Finite Group Databasebeta

← C22×#11762857 #17898016

Subgroups — C22×#11762857 #17898016

Order 4352.

2718 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#63223C211yesmaximal, characteristic
2C23×#63223C211yesmaximal, characteristic
3C2×#11762857C2124yesmaximal, direct product factor
4C22×#10587493C214yesmaximal, complemented
5C23×M16×C17C211yesmaximal, characteristic
6C22×#63223—21no—
7C22×#63223—26no—
8C22×#63223C412yes—
9C22×#63223C412yes—
10C22×#63223C4112yes—
11C22×#63223C2212yes—
12C22×#63223C22112yes—
13C22×#63223C22112yes—
14C23×#3622—21no—
15C23×#3622C2212yes—
16#11762857C22164yesdirect product factor
17C2×#10587493C22148yescomplemented
18C24×C4×C17C2211yescharacteristic
19C22×M16×C17—21no—
20C22×M16×C17—26no—
21C22×M16×C17C2212yes—
22C22×M16×C17C22112yes—
23C23×C8×C17—21no—
24C25×C17Q811yescharacteristic
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17C2×C416yes—
28C23×C4×C17D411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C23×C4×C17D416yes—
31C23×C4×C17Q811yescharacteristic
32C23×C4×C17Q816yes—
33C23×C4×C17C2311yescharacteristic
34C23×C4×C17C2316yes—
35C2×M16×C17—212no—
36C2×M16×C17—26no—
37C2×M16×C17—224no—
38C2×M16×C17C23112yes—
39C2×M16×C17C23116yes—
40C22×C8×C17—212no—
41C22×C8×C17—22no—
42C2×#63223—212no—
43C2×#63223—26no—
44C2×#63223—224no—
45C2×#63223C2×C4124yes—
46C2×#63223C2×C4124yes—
47C2×#63223C2×C4148yes—
48C2×#63223C23112yes—
49C2×#63223C23116yes—
50C2×#63223C23116yes—
51C22×#3622—212no—
52C22×#3622—22no—
53C22×#3622C2×C4112yes—
54C22×#3622C2×C414yes—
55C22×#3622C23112yes—
56#10587493C23164yescomplemented
57C24×C17—44no—
58C24×C17C4⋊C411yescharacteristic
59C24×C17C4⋊C411yescharacteristic
60C24×C17C4⋊C416yes—
61C24×C17C2×Q811yescharacteristic
62C24×C17C2×Q816yes—
63C22×C4×C17C4⋊C416yes—
64C22×C4×C17C4⋊C416yes—
65C22×C4×C17C4⋊C416yes—
66C22×C4×C17C4⋊C416yes—
67C22×C4×C17C4⋊C416yes—
68C22×C4×C17C4⋊C4124yes—
69C22×C4×C17C4⋊C411yescharacteristic
70C22×C4×C17C4⋊C411yescharacteristic
71C22×C4×C17C22×C413yes—
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C22×C416yes—
74C22×C4×C17C22×C416yes—
75C22×C4×C17C22×C4112yes—
76C22×C4×C17C2×D416yes—
77C22×C4×C17C2×D416yes—
78C22×C4×C17C2×D413yes—
79C22×C4×C17C2×D4112yes—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×Q813yes—
82C22×C4×C17C2×Q816yes—
83C22×C4×C17C2×Q8112yes—
84C22×C4×C17C2413yes—
85C22×C4×C17C2414yes—
86M16×C17—216no—
87M16×C17—212no—
88M16×C17C2418yes—
89C2×C8×C17—212no—
90C2×C8×C17—216no—
91#63223—216no—
92#63223—212no—
93#63223C22×C4132yes—
94#63223C22×C4132yes—
95#63223C2418yes—
96C2×#3622—212no—
97C2×#3622—216no—
98C2×#3622C22×C4124yes—
99C2×#3622C22×C4124yes—
100C2×#3622C2418yes—
101C22×#430—171nomaximal, Sylow
102C23×C17—24no—
103C23×C17—24no—
104C23×C17—24no—
105C23×C17—412no—
106C23×C17—412no—
107C23×C17C2×C4⋊C413yes—
