Finite Group Databasebeta

← C2×#11727464 #13325922

Subgroups — C2×#11727464 #13325922

Order 1792.

1136 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1136 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×#90174C211yesmaximal, characteristic
2C2×C7×#397C211yesmaximal, characteristic
3C2×#89743C212yesmaximal, complemented
4C2×#89320C211yesmaximal, characteristic
5#11727464C218yesmaximal, direct product factor
6C2×#89247C212yesmaximal, complemented
7C22×C7×C4⋊C4C2211yescharacteristic
8C22×C7×C22⋊C4—21no—
9C22×C7×C22⋊C4C2211yescharacteristic
10C22×C22⋊Dic7—21no—
11C2×#6487—21no—
12C2×#6487C2212yes—
13#90174—22no—
14#90174—24no—
15#90174C2214yes—
16C2×#6441—22no—
17C2×#6441—22no—
18C7×#397—22no—
19C7×#397C2214yes—
20C2×C7×#95—21no—
21C2×C7×#95C2211yescharacteristic
22C2×C22⋊(C7⋊C8)C2212yes—
23#89743—24no—
24#89743C2218yescomplemented
25C2×#6384—22no—
26C2×#6382—22no—
27#89320C2214yes—
28#89247C2218yescomplemented
29C2×#1608—71nomaximal, Sylow
30C25×C7D411yescharacteristic
31C23×C4×C7—21no—
32C23×C4×C7—21no—
33C23×C4×C7D411yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C23×Dic7—21no—
36C23×Dic7—21no—
37C22×Dic14—21no—
38C22×Dic14D412yes—
39C2×C7×C4⋊C4—22no—
40C2×C7×C4⋊C4—22no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4C2312yes—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—22no—
45C2×C7×C22⋊C4—24no—
46C2×C7×C22⋊C4—44no—
47C2×C7×C22⋊C4D412yes—
48C2×C7×C22⋊C4C2312yes—
49C2×C22⋊Dic7—22no—
50C2×C22⋊Dic7—24no—
51C2×C22⋊Dic7—24no—
52C2×C22⋊Dic7—22no—
53C2×C4⋊Dic7—21no—
54C2×C4⋊Dic7D412yes—
55#6487—24no—
56#6487—22no—
57#6487—22no—
58#6487—24no—
59#6487D414yes—
60#6487C2314yes—
61C2×Dic7⋊C4—22no—
62C2×Dic7⋊C4—22no—
63C22×C7⋊C8—22no—
64#6441—28no—
65#6441—28no—
66#6441—28no—
67#6441—28no—
68C7×#95—24no—
69C7×#95—21no—
70C7×#95C2312yes—
71C22⋊(C7⋊C8)—22no—
72C22⋊(C7⋊C8)C2314yes—
73#6384—28no—
74#6382—28no—
75C2×#543—71no—
76C2×#483—72no—
77C2×#397D711yescharacteristic
78#1608—78no—
79C2×#328—72no—
80C2×#323—71no—
81C24×C7—21no—
82C24×C7—21no—
83C24×C7—22no—
84C24×C7—42no—
85C24×C7—42no—
86C24×C7SD1614yes—
87C24×C7C2×D411yescharacteristic
88C24×C7C2×D412yes—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—22no—
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×C7—42no—
102C22×C4×C7C2×D412yes—
103C22×C4×C7C2×D412yes—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D412yes—
106C22×C4×C7C2411yescharacteristic
107C22×Dic7—22no—
108C22×Dic7—22no—
109C22×Dic7—24no—
110C22×Dic7—24no—
111C22×Dic7—82no—
112C22×Dic7—82no—
113C2×Dic14—24no—
114C2×Dic14—22no—
115C2×Dic14—44no—
116C2×Dic14—82no—
117C2×Dic14C2×D414yes—
118C7×C4⋊C4—24no—
119C7×C4⋊C4—42no—
120C7×C22⋊C4—24no—
121C7×C22⋊C4—24no—
122C7×C22⋊C4—48no—
