Finite Group Databasebeta

← C2×#23762972 #34270839

Subgroups — C2×#23762972 #34270839

Order 20224.

694 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×#19098850C211yesmaximal, characteristic
2C2×#19098840C211yesmaximal, characteristic
3C2×#19098642C211yesmaximal, characteristic
4#23762972C218yesmaximal, direct product factor
5C2×#19098314C211yesmaximal, characteristic, complemented
6C2×#19098312C211yesmaximal, characteristic, complemented
7C2×#19098309C211yesmaximal, characteristic, complemented
8C2×#487×C79C211yesmaximal, characteristic, complemented
9C22×#11929607—21no—
10C2×C4×D316C2211yescharacteristic
11#19098850—22no—
12#19098850C2214yes—
13#19098840—22no—
14#19098840C2214yes—
15#19098642—22no—
16#19098642C2214yes—
17C2×#14579094—21no—
18C2×#14579092—21no—
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22C2×#14579081—21no—
23C2×#14579081—21no—
24C2×(C2×D79)⋊C8—21no—
25#19098314C2214yescomplemented
26#19098312C2214yescomplemented
27#19098309C2214yescomplemented
28C2×C4⋊Q8×C79C2211yescharacteristic
29C2×C4⋊Q8×C79C2211yescharacteristic
30#487×C79—22no—
31#487×C79C2214yescomplemented
32C2×#103×C79—21no—
33C2×#103×C79—21no—
34C2×C4×C8×C79C2211yescharacteristic
35C22×D316D411yescharacteristic
36C22×C4×D79—21no—
37C2×#11929607—24no—
38C2×#11929607—44no—
39C22×C79⋊C8—21no—
40C22×C79⋊C8—21no—
41C22×C79⋊C8—21no—
42C4×D316—21no—
43C4×D316—21no—
44C4×D316D412yes—
45C4×D316C2312yes—
46C2×C4⋊Dic79D411yescharacteristic
47C2×(C2×D79)⋊C4—21no—
48#14579094—24no—
49#14579092—24no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)—21no—
52C4⋊(C79⋊C8)—21no—
53C4⋊(C79⋊C8)C2312yes—
54C4⋊(C79⋊C8)C2312yes—
55C4⋊(C79⋊C8)C2312yes—
56#14579081—24no—
57#14579081—24no—
58(C2×D79)⋊C8—24no—
59C22×Q8×C79—21no—
60C22×Q8×C79—21no—
61C22×C8×C79D411yescharacteristic
62C22×C8×C79D411yescharacteristic
63C4⋊Q8×C79—21no—
64C4⋊Q8×C79—21no—
65C4⋊Q8×C79—21no—
66C4⋊Q8×C79—21no—
67C4⋊Q8×C79—21no—
68C4⋊Q8×C79—21no—
69C4⋊Q8×C79C2312yes—
70C4⋊Q8×C79C2312yes—
71C2×C4⋊C4×C79—21no—
72C2×C4⋊C4×C79—21no—
73C2×C4⋊C4×C79—21no—
74C2×C4⋊C4×C79—21no—
75C2×C42×C79C2311yescharacteristic
76#103×C79—24no—
77#103×C79—24no—
78C4×C8×C79D412yes—
79C4×C8×C79C2312yes—
80C23×D79—21no—
81C2×D316—21no—
82C2×D316—22no—
83C2×D316—22no—
84C2×D316C2×D412yes—
85C22×Dic79—21no—
86C2×C4×D79—22no—
87C2×C4×D79—22no—
88C2×C4×D79—42no—
89C2×C4×D79—42no—
90#11929607—48no—
91C2×C79⋊C8—22no—
92C2×C79⋊C8—22no—
93C2×C79⋊C8—22no—
94C2×C79⋊C8—42no—
95C2×C79⋊C8—42no—
96C2×C79⋊C8—42no—
97C4⋊Dic79—21no—
98C4⋊Dic79C2×D412yes—
99(C2×D79)⋊C4—22no—
100(C2×D79)⋊C4—22no—
101C2×Q8×C79—22no—
102C2×Q8×C79—22no—
103C2×Q8×C79—41no—
104C2×Q8×C79—41no—
105C2×Q8×C79—42no—
106C2×Q8×C79—42no—
107C2×Q8×C79—82no—
108C22×C4×C79—41no—
109C22×C4×C79—41no—
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79#4111yescharacteristic
113C2×C8×C79—22no—
114C2×C8×C79—22no—
115C2×C8×C79C2×D412yes—
116C2×C8×C79C2×D412yes—
117C4⋊C4×C79—22no—
118C4⋊C4×C79—22no—
119C4⋊C4×C79—22no—
120C4⋊C4×C79—22no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79—41no—
