Finite Group Databasebeta

← C2×#23764542 #34315552

Subgroups — C2×#23764542 #34315552

Order 20224.

1518 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579161C211yesmaximal, characteristic
2C22×#14579086C212yesmaximal
3C2×#19098469C211yesmaximal, characteristic
4#23764542C218yesmaximal, direct product factor
5C2×#19098326C212yesmaximal
6C2×#336×C79C211yesmaximal, characteristic
7C23×Dic158—21no—
8C23×Dic158C2211yescharacteristic
9C2×#14579161—28no—
10C2×#14579161—22no—
11C2×#14579161C2214yes—
12C22×C4⋊Dic79—21no—
13C22×C4⋊Dic79C2211yescharacteristic
14C22×C4⋊Dic79C2212yes—
15C2×#14579086—28no—
16C2×#14579086C2218yes—
17#19098469—22no—
18#19098469C2214yes—
19C2×C79⋊#95—21no—
20C2×C79⋊#95—21no—
21#19098326C2218yes—
22C23×C8×C79C2212yes—
23C22×C42×C79C2211yescharacteristic
24#336×C79C2214yes—
25C22×Dic158—28no—
26C22×Dic158—21no—
27C22×Dic158—24no—
28C22×Dic158—22no—
29C22×Dic158—24no—
30C22×Dic158—24no—
31C22×Dic158—44no—
32C22×Dic158—44no—
33C22×Dic158Q814yescomplemented
34C22×Dic158C2312yes—
35C23×Dic79—21no—
36C23×Dic79—22no—
37C23×Dic79—21no—
38#14579161—216no—
39#14579161—216no—
40C2×C4⋊Dic79—28no—
41C2×C4⋊Dic79—24no—
42C2×C4⋊Dic79—22no—
43C2×C4⋊Dic79—24no—
44C2×C4⋊Dic79—24no—
45C2×C4⋊Dic79—21no—
46C2×C4⋊Dic79D418yes—
47C2×C4⋊Dic79C2312yes—
48C2×C4⋊Dic79C2314yes—
49#14579086—216no—
50C79⋊#95—24no—
51C79⋊#95—24no—
52C23×C4×C79D411yescharacteristic
53C23×C4×C79D411yescharacteristic
54C23×C4×C79C2311yescharacteristic
55C22×C8×C79—24no—
56C22×C8×C79—24no—
57C22×C8×C79—24no—
58C22×C8×C79C2314yes—
59C2×C42×C79—24no—
60C2×C42×C79D412yes—
61C2×C42×C79C2312yes—
62C2×Dic158—216no—
63C2×Dic158—216no—
64C2×Dic158—24no—
65C2×Dic158—24no—
66C2×Dic158—24no—
67C2×Dic158—22no—
68C2×Dic158—48no—
69C2×Dic158—432no—
70C2×Dic158—42no—
71C2×Dic158—48no—
72C2×Dic158—42no—
73C2×Dic158—48no—
74C2×Dic158C2×Q814yescomplemented
75C22×Dic79—28no—
76C22×Dic79—24no—
77C22×Dic79—24no—
78C22×Dic79—22no—
79C22×Dic79—22no—
80C22×Dic79—42no—
81C22×Dic79—42no—
82C22×Dic79—42no—
83C22×Dic79—44no—
84C22×Dic79—44no—
85C22×Dic79—42no—
86C22×Dic79—42no—
87C4⋊Dic79—216no—
88C4⋊Dic79—28no—
89C4⋊Dic79—24no—
90C4⋊Dic79—24no—
91C4⋊Dic79—42no—
92C4⋊Dic79C2×D418yes—
93C24×C79C2×D411yescharacteristic
94C22×C4×C79—22no—
95C22×C4×C79—22no—
96C22×C4×C79—22no—
97C22×C4×C79—22no—
98C22×C4×C79SD1614yes—
99C22×C4×C79Q1614yes—
100C22×C4×C79C2×D412yes—
101C22×C4×C79C2×D412yes—
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×D414yes—
104C22×C4×C79C2×Q812yes—
105C22×C4×C79#4114yes—
106C22×C4×C79#4112yes—
107C22×C4×C79C2411yescharacteristic
108C2×C8×C79—24no—
109C2×C8×C79—24no—
110C2×C8×C79—24no—
111C2×C8×C79—48no—
112C42×C79—24no—
113C42×C79—24no—
114C23×C79#12012yes—
115C23×C79C22⋊Q812yes—
116C23×C79#12412yes—
117C23×C79C2×SD1612yes—
118C23×C79C2×Q1612yes—
119C23×C79AGL(1,ℤ/8)12yes—
120C23×C79#13712yes—
121C23×C79C22×D411yescharacteristic
122C2×C4×C79—22no—
123C2×C4×C79—22no—
124C2×C4×C79—22no—
125C2×C4×C79—22no—
126C2×C4×C79—28no—