123C22⋊Dic7—28no—
124C22⋊Dic7—24no—
125C22⋊Dic7—24no—
126C4⋊Dic7—24no—
127C4⋊Dic7—22no—
128C4⋊Dic7C2×D414yes—
129Dic7⋊C4—28no—
130Dic7⋊C4—24no—
131Dic7⋊C4—24no—
132C2×C7⋊C8—24no—
133C2×C7⋊C8—44no—
134#543—74no—
135#543—144no—
136#543—142no—
137C2×C22⋊Q8—72no—
138C2×C22⋊Q8—142no—
139C2×C22⋊Q8—142no—
140C2×C22⋊Q8—141no—
141C22×C4⋊C4C2×D711yescharacteristic
142C22×C22⋊C4—21no—
143C22×C22⋊C4—141no—
144C22×C22⋊C4C2×D711yescharacteristic
145#483—78no—
146#483—144no—
147C2×#106—142no—
148C2×#103—142no—
149C2×C22⋊C8—72no—
150#397—22no—
151#397C2×D714yes—
152C2×#95—21no—
153C2×#95C2×D711yescharacteristic
154#328—78no—
155#323—74no—
156C23×C7—22no—
157C23×C7—22no—
158C23×C7—22no—
159C23×C7—24no—
160C23×C7—22no—
161C23×C7—24no—
162C23×C7—48no—
163C23×C7—42no—
164C23×C7—42no—
165C23×C7—42no—
166C23×C7—42no—
167C23×C7—84no—
168C23×C7—81no—
169C23×C7—81no—
170C23×C7#12011yescharacteristic
171C23×C7#12011yescharacteristic
172C23×C7C2×SD1612yes—
173C23×C7C2×SD1614yes—
174C23×C7#13712yes—
175C23×C7C22×D411yescharacteristic
176C2×C4×C7—24no—
177C2×C4×C7—22no—
178C2×C4×C7—42no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—81no—
190C2×C4×C7—84no—
191C2×C4×C7—82no—
192C2×C4×C7—82no—
193C2×C4×C7#12012yes—
194C2×C4×C7C22×D412yes—
195C2×Dic7—24no—
196C2×Dic7—24no—
197C2×Dic7—44no—
198C2×Dic7—44no—
199C2×Dic7—84no—
200C2×Dic7—84no—
201Dic14—44no—
202Dic14—82no—
203Dic14—84no—
204C7⋊C8—44no—
205C25C22⋊D711yescharacteristic
206C22×Q8—72no—
207C22×Q8—141no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—141no—
211C23×C4—141no—
212C23×C4C22⋊D711yescharacteristic
213C23×C4C22×D711yescharacteristic
214C22×C8—142no—
215C22⋊Q8—74no—
216C22⋊Q8—74no—
217C22⋊Q8—148no—
218C22⋊Q8—148no—
219C22⋊Q8—148no—
220C22⋊Q8—148no—
221C22⋊Q8—144no—
222C22⋊Q8—142no—
223C22⋊Q8—142no—
224C22⋊Q8—144no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—72no—
229C2×C4⋊C4—142no—
230C2×C4⋊C4—141no—
231C2×C4⋊C4—142no—
232C2×C4⋊C4C22×D712yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—142no—
238C2×C22⋊C4—144no—
239C2×C22⋊C4—142no—
240C2×C22⋊C4—144no—
241C2×C22⋊C4C22⋊D712yes—
242C2×C22⋊C4C22×D712yes—
243#106—148no—
244#103—148no—
245C22⋊C8—74no—
246C22⋊C8—142no—
247#95—24no—
248#95—21no—
249#95C22×D712yes—
250C22×C7—24no—
251C22×C7—24no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—24no—
257C22×C7—44no—
258C22×C7—48no—
259C22×C7—42no—
260C22×C7—42no—
261C22×C7—82no—
262C22×C7—84no—
263C22×C7#44812yes—
264C22×C7#44812yes—
265C22×C7#45512yes—
266C22×C7#45812yes—
267C22×C7C2×#12011yescharacteristic
268C22×C7C22×SD1611yescharacteristic
269C22×C7C2×#13711yescharacteristic
270C4×C7—22no—
271C4×C7—22no—
272C4×C7—42no—
273C4×C7—82no—
274C4×C7—82no—
275Dic7—44no—
276Dic7—44no—
277C24—21no—