123C4⋊C4×C79—41no—
124C4⋊C4×C79—41no—
125C42×C79C2×D411yescharacteristic
126C42×C79C2×D411yescharacteristic
127C42×C79#4111yescharacteristic
128C42×C79C2411yescharacteristic
129C23×C79#12111yescharacteristic
130C22×D79—22no—
131C22×D79—42no—
132C22×D79—42no—
133C22×D79—42no—
134Q8×C79—42no—
135Q8×C79—42no—
136Q8×C79—82no—
137Q8×C79—82no—
138Q8×C79—82no—
139C2×C4×C79—22no—
140C2×C4×C79—42no—
141C2×C4×C79—42no—
142C2×C4×C79—82no—
143C2×C4×C79—82no—
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79#12111yescharacteristic
147C2×C4×C79#13512yes—
148C2×C4×C79AGL(1,ℤ/8)12yes—
149C2×C4×C79#13712yes—
150C2×C4×C79#13712yes—
151C2×C4×C79C22×D411yescharacteristic
152C2×C4×C79C22×D411yescharacteristic
153C2×C4×C79C2×#4111yescharacteristic
154C2×Dic79—22no—
155C2×Dic79—42no—
156D316—22no—
157D316—22no—
158D316—42no—
159C4×D79—42no—
160C4×D79—42no—
161C4×D79—44no—
162C8×C79—22no—
163C8×C79—22no—
164C79⋊C8—42no—
165C79⋊C8—42no—
166C79⋊C8—42no—
167C22×C79#46312yes—
168C22×C79#47012yes—
169C22×C79#47112yes—
170C22×C79C2×#12111yescharacteristic
171C2×D79—42no—
172C2×D79—42no—
173C2×D79—42no—
174C2×D79—84no—
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—82no—
178C4×C79—82no—
179C4×C79#46312yes—
180C4×C79#47012yes—
181C4×C79C2×#13511yescharacteristic
182C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
183C4×C79C2×#13711yescharacteristic
184C4×C79C2×#13711yescharacteristic
185Dic79—42no—
186C2×#1703—791nomaximal, Sylow
187C2×C79#170314yescomplemented
188C2×C79C2×#46311yescharacteristic
189C2×C79C2×#47011yescharacteristic
190C2×C79C2×#47111yescharacteristic
191D79—82no—
192C2×#487D7911yescharacteristic, complemented
193C2×#476—791no—
194C2×#474—791no—
195C2×#436—791no—
196#1703—798no—
197C2×#334—791no—
198C2×#333—791no—
199C2×#332—791no—
200C79C2×#170311yescharacteristic, complemented, Sylow
201C22×M16—1581no—
202C2×C4⋊Q8C2×D7911yescharacteristic
203C2×C4⋊Q8C2×D7911yescharacteristic
204C2×C4×D4—791no—
205#487—22no—
206#487C2×D7914yescomplemented
207#476—794no—
208#476—1582no—
209#474—794no—
210#474—1582no—
211#436—794no—
212#436—1582no—
213C2×#107—1581no—
214C2×#106—1581no—
215C2×C4⋊C8—791no—
216C2×C4⋊C8—791no—
217C2×C4⋊C8—791no—
218C2×#103—21no—
219C2×#103—21no—
220C2×#102—1581no—
221C2×#102—1581no—
222C2×C22⋊C8—1581no—
223C2×C4×C8C2×D7911yescharacteristic
224#334—794no—
225#333—794no—
226#332—794no—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—791no—
230C23×C4—1581no—
231C2×M16—3164no—
232C2×M16—1584no—
233C22×C8—1581no—
234C22×C8—1581no—
235C22×C8—1581no—
236C22×C8C22⋊D7911yescharacteristic
237C22×C8C22⋊D7911yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22×D7912yes—
245C4⋊Q8C22×D7912yes—
246C4×D4—792no—
247C4×D4—792no—
248C4×D4—1581no—
249C4×D4—1581no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—791no—
255C2×C22⋊C4—1581no—
256C2×C42C22×D7911yescharacteristic
257#107—1584no—
258#106—1584no—
259C4⋊C8—792no—
260C4⋊C8—792no—
261C4⋊C8—792no—
262C4⋊C8—1581no—
263C4⋊C8—1581no—
264C4⋊C8—1581no—
265#103—24no—
266#103—24no—
