Finite Group Databasebeta

← C2×#23752391 #34078441

Subgroups — C2×#23752391 #34078441

Order 19328.

760 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18012785C211yesmaximal, characteristic
2C2×#18012772C211yesmaximal, characteristic, complemented
3C2×#18012769C211yesmaximal, characteristic, complemented
4C2×#18012727C211yesmaximal, characteristic, complemented
5#23752391C218yesmaximal, direct product factor
6C2×#18012667C211yesmaximal, characteristic, complemented
7C2×#18012655C211yesmaximal, characteristic, complemented
8C2×#104×C151C211yesmaximal, characteristic, complemented
9#18012785—22no—
10#18012785—22no—
11#18012785—22no—
12#18012785C2214yes—
13C2×#14559005—21no—
14C2×#14559004—21no—
15C2×#14559004C2211yescharacteristic
16C22×Dic302C2211yescharacteristic, complemented
17C22×D604—21no—
18C22×D604C2211yescharacteristic
19C2×D4×D151—21no—
20C2×D4×D151—22no—
21#18012772—22no—
22#18012772C2214yescomplemented
23#18012769—22no—
24#18012769C2214yescomplemented
25C2×Dic604—21no—
26C2×#14558994—21no—
27C2×#14558993—21no—
28C2×#14558992—21no—
29C2×#14558988C2211yescharacteristic, complemented
30#18012727—22no—
31#18012727C2214yescomplemented
32C2×(C2×D151)⋊C4—22no—
33#18012667C2214yescomplemented
34#18012655C2214yescomplemented
35C2×#41×C151C2211yescharacteristic
36C2×M16×C151C2211yescharacteristic, complemented
37C2×C42×C151C2211yescharacteristic, complemented
38#104×C151C2214yescomplemented
39C23×D151—22no—
40C23×D151D412yes—
41#14559005—24no—
42#14559005—24no—
43#14559004—24no—
44#14559004—24no—
45#14559004—21no—
46#14559004—21no—
47#14559004D412yes—
48#14559004C2312yes—
49C2×Dic302—21no—
50C2×Dic302—21no—
51C2×Dic302—21no—
52C2×Dic302—21no—
53C2×Dic302—21no—
54C2×Dic302D411yescharacteristic
55C2×Dic302C2312yescomplemented
56C2×D604—22no—
57C2×D604—24no—
58C2×D604—24no—
59C2×D604—21no—
60C2×D604—21no—
61C2×D604—21no—
62C2×D604—21no—
63C2×D604D412yes—
64C2×D604D411yescharacteristic
65C2×D604C2312yes—
66C2×C22⋊D151—22no—
67C2×C22⋊D151—21no—
68D4×D151—28no—
69D4×D151—24no—
70D4×D151—28no—
71D4×D151—24no—
72C22×Dic151—21no—
73C2×C4×D151—22no—
74C2×C4×D151—21no—
75Dic604—24no—
76#14558994—24no—
77#14558993—24no—
78#14558992—24no—
79#14558988—21no—
80#14558988C2312yescomplemented
81C2×C151⋊C8—21no—
82(C2×D151)⋊C4—28no—
83#41×C151—21no—
84#41×C151—21no—
85#41×C151D412yes—
86#41×C151C2312yes—
87C2×Q8×C151D411yescharacteristic
88C2×D4×C151—21no—
89C2×D4×C151D411yescharacteristic
90C22×C4×C151—21no—
91C22×C4×C151—21no—
92C22×C4×C151C2311yescharacteristic
93M16×C151—21no—
94M16×C151C2312yescomplemented
95C2×C8×C151—21no—
96C42×C151—21no—
97C42×C151C2312yescomplemented
98C22×D151—22no—
99C22×D151—21no—
100C22×D151—28no—
101C22×D151—22no—
102C22×D151—44no—
103C22×D151—84no—
104C22×D151C2×D414yes—
105Dic302—21no—
106Dic302—22no—
107Dic302—22no—
108Dic302—41no—
109Dic302—41no—
110Dic302C2×D412yes—
111D604—28no—
112D604—22no—
113D604—22no—
114D604—21no—
115D604—22no—
116D604—22no—
117D604C2×D412yes—
118C22⋊D151—28no—
