Finite Group Databasebeta

← C2×#11731689 #13520807

Subgroups — C2×#11731689 #13520807

Order 1792.

4342 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6458C212yesmaximal, complemented
2C22×#6437C211yesmaximal, characteristic, complemented
3C22×#6435C211yesmaximal, characteristic, complemented
4C2×C7×#382C211yesmaximal, characteristic, complemented
5C2×#89760C212yesmaximal, complemented
6#11731689C2116yesmaximal, direct product factor
7C2×#89443C214yesmaximal, complemented
8C2×#89444C214yesmaximal, complemented
9C23×D28C2212yes—
10C23×D28C2214yes—
11C23×D28C2214yes—
12C23×C4×D7C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C22×C42×C7C2211yescharacteristic
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2214yes—
17C22×(C2×D7)⋊C4C2218yes—
18C22×Dic7⋊C4C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C2×#6458—216no—
21C2×#6458C22116yescomplemented
22C2×#6437—28no—
23C2×#6437C2218yescomplemented
24C2×#6435—28no—
25C2×#6435C2218yescomplemented
26C7×#382C2218yescomplemented
27C2×C7×#95C2214yes—
28#89760C22116yescomplemented
29#89444C22132yescomplemented
30#89443C22132yescomplemented
31C2×#2619—71nomaximal, Sylow
32C24×D7C2314yes—
33C24×D7C2311yescharacteristic
34C23×C4×C7C2314yes—
35C23×C4×C7C2312yes—
36C23×C4×C7C2311yescharacteristic
37C23×Dic7C2312yes—
38C23×Dic7C2311yescharacteristic
39C22×D28—216no—
40C22×D28—216no—
41C22×D28—28no—
42C22×D28—28no—
43C22×D28—28no—
44C22×D28—416no—
45C22×D28—416no—
46C22×D28C2318yes—
47C22×D28C23116yes—
48C22×D28C23116yes—
49C22×C4×D7—24no—
50C22×C4×D7—24no—
51C22×C4×D7D418yescomplemented
52C22×C4×D7C2314yes—
53C2×C7×C4⋊C4—28no—
54C2×C7×C4⋊C4C2318yes—
55C2×C42×C7D418yescomplemented
56C2×C42×C7C2314yes—
57C2×(C2×D7)⋊C4—28no—
58C2×(C2×D7)⋊C4—216no—
59C2×(C2×D7)⋊C4—416no—
60C2×(C2×D7)⋊C4C2318yes—
61C2×(C2×D7)⋊C4C23116yes—
62C2×(C2×D7)⋊C4C23132yes—
63C2×Dic7⋊C4—28no—
64C2×Dic7⋊C4C2318yes—
65C2×C4×Dic7—24no—
66C2×C4×Dic7C2314yes—
67#6458—264no—
68#6437—232no—
69#6435—232no—
70C7×#95C23116yes—
71C22×#127—71no—
72C22×#121—72no—
73C22×#117—71no—
74#2619—716no—
75C2×#394—74no—
76C2×#392—74no—
77C2×#390—72no—
78C2×#382D711yescharacteristic, complemented
79C24×C7C2411yescharacteristic
80C23×D7—22no—
81C23×D7—22no—
82C23×D7—28no—
83C23×D7—28no—
84C23×D7—28no—
85C23×D7—28no—
86C23×D7—216no—
87C23×D7—216no—
88C23×D7C2×D414yes—
89C23×D7C2418yes—
90C23×D7C2412yes—
91C22×C4×C7—28no—
92C22×C4×C7—28no—
93C22×C4×C7—216no—
94C22×C4×C7C2×D418yes—
95C22×C4×C7C2×D414yes—
96C22×C4×C7C2×D414yes—
97C22×C4×C7#4114yes—
98C22×C4×C7C2418yes—
99C22×C4×C7C2414yes—
100C22×C4×C7C2412yes—
101C22×Dic7—22no—
102C22×Dic7—22no—
103C22×Dic7—24no—
104C22×Dic7—28no—
105C22×Dic7C2×D414yes—
106C22×Dic7C2414yes—
107C22×Dic7C2412yes—
108C2×D28—232no—
109C2×D28—232no—
110C2×D28—216no—
111C2×D28—216no—
112C2×D28—216no—
113C2×D28—432no—
114C2×D28—432no—
115C2×D28—464no—
116C2×D28—464no—
117C2×D28—432no—
118C2×C4×D7—232no—
119C2×C4×D7—28no—
120C2×C4×D7—28no—
121C2×C4×D7C2×D4116yescomplemented
122C7×C4⋊C4—216no—
123C42×C7C2×D4116yescomplemented
