Finite Group Databasebeta

← #17982956

Subgroups — #17982956

Order 4352.

1428 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#3633C212yesmaximal, complemented
2#14545934C216yesmaximal, complemented
3#14545924C213yesmaximal, complemented
4C4×C22⋊#3633C213yesmaximal, complemented
5C2×#41×#595C211yesmaximal, characteristic
6Q8×#3633—28no—
7C2×C4⋊#3633C2216yes—
8C2×C22⋊#3633C2216yes—
9C2×C22⋊#3633C2216yes—
10C2×C4×#3633C2216yes—
11(C2×#3633)⋊C4C2213yes—
12(C2×#3633)⋊C4C2211yescharacteristic
13#41×#595—28no—
14C2×Q8×#595C411yescharacteristic
15C2×Q8×#595C2211yescharacteristic
16C2×D4×#595C413yes—
17C2×D4×#595C2213yes—
18C2×#10587677C413yes—
19C22×C4×#595C2213yes—
20C2×#41×D17C411yescharacteristic
21C23×#595C2×C413yes—
22C23×#595C2313yes—
23C22×#3633C2316yes—
24C22×#3633C2312yes—
25C2×#41×C17C811yescharacteristic, complemented
26#41×D17—24no—
27#41×D17—24no—
28C2×#63256C811yescharacteristic, complemented
29C2×Q8×D17C2×C411yescharacteristic
30C2×#63254C813yescomplemented
31C2×D4×D17C2×C413yes—
32C2×#63252C813yescomplemented
33C22×C4×D17C2×C413yes—
34C2×C22⋊#595C2×C416yes—
35Q8×#595—24no—
36Q8×#595D418yes—
37D4×#595—212no—
38D4×#595—212no—
39#10587677—212no—
40C2×C4⋊#595C2×C413yes—
41C2×C4⋊#595C2×C413yes—
42C2×C4×#595—212no—
43C2×C4×#595C2×C413yes—
44C2×C4×#595C2×C413yes—
45C2×C4×#595C2311yescharacteristic
46C2×C4×#595C2313yes—
47C22⋊#3633—212no—
48C22⋊#3633—46no—
49C4⋊#3633—212no—
50C4×#3633—212no—
51C23×D17C22×C413yes—
52C22×#595—26no—
53C22×#595—26no—
54C22×#595—26no—
55C22×#595—26no—
56C22×#595—23no—
57C22×#595—23no—
58C22×#595—43no—
59C22×#595C22×C413yes—
60C22×#595C22×C413yes—
61C22×#595C22×C411yescharacteristic
62C22×#595C2411yescharacteristic
63C2×#3633—22no—
64C2×#3633—22no—
65C2×#3633—26no—
66C2×#3633—26no—
67C2×#3633—26no—
68C2×#3633#4114yescomplemented
69#41×C17—24no—
70C2×Q8×C17C2×C811yescharacteristic
71C2×D4×C17C2×C813yes—
72C22×C4×C17C2×C813yes—
73C2×C22⋊D17C2×C816yes—
74C22×Dic17C2×C813yes—
75#63256—24no—
76Q8×D17C22⋊C414yes—
77Q8×D17M1614yes—
78#63254—212no—
79D4×D17—26no—
80D4×D17—26no—
81#63252—212no—
82C2×D68C2×C813yes—
83C2×C4×D17—26no—
84C2×C4×D17—26no—
85C2×C4×D17C2×C813yes—
86C2×C4×D17C2×C813yes—
87C2×C4×D17C22×C411yescharacteristic
88C2×C4×D17C22×C413yes—
89C2×Dic34C2×C813yes—
90C22⋊#595—212no—
91C4⋊#595—26no—
92C4⋊#595C22⋊C4112yes—
93C4×#595—26no—
94C4×#595—22no—
95C4×#595—26no—
96C4×#595C2×D4112yes—
97#22530—171nomaximal, Sylow
98C23×C17C22×C813yes—
99C22×D17—23no—
100C22×D17—23no—
101C22×D17—23no—
102C22×D17—23no—
103C22×D17—23no—
104C22×D17—23no—
105C22×D17C22×C813yes—
106C22×D17C23×C411yescharacteristic
107C2×#595—21no—
108C2×#595—23no—
109C2×#595—21no—