108C23×C17C2×C4⋊C416yes—
109C23×C17C2×C4⋊C416yes—
110C23×C17C2×C4⋊C4112yes—
111C23×C17C2×C4⋊C411yescharacteristic
112C23×C17C22×Q813yes—
113C23×C17C22×Q814yes—
114C2×C4×C17C2×C4⋊C416yes—
115C2×C4×C17C2×C4⋊C4112yes—
116C2×C4×C17C2×C4⋊C4112yes—
117C2×C4×C17C2×C4⋊C4112yes—
118C2×C4×C17C2×C4⋊C4112yes—
119C2×C4×C17C2×C4⋊C4112yes—
120C2×C4×C17C2×C4⋊C413yes—
121C2×C4×C17C2×C4⋊C416yes—
122C2×C4×C17C2×C4⋊C416yes—
123C2×C4×C17C2×C4⋊C413yes—
124C2×C4×C17C23×C413yes—
125C2×C4×C17C23×C413yes—
126C2×C4×C17C23×C414yes—
127C2×C4×C17C23×C414yes—
128C2×C4×C17C22×D413yes—
129C2×C4×C17C22×D414yes—
130C2×C4×C17C22×D414yes—
131C2×C4×C17C22×D413yes—
132C2×C4×C17C22×Q813yes—
133C2×C4×C17C22×Q814yes—
134C2×C4×C17C2511yescharacteristic
135C8×C17—28no—
136#3622—28no—
137#3622C23×C4116yes—
138C23×M16—171no—
139C23×M16—171no—
140C23×M16D1711yescharacteristic
141C2×#430—1724no—
142C22×#108—174no—
143C22×C17—212no—
144C22×C17—212no—
145C22×C17—26no—
146C22×C17—26no—
147C22×C17—412no—
148C22×C17—44no—
149C22×C17#43014yescomplemented
150C22×C17C22×C4⋊C413yes—
151C22×C17C22×C4⋊C414yes—
152C22×C17C22×C4⋊C414yes—
153C22×C17C22×C4⋊C413yes—
154C22×C17C23×Q811yescharacteristic
155C4×C17C22×C4⋊C413yes—
156C4×C17C22×C4⋊C413yes—
157C4×C17C22×C4⋊C414yes—
158C4×C17C22×C4⋊C414yes—
159C4×C17C24×C411yescharacteristic
160C4×C17C23×D411yescharacteristic
161C24×C4C2×D1711yescharacteristic
162C22×M16—26no—
163C22×M16—21no—
164C22×M16—172no—
165C22×M16—172no—
166C22×M16—1712no—
167C22×M16—1712no—
168C22×M16—172no—
169C22×M16—1712no—
170C22×M16—346no—
171C22×M16—341no—
172C22×M16C2×D1712yes—
173C22×M16C2×D17112yes—
174C23×C8—21no—
175C23×C8—172no—
176C23×C8—341no—
177#430—1764no—
178C2×#108—1748no—
179C2×C17—24no—
180C2×C17—24no—
181C2×C17—23no—
182C2×C17—21no—
183C2×C17C2×#43016yescomplemented
184C2×C17C23×C4⋊C411yescharacteristic
185C25Dic3411yescharacteristic
186C23×C4Dic3411yescharacteristic
187C23×C4Dic3416yes—
188C23×C4C4×D1711yescharacteristic
189C23×C4C4×D1711yescharacteristic
190C23×C4C4×D1716yes—
191C23×C4C22⋊D1716yes—
192C23×C4C22⋊D1711yescharacteristic
193C23×C4C22⋊D1711yescharacteristic
194C23×C4C22×D1716yes—
195C23×C4C22×D1711yescharacteristic
196C2×M16—212no—
197C2×M16—26no—
198C2×M16—224no—
199C2×M16—1724no—
200C2×M16—1724no—
201C2×M16—1748no—
202C2×M16—1716no—
203C2×M16—1712no—
204C2×M16—1716no—
205C2×M16—3412no—
206C2×M16—346no—
207C2×M16—3424no—
208C2×M16C22×D17112yes—
209C2×M16C22×D17116yes—
210C22×C8—212no—
211C22×C8—22no—
212C22×C8—174no—
213C22×C8—1712no—
214C22×C8—1712no—
215C22×C8—3412no—
216C22×C8—342no—
217#108—1764no—
218C17C22×#43011yescharacteristic, complemented, Sylow
219C24—44no—
220C24#6322511yescharacteristic
221C24#6322511yescharacteristic
222C24#6322516yes—
223C24C2×Dic3411yescharacteristic
224C24C2×Dic3416yes—
225C22×C4#6322516yes—
226C22×C4#6322516yes—
227C22×C4#6322516yes—
