Finite Group Databasebeta

← C2×#11727466 #13326012

Subgroups — C2×#11727466 #13326012

Order 1792.

1308 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1308 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90201C211yesmaximal, characteristic
2C2×C7×#398C211yesmaximal, characteristic, complemented
3C2×#89739C212yesmaximal, complemented
4C2×#89323C211yesmaximal, characteristic, complemented
5#11727466C218yesmaximal, direct product factor
6C2×#89246C212yesmaximal, complemented
7C22×#1157—21no—
8C22×C7×C4⋊C4—21no—
9C22×C7×C4⋊C4C2211yescharacteristic
10C22×C7×C4⋊C4C2211yescharacteristic
11C2×#6493—21no—
12C2×#6493C2212yes—
13#90201—22no—
14#90201—24no—
15#90201C2214yes—
16C2×#6460—22no—
17C2×#6458—22no—
18C7×#398—22no—
19C7×#398C2214yescomplemented
20C2×C7×#95—21no—
21C2×C7×#95C2211yescharacteristic
22C2×C22⋊(C7⋊C8)C2212yes—
23#89739—24no—
24#89739C2218yescomplemented
25C2×#6383—22no—
26C2×#6381—22no—
27#89323C2214yescomplemented
28#89246C2218yescomplemented
29C2×#1610—71nomaximal, Sylow
30C23×C4×C7—21no—
31C23×C4×C7—21no—
32C23×C4×C7D411yescharacteristic
33C23×C4×C7D411yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C22×C22⋊D7—22no—
36C2×#1157—24no—
37C2×#1157—24no—
38C2×#1157—44no—
39C2×#1157—44no—
40C22×D28—21no—
41C22×D28—21no—
42C22×D28D412yes—
43C22×C4×D7—22no—
44C22×Dic14—21no—
45C2×C7×C4⋊C4—22no—
46C2×C7×C4⋊C4—22no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—24no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—44no—
52C2×C7×C4⋊C4D412yes—
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2312yes—
55#6493—24no—
56#6493—22no—
57#6493—22no—
58#6493—24no—
59#6493D414yes—
60#6493C2314yes—
61C2×(C2×D7)⋊C4—22no—
62C2×(C2×D7)⋊C4—22no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7D412yes—
65C22×C7⋊C8—22no—
66#6460—28no—
67#6460—28no—
68#6458—28no—
69#6458—28no—
70C7×#95—24no—
71C7×#95—21no—
72C7×#95C2312yes—
73C22⋊(C7⋊C8)—22no—
74C22⋊(C7⋊C8)C2314yes—
75#6383—28no—
76#6381—28no—
77C2×#537—71no—
78C2×#480—72no—
79C2×#398D711yescharacteristic, complemented
80#1610—78no—
81C2×#327—72no—
82C2×#324—71no—
83C24×C7C2×D411yescharacteristic
84C23×D7—22no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—22no—
88C22×C4×C7—24no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—21no—
92C22×C4×C7—22no—
93C22×C4×C7—42no—
94C22×C4×C7—42no—
95C22×C4×C7—42no—
96C22×C4×C7—42no—
97C22×C4×C7—41no—
98C22×C4×C7—41no—
99C22×C4×C7—42no—
100C22×C4×C7—42no—
101C22×C4×C7D814yes—
102C22×C4×C7C2×D412yes—
103C22×C4×C7C2×D412yes—
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D412yes—
107C22×C4×C7C2411yescharacteristic
108C2×C22⋊D7—24no—
109C2×C22⋊D7—24no—
110C2×C22⋊D7—44no—
111C2×C22⋊D7—44no—
112C22×Dic7—22no—
113#1157—48no—
114#1157—416no—
115#1157—48no—
116C2×D28—24no—
117C2×D28—24no—
118C2×D28—22no—
119C2×D28—44no—
