Finite Group Databasebeta

← C2×#11740423 #13520895

Subgroups — C2×#11740423 #13520895

Order 1792.

2806 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#124C211yesmaximal, characteristic
2C22×#6432C211yesmaximal, characteristic, complemented
3C2×#90002C212yesmaximal, complemented
4C2×#89993C212yesmaximal
5#11740423C2116yesmaximal, direct product factor
6C2×#89725C214yesmaximal, complemented
7C2×#89719C214yesmaximal, complemented
8C2×#89704C211yesmaximal, characteristic, complemented
9C23×C7×Q8C2211yescharacteristic
10C23×C7×D4C2211yescharacteristic
11C23×Dic14C2212yes—
12C2×C7×#124—28no—
13C2×C7×#124C2218yes—
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C42×C7C2211yescharacteristic
17C22×C22⋊Dic7C2218yes—
18C22×C4⋊Dic7C2212yes—
19C22×C4⋊Dic7C2212yes—
20C22×C4×Dic7C2212yes—
21C2×#6432—28no—
22C2×#6432C2218yescomplemented
23#90002C22116yescomplemented
24#89993C22116yes—
25C2×C7⋊#95C2218yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28#89725C22132yescomplemented
29#89719C22132yescomplemented
30#89704C2218yescomplemented
31C2×#2640—71nomaximal, Sylow
32C25×C7C2312yes—
33C22×C7×Q8—24no—
34C22×C7×Q8—28no—
35C22×C7×Q8C2314yes—
36C22×C7×D4—24no—
37C22×C7×D4—28no—
38C22×C7×D4C2314yes—
39C23×C4×C7C2312yes—
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×Dic7C2314yes—
44C23×Dic7C2314yes—
45C22×Dic14—28no—
46C22×Dic14—48no—
47C22×Dic14C2318yes—
48C7×#124—232no—
49C2×C7×C22⋊C4—28no—
50C2×C7×C22⋊C4—28no—
51C2×C7×C22⋊C4C2318yes—
52C2×C7×C22⋊C4C2318yes—
53C2×C42×C7D418yes—
54C2×C42×C7C2314yes—
55C2×C22⋊Dic7—416no—
56C2×C22⋊Dic7C23132yes—
57C2×C4⋊Dic7—28no—
58C2×C4⋊Dic7—28no—
59C2×C4⋊Dic7C2318yes—
60C2×C4⋊Dic7C2318yes—
61C2×C4×Dic7—28no—
62C2×C4×Dic7C2318yes—
63#6432—232no—
64C7⋊#95C2318yes—
65C7⋊#95C2318yes—
66C7⋊#95C23132yes—
67C22×C4⋊Q8—71no—
68C22×#124D711yescharacteristic
69#2640—716no—
70C2×#397—74no—
71C2×#393—74no—
72C2×#391—72no—
73C2×#390—72no—
74C2×#376—71no—
75C24×C7—24no—
76C24×C7—28no—
77C24×C7—28no—
78C24×C7—28no—
79C24×C7C2414yes—
80C24×C7C2411yescharacteristic
81C2×C7×Q8—28no—
82C2×C7×Q8—216no—
83C2×C7×Q8—416no—
84C2×C7×D4—28no—
85C2×C7×D4—216no—
86C2×C7×D4—416no—
87C22×C4×C7—24no—
88C22×C4×C7—24no—
89C22×C4×C7—24no—
90C22×C4×C7—28no—
91C22×C4×C7C2×D414yes—
92C22×C4×C7C2×D414yes—
93C22×C4×C7C2×D414yes—
94C22×C4×C7#4118yes—
95C22×C4×C7C2414yes—
96C22×C4×C7C2412yes—
97C22×C4×C7C2412yes—
98C22×C4×C7C2412yes—
99C22×Dic7—28no—
100C22×Dic7—28no—
101C22×Dic7—28no—
102C22×Dic7—216no—
103C22×Dic7—28no—
104C22×Dic7C2418yes—
105C22×Dic7C2418yes—
106C2×Dic14—216no—