110C2×#595—23no—
111C2×#595—23no—
112C2×#595—23no—
113C2×#595—23no—
114C2×#595—412no—
115C2×#595—46no—
116C2×#595—46no—
117C2×#595C2×C22⋊C416yes—
118C2×#595#13112yes—
119C2×#595C22×D412yes—
120C2×#595C2×#4112yes—
121C2×#595H(5,𝔽2)11yescharacteristic
122C2×#5952−1+411yescharacteristic
123#3633—24no—
124#3633—46no—
125Q8×C17C22⋊C814yescomplemented
126D4×C17—26no—
127C2×C4×C17—26no—
128C2×C4×C17C22×C811yescharacteristic
129C2×C4×C17C22×C813yes—
130C22⋊D17—212no—
131C2×Dic17—26no—
132C2×Dic17C22×C811yescharacteristic
133C2×Dic17C22×C813yes—
134D68—26no—
135C4×D17—21no—
136C4×D17—21no—
137C4×D17—26no—
138C4×D17—26no—
139C4×D17C2×C22⋊C416yes—
140C4×D17C2×M1616yes—
141Dic34C22⋊C8112yescomplemented
142C2×C4×#41—171no—
143C2×C8×Q8—172no—
144#1954—176no—
145#1797—173no—
146C4×C22⋊C8—173no—
147C22×C17—23no—
148C22×C17—23no—
149C22×C17—23no—
150C22×C17C23×C811yescharacteristic
151C2×D17—23no—
152C2×D17—23no—
153C2×D17—23no—
154C2×D17—43no—
155C2×D17—43no—
156C2×D17—43no—
157C2×D17—43no—
158C2×D17C22×C22⋊C411yescharacteristic
159C2×D17C4×#4111yescharacteristic
160C2×D17#52011yescharacteristic
161C2×D17C22×M1611yescharacteristic
162C2×D17C2×#13111yescharacteristic
163C2×D17#56811yescharacteristic
164#595—22no—
165#595—26no—
166#595—43no—
167#595#54812yes—
168#595#55012yes—
169C4×C17—21no—
170C4×C17C2×C22⋊C816yes—
171Dic17—21no—
172Dic17C2×C22⋊C816yes—
173C22×#41—171no—
174C4×#41—348no—
175C2×C4×Q8—171no—
176C2×C4×Q8—171no—
177C2×C4×D4—173no—
178C2×C4×D4—173no—
179C2×#117—173no—
180C22×C42—173no—
181C8×Q8—348no—
182C2×C4⋊C8—176no—
183C2×C22⋊C8—176no—
184C2×C22⋊C8—176no—
185C2×C4×C8—176no—
186#336—173no—
187#336—171no—
188C2×C17—43no—
189C2×C17C22×C22⋊C811yescharacteristic
190C2×C17C8×#4111yescharacteristic
191C2×C17#298811yescharacteristic
192D17—43no—
193D17#234111yescharacteristic
194D17#288011yescharacteristic
195D17#297111yescharacteristic
196D17#299111yescharacteristic
197C2×#41—171no—
198C2×#41—173no—
199C2×#41—173no—
200C2×#41—344no—
201C2×#41—344no—
202C2×#41#363311yescharacteristic, complemented
203C22×Q8—171no—
204C22×D4—173no—
205C23×C4—173no—
206C23×C4—173no—
207C23×C4—173no—
208C22×C8—172no—
209C22×C8—176no—
210C4×Q8—178no—
211C4×Q8—344no—
212C4×D4—3412no—
213C4×D4—3412no—
214#117—3412no—
215C2×C4⋊C4—173no—
216C2×C4⋊C4—173no—
217C2×C22⋊C4—176no—
218C2×C42—173no—
219C2×C42—173no—
220C2×C42—173no—
221C2×C42—171no—
222C2×C42—3412no—
223C4⋊C8—3412no—
224C22⋊C8—3412no—
225C22⋊C8—686no—
226C4×C8—3412no—
227C17#2253011yescharacteristic, complemented, Sylow
228C24—173no—
229#41—24no—
230#41—3412no—
231#41—3412no—
232#41—344no—
233C2×Q8—174no—
234C2×Q8—174no—
235C2×Q8—173no—
236C2×Q8C2×#363311yescharacteristic