278C24—21no—
279C24—22no—
280C24—42no—
281C24—42no—
282C24#114214yes—
283C24C2×C22⋊D711yescharacteristic
284C24C2×C22⋊D712yes—
285C2×Q8—74no—
286C2×Q8—144no—
287C2×Q8—142no—
288C2×Q8—284no—
289C2×Q8—562no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—142no—
304C22×C4—142no—
305C22×C4—144no—
306C22×C4—144no—
307C22×C4—562no—
308C22×C4—562no—
309C22×C4D4×D712yes—
310C22×C4D4×D712yes—
311C22×C4C2×C22⋊D712yes—
312C22×C4C2×C22⋊D711yescharacteristic
313C22×C4C23×D711yescharacteristic
314C2×C8—144no—
315C2×C8—284no—
316C4⋊C4—24no—
317C4⋊C4—42no—
318C4⋊C4—74no—
319C4⋊C4—148no—
320C4⋊C4—144no—
321C4⋊C4—144no—
322C4⋊C4—142no—
323C4⋊C4—144no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—48no—
327C22⋊C4—148no—
328C22⋊C4—144no—
329C22⋊C4—144no—
330C2×C7—21no—
331C2×C7—21no—
332C2×C7—22no—
333C2×C7—44no—
334C2×C7#160814yescomplemented
335C2×C7C2×#44811yescharacteristic
336C2×C7C2×#45511yescharacteristic
337C2×C7C2×#45811yescharacteristic
338C23—22no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—22no—
343C23—24no—
344C23—48no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—84no—
350C23—81no—
351C23—81no—
352C23C2×#114212yes—
353C23C2×#114214yes—
354C23#650011yescharacteristic
355C23#650011yescharacteristic
356C23#651412yes—
357C23C22×C22⋊D711yescharacteristic
358Q8—284no—
359Q8—562no—
360Q8—564no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—144no—
379C2×C4—144no—
380C2×C4—284no—
381C2×C4—284no—
382C2×C4—564no—
383C2×C4—564no—
384C2×C4#650012yes—
385C2×C4C2×D4×D712yes—
386C8—284no—
387C7C2×#160811yescharacteristic, complemented, Sylow
388C22—24no—
389C22—24no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—44no—
396C22—48no—
397C22—42no—
398C22—42no—
399C22—82no—
400C22—84no—
401C22#8981212yes—
402C22#8981212yes—
403C22#8982512yes—
404C22#8983312yes—
405C22C22×#114211yescharacteristic
406C22C2×#650011yescharacteristic
407C22C2×#651411yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—82no—
412C4—82no—
413C4—284no—
414C4—284no—
415C2—21no—
416C2—21no—
417C2—22no—
418C2—44no—
419C2#1172746414yesdirect product factor
420C2C2×#8981211yescharacteristic
421C2C2×#8982511yescharacteristic
422C2C2×#8983311yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#397C2
socleC23×C7#120
prime radicalC25×C7D4
hypercenterC2×#397D7
hypocenterC7C2×#1608
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
3. C2×#89743C2semidirect
5. #11727464C2direct (N × H)
6. C2×#89247C2semidirect
24. #89743C22semidirect
28. #89247C22semidirect
334. C2×C7#1608semidirect
387. C7C2×#1608semidirect
419. C2#11727464direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#901742
C2×C7×#3972
C2×#897432
C2×#893202
#117274642
C2×#892472
C2×#16087