Finite Group Databasebeta

← C2×#11726325 #13322813

Subgroups — C2×#11726325 #13322813

Order 1792.

758 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89609C211yesmaximal, characteristic
2C2×#89607C211yesmaximal, characteristic
3C2×C4×#1131C211yesmaximal, characteristic
4C2×C7×#335C211yesmaximal, characteristic, complemented
5C2×#89274C211yesmaximal, characteristic, complemented
6C2×#89275C211yesmaximal, characteristic, complemented
7C2×#89252C211yesmaximal, characteristic, complemented
8#11726325C218yesmaximal, direct product factor
9C22×#1131—21no—
10C22×#1131C411yescharacteristic
11C22×#1131C411yescharacteristic
12C2×C42×D7C2211yescharacteristic
13C2×C7×C4⋊C8C2211yescharacteristic
14C2×C7×C4⋊C8C2211yescharacteristic
15C2×C4×C7×C8C2211yescharacteristic
16C2×(C2×D7)⋊C8—21no—
17C2×#6389—21no—
18C2×#6389C411yescharacteristic
19C2×#6389C411yescharacteristic
20C2×Dic7⋊C8—21no—
21C2×C4⋊(C7⋊C8)C2211yescharacteristic
22C2×C4⋊(C7⋊C8)C2211yescharacteristic
23C2×C4×C7⋊C8C2211yescharacteristic
24#89609—22no—
25#89609C2214yes—
26#89607—22no—
27#89607C2214yes—
28C4×#1131C414yes—
29C4×#1131C2214yes—
30C7×#335C2214yescomplemented
31#89275C2214yescomplemented
32#89274C2214yescomplemented
33#89252C2214yescomplemented
34C2×#1569—71nomaximal, Sylow
35C22×C4×D7C2×C411yescharacteristic
36C22×C4×D7D411yescharacteristic
37C22×C4×D7Q811yescharacteristic
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C22×C7×C8C2×C411yescharacteristic
41C22×C7×C8C2×C411yescharacteristic
42C2×C42×C7C2311yescharacteristic
43C2×C4×Dic7C2×C411yescharacteristic
44C2×C4×Dic7D411yescharacteristic
45C2×C4×Dic7Q811yescharacteristic
46C22×C7⋊C8—21no—
47C22×C7⋊C8—21no—
48C22×C7⋊C8C2×C411yescharacteristic
49C22×C7⋊C8C2×C411yescharacteristic
50C2×#1131—24no—
51C2×#1131—24no—
52C2×#1131—24no—
53C2×#1131—44no—
54C2×#1131C2×C414yes—
55C2×#1131C2×C414yes—
56C42×D7C2×C412yes—
57C42×D7D412yes—
58C42×D7Q812yes—
59C42×D7C2312yes—
60C7×C4⋊C8—21no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8C2312yes—
63C7×C4⋊C8C2312yes—
64C4×C7×C8C2×C412yes—
65C4×C7×C8C2312yes—
66(C2×D7)⋊C8—24no—
67#6389—24no—
68#6389C2×C414yes—
69#6389C2×C414yes—
70Dic7⋊C8—24no—
71C4⋊(C7⋊C8)—21no—
72C4⋊(C7⋊C8)—21no—
73C4⋊(C7⋊C8)C2312yes—
74C4⋊(C7⋊C8)C2312yes—
75C4×C7⋊C8C2×C412yes—
76C4×C7⋊C8C2312yes—
77C2×#432—71no—
78C2×#423—71no—
79C2×C4×M16—71no—
80#1569—78no—
81C2×#335—71no—
82C2×#335D711yescharacteristic, complemented
83C2×#334—71no—
84C2×#334—71no—
85C23×D7C4⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×Q811yescharacteristic
89C22×Dic7C4⋊C411yescharacteristic
90C22×Dic7C4⋊C411yescharacteristic
91C22×Dic7C4⋊C411yescharacteristic
92C2×C4×D7—22no—
93C2×C4×D7—22no—
94C2×C4×D7—22no—
95C2×C4×D7—22no—
96C2×C4×D7—42no—
97C2×C4×D7C4⋊C412yes—
98C2×C4×D7C4⋊C412yes—
99C2×C4×D7C4⋊C412yes—
100C2×C4×D7C22×C412yes—
101C2×C4×D7C2×D412yes—
102C2×C4×D7C2×Q812yes—
103C2×C7×C8—22no—
104C2×C7×C8—22no—
105C2×C7×C8—22no—
106C2×C7×C8—42no—
107C2×C7×C8—42no—
108C2×C7×C8M1614yescomplemented
109C2×C7×C8C22×C412yes—
110C2×C7×C8C22×C412yes—
111C42×C7C22×C411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7C2×Q811yescharacteristic
