Finite Group Databasebeta

← C2×#22550057 #30648070

Subgroups — C2×#22550057 #30648070

Order 16256.

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

normal subgroups only

This group has 708 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×#16861256C211yesmaximal, characteristic
2C2×#16861240C211yesmaximal, characteristic, complemented
3C2×#16861207C211yesmaximal, characteristic, complemented
4#22550057C218yesmaximal, direct product factor
5C2×#16861136C211yesmaximal, characteristic, complemented
6C2×#16861135C211yesmaximal, characteristic, complemented
7C2×#16861127C211yesmaximal, characteristic, complemented
8C2×#137×C127C211yesmaximal, characteristic, complemented
9#16861256—22no—
10#16861256—22no—
11#16861256—22no—
12#16861256C2214yes—
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C2×#14523247—22no—
16C2×#14523247—21no—
17C22×Dic254—21no—
18C22×Dic254C2211yescharacteristic
19C2×Q8×D127—21no—
20#16861240—22no—
21#16861240C2214yescomplemented
22C2×#14523238—21no—
23C2×#14523235—21no—
24C2×#14523232C2211yescharacteristic
25#16861207—22no—
26#16861207C2214yescomplemented
27C2×Dic127⋊C4—22no—
28C2×C4×Dic127C2211yescharacteristic
29#16861136C2214yescomplemented
30#16861135C2214yescomplemented
31#16861127C2214yescomplemented
32C2×#41×C127C2211yescharacteristic
33C22×Q8×C127C2211yescharacteristic
34#137×C127—22no—
35#137×C127C2214yescomplemented
36C2×Q16×C127—21no—
37C2×SD16×C127—21no—
38C2×M16×C127C2211yescharacteristic
39#14523248—24no—
40#14523248—24no—
41#14523248—21no—
42#14523248—21no—
43#14523248D412yes—
44#14523248C2312yes—
45#14523247—28no—
46#14523247—24no—
47#14523247—28no—
48#14523247—24no—
49C2×Dic254—22no—
50C2×Dic254—24no—
51C2×Dic254—24no—
52C2×Dic254—21no—
53C2×Dic254—21no—
54C2×Dic254—21no—
55C2×Dic254—21no—
56C2×Dic254D411yescharacteristic
57C2×Dic254D412yes—
58C2×Dic254C2312yes—
59Q8×D127—24no—
60Q8×D127—24no—
61C2×D508D411yescharacteristic
62C2×C22⋊D127—22no—
63C2×C22⋊D127—21no—
64C22×Dic127—22no—
65C22×Dic127—21no—
66C22×Dic127D412yes—
67C2×C4×D127—22no—
68C2×C4×D127—21no—
69#14523238—24no—
70#14523235—24no—
71#14523232—21no—
72#14523232C2312yes—
73C2×C127⋊C8—21no—
74Dic127⋊C4—28no—
75C4×Dic127—21no—
76C4×Dic127C2312yes—
77#41×C127—21no—
78#41×C127—21no—
79#41×C127D412yes—
80#41×C127C2312yes—
81C2×Q8×C127—21no—
82C2×Q8×C127—21no—
83C2×Q8×C127—21no—
84C2×Q8×C127—21no—
85C2×Q8×C127—21no—
86C2×Q8×C127D411yescharacteristic
87C2×Q8×C127C2312yes—
88C2×D4×C127—21no—
89C2×D4×C127D411yescharacteristic
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93Q16×C127—24no—
94SD16×C127—24no—
95M16×C127—21no—
96M16×C127C2312yes—
97C2×C8×C127—21no—
98C22×D127—21no—
99Dic254—28no—
100Dic254—22no—
101Dic254—21no—
102Dic254—22no—
103Dic254—22no—
104Dic254—22no—
105Dic254C2×D412yes—
106D508—21no—
107D508C2×D412yes—
108C22⋊D127—28no—
109C22⋊D127—22no—
110C22⋊D127—22no—
111C2×Dic127—24no—
112C2×Dic127—28no—
113C2×Dic127—21no—
114C2×Dic127—22no—
115C2×Dic127—22no—
116C2×Dic127—22no—
117C2×Dic127C2×D414yes—
