Finite Group Databasebeta

← C22×#89896 #14217488

Subgroups — C22×#89896 #14217488

Order 1792.

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×C7×M16C211yesmaximal, characteristic
2C23×#1136C211yesmaximal, characteristic
3C23×#1136C211yesmaximal, characteristic
4C2×#89896C2124yesmaximal, direct product factor
5C22×#6396C214yesmaximal, complemented
6C24×C4×C7C2211yescharacteristic
7C22×C7×M16—21no—
8C22×C7×M16—26no—
9C22×C7×M16C2212yes—
10C22×C7×M16C22112yes—
11C23×C7×C8—21no—
12C22×#1136—21no—
13C22×#1136—26no—
14C22×#1136C412yes—
15C22×#1136C412yes—
16C22×#1136C4112yes—
17C22×#1136C2212yes—
18C22×#1136C22112yes—
19C22×#1136C22112yes—
20C23×C7⋊C8—21no—
21C23×C7⋊C8C2212yes—
22#89896C22164yesdirect product factor
23C2×#6396C22148yescomplemented
24C22×#430—71nomaximal, Sylow
25C25×C7Q811yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7C2×C416yes—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D416yes—
32C23×C4×C7Q811yescharacteristic
33C23×C4×C7Q816yes—
34C23×C4×C7C2311yescharacteristic
35C23×C4×C7C2316yes—
36C2×C7×M16—212no—
37C2×C7×M16—26no—
38C2×C7×M16—224no—
39C2×C7×M16C23112yes—
40C2×C7×M16C23116yes—
41C22×C7×C8—212no—
42C22×C7×C8—22no—
43C2×#1136—212no—
44C2×#1136—26no—
45C2×#1136—224no—
46C2×#1136C2×C4124yes—
47C2×#1136C2×C4124yes—
48C2×#1136C2×C4148yes—
49C2×#1136C23112yes—
50C2×#1136C23116yes—
51C2×#1136C23116yes—
52C22×C7⋊C8—212no—
53C22×C7⋊C8—22no—
54C22×C7⋊C8C2×C4112yes—
55C22×C7⋊C8C2×C414yes—
56C22×C7⋊C8C23112yes—
57#6396C23164yescomplemented
58C23×M16—71no—
59C23×M16—71no—
60C23×M16D711yescharacteristic
61C2×#430—724no—
62C22×#108—74no—
63C24×C7—44no—
64C24×C7C4⋊C411yescharacteristic
65C24×C7C4⋊C411yescharacteristic
66C24×C7C4⋊C416yes—
67C24×C7C2×Q811yescharacteristic
68C24×C7C2×Q816yes—
69C22×C4×C7C4⋊C416yes—
70C22×C4×C7C4⋊C416yes—
71C22×C4×C7C4⋊C416yes—
72C22×C4×C7C4⋊C416yes—
73C22×C4×C7C4⋊C416yes—
74C22×C4×C7C4⋊C4124yes—
75C22×C4×C7C4⋊C411yescharacteristic
76C22×C4×C7C4⋊C411yescharacteristic
77C22×C4×C7C22×C413yes—
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C22×C416yes—
80C22×C4×C7C22×C416yes—
81C22×C4×C7C22×C4112yes—
82C22×C4×C7C2×D416yes—
83C22×C4×C7C2×D416yes—
84C22×C4×C7C2×D413yes—
85C22×C4×C7C2×D4112yes—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×Q813yes—
88C22×C4×C7C2×Q816yes—
89C22×C4×C7C2×Q8112yes—
90C22×C4×C7C2413yes—
91C22×C4×C7C2414yes—
92C7×M16—216no—
93C7×M16—212no—
94C7×M16C2418yes—
95C2×C7×C8—212no—
96C2×C7×C8—216no—
97#1136—216no—
98#1136—212no—
99#1136C22×C4132yes—
100#1136C22×C4132yes—
101#1136C2418yes—
102C2×C7⋊C8—212no—
103C2×C7⋊C8—216no—
104C2×C7⋊C8C22×C4124yes—
105C2×C7⋊C8C22×C4124yes—
106C2×C7⋊C8C2418yes—
107C24×C4C2×D711yescharacteristic
108C22×M16—26no—
109C22×M16—21no—