267#102—1584no—
268#102—1584no—
269C22⋊C8—1584no—
270C4×C8C22⋊D7912yes—
271C4×C8C22×D7912yes—
272C24—1581no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—82no—
280C2×D4—792no—
281C2×D4—1581no—
282C2×D4—1582no—
283C2×D4—1582no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—3162no—
287C22×C4—3162no—
288C22×C4—1581no—
289C22×C4—1582no—
290C22×C4—1582no—
291C22×C4D4×D7911yescharacteristic
292C22×C4C2×C22⋊D7911yescharacteristic
293C22×C4#1192962311yescharacteristic
294M16—3168no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—3162no—
298C2×C8—3162no—
299C2×C8—3162no—
300C2×C8—1582no—
301C2×C8—1582no—
302C2×C8—1582no—
303C2×C8C2×C22⋊D7912yes—
304C2×C8C2×C22⋊D7912yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—792no—
314C4⋊C4—1581no—
315C22⋊C4—1582no—
316C22⋊C4—1582no—
317C42D4×D7911yescharacteristic
318C42C2×C22⋊D7911yescharacteristic
319C42#1192962311yescharacteristic
320C42C23×D7911yescharacteristic
321C23—3162no—
322C23—3162no—
323C23—3162no—
324C23—1582no—
325C23#1457913211yescharacteristic
326Q8—42no—
327Q8—42no—
328Q8—82no—
329Q8—82no—
330Q8—82no—
331D4—3162no—
332D4—1582no—
333D4—1582no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—3162no—
340C2×C4—3164no—
341C2×C4—3162no—
342C2×C4—3162no—
343C2×C4—1582no—
344C2×C4#1457913211yescharacteristic
345C2×C4#1457913211yescharacteristic
346C2×C4#1457913211yescharacteristic
347C2×C4#1457918112yes—
348C2×C4#1457918612yes—
349C2×C4#1457919312yes—
350C2×C4#1457919312yes—
351C2×C4C2×D4×D7911yescharacteristic
352C2×C4C22×C22⋊D7911yescharacteristic
353C2×C4C2×#1192962311yescharacteristic
354C8—22no—
355C8—22no—
356C8—3162no—
357C8—3162no—
358C8—3162no—
359C22—3162no—
360C22—3162no—
361C22—3162no—
362C22—6324no—
363C22#1909877512yes—
364C22#1909881812yes—
365C22#1909882312yes—
366C22C2×#1457913211yescharacteristic
367C4—21no—
368C4—21no—
369C4—82no—
370C4—82no—
371C4—3162no—
372C4#1909877512yes—
373C4#1909881812yes—
374C4C2×#1457918111yescharacteristic
375C4C2×#1457918611yescharacteristic
376C4C2×#1457919311yescharacteristic
377C4C2×#1457919311yescharacteristic
378C2—6322no—
379C2#2376297214yesdirect product factor
380C2C2×#1909877511yescharacteristic
381C2C2×#1909881811yescharacteristic
382C2C2×#1909882311yescharacteristic
subgroupG/H
centerC23#14579132
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC2×#487×C79C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenterC2×#487D79
hypocenterC79C2×#1703
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #23762972C2direct (N × H)
5. C2×#19098314C2semidirect
6. C2×#19098312C2semidirect
7. C2×#19098309C2semidirect
8. C2×#487×C79C2semidirect
25. #19098314C22semidirect
26. #19098312C22semidirect
27. #19098309C22semidirect
31. #487×C79C22semidirect
187. C2×C79#1703semidirect
192. C2×#487D79semidirect
200. C79C2×#1703semidirect
206. #487C2×D79semidirect
379. C2#23762972direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190988502
C2×#190988402
C2×#190986422
#237629722
C2×#190983142
C2×#190983122
C2×#190983092
C2×#487×C792
C2×#170379