124(C2×D7)⋊C4—216no—
125(C2×D7)⋊C4—232no—
126(C2×D7)⋊C4—432no—
127Dic7⋊C4—216no—
128C4×Dic7—28no—
129C23×D4—72no—
130C23×D4—74no—
131C23×D4—74no—
132C24×C4—71no—
133C2×#127—78no—
134C2×#127—148no—
135C2×#121—716no—
136C2×#121—1416no—
137C2×#117—78no—
138C2×#117—148no—
139C22×C4⋊C4—72no—
140C22×C4⋊C4C2×D712yes—
141C22×C22⋊C4—72no—
142C22×C22⋊C4—74no—
143C22×C22⋊C4—78no—
144C22×C42—71no—
145C22×C42C2×D711yescharacteristic
146#394—732no—
147#392—732no—
148#390—716no—
149#382C2×D718yescomplemented
150C2×#95C2×D714yes—
151C23×C7C22×D412yes—
152C23×C7C22×D412yes—
153C23×C7C2×#4112yes—
154C23×C7H(5,𝔽2)18yes—
155C23×C7C2511yescharacteristic
156C22×D7—28no—
157C22×D7—216no—
158C22×D7—22no—
159C22×D7—22no—
160C22×D7—28no—
161C22×D7—28no—
162C22×D7—28no—
163C22×D7—28no—
164C22×D7—216no—
165C22×D7—216no—
166C22×D7—48no—
167C22×D7—48no—
168C22×D7—416no—
169C22×D7—432no—
170C22×D7—416no—
171C22×D7—416no—
172C22×D7—432no—
173C22×D7C22×D414yes—
174C2×C4×C7—216no—
175C2×C4×C7—28no—
176C2×C4×C7—28no—
177C2×C4×C7—216no—
178C2×C4×C7—416no—
179C2×C4×C7#121116yes—
180C2×C4×C7C22×D418yes—
181C2×C4×C7C22×D414yes—
182C2×C4×C7C22×D414yes—
183C2×C4×C7C2×#4114yes—
184C2×Dic7—28no—
185C2×Dic7—22no—
186C2×Dic7—22no—
187C2×Dic7—24no—
188C2×Dic7—28no—
189C2×Dic7—48no—
190C2×Dic7C22×D414yes—
191D28—464no—
192D28—464no—
193D28—432no—
194C4×D7—232no—
195C25—71no—
196C25—74no—
197C22×D4—78no—
198C22×D4—716no—
199C22×D4—716no—
200C22×D4—1416no—
201C22×D4—1416no—
202C22×D4—148no—
203C22×D4—148no—
204C22×D4—148no—
205C22×D4—2816no—
206C22×D4—2816no—
207C23×C4—71no—
208C23×C4—78no—
209C23×C4—74no—
210C23×C4—72no—
211C23×C4—144no—
212C23×C4—144no—
213C23×C4C22×D714yes—
214C23×C4C22×D712yes—
215C23×C4C22×D711yescharacteristic
216#127—1432no—
217#121—1464no—
218#117—1432no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—78no—
221C2×C4⋊C4—148no—
222C2×C4⋊C4C22×D718yes—
223C2×C22⋊C4—78no—
224C2×C22⋊C4—716no—
225C2×C22⋊C4—732no—
226C2×C22⋊C4—1416no—
227C2×C22⋊C4—148no—
228C2×C22⋊C4—2816no—
229C2×C42—74no—
230C2×C42—144no—
231C2×C42D2818yescomplemented
232C2×C42C22×D714yes—
233#95C22×D7116yes—
234C22×C7C2×#12114yes—
235C22×C7#53518yes—
236C22×C7#53518yes—
237C22×C7#54018yes—
238C22×C7C23×D411yescharacteristic
239C22×C7C23×D411yescharacteristic
240C22×C7C22×#4111yescharacteristic
241C22×C7C2×H(5,𝔽2)14yes—
242C2×D7—24no—
243C2×D7—28no—
244C2×D7—416no—
245C2×D7—44no—
246C2×D7—44no—
247C2×D7—48no—
248C2×D7—416no—
249C2×D7—48no—
250C2×D7—48no—
251C2×D7—416no—
252C2×D7—832no—
253C4×C7—28no—
254C4×C7—48no—
255C4×C7C2×#12118yes—
256Dic7—24no—
257Dic7—44no—
258C24—74no—
259C24—72no—
260C24—78no—
261C24—142no—
262C24—142no—
263C24—148no—
264C24—148no—
265C24—148no—
266C24—148no—
267C24—1416no—
268C24—1416no—
269C24C23×D711yescharacteristic
270C2×D4—1432no—
271C2×D4—1432no—
272C2×D4—1416no—
273C2×D4—1416no—
274C2×D4—1416no—
275C2×D4—2832no—
276C2×D4—2832no—
277C2×D4—2864no—
278C2×D4—2864no—
279C2×D4—2832no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—216no—