107C2×Dic14—416no—
108C2×Dic14—432no—
109C7×C22⋊C4—216no—
110C7×C22⋊C4—216no—
111C42×C7C2×D4116yes—
112C22⋊Dic7—432no—
113C4⋊Dic7—216no—
114C4⋊Dic7—216no—
115C4×Dic7—216no—
116C23×Q8—72no—
117C23×Q8C2×D711yescharacteristic
118C23×D4C2×D711yescharacteristic
119C2×C4⋊Q8—78no—
120C2×C4⋊Q8—148no—
121C2×#124—28no—
122C2×#124C2×D718yes—
123C22×C4⋊C4—72no—
124C22×C4⋊C4—72no—
125C22×C22⋊C4—78no—
126C22×C22⋊C4C2×D712yes—
127C22×C22⋊C4C2×D712yes—
128C22×C42—72no—
129C22×C42C2×D711yescharacteristic
130#397—732no—
131#393—732no—
132#391—716no—
133#390—716no—
134#376—78no—
135C2×#95—72no—
136C2×#95—78no—
137C2×#95—72no—
138C23×C7—24no—
139C23×C7—28no—
140C23×C7—28no—
141C23×C7—28no—
142C23×C7—48no—
143C23×C7—48no—
144C23×C7—48no—
145C23×C7—432no—
146C23×C7C22×D412yes—
147C23×C7C2×#4114yes—
148C23×C7H(5,𝔽2)14yes—
149C23×C72−1+414yes—
150C23×C7C2511yescharacteristic
151C7×Q8—416no—
152C7×D4—416no—
153C2×C4×C7—28no—
154C2×C4×C7—28no—
155C2×C4×C7—24no—
156C2×C4×C7—24no—
157C2×C4×C7—24no—
158C2×C4×C7—28no—
159C2×C4×C7—48no—
160C2×C4×C7#124116yes—
161C2×C4×C7C22×D414yes—
162C2×C4×C7C22×D414yes—
163C2×C4×C7C22×D414yes—
164C2×C4×C7C2×#4118yes—
165C2×Dic7—28no—
166C2×Dic7—28no—
167C2×Dic7—28no—
168C2×Dic7—216no—
169C2×Dic7—28no—
170C2×Dic7—416no—
171C2×Dic7—416no—
172Dic14—432no—
173C25C22×D712yes—
174C22×Q8—28no—
175C22×Q8—24no—
176C22×Q8—78no—
177C22×Q8—148no—
178C22×Q8—288no—
179C22×Q8C22×D714yes—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4C22×D714yes—
183C23×C4—74no—
184C23×C4—74no—
185C23×C4C22×D712yes—
186C23×C4C22×D711yescharacteristic
187C23×C4C22×D711yescharacteristic
188C23×C4C22×D711yescharacteristic
189C4⋊Q8—1432no—
190#124—232no—
191C2×C4⋊C4—78no—
192C2×C4⋊C4—78no—
193C2×C4⋊C4—148no—
194C2×C4⋊C4—148no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—732no—
198C2×C22⋊C4—2816no—
199C2×C22⋊C4C22×D718yes—
200C2×C22⋊C4C22×D718yes—
201C2×C42—78no—
202C2×C42—148no—
203C2×C42C22⋊D718yes—
204C2×C42C22×D714yes—
205#95—78no—
206#95—78no—
207#95—732no—
208C22×C7—44no—
209C22×C7—44no—
210C22×C7—44no—
211C22×C7—416no—
212C22×C7—816no—
213C22×C7C2×#12414yes—
214C22×C7#53514yes—
215C22×C7#53614yes—
216C22×C7#542116yes—
217C22×C7C23×D411yescharacteristic
218C22×C7C22×#4112yes—
219C22×C7C2×H(5,𝔽2)12yes—
220C22×C7C2×2−1+412yes—
221C4×C7—24no—
222C4×C7—24no—
223C4×C7—44no—
224C4×C7C2×#12418yes—
225Dic7—48no—
226Dic7—48no—
227C24—28no—
228C24—28no—
229C24—28no—
230C24—24no—
231C24C23×D714yes—
232C24C23×D711yescharacteristic
233C2×Q8—216no—
234C2×Q8—28no—
235C2×Q8—416no—
236C2×Q8—1416no—
237C2×Q8—2816no—