120C2×D28—44no—
121C2×D28—82no—
122C2×D28C2×D414yes—
123C2×C4×D7—28no—
124C2×C4×D7—48no—
125C2×Dic14—24no—
126C2×Dic14—44no—
127C7×C4⋊C4—24no—
128C7×C4⋊C4—24no—
129C7×C4⋊C4—24no—
130C7×C4⋊C4—42no—
131C7×C4⋊C4—48no—
132(C2×D7)⋊C4—28no—
133(C2×D7)⋊C4—24no—
134(C2×D7)⋊C4—24no—
135C4⋊Dic7—24no—
136C4⋊Dic7—22no—
137C4⋊Dic7C2×D414yes—
138C2×C7⋊C8—24no—
139C2×C7⋊C8—44no—
140C22×#41—141no—
141#537—74no—
142#537—144no—
143#537—142no—
144C2×C22⋊Q8—142no—
145C2×#121—72no—
146C2×#121—142no—
147C2×#121—141no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4C2×D711yescharacteristic
150C22×C4⋊C4C2×D711yescharacteristic
151#480—78no—
152#480—144no—
153C2×#107—142no—
154C2×#102—142no—
155C2×C22⋊C8—72no—
156#398—22no—
157#398C2×D714yescomplemented
158C2×#95—21no—
159C2×#95C2×D711yescharacteristic
160#327—78no—
161#324—74no—
162C23×C7—22no—
163C23×C7—22no—
164C23×C7#12011yescharacteristic
165C23×C7#12011yescharacteristic
166C23×C7C2×D812yes—
167C23×C7AGL(1,ℤ/8)12yes—
168C23×C7C22×D411yescharacteristic
169C22×D7—24no—
170C22×D7—44no—
171C22×D7—44no—
172C22×D7—44no—
173C2×C4×C7—22no—
174C2×C4×C7—22no—
175C2×C4×C7—24no—
176C2×C4×C7—24no—
177C2×C4×C7—24no—
178C2×C4×C7—24no—
179C2×C4×C7—22no—
180C2×C4×C7—42no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—48no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—81no—
192C2×C4×C7—82no—
193C2×C4×C7—82no—
194C2×C4×C7#12012yes—
195C2×C4×C7C2×D814yes—
196C2×C4×C7C22×D412yes—
197C22⋊D7—44no—
198C22⋊D7—44no—
199C22⋊D7—48no—
200C2×Dic7—24no—
201C2×Dic7—44no—
202D28—44no—
203D28—48no—
204D28—82no—
205D28—84no—
206C4×D7—416no—
207Dic14—48no—
208C7⋊C8—44no—
209C2×#41—144no—
210C2×#41—144no—
211C2×#41—284no—
212C2×#41—284no—
213C22×Q8—141no—
214C22×D4—72no—
215C22×D4—141no—
216C22×D4—141no—
217C22×D4—142no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—142no—
221C23×C4C22⋊D711yescharacteristic
222C23×C4C22⋊D711yescharacteristic
223C23×C4C22×D711yescharacteristic
224C22×C8—142no—
225C22⋊Q8—148no—
226C22⋊Q8—148no—
227#121—74no—
228#121—74no—
229#121—148no—
230#121—148no—
231#121—144no—
232#121—142no—
233#121—142no—
234#121—144no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—72no—
243C2×C4⋊C4—141no—
244C2×C4⋊C4C22⋊D712yes—
245C2×C4⋊C4C22×D712yes—
246C2×C4⋊C4C22×D712yes—
247C2×C22⋊C4—142no—
248C2×C22⋊C4—142no—
249#107—148no—
250#102—148no—
251C22⋊C8—74no—
252C22⋊C8—142no—
253#95—24no—
254#95—21no—
255#95C22×D712yes—
256C22×C7—24no—
257C22×C7—24no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7#44712yes—
263C22×C7#44712yes—
264C22×C7#45612yes—
265C22×C7#45812yes—
266C22×C7C2×#12011yescharacteristic
267C22×C7C22×D811yescharacteristic
268C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
269C2×D7—44no—
270C2×D7—44no—
271C2×D7—44no—
272C2×D7—88no—
273C4×C7—24no—
274C4×C7—22no—
275C4×C7—22no—
276C4×C7—44no—