283C22×C4—74no—
284C22×C4—72no—
285C22×C4—716no—
286C22×C4—74no—
287C22×C4—142no—
288C22×C4—142no—
289C22×C4—1432no—
290C22×C4—148no—
291C22×C4—148no—
292C22×C4—144no—
293C22×C4—148no—
294C22×C4C2×D2818yes—
295C22×C4C2×D2814yes—
296C22×C4D4×D714yes—
297C22×C4#116114yes—
298C22×C4C23×D718yes—
299C22×C4C23×D714yes—
300C22×C4C23×D712yes—
301C4⋊C4—216no—
302C4⋊C4—1416no—
303C22⋊C4—1432no—
304C22⋊C4—1416no—
305C22⋊C4—2832no—
306C42—148no—
307C42C2×D28116yescomplemented
308C2×C7#261918yescomplemented
309C2×C7C22×#12111yescharacteristic
310C2×C7C2×#53512yes—
311C2×C7C2×#53512yes—
312C2×C7C2×#54012yes—
313D7—44no—
314D7—88no—
315C23—74no—
316C23—148no—
317C23—142no—
318C23—142no—
319C23—148no—
320C23—1416no—
321C23—148no—
322C23—148no—
323C23—148no—
324C23—1416no—
325C23—1416no—
326C23—288no—
327C23—288no—
328C23—2832no—
329C23—2816no—
330C23—2832no—
331C23—2816no—
332C23—2816no—
333C23C22×D2812yes—
334C23C2×D4×D712yes—
335C23C2×#116112yes—
336C23#655318yes—
337C23C24×D711yescharacteristic
338D4—2864no—
339D4—2864no—
340D4—2832no—
341C2×C4—216no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—216no—
345C2×C4—416no—
346C2×C4—74no—
347C2×C4—148no—
348C2×C4—142no—
349C2×C4—142no—
350C2×C4—1432no—
351C2×C4—144no—
352C2×C4—148no—
353C2×C4—288no—
354C2×C4#6458116yes—
355C2×C4C22×D2818yes—
356C2×C4C22×D2814yes—
357C2×C4C2×D4×D714yes—
358C2×C4C2×#116114yes—
359C7C2×#261911yescharacteristic, complemented, Sylow
360C22—144no—
361C22—148no—
362C22—284no—
363C22—284no—
364C22—2816no—
365C22—288no—
366C22—2816no—
367C22—288no—
368C22—288no—
369C22—2816no—
370C22—5632no—
371C22C2×#645814yes—
372C22#9021618yes—
373C22#9036518yes—
374C22#9038918yes—
375C22C23×D2811yescharacteristic
376C22C22×D4×D711yescharacteristic
377C22C22×#116111yescharacteristic
378C22C2×#655314yes—
379C4—28no—
380C4—48no—
381C4—144no—
382C4—284no—
383C4C2×#645818yes—
384C2—284no—
385C2—568no—
386C2#1173168918yesdirect product factor
387C2C22×#645811yescharacteristic
388C2C2×#9021612yes—
389C2C2×#9036512yes—
390C2C2×#9038912yes—
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#382C2
socleC24×C7C24
hypercenterC2×#382D7
hypocenterC7C2×#2619
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
1. C22×#6458C2semidirect
2. C22×#6437C2semidirect
3. C22×#6435C2semidirect
4. C2×C7×#382C2semidirect
5. C2×#89760C2semidirect
6. #11731689C2direct (N × H)
7. C2×#89443C2semidirect
8. C2×#89444C2semidirect
21. C2×#6458C22semidirect
23. C2×#6437C22semidirect
25. C2×#6435C22semidirect
26. C7×#382C22semidirect
28. #89760C22semidirect
29. #89444C22semidirect
30. #89443C22semidirect
51. C22×C4×D7D4semidirect
55. C2×C42×C7D4semidirect
78. C2×#382D7semidirect
121. C2×C4×D7C2×D4semidirect
123. C42×C7C2×D4semidirect
149. #382C2×D7semidirect
231. C2×C42D28semidirect
307. C42C2×D28semidirect
308. C2×C7#2619semidirect
359. C7C2×#2619semidirect
386. C2#11731689direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64582
C22×#64372
C22×#64352
C2×C7×#3822
C2×#897602
#117316892
C2×#894432
C2×#894442
C2×#26197