Finite Group Databasebeta

← C2×#22587794 #31979756

Subgroups — C2×#22587794 #31979756

Order 18176.

726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986351C211yesmaximal, characteristic, complemented
2C2×#17986286C211yesmaximal, characteristic
3C2×#17986280C211yesmaximal, characteristic
4C2×#17986064C211yesmaximal, characteristic
5#22587794C218yesmaximal, direct product factor
6C2×#17985751C211yesmaximal, characteristic, complemented
7C2×#17985746C211yesmaximal, characteristic, complemented
8C2×#333×C71C211yesmaximal, characteristic, complemented
9C22×C8×D71—21no—
10C2×#14556128C2211yescharacteristic
11C2×C4×D284C2211yescharacteristic
12#17986351—22no—
13#17986351C2214yescomplemented
14#17986286—22no—
15#17986286C2214yes—
16#17986280—22no—
17#17986280C2214yes—
18#17986064—22no—
19#17986064C2214yes—
20C2×#14556062—21no—
21C2×#14556059—21no—
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×#14556051—21no—
24C2×#14556051—21no—
25C2×#14556050—21no—
26C2×#14556048—21no—
27C2×(C2×D71)⋊C8—21no—
28C2×C4×C71⋊C8C2211yescharacteristic
29#17985751C2214yescomplemented
30#17985746C2214yescomplemented
31C2×C4⋊Q8×C71C2211yescharacteristic
32C2×C4⋊C8×C71C2211yescharacteristic
33C2×C4⋊C8×C71C2211yescharacteristic
34#333×C71C2214yescomplemented
35C22×Dic142—21no—
36C22×D284D411yescharacteristic
37C22×C4×D71—21no—
38C22×C71⋊C8—21no—
39C22×C71⋊C8D411yescharacteristic
40C22×C71⋊C8D411yescharacteristic
41C2×C8×D71—24no—
42C2×C8×D71—44no—
43#14556128—21no—
44#14556128—21no—
45#14556128—21no—
46#14556128C2312yes—
47C4×D284—21no—
48C4×D284—21no—
49C4×D284D412yes—
50C4×D284C2312yes—
51C2×C4⋊Dic71—21no—
52C2×C4⋊Dic71—21no—
53C2×C4⋊Dic71D411yescharacteristic
54C2×(C2×D71)⋊C4—21no—
55#14556062—24no—
56#14556059—24no—
57C4⋊(C71⋊C8)—21no—
58C4⋊(C71⋊C8)C2312yes—
59#14556051—24no—
60#14556051—24no—
61#14556050—24no—
62#14556048—24no—
63(C2×D71)⋊C8—24no—
64C4×C71⋊C8D412yes—
65C4×C71⋊C8C2312yes—
66C22×Q8×C71—21no—
67C22×C8×C71—21no—
68C22×C8×C71—21no—
69C4⋊Q8×C71—21no—
70C4⋊Q8×C71—21no—
71C4⋊Q8×C71—21no—
72C4⋊Q8×C71C2312yes—
73C2×C4⋊C4×C71—21no—
74C2×C4⋊C4×C71—21no—
75C2×C42×C71C2311yescharacteristic
76C4⋊C8×C71—21no—
77C4⋊C8×C71—21no—
78C4⋊C8×C71C2312yes—
79C4⋊C8×C71C2312yes—
80C23×D71—21no—
81C2×Dic142—22no—
82C2×Dic142—42no—
83C2×Dic142—42no—
84C2×Dic142—41no—
85C2×D284—21no—
86C2×D284—22no—
87C2×D284Q1614yes—
88C2×D284C2×D412yes—
89C22×Dic71—21no—
90C22×Dic71—41no—
91C2×C4×D71—22no—
92C2×C4×D71—22no—
93C2×C4×D71—42no—
94C2×C4×D71—42no—
95C2×C71⋊C8—22no—
96C2×C71⋊C8—22no—
97C2×C71⋊C8—22no—
98C2×C71⋊C8—42no—
99C2×C71⋊C8C2×D412yes—
100C2×C71⋊C8C2×D412yes—
101C8×D71—48no—
102C4⋊Dic71—21no—
103C4⋊Dic71—22no—
104C4⋊Dic71—22no—
105C4⋊Dic71—41no—
106C4⋊Dic71—41no—
107C4⋊Dic71C2×D412yes—
108(C2×D71)⋊C4—22no—
109(C2×D71)⋊C4—22no—
110C2×Q8×C71—22no—
111C2×Q8×C71—41no—
112C2×Q8×C71—82no—
113C22×C4×C71—41no—
114C22×C4×C71C2×D411yescharacteristic
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71#4111yescharacteristic
117C2×C8×C71—22no—
118C2×C8×C71—22no—
119C2×C8×C71—42no—
120C2×C8×C71—42no—
121C4⋊C4×C71—22no—
122C4⋊C4×C71—22no—
123C4⋊C4×C71—41no—