119C22⋊D151—22no—
120C22⋊D151—22no—
121C2×Dic151—21no—
122C2×Dic151—22no—
123C2×Dic151—41no—
124C2×Dic151—41no—
125C4×D151—28no—
126C4×D151—22no—
127C4×D151—22no—
128C151⋊C8—22no—
129C23×C151—21no—
130C23×C151C2×D411yescharacteristic
131Q8×C151—21no—
132Q8×C151C2×D412yes—
133D4×C151—22no—
134D4×C151—22no—
135D4×C151—21no—
136D4×C151C2×D412yes—
137C2×C4×C151—21no—
138C2×C4×C151—22no—
139C2×C4×C151—22no—
140C2×C4×C151—22no—
141C2×C4×C151—42no—
142C2×C4×C151C2×D411yescharacteristic
143C2×C4×C151C2×D411yescharacteristic
144C2×C4×C151C2×D411yescharacteristic
145C2×C4×C151C2×D411yescharacteristic
146C2×C4×C151C2×D411yescharacteristic
147C2×C4×C151C2411yescharacteristic
148C8×C151—22no—
149C22×C151—22no—
150C22×C151—22no—
151C22×C151—42no—
152C22×C151#12011yescharacteristic
153C22×C151#12011yescharacteristic
154C22×C151C22×D411yescharacteristic
155C2×D151—24no—
156C2×D151—22no—
157C2×D151—44no—
158C2×D151—42no—
159C2×D151—44no—
160C2×D151—82no—
161C2×D151—88no—
162C4×C151—22no—
163C4×C151—42no—
164C4×C151#12011yescharacteristic
165C4×C151#12011yescharacteristic
166C4×C151C22×D411yescharacteristic
167C4×C151C22×D411yescharacteristic
168Dic151—22no—
169Dic151—42no—
170C2×C151—21no—
171C2×C151—21no—
172C2×C151—42no—
173C2×C151#45512yescomplemented
174C2×C151C2×#12011yescharacteristic
175D151—44no—
176D151—42no—
177C151C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1511nomaximal, Sylow
179C2×H(5,𝔽2)—1511no—
180C2×#137—1511no—
181C2×#137—1511no—
182C2×#124—1511no—
183#455—1518no—
184C2×#104—1511no—
185C2×#104D15111yescharacteristic, complemented
186C2×#100—1511no—
187H(5,𝔽2)—3022no—
188H(5,𝔽2)—3022no—
189H(5,𝔽2)—3022no—
190H(5,𝔽2)—1514no—
191C2×#41—3021no—
192C2×#41—3021no—
193C2×#41—1511no—
194C2×#41C2×D15111yescharacteristic
195C22×Q8—1511no—
196C22×D4—3021no—
197C22×D4—3022no—
198C22×D4—3021no—
199C22×D4—1511no—
200#137—3022no—
201#137—3022no—
202#137—1514no—
203#137—1514no—
204C2×Q16—3021no—
205C2×Q16—3021no—
206C2×SD16—3021no—
207C2×SD16—3021no—
208C2×M16—1511no—
209C2×M16C2×D15111yescharacteristic, complemented
210#124—3022no—
211#124—1514no—
212C2×C22⋊C4—3022no—
213C2×C42C2×D15111yescharacteristic, complemented
214#104—1514no—
215#104C2×D15114yescomplemented
216#100—1514no—
217C24—3022no—
218C24—1512no—
219#41—21no—
220#41—21no—
221#41—3024no—
222#41—3024no—
223#41—3024no—
224#41—3024no—
225#41—3021no—
226#41—3021no—
227#41—1512no—
228#41—1512no—
229#41D60412yes—
230#41C22×D15112yes—
231C2×Q8—3021no—
232C2×Q8—3021no—
233C2×Q8—3021no—
234C2×Q8—3021no—
235C2×Q8—3021no—
236C2×Q8—1511no—
237C2×Q8—1512no—
238C2×Q8D60411yescharacteristic
239C2×D4—21no—
240C2×D4—3022no—
241C2×D4—3024no—
242C2×D4—3024no—
243C2×D4—3028no—
244C2×D4—3024no—
245C2×D4—3028no—
246C2×D4—3022no—
247C2×D4—3021no—
248C2×D4—3024no—
249C2×D4—3021no—
250C2×D4—3021no—
251C2×D4—3021no—
252C2×D4—3021no—
253C2×D4—1512no—
254C2×D4—1511no—
255C2×D4—1512no—
256C2×D4D60411yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3022no—
260C22×C4—3021no—