114C42×C7C2411yescharacteristic
115C4×Dic7C4⋊C412yes—
116C4×Dic7C4⋊C412yes—
117C4×Dic7C4⋊C412yes—
118C4×Dic7C22×C412yes—
119C4×Dic7C2×D412yes—
120C4×Dic7C2×Q812yes—
121C2×C7⋊C8—22no—
122C2×C7⋊C8—22no—
123C2×C7⋊C8—22no—
124C2×C7⋊C8—42no—
125C2×C7⋊C8—42no—
126C2×C7⋊C8M1614yescomplemented
127C2×C7⋊C8C22×C412yes—
128C2×C7⋊C8C22×C412yes—
129#1131—28no—
130#1131—28no—
131#1131—48no—
132C22×M16—71no—
133C22×M16—71no—
134C22×M16—141no—
135C22×C42—71no—
136#432—74no—
137#432—142no—
138#423—74no—
139#423—142no—
140C2×C4⋊C8—71no—
141C2×C4⋊C8—71no—
142C2×C4⋊C8—141no—
143C2×C4⋊C8C2×D711yescharacteristic
144C2×C4⋊C8C2×D711yescharacteristic
145C2×C22⋊C8—141no—
146C4×M16—74no—
147C4×M16—74no—
148C2×#97—71no—
149C2×#97—71no—
150C2×#97—141no—
151C2×C4×C8—71no—
152C2×C4×C8C2×D711yescharacteristic
153#335—74no—
154#335C2×D714yescomplemented
155#334—74no—
156#334—74no—
157C23×C7C2×C4⋊C411yescharacteristic
158C22×D7—22no—
159C22×D7—22no—
160C22×D7—22no—
161C22×D7C2×C4⋊C412yes—
162C2×C4×C7—22no—
163C2×C4×C7C2×C4⋊C411yescharacteristic
164C2×C4×C7C2×C4⋊C411yescharacteristic
165C2×C4×C7C2×C4⋊C411yescharacteristic
166C2×C4×C7C2×M1612yes—
167C2×C4×C7C2×M1612yes—
168C2×C4×C7AGL(1,ℤ/8)12yes—
169C2×C4×C7#13712yes—
170C2×C4×C7C23×C411yescharacteristic
171C2×C4×C7C22×D411yescharacteristic
172C2×C4×C7C22×Q811yescharacteristic
173C2×Dic7—22no—
174C2×Dic7—22no—
175C2×Dic7—22no—
176C2×Dic7C2×C4⋊C412yes—
177C2×Dic7C2×C4⋊C412yes—
178C2×Dic7C2×C4⋊C412yes—
179C4×D7—22no—
180C4×D7—22no—
181C4×D7—22no—
182C4×D7—22no—
183C4×D7—24no—
184C4×D7—24no—
185C4×D7—42no—
186C4×D7—42no—
187C7×C8—22no—
188C7×C8—42no—
189C7×C8—42no—
190C7×C8C2×M1614yescomplemented
191C7⋊C8—22no—
192C7⋊C8—42no—
193C7⋊C8—42no—
194C7⋊C8C2×M1614yescomplemented
195C23×C4—71no—
196C23×C4—71no—
197C23×C4—71no—
198C2×M16—74no—
199C2×M16—74no—
200C2×M16—144no—
201C2×M16—144no—
202C2×M16—144no—
203C2×M16—284no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—71no—
207C22×C8—71no—
208C22×C8—141no—
209C22×C8—141no—
210C22×C8C4×D711yescharacteristic
211C22×C8C4×D711yescharacteristic
212C2×C42—71no—
213C2×C42—71no—
214C2×C42—71no—
215C2×C42—72no—
216C2×C42—72no—
217C2×C42—72no—
218C2×C42—72no—
219C2×C42C22×D711yescharacteristic
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—72no—
223C4⋊C8—72no—
224C4⋊C8—144no—
225C4⋊C8—141no—
226C4⋊C8—141no—
227C4⋊C8C22×D712yes—
228C4⋊C8C22×D712yes—
229C22⋊C8—144no—
230#97—74no—
231#97—74no—
232#97—144no—
233C4×C8—72no—
234C4×C8—72no—
235C4×C8C4×D712yes—
236C4×C8C22×D712yes—
237C22×C7#42312yes—
238C22×C7#42812yes—
239C22×C7#43012yes—
240C22×C7C22×C4⋊C411yescharacteristic
241C2×D7—22no—
242C2×D7—22no—
243C2×D7—22no—
244C2×D7—44no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7#42312yes—
248C4×C7#42812yes—
249C4×C7C22×M1611yescharacteristic
250C4×C7C22×M1611yescharacteristic
251C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C7C2×#13711yescharacteristic
253Dic7—22no—
254Dic7—22no—
255Dic7—22no—
256C24—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—72no—
261C22×C4—72no—
262C22×C4—72no—
263C22×C4—72no—