110C22×M16—72no—
111C22×M16—72no—
112C22×M16—712no—
113C22×M16—712no—
114C22×M16—72no—
115C22×M16—712no—
116C22×M16—146no—
117C22×M16—141no—
118C22×M16C2×D712yes—
119C22×M16C2×D7112yes—
120C23×C8—21no—
121C23×C8—72no—
122C23×C8—141no—
123#430—764no—
124C2×#108—748no—
125C23×C7—24no—
126C23×C7—24no—
127C23×C7—24no—
128C23×C7—412no—
129C23×C7—412no—
130C23×C7C2×C4⋊C413yes—
131C23×C7C2×C4⋊C416yes—
132C23×C7C2×C4⋊C416yes—
133C23×C7C2×C4⋊C4112yes—
134C23×C7C2×C4⋊C411yescharacteristic
135C23×C7C22×Q813yes—
136C23×C7C22×Q814yes—
137C2×C4×C7C2×C4⋊C416yes—
138C2×C4×C7C2×C4⋊C4112yes—
139C2×C4×C7C2×C4⋊C4112yes—
140C2×C4×C7C2×C4⋊C4112yes—
141C2×C4×C7C2×C4⋊C4112yes—
142C2×C4×C7C2×C4⋊C4112yes—
143C2×C4×C7C2×C4⋊C413yes—
144C2×C4×C7C2×C4⋊C416yes—
145C2×C4×C7C2×C4⋊C416yes—
146C2×C4×C7C2×C4⋊C413yes—
147C2×C4×C7C23×C413yes—
148C2×C4×C7C23×C413yes—
149C2×C4×C7C23×C414yes—
150C2×C4×C7C23×C414yes—
151C2×C4×C7C22×D413yes—
152C2×C4×C7C22×D414yes—
153C2×C4×C7C22×D414yes—
154C2×C4×C7C22×D413yes—
155C2×C4×C7C22×Q813yes—
156C2×C4×C7C22×Q814yes—
157C2×C4×C7C2511yescharacteristic
158C7×C8—28no—
159C7⋊C8—28no—
160C7⋊C8C23×C4116yes—
161C25Dic1411yescharacteristic
162C23×C4Dic1411yescharacteristic
163C23×C4Dic1416yes—
164C23×C4C4×D711yescharacteristic
165C23×C4C4×D711yescharacteristic
166C23×C4C4×D716yes—
167C23×C4C22⋊D716yes—
168C23×C4C22⋊D711yescharacteristic
169C23×C4C22⋊D711yescharacteristic
170C23×C4C22×D716yes—
171C23×C4C22×D711yescharacteristic
172C2×M16—212no—
173C2×M16—26no—
174C2×M16—224no—
175C2×M16—724no—
176C2×M16—724no—
177C2×M16—748no—
178C2×M16—716no—
179C2×M16—712no—
180C2×M16—716no—
181C2×M16—1412no—
182C2×M16—146no—
183C2×M16—1424no—
184C2×M16C22×D7112yes—
185C2×M16C22×D7116yes—
186C22×C8—212no—
187C22×C8—22no—
188C22×C8—74no—
189C22×C8—712no—
190C22×C8—712no—
191C22×C8—1412no—
192C22×C8—142no—
193#108—764no—
194C22×C7—212no—
195C22×C7—212no—
196C22×C7—26no—
197C22×C7—26no—
198C22×C7—412no—
199C22×C7—44no—
200C22×C7#43014yescomplemented
201C22×C7C22×C4⋊C413yes—
202C22×C7C22×C4⋊C414yes—
203C22×C7C22×C4⋊C414yes—
204C22×C7C22×C4⋊C413yes—
205C22×C7C23×Q811yescharacteristic
206C4×C7C22×C4⋊C413yes—
207C4×C7C22×C4⋊C413yes—
208C4×C7C22×C4⋊C414yes—
209C4×C7C22×C4⋊C414yes—
210C4×C7C24×C411yescharacteristic
211C4×C7C23×D411yescharacteristic
212C24—44no—
213C24Dic7⋊C411yescharacteristic
214C24Dic7⋊C411yescharacteristic
215C24Dic7⋊C416yes—
216C24C2×Dic1411yescharacteristic
217C24C2×Dic1416yes—
218C22×C4Dic7⋊C416yes—
219C22×C4Dic7⋊C416yes—
220C22×C4Dic7⋊C416yes—
221C22×C4Dic7⋊C4124yes—
222C22×C4Dic7⋊C411yescharacteristic
223C22×C4Dic7⋊C411yescharacteristic
224C22×C4Dic7⋊C416yes—
225C22×C4Dic7⋊C416yes—
226C22×C4C2×Dic1416yes—
227C22×C4C2×Dic14112yes—
228C22×C4C2×Dic1413yes—