127C2×C4×C79—28no—
128C2×C4×C79#12012yes—
129C2×C4×C79C22⋊Q818yes—
130C2×C4×C79C22⋊Q812yes—
131C2×C4×C79#12418yes—
132C2×C4×C79#12412yes—
133C2×C4×C79C2×SD1614yes—
134C2×C4×C79C2×Q1614yes—
135C2×C4×C79C22×D412yes—
136C2×C4×C79C22×Q811yescharacteristic
137C2×C4×C79C2×#4111yescharacteristic
138C2×C4×C79C2×#4112yes—
139C2×Dic79—28no—
140C2×Dic79—24no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C2×Dic79—44no—
144C2×Dic79—44no—
145C2×Dic79—48no—
146C2×Dic79—416no—
147C2×Dic79—42no—
148C2×Dic79—42no—
149C2×Dic79—42no—
150C2×Dic79—84no—
151Dic158—28no—
152Dic158—432no—
153Dic158—44no—
154Dic158—48no—
155C8×C79—44no—
156C22×C79#39314yes—
157C22×C79#45014yes—
158C22×C79#45114yes—
159C22×C79#47514yes—
160C22×C79#47714yes—
161C22×C79#48714yes—
162C22×C79#48914yes—
163C22×C79C2×#12011yescharacteristic
164C22×C79C2×C22⋊Q811yescharacteristic
165C22×C79C2×#12411yescharacteristic
166C22×C79C22×SD1611yescharacteristic
167C22×C79C22×Q1611yescharacteristic
168C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
169C22×C79C2×#13711yescharacteristic
170C4×C79—24no—
171C4×C79—24no—
172C4×C79#39314yes—
173C4×C79C2×C22⋊Q812yes—
174C4×C79C2×#12412yes—
175Dic79—44no—
176Dic79—48no—
177Dic79—82no—
178C2×#2024—791nomaximal, Sylow
179C2×C79#202418yescomplemented
180C2×C79C2×#39311yescharacteristic
181C2×C79C2×#45011yescharacteristic
182C2×C79C2×#45111yescharacteristic
183C2×C79C2×#47511yescharacteristic
184C2×C79C2×#47711yescharacteristic
185C2×C79C2×#48711yescharacteristic
186C2×C79C2×#48911yescharacteristic
187C22×C4⋊Q8—791no—
188C22×#103—792no—
189C2×#391—791no—
190#2024—798no—
191C2×#340—792no—
192C2×#336D7911yescharacteristic
193C79C2×#202411yescharacteristic, complemented, Sylow
194C23×Q8—791no—
195C23×Q8—1581no—
196C23×C8C2×D7912yes—
197C2×C4⋊Q8—794no—
198C2×C4⋊Q8—1588no—
199C2×C4⋊Q8—1582no—
200C22×C4⋊C4—792no—
201C22×C4⋊C4—791no—
202C22×C4⋊C4—1581no—
203C22×C42C2×D7911yescharacteristic
204C2×#103—798no—
205C2×#103—1588no—
206#391—794no—
207#391—1582no—
208C2×#95—1581no—
209C2×#95—1581no—
210#340—798no—
211#336C2×D7914yes—
212C22×Q8—3164no—
213C22×Q8—3164no—
214C22×Q8—794no—
215C22×Q8—792no—
216C22×Q8—1588no—
217C22×Q8—1584no—
218C22×Q8—1584no—
219C22×Q8—1584no—
220C22×Q8—1581no—
221C22×Q8—1582no—
222C23×C4—1581no—
223C23×C4—1582no—
224C23×C4—1581no—
225C23×C4D31611yescharacteristic
226C23×C4D31611yescharacteristic
227C23×C4C22×D7911yescharacteristic
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C22×D7914yes—
232C4⋊Q8—15816no—
233C4⋊Q8—15816no—
234C2×C4⋊C4—798no—
235C2×C4⋊C4—794no—
236C2×C4⋊C4—792no—
237C2×C4⋊C4—1581no—
238C2×C4⋊C4—1584no—
239C2×C4⋊C4—1588no—
240C2×C4⋊C4—1584no—
241C2×C4⋊C4—1582no—
242C2×C4⋊C4—1584no—
243C2×C42—24no—
244C2×C42D31612yes—
245C2×C42C22×D7912yes—
246#103—15816no—
247#95—1584no—
248#95—1584no—
249C24C2×D31611yescharacteristic
250C2×Q8—31632no—
251C2×Q8—3168no—
252C2×Q8—3162no—
253C2×Q8—3168no—
254C2×Q8—3162no—
255C2×Q8—3168no—
256C2×Q8—794no—
257C2×Q8—15816no—
258C2×Q8—1584no—