264C22×C4—72no—
265C22×C4—72no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—282no—
271C22×C4C2×C4×D711yescharacteristic
272C22×C4D4×D711yescharacteristic
273C22×C4Q8×D711yescharacteristic
274M16—148no—
275M16—148no—
276M16—288no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—74no—
283C2×C8—72no—
284C2×C8—72no—
285C2×C8—142no—
286C2×C8—142no—
287C2×C8—142no—
288C2×C8—282no—
289C2×C8—282no—
290C2×C8#113114yescomplemented
291C2×C8C2×C4×D712yes—
292C2×C8C2×C4×D712yes—
293C42—72no—
294C42—72no—
295C42—72no—
296C42—72no—
297C42—72no—
298C42—72no—
299C42C2×C4×D711yescharacteristic
300C42D4×D711yescharacteristic
301C42Q8×D711yescharacteristic
302C42C23×D711yescharacteristic
303C2×C7#156914yescomplemented
304C2×C7C2×#42311yescharacteristic
305C2×C7C2×#42811yescharacteristic
306C2×C7C2×#43011yescharacteristic
307D7—42no—
308C23—72no—
309C23—142no—
310C23—142no—
311C23—142no—
312C23D7×C4⋊C411yescharacteristic
313C2×C4—22no—
314C2×C4—72no—
315C2×C4—72no—
316C2×C4—72no—
317C2×C4—142no—
318C2×C4—142no—
319C2×C4—142no—
320C2×C4—144no—
321C2×C4—144no—
322C2×C4—142no—
323C2×C4—142no—
324C2×C4—142no—
325C2×C4—142no—
326C2×C4—282no—
327C2×C4—282no—
328C2×C4D7×C4⋊C411yescharacteristic
329C2×C4D7×C4⋊C411yescharacteristic
330C2×C4D7×C4⋊C411yescharacteristic
331C2×C4C2×#113112yes—
332C2×C4D7×M1612yes—
333C2×C4#647412yes—
334C2×C4#648112yes—
335C2×C4C22×C4×D711yescharacteristic
336C2×C4C2×D4×D711yescharacteristic
337C2×C4C2×Q8×D711yescharacteristic
338C8—22no—
339C8—42no—
340C8—42no—
341C8—74no—
342C8—142no—
343C8—282no—
344C8—282no—
345C8C2×#113114yescomplemented
346C7C2×#156911yescharacteristic, complemented, Sylow
347C22—142no—
348C22—142no—
349C22—142no—
350C22—284no—
351C22#8960912yes—
352C22#8965312yes—
353C22#8966512yes—
354C22C2×D7×C4⋊C411yescharacteristic
355C4—21no—
356C4—21no—
357C4—142no—
358C4—142no—
359C4—142no—
360C4#8960912yes—
361C4#8965312yes—
362C4C22×#113111yescharacteristic
363C4C2×D7×M1611yescharacteristic
364C4C2×#647411yescharacteristic
365C4C2×#648111yescharacteristic
366C2—282no—
367C2#1172632514yesdirect product factor
368C2C2×#8960911yescharacteristic
369C2C2×#8965311yescharacteristic
370C2C2×#8966511yescharacteristic
subgroupG/H
centerC22×C4D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC2×C7×#335C2
socleC23×C7C2×C4⋊C4
prime radicalC23×D7C4⋊C4
cosocleC42×C7C24
hypercenterC2×#335D7
hypocenterC7C2×#1569
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
4. C2×C7×#335C2semidirect
5. C2×#89274C2semidirect
6. C2×#89275C2semidirect
7. C2×#89252C2semidirect
8. #11726325C2direct (N × H)
30. C7×#335C22semidirect
31. #89275C22semidirect
32. #89274C22semidirect
33. #89252C22semidirect
82. C2×#335D7semidirect
108. C2×C7×C8M16semidirect
126. C2×C7⋊C8M16semidirect
154. #335C2×D7semidirect
190. C7×C8C2×M16semidirect
194. C7⋊C8C2×M16semidirect
290. C2×C8#1131semidirect
303. C2×C7#1569semidirect
345. C8C2×#1131semidirect
346. C7C2×#1569semidirect
367. C2#11726325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#896092
C2×#896072
C2×C4×#11312
C2×C7×#3352
C2×#892742
C2×#892752
C2×#892522
#117263252
C2×#15697