Finite Group Databasebeta

← C2×#23762402 #34260485

Subgroups — C2×#23762402 #34260485

Order 20224.

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×#19099144C211yesmaximal, characteristic
2C2×#19098876C212yesmaximal, complemented
3#23762402C218yesmaximal, direct product factor
4C2×#19098296C212yesmaximal, complemented
5C2×#19098289C211yesmaximal, characteristic, complemented
6C2×#398×C79C211yesmaximal, characteristic, complemented
7C22×#11929624—21no—
8#19099144—24no—
9#19099144—22no—
10#19099144C2214yes—
11C2×#14579141—22no—
12C2×#14579136—22no—
13C2×#14579135—21no—
14C2×#14579135C2212yes—
15#19098876—24no—
16#19098876C2218yescomplemented
17C2×#14579094—22no—
18C2×#14579081—22no—
19C2×C22⋊(C79⋊C8)C2212yes—
20#19098296C2218yescomplemented
21#19098289C2214yescomplemented
22C22×C4⋊C4×C79—21no—
23C22×C4⋊C4×C79C2211yescharacteristic
24C22×C4⋊C4×C79C2211yescharacteristic
25#398×C79—22no—
26#398×C79C2214yescomplemented
27C2×#95×C79—21no—
28C2×#95×C79C2211yescharacteristic
29C2×#11929624—24no—
30C2×#11929624—24no—
31C2×#11929624—44no—
32C2×#11929624—44no—
33C22×Dic158—21no—
34C22×D316—21no—
35C22×D316—21no—
36C22×D316D412yes—
37C22×C22⋊D79—22no—
38C22×C4×D79—22no—
39C22×C79⋊C8—22no—
40#14579141—28no—
41#14579141—28no—
42#14579136—28no—
43#14579136—28no—
44#14579135—24no—
45#14579135—22no—
46#14579135—22no—
47#14579135—24no—
48#14579135D414yes—
49#14579135C2314yes—
50C2×C4⋊Dic79—21no—
51C2×C4⋊Dic79D412yes—
52C2×(C2×D79)⋊C4—22no—
53C2×(C2×D79)⋊C4—22no—
54#14579094—28no—
55#14579081—28no—
56C22⋊(C79⋊C8)—22no—
57C22⋊(C79⋊C8)C2314yes—
58C23×C4×C79—21no—
59C23×C4×C79—21no—
60C23×C4×C79D411yescharacteristic
61C23×C4×C79D411yescharacteristic
62C23×C4×C79C2311yescharacteristic
63C2×C4⋊C4×C79—22no—
64C2×C4⋊C4×C79—22no—
65C2×C4⋊C4×C79—22no—
66C2×C4⋊C4×C79—22no—
67C2×C4⋊C4×C79—24no—
68C2×C4⋊C4×C79—21no—
69C2×C4⋊C4×C79—44no—
70C2×C4⋊C4×C79D412yes—
71C2×C4⋊C4×C79C2312yes—
72C2×C4⋊C4×C79C2312yes—
73#95×C79—24no—
74#95×C79—21no—
75#95×C79C2312yes—
76C23×D79—22no—
77#11929624—48no—
78#11929624—416no—
79#11929624—48no—
80C2×Dic158—24no—
81C2×Dic158—44no—
82C2×D316—24no—
83C2×D316—24no—
84C2×D316—22no—
85C2×D316—44no—
86C2×D316—44no—
87C2×D316—82no—
88C2×D316C2×D414yes—
89C2×C22⋊D79—24no—
90C2×C22⋊D79—24no—
91C2×C22⋊D79—44no—
92C2×C22⋊D79—44no—
93C22×Dic79—22no—
94C2×C4×D79—28no—
95C2×C4×D79—48no—
96C2×C79⋊C8—24no—
97C2×C79⋊C8—44no—
98C4⋊Dic79—24no—
99C4⋊Dic79—22no—
100C4⋊Dic79C2×D414yes—
101(C2×D79)⋊C4—28no—
102(C2×D79)⋊C4—24no—
103(C2×D79)⋊C4—24no—
104C24×C79C2×D411yescharacteristic
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—22no—
108C22×C4×C79—24no—
109C22×C4×C79—21no—
110C22×C4×C79—22no—
111C22×C4×C79—21no—
112C22×C4×C79—22no—
113C22×C4×C79—42no—
114C22×C4×C79—42no—
115C22×C4×C79—42no—
116C22×C4×C79—42no—
117C22×C4×C79—41no—
118C22×C4×C79—41no—
119C22×C4×C79—42no—
120C22×C4×C79—42no—
121C22×C4×C79D814yes—
122C22×C4×C79C2×D412yes—
123C22×C4×C79C2×D412yes—
124C22×C4×C79C2×D412yes—
125C22×C4×C79C2×D411yescharacteristic
