Finite Group Databasebeta

← C2×#21422027 #28497787

Subgroups — C2×#21422027 #28497787

Order 15104.

834 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16837303C211yesmaximal, characteristic
2C2×#16837128C211yesmaximal, characteristic, complemented
3#21422027C214yesmaximal, direct product factor
4C2×C59×#324C211yesmaximal, characteristic, complemented
5C23×D236C411yescharacteristic
6C22×(C2×D59)⋊C4—21no—
7C22×(C2×D59)⋊C4—21no—
8C22×C4×Dic59C411yescharacteristic
9C2×C22⋊(C59⋊C8)—21no—
10#16837303—21no—
11#16837303—21no—
12#16837303C412yes—
13#16837303C2212yes—
14#16837128—21no—
15#16837128C2212yescomplemented
16C22×C4⋊C4×C59C2211yescharacteristic
17C2×C22⋊C8×C59—21no—
18C59×#324—21no—
19C59×#324C2212yescomplemented
20C24×D59—21no—
21C22×D236—21no—
22C22×D236—22no—
23C22×D236—41no—
24C22×D236—41no—
25C22×D236—42no—
26C22×D236C814yescomplemented
27C22×D236C2×C412yes—
28C23×Dic59—21no—
29C22×C59⋊C8—41no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—44no—
35C2×(C2×D59)⋊C4—44no—
36C2×C4×Dic59—21no—
37C2×C4×Dic59—42no—
38C2×C4×Dic59C2×C412yes—
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—41no—
41C23×C4×C59C2×C411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C23×C4×C59D411yescharacteristic
44C22×C8×C59—41no—
45C2×C4⋊C4×C59—41no—
46C2×C4⋊C4×C59—41no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C22⋊C8×C59—22no—
52C22⋊C8×C59—41no—
53C24×C59C22⋊C411yescharacteristic
54C23×D59—24no—
55C23×D59—22no—
56C23×D59—42no—
57C23×D59—42no—
58C23×D59—42no—
59C23×D59—42no—
60C23×D59—41no—
61C23×D59—41no—
62C23×D59—42no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—42no—
71C22×C4×C59—42no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C2×C812yes—
76C22×C4×C59M1612yes—
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×Dic59—24no—
81C22×Dic59—22no—
82C22×Dic59—42no—
83C22×Dic59—42no—
84C2×D236—24no—
85C2×D236—22no—
86C2×D236—44no—
87C2×D236—44no—
88C2×D236—42no—
89C2×D236—42no—
90C2×D236—44no—
91C2×D236—44no—
92C2×D236—42no—
93C2×D236—42no—
94C2×D236C2×C814yescomplemented
95C2×C8×C59—42no—
96C2×C8×C59—82no—
97C4⋊C4×C59—41no—
98C4⋊C4×C59—41no—
99C4⋊C4×C59—41no—
100C4⋊C4×C59—41no—
101(C2×D59)⋊C4—44no—
102(C2×D59)⋊C4—44no—
103(C2×D59)⋊C4—44no—
104(C2×D59)⋊C4—44no—
105C4×Dic59—42no—
106C4×Dic59—42no—
107C2×C59⋊C8—42no—
108C2×C59⋊C8—82no—
109C23×C59—22no—
110C23×C59—22no—
111C23×C59C22⋊C812yes—
112C23×C59#9911yescharacteristic
113C23×C59#9911yescharacteristic
114C23×C59#10012yes—
115C23×C59C2×C22⋊C411yescharacteristic
116C22×D59—24no—
117C22×D59—42no—
118C22×D59—42no—
119C22×D59—48no—
120C22×D59—48no—
121C22×D59—42no—
122C22×D59—42no—
123C22×D59—44no—
124C22×D59—44no—
125C22×D59—42no—
126C22×D59—42no—
127C22×D59—84no—
128C22×D59—84no—
129C22×D59—82no—
130C22×D59—82no—
131C22×D59—82no—
132C22×D59—82no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—42no—
146C2×C4×C59—42no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—81no—
152C2×C4×C59—81no—
153C2×C4×C59C22⋊C812yes—
154C2×C4×C59C22×C811yescharacteristic
155C2×C4×C59C2×M1611yescharacteristic
156C2×Dic59—24no—
157C2×Dic59—48no—
158C2×Dic59—42no—
159C2×Dic59—42no—
160D236—44no—
161D236—48no—
162D236—44no—
163C8×C59—82no—
164C59⋊C8—82no—
165C2×#1318—591nomaximal, Sylow
166C22×C59—24no—
167C22×C59—24no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59#32312yes—
173C22×C59#32312yes—
174C22×C59#35312yes—
175C22×C59#35512yes—
176C22×C59C2×C22⋊C811yescharacteristic
177C22×C59C2×#9911yescharacteristic
178C22×C59C2×#10011yescharacteristic
179C2×D59—44no—
180C2×D59—44no—
181C2×D59—44no—
182C2×D59—82no—
183C2×D59—82no—
184C2×D59—88no—
185C2×D59—88no—
186C2×D59—82no—
187C2×D59—82no—
188C4×C59—22no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59—41no—
194C4×C59—41no—
195Dic59—44no—
196C2×#390—591no—
197C2×#324—591no—
198C2×#324D5911yescharacteristic, complemented
199#1318—594no—
200C2×C59—21no—
201C2×C59—21no—
202C2×C59—22no—
203C2×C59#131814yescomplemented
204C2×C59C2×#32311yescharacteristic
205C2×C59C2×#35311yescharacteristic
206C2×C59C2×#35511yescharacteristic
207D59—84no—
208C23×D4—591no—