238C2×Q8—2832no—
239C2×D4—216no—
240C2×D4—28no—
241C2×D4—416no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—78no—
247C22×C4—78no—
248C22×C4—148no—
249C22×C4—148no—
250C22×C4—1416no—
251C22×C4—148no—
252C22×C4—148no—
253C22×C4#115918yes—
254C22×C4C2×C22⋊D714yes—
255C22×C4C2×C22⋊D714yes—
256C22×C4C2×C22⋊D714yes—
257C22×C4C23×D714yes—
258C22×C4C23×D712yes—
259C22×C4C23×D712yes—
260C22×C4C23×D712yes—
261C4⋊C4—1416no—
262C4⋊C4—1416no—
263C22⋊C4—216no—
264C22⋊C4—216no—
265C22⋊C4—2832no—
266C42—1416no—
267C42C2×C22⋊D7116yes—
268C2×C7—84no—
269C2×C7#264018yescomplemented
270C2×C7C22×#12411yescharacteristic
271C2×C7C2×#53511yescharacteristic
272C2×C7C2×#53611yescharacteristic
273C2×C7C2×#54214yes—
274C23—28no—
275C23—28no—
276C23—28no—
277C23—24no—
278C23—432no—
279C23—48no—
280C23—48no—
281C23—48no—
282C23C2×#115914yes—
283C23#655314yes—
284C23#655414yes—
285C23C22×C22⋊D712yes—
286C23C24×D711yescharacteristic
287Q8—416no—
288Q8—2832no—
289D4—416no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—28no—
295C2×C4—24no—
296C2×C4—48no—
297C2×C4—148no—
298C2×C4—148no—
299C2×C4—1416no—
300C2×C4—148no—
301C2×C4—148no—
302C2×C4—2816no—
303C2×C4—2816no—
304C2×C4#6499116yes—
305C2×C4C2×#115918yes—
306C2×C4C22×C22⋊D714yes—
307C2×C4C22×C22⋊D714yes—
308C2×C4C22×C22⋊D714yes—
309C7C2×#264011yescharacteristic, complemented, Sylow
310C22—44no—
311C22—44no—
312C22—44no—
313C22—416no—
314C22—816no—
315C22#90373116yes—
316C22C2×#649914yes—
317C22#9051614yes—
318C22#9052114yes—
319C22C22×#115912yes—
320C22C2×#655312yes—
321C22C2×#655412yes—
322C22C23×C22⋊D711yescharacteristic
323C4—24no—
324C4—24no—
325C4—44no—
326C4—288no—
327C4—288no—
328C4C2×#649918yes—
329C2—84no—
330C2#1174042318yesdirect product factor
331C2C2×#9037314yes—
332C2C22×#649911yescharacteristic
333C2C2×#9051611yescharacteristic
334C2C2×#9052111yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC22×C7×#124C2
socleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC22×#124D7
hypocenterC7C2×#2640
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
2. C22×#6432C2semidirect
3. C2×#90002C2semidirect
5. #11740423C2direct (N × H)
6. C2×#89725C2semidirect
7. C2×#89719C2semidirect
8. C2×#89704C2semidirect
22. C2×#6432C22semidirect
23. #90002C22semidirect
28. #89725C22semidirect
29. #89719C22semidirect
30. #89704C22semidirect
269. C2×C7#2640semidirect
309. C7C2×#2640semidirect
330. C2#11740423direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1242
C22×#64322
C2×#900022
C2×#899932
#117404232
C2×#897252
C2×#897192
C2×#897042
C2×#26407