124C4⋊C4×C71—41no—
125C42×C71C2×D411yescharacteristic
126C42×C71C2×D411yescharacteristic
127C42×C71#4111yescharacteristic
128C42×C71C2411yescharacteristic
129C23×C71#12111yescharacteristic
130C22×D71—22no—
131C22×D71—24no—
132C22×D71—42no—
133C22×D71—42no—
134Q8×C71—82no—
135Q8×C71—82no—
136C2×C4×C71—22no—
137C2×C4×C71—42no—
138C2×C4×C71—82no—
139C2×C4×C71#12111yescharacteristic
140C2×C4×C71#12111yescharacteristic
141C2×C4×C71#12111yescharacteristic
142C2×C4×C71C2×Q1612yes—
143C2×C4×C71#13512yes—
144C2×C4×C71#13512yes—
145C2×C4×C71AGL(1,ℤ/8)12yes—
146C2×C4×C71C22×D411yescharacteristic
147C2×C4×C71C22×D411yescharacteristic
148C2×C4×C71C2×#4111yescharacteristic
149C2×Dic71—22no—
150C2×Dic71—42no—
151C2×Dic71—42no—
152C2×Dic71—82no—
153D284—22no—
154D284—24no—
155D284C2×Q1614yes—
156C4×D71—42no—
157C4×D71—42no—
158C4×D71—44no—
159Dic142—42no—
160Dic142—42no—
161Dic142—82no—
162C8×C71—42no—
163C8×C71—42no—
164C71⋊C8—22no—
165C71⋊C8—22no—
166C71⋊C8—42no—
167C22×C71#46312yes—
168C22×C71#46712yes—
169C22×C71#47112yes—
170C22×C71C2×#12111yescharacteristic
171C2×D71—24no—
172C2×D71—48no—
173C2×D71—42no—
174C2×D71—42no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—82no—
178C4×C71#46312yes—
179C4×C71#46712yes—
180C4×C71C22×Q1611yescharacteristic
181C4×C71C2×#13511yescharacteristic
182C4×C71C2×#13511yescharacteristic
183C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
184Dic71—42no—
185Dic71—82no—
186C2×#1683—711nomaximal, Sylow
187C2×C71#168314yescomplemented
188C2×C71C2×#46311yescharacteristic
189C2×C71C2×#46711yescharacteristic
190C2×C71C2×#47111yescharacteristic
191D71—44no—
192C2×#487—711no—
193C2×#476—711no—
194C2×#474—711no—
195C2×C8×D4—711no—
196#1683—718no—
197C2×#335—711no—
198C2×#333D7111yescharacteristic, complemented
199C2×#332—711no—
200C71C2×#168311yescharacteristic, complemented, Sylow
201C23×C8—1421no—
202C2×C4⋊Q8—711no—
203C2×C4⋊Q8C2×D7111yescharacteristic
204C2×C4×D4—711no—
205#487—714no—
206#487—1422no—
207#476—714no—
208#476—1422no—
209#474—714no—
210#474—1422no—
211C8×D4—714no—
212C8×D4—1422no—
213C2×#107—1421no—
214C2×#106—1421no—
215C2×C4⋊C8—711no—
216C2×C4⋊C8C2×D7111yescharacteristic
217C2×C4⋊C8C2×D7111yescharacteristic
218C2×#103—1421no—
219C2×#103—1421no—
220C2×#102—1421no—
221C2×#102—1421no—
222C2×C22⋊C8—1421no—
223C2×C4×C8—711no—
224#335—714no—
225#333C2×D7114yescomplemented
226#332—714no—
227C22×Q8—21no—
228C22×Q8—1421no—
229C22×D4—711no—
230C23×C4—1421no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—2844no—
234C22×C8—711no—
235C22×C8—711no—
236C22×C8—1424no—
237C22×C8—1421no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—712no—
242C4⋊Q8—1421no—
243C4⋊Q8—1421no—
244C4⋊Q8—1421no—
245C4⋊Q8C22×D7112yes—
246C4×D4—712no—
247C4×D4—712no—
248C4×D4—1421no—
249C4×D4—1421no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—711no—
253C2×C4⋊C4—1421no—
254C2×C4⋊C4—1421no—
255C2×C22⋊C4—1421no—
256C2×C42C22×D7111yescharacteristic
257#107—1424no—
258#106—1424no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—712no—
262C4⋊C8—1421no—
263C4⋊C8C22×D7112yes—
264C4⋊C8C22×D7112yes—
265#103—1424no—