209C22×C4⋊C4C2×D5911yescharacteristic
210C22×C22⋊C4—1181no—
211C22×C22⋊C4—1181no—
212C22×C42—591no—
213C2×C22⋊C8—21no—
214C2×C22⋊C8—1181no—
215#390—592no—
216#390—592no—
217#390—1181no—
218#390—1181no—
219#324—21no—
220#324—592no—
221#324—1181no—
222#324C2×D5912yescomplemented
223C59C2×#131811yescharacteristic, complemented, Sylow
224C25—1181no—
225C22×D4—594no—
226C22×D4—592no—
227C22×D4—1182no—
228C22×D4—1181no—
229C22×D4—2361no—
230C22×D4—2361no—
231C22×D4—2362no—
232C23×C4—1181no—
233C23×C4C4×D5911yescharacteristic
234C23×C4D23611yescharacteristic
235C23×C4C22⋊D5911yescharacteristic
236C22×C8—41no—
237C22×C8—2361no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C4×D5911yescharacteristic
241C2×C4⋊C4D23611yescharacteristic
242C2×C4⋊C4C22⋊D5911yescharacteristic
243C2×C4⋊C4C22×D5911yescharacteristic
244C2×C22⋊C4—1182no—
245C2×C22⋊C4—1182no—
246C2×C22⋊C4—1182no—
247C2×C22⋊C4—1182no—
248C2×C22⋊C4—2364no—
249C2×C22⋊C4—2364no—
250C2×C42—592no—
251C2×C42—1181no—
252C2×C42—2362no—
253C22⋊C8—22no—
254C22⋊C8—41no—
255C22⋊C8—1182no—
256C22⋊C8—2361no—
257C24—1184no—
258C24—1182no—
259C24—2362no—
260C24—2362no—
261C24—2362no—
262C24—2362no—
263C24—2361no—
264C24—2361no—
265C24—2362no—
266C24(C2×D59)⋊C411yescharacteristic
267C2×D4—594no—
268C2×D4—1184no—
269C2×D4—1182no—
270C2×D4—2362no—
271C2×D4—2364no—
272C2×D4—2364no—
273C2×D4—2362no—
274C2×D4—2362no—
275C2×D4—2364no—
276C2×D4—2364no—
277C2×D4—2362no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—1184no—
288C22×C4—1182no—
289C22×C4—2362no—
290C22×C4—2362no—
291C22×C4C8×D5912yes—
292C22×C4#1174565912yes—
293C22×C4(C2×D59)⋊C411yescharacteristic
294C22×C4(C2×D59)⋊C411yescharacteristic
295C22×C4(C2×D59)⋊C411yescharacteristic
296C22×C4C2×C4×D5911yescharacteristic
297C22×C4C2×D23611yescharacteristic
298C22×C4C2×C22⋊D5911yescharacteristic
299C2×C8—42no—
300C2×C8—82no—
301C2×C8—4722no—
302C2×C8—2362no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—2364no—
308C22⋊C4—2364no—
309C22⋊C4—2364no—
310C22⋊C4—2364no—
311C42—2362no—
312C42—2362no—
313C23—22no—
314C23—22no—
315C23—1184no—
316C23—4722no—
317C23—4722no—
318C23—4724no—
319C23—4724no—
320C23—4722no—
321C23—4722no—
322C23—2362no—
323C23—2362no—
324C23—2362no—
325C23—2362no—
326C23—2368no—
327C23—2368no—
328C23—2362no—
329C23—2364no—
330C23—2364no—
331C23—2362no—
332C23(C2×D59)⋊C812yes—
333C23#1427364111yescharacteristic
334C23#1427364111yescharacteristic
335C23#1427364312yes—
336C23C2×(C2×D59)⋊C411yescharacteristic
337D4—2364no—
338D4—2368no—
339D4—2364no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—1184no—
361C2×C4—2362no—
362C2×C4—2368no—
363C2×C4—2362no—
364C2×C4(C2×D59)⋊C812yes—
365C2×C4C2×C8×D5911yescharacteristic
366C2×C4C2×#1174565911yescharacteristic
367C8—82no—
368C8—4722no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22—4722no—
376C22—4722no—
377C22—4728no—
378C22—4722no—
379C22—4722no—
380C22—4728no—
381C22—2364no—
382C22—2364no—
383C22—2364no—
384C22#1683712512yes—
385C22#1683712512yes—
386C22#1683719412yes—
387C22#1683719812yes—
388C22C2×(C2×D59)⋊C811yescharacteristic
389C22C2×#1427364111yescharacteristic
390C22C2×#1427364311yescharacteristic
391C4—22no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—2364no—
399C2—21no—
400C2—22no—
401C2—21no—
402C2—4724no—
403C2#2142202714yesdirect product factor
404C2C2×#1683712511yescharacteristic
405C2C2×#1683719411yescharacteristic
406C2C2×#1683719811yescharacteristic
subgroupG/H
centerC23#14273641
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC2×C59×#324C2
socleC23×C59#99
prime radicalC23×D236C4
cosocleC2×C4⋊C4×C59C23
hypercenterC2×#324D59
hypocenterC59C2×#1318
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#16837128C2semidirect
3. #21422027C2direct (N × H)
4. C2×C59×#324C2semidirect
15. #16837128C22semidirect
19. C59×#324C22semidirect
26. C22×D236C8semidirect
94. C2×D236C2×C8semidirect
198. C2×#324D59semidirect
203. C2×C59#1318semidirect
222. #324C2×D59semidirect
223. C59C2×#1318semidirect
403. C2#21422027direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168373032
C2×#168371282
#214220272
C2×C59×#3242
C2×#131859