126C22×C4×C79C2×D412yes—
127C22×C4×C79C2411yescharacteristic
128C4⋊C4×C79—24no—
129C4⋊C4×C79—24no—
130C4⋊C4×C79—24no—
131C4⋊C4×C79—42no—
132C4⋊C4×C79—48no—
133C23×C79—22no—
134C23×C79—22no—
135C23×C79#12011yescharacteristic
136C23×C79#12011yescharacteristic
137C23×C79C2×D812yes—
138C23×C79AGL(1,ℤ/8)12yes—
139C23×C79C22×D411yescharacteristic
140C22×D79—24no—
141C22×D79—44no—
142C22×D79—44no—
143C22×D79—44no—
144C2×C4×C79—22no—
145C2×C4×C79—22no—
146C2×C4×C79—24no—
147C2×C4×C79—24no—
148C2×C4×C79—24no—
149C2×C4×C79—24no—
150C2×C4×C79—22no—
151C2×C4×C79—42no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—48no—
155C2×C4×C79—42no—
156C2×C4×C79—42no—
157C2×C4×C79—42no—
158C2×C4×C79—42no—
159C2×C4×C79—42no—
160C2×C4×C79—42no—
161C2×C4×C79—42no—
162C2×C4×C79—81no—
163C2×C4×C79—82no—
164C2×C4×C79—82no—
165C2×C4×C79#12012yes—
166C2×C4×C79C2×D814yes—
167C2×C4×C79C22×D412yes—
168C22⋊D79—44no—
169C22⋊D79—44no—
170C22⋊D79—48no—
171C2×Dic79—24no—
172C2×Dic79—44no—
173D316—44no—
174D316—48no—
175D316—82no—
176D316—84no—
177C4×D79—416no—
178Dic158—48no—
179C79⋊C8—44no—
180C22×C79—24no—
181C22×C79—24no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79#44712yes—
187C22×C79#44712yes—
188C22×C79#45612yes—
189C22×C79#45812yes—
190C22×C79C2×#12011yescharacteristic
191C22×C79C22×D811yescharacteristic
192C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
193C2×D79—44no—
194C2×D79—44no—
195C2×D79—44no—
196C2×D79—88no—
197C4×C79—24no—
198C4×C79—22no—
199C4×C79—22no—
200C4×C79—44no—
201C4×C79—42no—
202C4×C79—82no—
203Dic79—44no—
204C2×#1610—791nomaximal, Sylow
205C2×C79—21no—
206C2×C79—21no—
207C2×C79—22no—
208C2×C79#161014yescomplemented
209C2×C79C2×#44711yescharacteristic
210C2×C79C2×#45611yescharacteristic
211C2×C79C2×#45811yescharacteristic
212D79—84no—
213C2×#537—791no—
214C2×#480—792no—
215C2×#398D7911yescharacteristic, complemented
216#1610—798no—
217C2×#327—792no—
218C2×#324—791no—
219C79C2×#161011yescharacteristic, complemented, Sylow
220C22×#41—1581no—
221#537—794no—
222#537—1584no—
223#537—1582no—
224C2×C22⋊Q8—1582no—
225C2×#121—792no—
226C2×#121—1582no—
227C2×#121—1581no—
228C22×C4⋊C4—21no—
229C22×C4⋊C4C2×D7911yescharacteristic
230C22×C4⋊C4C2×D7911yescharacteristic
231#480—798no—
232#480—1584no—
233C2×#107—1582no—
234C2×#102—1582no—
235C2×C22⋊C8—792no—
236#398—22no—
237#398C2×D7914yescomplemented
238C2×#95—21no—
239C2×#95C2×D7911yescharacteristic
240#327—798no—
241#324—794no—
242C2×#41—3164no—
243C2×#41—3164no—
244C2×#41—1584no—
245C2×#41—1584no—
246C22×Q8—1581no—
247C22×D4—792no—
248C22×D4—1581no—
249C22×D4—1581no—
250C22×D4—1582no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—1582no—
254C23×C4C22⋊D7911yescharacteristic
255C23×C4C22⋊D7911yescharacteristic
256C23×C4C22×D7911yescharacteristic
257C22×C8—1582no—
258C22⋊Q8—1588no—
259C22⋊Q8—1588no—
260#121—794no—
261#121—794no—
262#121—1588no—
263#121—1588no—
264#121—1584no—
265#121—1582no—
266#121—1582no—