266#103—1424no—
267#102—1424no—
268#102—1424no—
269C22⋊C8—1424no—
270C4×C8—712no—
271C4×C8—712no—
272C24—1421no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×Q8—2841no—
277C2×Q8—2842no—
278C2×Q8—2842no—
279C2×Q8—1422no—
280C2×D4—712no—
281C2×D4—714no—
282C2×D4—1421no—
283C2×D4—1422no—
284C22×C4—41no—
285C22×C4—2842no—
286C22×C4—2842no—
287C22×C4—2841no—
288C22×C4—1421no—
289C22×C4—1422no—
290C22×C4—1422no—
291C22×C4D4×D7111yescharacteristic
292C22×C4D4×D7111yescharacteristic
293C22×C4#1176586511yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—2848no—
299C2×C8—2842no—
300C2×C8—712no—
301C2×C8—712no—
302C2×C8—1422no—
303C2×C8—1422no—
304C2×C8—1422no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—2841no—
310C4⋊C4—2841no—
311C4⋊C4—712no—
312C4⋊C4—1422no—
313C4⋊C4—1422no—
314C4⋊C4—1421no—
315C22⋊C4—1422no—
316C22⋊C4—1422no—
317C42D4×D7111yescharacteristic
318C42D4×D7111yescharacteristic
319C42#1176586511yescharacteristic
320C42C23×D7111yescharacteristic
321C23—2842no—
322C23—2842no—
323C23—1422no—
324C23—1424no—
325C23#1455610111yescharacteristic
326Q8—82no—
327Q8—82no—
328Q8—2842no—
329Q8—2842no—
330Q8—5682no—
331D4—714no—
332D4—1424no—
333D4—1422no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—2842no—
338C2×C4—2844no—
339C2×C4—2842no—
340C2×C4—2842no—
341C2×C4—2842no—
342C2×C4—5682no—
343C2×C4—1422no—
344C2×C4#1455610111yescharacteristic
345C2×C4#1455610111yescharacteristic
346C2×C4#1455610111yescharacteristic
347C2×C4Q16×D7112yes—
348C2×C4#1455614912yes—
349C2×C4#1455615112yes—
350C2×C4#1455615512yes—
351C2×C4C2×D4×D7111yescharacteristic
352C2×C4C2×D4×D7111yescharacteristic
353C2×C4C2×#1176586511yescharacteristic
354C8—42no—
355C8—42no—
356C8—2842no—
357C8—1422no—
358C8—1422no—
359C22—2842no—
360C22—2848no—
361C22—2842no—
362C22—1424no—
363C22#1798621712yes—
364C22#1798623912yes—
365C22#1798626112yes—
366C22C2×#1455610111yescharacteristic
367C4—21no—
368C4—21no—
369C4—82no—
370C4—2842no—
371C4—5682no—
372C4#1798621712yes—
373C4#1798623912yes—
374C4C2×Q16×D7111yescharacteristic
375C4C2×#1455614911yescharacteristic
376C4C2×#1455615111yescharacteristic
377C4C2×#1455615511yescharacteristic
378C2—2844no—
379C2#2258779414yesdirect product factor
380C2C2×#1798621711yescharacteristic
381C2C2×#1798623911yescharacteristic
382C2C2×#1798626111yescharacteristic
subgroupG/H
centerC23#14556101
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC2×#333×C71C2
socleC23×C71#121
prime radicalC22×D284D4
hypercenterC2×#333D71
hypocenterC71C2×#1683
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
1. C2×#17986351C2semidirect
5. #22587794C2direct (N × H)
6. C2×#17985751C2semidirect
7. C2×#17985746C2semidirect
8. C2×#333×C71C2semidirect
13. #17986351C22semidirect
29. #17985751C22semidirect
30. #17985746C22semidirect
34. #333×C71C22semidirect
187. C2×C71#1683semidirect
198. C2×#333D71semidirect
200. C71C2×#1683semidirect
225. #333C2×D71semidirect
379. C2#22587794direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179863512
C2×#179862862
C2×#179862802
C2×#179860642
#225877942
C2×#179857512
C2×#179857462
C2×#333×C712
C2×#168371