229C22×C4C2×C4×D711yescharacteristic
230C22×C4C2×C4×D713yes—
231C22×C4C2×C4×D716yes—
232C22×C4C2×C4×D716yes—
233C22×C4C2×C4×D7112yes—
234C22×C4C2×C22⋊D716yes—
235C22×C4C2×C22⋊D716yes—
236C22×C4C2×C22⋊D713yes—
237C22×C4C2×C22⋊D7112yes—
238C22×C4C2×C22⋊D711yescharacteristic
239C22×C4C23×D713yes—
240C22×C4C23×D714yes—
241M16—216no—
242M16—212no—
243M16—732no—
244M16—732no—
245M16—78no—
246M16—1416no—
247M16—1412no—
248M16C23×D718yes—
249C2×C8—212no—
250C2×C8—216no—
251C2×C8—724no—
252C2×C8—724no—
253C2×C8—78no—
254C2×C8—1412no—
255C2×C8—1416no—
256C2×C7—24no—
257C2×C7—24no—
258C2×C7—23no—
259C2×C7—21no—
260C2×C7C2×#43016yescomplemented
261C2×C7C23×C4⋊C411yescharacteristic
262C23—24no—
263C23—24no—
264C23—24no—
265C23—412no—
266C23—412no—
267C23C2×Dic7⋊C413yes—
268C23C2×Dic7⋊C416yes—
269C23C2×Dic7⋊C416yes—
270C23C2×Dic7⋊C4112yes—
271C23C2×Dic7⋊C411yescharacteristic
272C23C22×Dic1413yes—
273C23C22×Dic1414yes—
274C2×C4C2×Dic7⋊C416yes—
275C2×C4C2×Dic7⋊C4112yes—
276C2×C4C2×Dic7⋊C4112yes—
277C2×C4C2×Dic7⋊C4112yes—
278C2×C4C2×Dic7⋊C4112yes—
279C2×C4C2×Dic7⋊C4112yes—
280C2×C4C2×Dic7⋊C413yes—
281C2×C4C2×Dic7⋊C416yes—
282C2×C4C2×Dic7⋊C416yes—
283C2×C4C2×Dic7⋊C413yes—
284C2×C4C22×Dic1413yes—
285C2×C4C22×Dic1414yes—
286C2×C4C22×C4×D713yes—
287C2×C4C22×C4×D713yes—
288C2×C4C22×C4×D714yes—
289C2×C4C22×C4×D714yes—
290C2×C4C22×C22⋊D713yes—
291C2×C4C22×C22⋊D714yes—
292C2×C4C22×C22⋊D714yes—
293C2×C4C22×C22⋊D713yes—
294C2×C4C24×D711yescharacteristic
295C8—28no—
296C8—716no—
297C8—148no—
298C7C22×#43011yescharacteristic, complemented, Sylow
299C22—212no—
300C22—212no—
301C22—26no—
302C22—26no—
303C22—412no—
304C22—44no—
305C22#8989614yesdirect product factor
306C22C22×Dic7⋊C413yes—
307C22C22×Dic7⋊C414yes—
308C22C22×Dic7⋊C414yes—
309C22C22×Dic7⋊C413yes—
310C22C23×Dic1411yescharacteristic
311C4C22×Dic7⋊C413yes—
312C4C22×Dic7⋊C414yes—
313C4C22×Dic7⋊C414yes—
314C4C22×Dic7⋊C413yes—
315C4C23×C4×D711yescharacteristic
316C4C23×C22⋊D711yescharacteristic
317C2—24no—
318C2—24no—
319C2—23no—
320C2—21no—
321C2C2×#8989616yesdirect product factor
322C2C23×Dic7⋊C411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D7
commutatorC4×C7C24×C4
FrattiniC2×C4C24×D7
FittingC23×C7×M16C2
socleC23×C7C2×C4⋊C4
prime radicalC25×C7Q8
cosocleC2×C4×C7C25
hypercenterC23×M16D7
hypocenterC7C22×#430
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
4. C2×#89896C2direct (N × H)
5. C22×#6396C2semidirect
22. #89896C22direct (N × H)
23. C2×#6396C22semidirect
57. #6396C23semidirect
200. C22×C7#430semidirect
260. C2×C7C2×#430semidirect
298. C7C22×#430semidirect
305. C22#89896direct (N × H)
321. C2C2×#89896direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C7×M162
C23×#11362
C2×#898962
C22×#63962
C22×#4307