267#121—1584no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—792no—
276C2×C4⋊C4—1581no—
277C2×C4⋊C4C22⋊D7912yes—
278C2×C4⋊C4C22×D7912yes—
279C2×C4⋊C4C22×D7912yes—
280C2×C22⋊C4—1582no—
281C2×C22⋊C4—1582no—
282#107—1588no—
283#102—1588no—
284C22⋊C8—794no—
285C22⋊C8—1582no—
286#95—24no—
287#95—21no—
288#95C22×D7912yes—
289C24—1582no—
290C24C2×C22⋊D7911yescharacteristic
291#41—31616no—
292#41—3168no—
293#41—3168no—
294C2×Q8—3164no—
295C2×Q8—1584no—
296C2×D4—3164no—
297C2×D4—3164no—
298C2×D4—3164no—
299C2×D4—3164no—
300C2×D4—794no—
301C2×D4—6322no—
302C2×D4—1584no—
303C2×D4—1584no—
304C2×D4—1584no—
305C2×D4—1584no—
306C2×D4—1582no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—3168no—
324C22×C4—1588no—
325C22×C4—1582no—
326C22×C4#1192960914yes—
327C22×C4D4×D7912yes—
328C22×C4D4×D7912yes—
329C22×C4C2×C22⋊D7912yes—
330C22×C4C2×C22⋊D7912yes—
331C22×C4C2×C22⋊D7911yescharacteristic
332C22×C4C23×D7911yescharacteristic
333C2×C8—3164no—
334C2×C8—1584no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—48no—
340C4⋊C4—794no—
341C4⋊C4—1584no—
342C4⋊C4—1582no—
343C22⋊C4—1588no—
344C22⋊C4—1584no—
345C22⋊C4—1584no—
346C23—22no—
347C23—22no—
348C23—3164no—
349C23—3164no—
350C23—3164no—
351C23—1584no—
352C23#1457912911yescharacteristic
353C23#1457912911yescharacteristic
354C23C2×#1192960912yes—
355C23#1457918612yes—
356C23C22×C22⋊D7911yescharacteristic
357Q8—3168no—
358D4—3164no—
359D4—3168no—
360D4—3168no—
361D4—3164no—
362D4—3164no—
363D4—6322no—
364D4—6324no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—31616no—
387C2×C4—3164no—
388C2×C4—1584no—
389C2×C4#1457912912yes—
390C2×C4C2×#1192960914yes—
391C2×C4C2×D4×D7912yes—
392C8—3164no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—24no—
399C22—3164no—
400C22—3164no—
401C22—3164no—
402C22—6328no—
403C22#1909869612yes—
404C22#1909869612yes—
405C22#1909874512yes—
406C22#1909875212yes—
407C22C2×#1457912911yescharacteristic
408C22C22×#1192960911yescharacteristic
409C22C2×#1457918611yescharacteristic
410C4—22no—
411C4—22no—
412C4—24no—
413C4—44no—
414C4—42no—
415C4—82no—
416C4—3164no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—6324no—
421C2#2376240214yesdirect product factor
422C2C2×#1909869611yescharacteristic
423C2C2×#1909874511yescharacteristic
424C2C2×#1909875211yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#398×C79C2
socleC23×C79#120
prime radicalC2×#19099144C2
hypercenterC2×#398D79
hypocenterC79C2×#1610
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
2. C2×#19098876C2semidirect
3. #23762402C2direct (N × H)
4. C2×#19098296C2semidirect
5. C2×#19098289C2semidirect
6. C2×#398×C79C2semidirect
16. #19098876C22semidirect
20. #19098296C22semidirect
21. #19098289C22semidirect
26. #398×C79C22semidirect
208. C2×C79#1610semidirect
215. C2×#398D79semidirect
219. C79C2×#1610semidirect
237. #398C2×D79semidirect
421. C2#23762402direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190991442
C2×#190988762
#237624022
C2×#190982962
C2×#190982892
C2×#398×C792
C2×#161079