237C2×D4—176no—
238C2×D4—173no—
239C2×D4—346no—
240C2×D4—346no—
241C2×D4C2×#363313yes—
242C22×C4—171no—
243C22×C4—173no—
244C22×C4—171no—
245C22×C4—173no—
246C22×C4—173no—
247C22×C4—173no—
248C22×C4—173no—
249C22×C4—171no—
250C22×C4—173no—
251C22×C4—346no—
252C22×C4—346no—
253C22×C4—346no—
254C22×C4—343no—
255C22×C4—346no—
256C22×C4—346no—
257C22×C4—343no—
258C22×C4—346no—
259C22×C4—683no—
260C22×C4C2×#363313yes—
261C2×C8—174no—
262C2×C8—342no—
263C2×C8—342no—
264C2×C8—346no—
265C2×C8—346no—
266C2×C8—346no—
267C4⋊C4—1712no—
268C4⋊C4—346no—
269C22⋊C4—3412no—
270C42—1712no—
271C42—342no—
272C42—346no—
273C42—346no—
274C23—171no—
275C23—173no—
276C23—343no—
277C23—343no—
278C23—343no—
279C23—343no—
280C23—343no—
281C23—343no—
282C23C22×#363313yes—
283Q8—1712no—
284Q8C22⋊#363314yescomplemented
285D4—26no—
286D4—3412no—
287D4—346no—
288C2×C4—26no—
289C2×C4—172no—
290C2×C4—172no—
291C2×C4—171no—
292C2×C4—171no—
293C2×C4—173no—
294C2×C4—176no—
295C2×C4—176no—
296C2×C4—176no—
297C2×C4—172no—
298C2×C4—171no—
299C2×C4—346no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—346no—
303C2×C4—341no—
304C2×C4—343no—
305C2×C4—341no—
306C2×C4—343no—
307C2×C4—346no—
308C2×C4—343no—
309C2×C4—343no—
310C2×C4—343no—
311C2×C4—686no—
312C2×C4—686no—
313C2×C4—6812no—
314C2×C4C22×#363311yescharacteristic
315C2×C4C22×#363313yes—
316C8—344no—
317C8—686no—
318C22—23no—
319C22—23no—
320C22—23no—
321C22—171no—
322C22—171no—
323C22—171no—
324C22—171no—
325C22—171no—
326C22—171no—
327C22—343no—
328C22—343no—
329C22—343no—
330C22—683no—
331C22—683no—
332C22—683no—
333C22—683no—
334C22C23×#363311yescharacteristic
335C4—21no—
336C4—172no—
337C4—172no—
338C4—176no—
339C4—342no—
340C4—346no—
341C4—341no—
342C4—683no—
343C4C2×C22⋊#363316yes—
344C2—43no—
345C2—171no—
346C2—171no—
347C2—171no—
348C2—171no—
349C2—683no—
350C2C22×C22⋊#363311yescharacteristic
351C2#41×#363311yescharacteristic
352C2#1454602811yescharacteristic
subgroupG/H
center, FrattiniC22C23×#3633
commutator, socleC22×C17C23×C8
FittingC2×#41×C17C8
prime radicalC2×#41×D17C4
cosocleC22×#595C24
hypercenterC2×#41#3633
hypocenterC17#22530
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#3633C2semidirect
2. #14545934C2semidirect
3. #14545924C2semidirect
4. C4×C22⋊#3633C2semidirect
25. C2×#41×C17C8semidirect
28. C2×#63256C8semidirect
30. C2×#63254C8semidirect
32. C2×#63252C8semidirect
68. C2×#3633#41semidirect
125. Q8×C17C22⋊C8semidirect
141. Dic34C22⋊C8semidirect
202. C2×#41#3633semidirect
227. C17#22530semidirect
284. Q8C22⋊#3633semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#36332
#145459342
#145459242
C4×C22⋊#36332
C2×#41×#5952
#2253017