic
207D17—82no—
208C22×D4—171no—
209C23×C4—341no—
210C22×C8—342no—
211C22×C8—342no—
212C4×D4—174no—
213C4×D4—342no—
214#117—682no—
215#117—682no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4C2×#59511yescharacteristic
220C2×C4⋊C4C2×#59511yescharacteristic
221C2×C22⋊C4—341no—
222C2×C22⋊C4—341no—
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C22⋊C4—341no—
226C2×C42—171no—
227C2×C42—171no—
228C2×C42—171no—
229C2×C42—171no—
230C2×C42—171no—
231C2×C42—171no—
232C2×C42—341no—
233C2×C42C2×#59511yescharacteristic
234C4⋊C8—344no—
235C4⋊C8—682no—
236C4×C8—682no—
237#95—21no—
238#95—341no—
239#95—341no—
240C17#757411yescharacteristic, complemented, Sylow
241C24—341no—
242C2×D4—172no—
243C2×D4—341no—
244C2×D4—681no—
245C2×D4—681no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—171no—
249C22×C4—171no—
250C22×C4—171no—
251C22×C4—171no—
252C22×C4—344no—
253C22×C4—341no—
254C22×C4—682no—
255C22×C4—682no—
256C22×C4C22⋊#59511yescharacteristic
257C22×C4C22⋊#59511yescharacteristic
258C22×C4C22×#59511yescharacteristic
259C2×C8—344no—
260C2×C8—682no—
261C2×C8—682no—
262C2×C8—682no—
263C2×C8—682no—
264C2×C8—682no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—681no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4#6324212yes—
272C22⋊C4—344no—
273C22⋊C4—682no—
274C22⋊C4—681no—
275C22⋊C4—681no—
276C22⋊C4—681no—
277C22⋊C4—681no—
278C22⋊C4—682no—
279C42—41no—
280C42—172no—
281C42—172no—
282C42—172no—
283C42—172no—
284C42—341no—
285C42—342no—
286C42—341no—
287C42—681no—
288C42—681no—
289C42—682no—
290C23—342no—
291C23—681no—
292C23—681no—
293C23—681no—
294C23—681no—
295C23—681no—
296C23—681no—
297C23C2×C22⋊#59511yescharacteristic
298D4—682no—
299D4—682no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—172no—
314C2×C4—172no—
315C2×C4—172no—
316C2×C4—172no—
317C2×C4—171no—
318C2×C4—171no—
319C2×C4—171no—
320C2×C4—171no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—688no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4#1058754712yes—
332C2×C4C2×#6324211yescharacteristic
333C2×C4D17⋊M1611yescharacteristic
334C8—684no—
335C8—1362no—
336C22—682no—
337C22—682no—
338C22—682no—
339C22—1361no—
340C22—1361no—
341C22—1361no—
342C22—1361no—
343C22#1176377711yescharacteristic
344C22#1176379011yescharacteristic
345C22#1176379611yescharacteristic
346C22C2×#1058754711yescharacteristic
347C22#1176380011yescharacteristic
348C22D17⋊#10411yescharacteristic
349C22#1176380811yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—41no—
354C4—42no—
355C4—81no—
356C4—174no—
357C4—341no—
358C4—341no—
359C4—341no—
360C4—341no—
361C4—342no—
362C4—1361no—
363C2—1362no—
364C2#1454401811yescharacteristic
365C2#1454407011yescharacteristic
366C2#1454408211yescharacteristic
367C2#1454418711yescharacteristic
368C2#1454421511yescharacteristic
369C2#1454425211yescharacteristic
370C2#1454429611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D68C2×C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7574
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #14544700C2semidirect
2. C4×#10587544C2semidirect
3. #14543602C2semidirect
46. #10587579C8semidirect
240. C17#7574semidirect

Maximal subgroups

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

subgroupindex
#145447002
C4×#105875442
#145436022
#145436012
#145312422
#757417