261C22×C4—3021no—
262C22×C4C22×D15111yescharacteristic
263Q16—3024no—
264Q16—3024no—
265SD16—3024no—
266SD16—3024no—
267M16—21no—
268M16—3021no—
269M16—1512no—
270M16C22×D15112yescomplemented
271C2×C8—21no—
272C2×C8—3021no—
273C22⋊C4—3028no—
274C42—21no—
275C42C22×D15112yescomplemented
276C23—21no—
277C23—3028no—
278C23—3022no—
279C23—3021no—
280C23—3022no—
281C23—6044no—
282C23—1514no—
283C23—12084no—
284C23D4×D15111yescharacteristic
285Q8—21no—
286Q8—3021no—
287Q8—3022no—
288Q8—3022no—
289Q8—6041no—
290Q8—6041no—
291Q8—1512no—
292Q8C2×D60412yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—3028no—
297D4—3028no—
298D4—3022no—
299D4—3022no—
300D4—3022no—
301D4—3021no—
302D4—3022no—
303D4—3022no—
304D4—3022no—
305D4—1512no—
306D4C2×D60412yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—3028no—
313C2×C4—3021no—
314C2×C4—3022no—
315C2×C4—3022no—
316C2×C4—3022no—
317C2×C4—6041no—
318C2×C4—6041no—
319C2×C4D4×D15111yescharacteristic
320C2×C4D4×D15111yescharacteristic
321C2×C4D4×D15111yescharacteristic
322C2×C4C2×D60411yescharacteristic
323C2×C4C2×D60411yescharacteristic
324C2×C4C23×D15111yescharacteristic
325C8—22no—
326C8—3022no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—3024no—
331C22—3022no—
332C22—6044no—
333C22—6044no—
334C22—6042no—
335C22—12082no—
336C22—12088no—
337C22#1801270911yescharacteristic
338C22#1801270911yescharacteristic
339C22C2×D4×D15111yescharacteristic
340C4—22no—
341C4—42no—
342C4—3022no—
343C4—6042no—
344C4#1801270911yescharacteristic
345C4#1801270911yescharacteristic
346C4C2×D4×D15111yescharacteristic
347C4C22×D60411yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—6044no—
352C2—6042no—
353C2#2375239112yesdirect product factor
354C2C2×#1801270911yescharacteristic
subgroupG/H
centerC22#18012709
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×#104×C151C2
socleC22×C151#120
prime radicalC2×#18012785C2
hypercenterC2×#104D151
hypocenterC151C2×#455
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#18012772C2semidirect
3. C2×#18012769C2semidirect
4. C2×#18012727C2semidirect
5. #23752391C2direct (N × H)
6. C2×#18012667C2semidirect
7. C2×#18012655C2semidirect
8. C2×#104×C151C2semidirect
16. C22×Dic302C22semidirect
22. #18012772C22semidirect
24. #18012769C22semidirect
29. C2×#14558988C22semidirect
31. #18012727C22semidirect
33. #18012667C22semidirect
34. #18012655C22semidirect
36. C2×M16×C151C22semidirect
37. C2×C42×C151C22semidirect
38. #104×C151C22semidirect
55. C2×Dic302C23semidirect
80. #14558988C23semidirect
94. M16×C151C23semidirect
97. C42×C151C23semidirect
173. C2×C151#455semidirect
177. C151C2×#455semidirect
185. C2×#104D151semidirect
209. C2×M16C2×D151semidirect
213. C2×C42C2×D151semidirect
215. #104C2×D151semidirect
270. M16C22×D151semidirect
275. C42C22×D151semidirect
353. C2#23752391direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#180127852
C2×#180127722
C2×#180127692
C2×#180127272
#237523912
C2×#180126672
C2×#180126552
C2×#104×C1512
C2×#455151