277C4×C7—42no—
278C4×C7—82no—
279Dic7—44no—
280C24—142no—
281C24C2×C22⋊D711yescharacteristic
282#41—288no—
283#41—2816no—
284#41—288no—
285C2×Q8—144no—
286C2×Q8—284no—
287C2×D4—74no—
288C2×D4—144no—
289C2×D4—144no—
290C2×D4—144no—
291C2×D4—144no—
292C2×D4—142no—
293C2×D4—284no—
294C2×D4—284no—
295C2×D4—284no—
296C2×D4—284no—
297C2×D4—562no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—148no—
315C22×C4—142no—
316C22×C4—288no—
317C22×C4#114114yes—
318C22×C4D4×D712yes—
319C22×C4D4×D712yes—
320C22×C4C2×C22⋊D712yes—
321C22×C4C2×C22⋊D712yes—
322C22×C4C2×C22⋊D711yescharacteristic
323C22×C4C23×D711yescharacteristic
324C2×C8—144no—
325C2×C8—284no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—48no—
330C4⋊C4—42no—
331C4⋊C4—74no—
332C4⋊C4—144no—
333C4⋊C4—142no—
334C22⋊C4—148no—
335C22⋊C4—144no—
336C22⋊C4—144no—
337C2×C7—21no—
338C2×C7—21no—
339C2×C7—22no—
340C2×C7#161014yescomplemented
341C2×C7C2×#44711yescharacteristic
342C2×C7C2×#45611yescharacteristic
343C2×C7C2×#45811yescharacteristic
344D7—84no—
345C23—22no—
346C23—22no—
347C23—144no—
348C23—284no—
349C23—284no—
350C23—284no—
351C23C2×#114112yes—
352C23#650011yescharacteristic
353C23#650011yescharacteristic
354C23#651212yes—
355C23C22×C22⋊D711yescharacteristic
356Q8—288no—
357D4—284no—
358D4—288no—
359D4—284no—
360D4—284no—
361D4—288no—
362D4—562no—
363D4—564no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—144no—
386C2×C4—2816no—
387C2×C4—284no—
388C2×C4C2×#114114yes—
389C2×C4#650012yes—
390C2×C4C2×D4×D712yes—
391C8—284no—
392C7C2×#161011yescharacteristic, complemented, Sylow
393C22—24no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—284no—
400C22—284no—
401C22—284no—
402C22—568no—
403C22#8980812yes—
404C22#8980812yes—
405C22#8982512yes—
406C22#8986712yes—
407C22C22×#114111yescharacteristic
408C22C2×#650011yescharacteristic
409C22C2×#651211yescharacteristic
410C4—24no—
411C4—22no—
412C4—22no—
413C4—44no—
414C4—42no—
415C4—82no—
416C4—284no—
417C2—21no—
418C2—21no—
419C2—22no—
420C2—564no—
421C2#1172746614yesdirect product factor
422C2C2×#8980811yescharacteristic
423C2C2×#8982511yescharacteristic
424C2C2×#8986711yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#398C2
socleC23×C7#120
prime radicalC2×#90201C2
hypercenterC2×#398D7
hypocenterC7C2×#1610
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
2. C2×C7×#398C2semidirect
3. C2×#89739C2semidirect
4. C2×#89323C2semidirect
5. #11727466C2direct (N × H)
6. C2×#89246C2semidirect
19. C7×#398C22semidirect
24. #89739C22semidirect
27. #89323C22semidirect
28. #89246C22semidirect
79. C2×#398D7semidirect
157. #398C2×D7semidirect
340. C2×C7#1610semidirect
392. C7C2×#1610semidirect
421. C2#11727466direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902012
C2×C7×#3982
C2×#897392
C2×#893232
#117274662
C2×#892462
C2×#16107