259C2×Q8—15816no—
260C2×Q8—1584no—
261C2×Q8—1582no—
262C2×Q8—1584no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—3162no—
268C22×C4—3162no—
269C22×C4—3164no—
270C22×C4—3164no—
271C22×C4—3162no—
272C22×C4—3162no—
273C22×C4—3162no—
274C22×C4—1582no—
275C22×C4—1588no—
276C22×C4—1584no—
277C22×C4—1582no—
278C22×C4—1584no—
279C22×C4#1192961314yes—
280C22×C4Dic31614yes—
281C22×C4D4×D7914yes—
282C22×C4C2×D31612yes—
283C22×C4C2×D31612yes—
284C22×C4C2×D31611yescharacteristic
285C22×C4Q8×D7912yes—
286C22×C4#1192962312yes—
287C22×C4#1192962414yes—
288C22×C4C23×D7911yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—48no—
293C4⋊C4—3162no—
294C4⋊C4—798no—
295C4⋊C4—15816no—
296C4⋊C4—1584no—
297C4⋊C4—1588no—
298C4⋊C4—1584no—
299C42—24no—
300C42—24no—
301C23#1457913112yes—
302C23#1457914112yes—
303C23#1457914912yes—
304C23C2×#1192961312yes—
305C23C2×Dic31612yes—
306C23#1457918912yes—
307C23#1457919412yes—
308C23C22×D31611yescharacteristic
309Q8—31632no—
310Q8—3168no—
311Q8—3164no—
312Q8—1588no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—31616no—
320C2×C4—3168no—
321C2×C4—3162no—
322C2×C4—3162no—
323C2×C4—3164no—
324C2×C4—3164no—
325C2×C4—3162no—
326C2×C4—3162no—
327C2×C4—3162no—
328C2×C4—6324no—
329C2×C4—1588no—
330C2×C4—1584no—
331C2×C4#1457913112yes—
332C2×C4#1457913918yes—
333C2×C4#1457914112yes—
334C2×C4#1457914718yes—
335C2×C4#1457914912yes—
336C2×C4C2×#1192961314yes—
337C2×C4C2×Dic31614yes—
338C2×C4C2×D4×D7912yes—
339C2×C4C2×Q8×D7911yescharacteristic
340C2×C4C2×#1192962311yescharacteristic
341C2×C4C2×#1192962412yes—
342C8—44no—
343C22#1909847614yes—
344C22#1909871614yes—
345C22#1909872114yes—
346C22#1909884914yes—
347C22#1909885914yes—
348C22#1909891414yes—
349C22#1909892314yes—
350C22C2×#1457913111yescharacteristic
351C22C2×#1457914111yescharacteristic
352C22C2×#1457914911yescharacteristic
353C22C22×#1192961311yescharacteristic
354C22C22×Dic31611yescharacteristic
355C22C2×#1457918911yescharacteristic
356C22C2×#1457919411yescharacteristic
357C4—24no—
358C4—24no—
359C4—3164no—
360C4—3168no—
361C4—6322no—
362C4#1909847614yes—
363C4C2×#1457913912yes—
364C4C2×#1457914712yes—
365C2#2376454218yesdirect product factor
366C2C2×#1909847611yescharacteristic
367C2C2×#1909871611yescharacteristic
368C2C2×#1909872111yescharacteristic
369C2C2×#1909884911yescharacteristic
370C2C2×#1909885911yescharacteristic
371C2C2×#1909891411yescharacteristic
372C2C2×#1909892311yescharacteristic
subgroupG/H
centerC24C2×D316
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#336×C79C2
socle, prime radicalC24×C79C2×D4
hypercenterC2×#336D79
hypocenterC79C2×#2024
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
4. #23764542C2direct (N × H)
33. C22×Dic158Q8semidirect
74. C2×Dic158C2×Q8semidirect
179. C2×C79#2024semidirect
193. C79C2×#2024semidirect
365. C2#23764542direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791612
C22×#145790862
C2×#190984692
#237645422
C2×#190983262
C2×#336×C792
C2×#202479