228C22×C4#63225124yes—
229C22×C4#6322511yescharacteristic
230C22×C4#6322511yescharacteristic
231C22×C4#6322516yes—
232C22×C4#6322516yes—
233C22×C4C2×Dic3416yes—
234C22×C4C2×Dic34112yes—
235C22×C4C2×Dic3413yes—
236C22×C4C2×C4×D1711yescharacteristic
237C22×C4C2×C4×D1713yes—
238C22×C4C2×C4×D1716yes—
239C22×C4C2×C4×D1716yes—
240C22×C4C2×C4×D17112yes—
241C22×C4C2×C22⋊D1716yes—
242C22×C4C2×C22⋊D1716yes—
243C22×C4C2×C22⋊D1713yes—
244C22×C4C2×C22⋊D17112yes—
245C22×C4C2×C22⋊D1711yescharacteristic
246C22×C4C23×D1713yes—
247C22×C4C23×D1714yes—
248M16—216no—
249M16—212no—
250M16—1732no—
251M16—1732no—
252M16—178no—
253M16—3416no—
254M16—3412no—
255M16C23×D1718yes—
256C2×C8—212no—
257C2×C8—216no—
258C2×C8—1724no—
259C2×C8—1724no—
260C2×C8—178no—
261C2×C8—3412no—
262C2×C8—3416no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—412no—
267C23—412no—
268C23C2×#6322513yes—
269C23C2×#6322516yes—
270C23C2×#6322516yes—
271C23C2×#63225112yes—
272C23C2×#6322511yescharacteristic
273C23C22×Dic3413yes—
274C23C22×Dic3414yes—
275C2×C4C2×#6322516yes—
276C2×C4C2×#63225112yes—
277C2×C4C2×#63225112yes—
278C2×C4C2×#63225112yes—
279C2×C4C2×#63225112yes—
280C2×C4C2×#63225112yes—
281C2×C4C2×#6322513yes—
282C2×C4C2×#6322516yes—
283C2×C4C2×#6322516yes—
284C2×C4C2×#6322513yes—
285C2×C4C22×Dic3413yes—
286C2×C4C22×Dic3414yes—
287C2×C4C22×C4×D1713yes—
288C2×C4C22×C4×D1713yes—
289C2×C4C22×C4×D1714yes—
290C2×C4C22×C4×D1714yes—
291C2×C4C22×C22⋊D1713yes—
292C2×C4C22×C22⋊D1714yes—
293C2×C4C22×C22⋊D1714yes—
294C2×C4C22×C22⋊D1713yes—
295C2×C4C24×D1711yescharacteristic
296C8—28no—
297C8—1716no—
298C8—348no—
299C22—26no—
300C22—26no—
301C22—212no—
302C22—212no—
303C22—44no—
304C22—412no—
305C22#1176285714yesdirect product factor
306C22C22×#6322513yes—
307C22C22×#6322514yes—
308C22C22×#6322514yes—
309C22C22×#6322513yes—
310C22C23×Dic3411yescharacteristic
311C4C22×#6322513yes—
312C4C22×#6322513yes—
313C4C22×#6322514yes—
314C4C22×#6322514yes—
315C4C23×C4×D1711yescharacteristic
316C4C23×C22⋊D1711yescharacteristic
317C2—23no—
318C2—24no—
319C2—24no—
320C2—21no—
321C2C2×#1176285716yesdirect product factor
322C2C23×#6322511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D17
commutatorC4×C17C24×C4
FrattiniC2×C4C24×D17
FittingC23×M16×C17C2
socleC23×C17C2×C4⋊C4
prime radicalC25×C17Q8
cosocleC2×C4×C17C25
hypercenterC23×M16D17
hypocenterC17C22×#430
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#11762857C2direct (N × H)
4. C22×#10587493C2semidirect
16. #11762857C22direct (N × H)
17. C2×#10587493C22semidirect
56. #10587493C23semidirect
149. C22×C17#430semidirect
183. C2×C17C2×#430semidirect
218. C17C22×#430semidirect
305. C22#11762857direct (N × H)
321. C2C2×#11762857direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#632232
C2×#117628572
C22×#105874932
C23×M16×C172
C22×#43017