118C4×D127—28no—
119C4×D127—22no—
120C4×D127—22no—
121C127⋊C8—22no—
122C23×C127—21no—
123C23×C127C2×D411yescharacteristic
124Q8×C127—21no—
125Q8×C127—22no—
126Q8×C127—22no—
127Q8×C127—41no—
128Q8×C127—41no—
129Q8×C127C2×D412yes—
130D4×C127—22no—
131D4×C127—22no—
132D4×C127—21no—
133D4×C127C2×D412yes—
134C2×C4×C127—21no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—22no—
147C22×C127—22no—
148C22×C127—22no—
149C22×C127—42no—
150C22×C127#12011yescharacteristic
151C22×C127#12011yescharacteristic
152C22×C127C22×D411yescharacteristic
153C2×D127—22no—
154C2×D127—42no—
155C4×C127—22no—
156C4×C127—42no—
157C4×C127#12011yescharacteristic
158C4×C127#12011yescharacteristic
159C4×C127C22×D411yescharacteristic
160C4×C127C22×D411yescharacteristic
161Dic127—24no—
162Dic127—22no—
163Dic127—24no—
164C2×C127—21no—
165C2×C127—21no—
166C2×C127—42no—
167C2×C127#45612yescomplemented
168C2×C127C2×#12011yescharacteristic
169D127—42no—
170C2×#456—1271nomaximal, Sylow
171C127C2×#45611yescharacteristic, complemented, Sylow
172C2×2−1+4—1271no—
173C2×#137—1271no—
174C2×#137D12711yescharacteristic, complemented
175C2×C4⋊Q8—1271no—
176#456—1278no—
177C2×#104—1271no—
178C2×#104—1271no—
179C2×#101—1271no—
1802−1+4—1274no—
1812−1+4—2542no—
1822−1+4—2542no—
1832−1+4—2542no—
184C2×#41—1271no—
185C2×#41—2541no—
186C2×#41—2542no—
187C2×#41—2541no—
188C2×#41C2×D12711yescharacteristic
189C22×Q8—1271no—
190C22×Q8—2541no—
191C22×Q8—2541no—
192C22×Q8C2×D12711yescharacteristic
193#137—22no—
194#137—1274no—
195#137—2542no—
196#137C2×D12714yescomplemented
197C2×Q16—21no—
198C2×Q16—2541no—
199C2×SD16—21no—
200C2×SD16—2541no—
201C2×M16—1271no—
202C2×M16C2×D12711yescharacteristic
203C4⋊Q8—1274no—
204C4⋊Q8—2542no—
205C2×C4⋊C4—2542no—
206C2×C42—1271no—
207#104—1274no—
208#104—1274no—
209#101—1274no—
210#41—21no—
211#41—21no—
212#41—1272no—
213#41—1272no—
214#41—2544no—
215#41—2544no—
216#41—2548no—
217#41—2548no—
218#41—2544no—
219#41—2544no—
220#41—2541no—
221#41—2541no—
222#41C22⋊D12712yes—
223#41C22×D12712yes—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—1272no—
230C2×Q8—1272no—
231C2×Q8—1271no—
232C2×Q8—2544no—
233C2×Q8—2542no—
234C2×Q8—2544no—
235C2×Q8—2544no—
236C2×Q8—2544no—
237C2×Q8—2541no—
238C2×Q8—2541no—
239C2×Q8—2541no—
240C2×Q8—2541no—
241C2×Q8C22⋊D12711yescharacteristic
242C2×Q8C22×D12712yes—
243C2×D4—21no—
244C2×D4—1271no—
245C2×D4—2542no—
246C2×D4—2541no—
247C2×D4C22⋊D12711yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—1272no—
251C22×C4—2542no—
252C22×C4—2542no—
253C22×C4—2541no—
254C22×C4—2541no—
255C22×C4C22×D12711yescharacteristic
256Q16—24no—
257Q16—2544no—
258SD16—24no—
259SD16—2544no—
260M16—21no—
261M16—1272no—
262M16—2541no—
263M16C22×D12712yes—
264C2×C8—21no—
265C2×C8—2541no—
266C4⋊C4—2548no—
267C42—1272no—
268C42—2541no—
269C23—21no—
270C23—2541no—
271C23D4×D12711yescharacteristic
272Q8—21no—
273Q8—22no—
274Q8—22no—
275Q8—41no—
276Q8—41no—
277Q8—1272no—
278Q8—2548no—
279Q8—2542no—
280Q8—2542no—
281Q8—2542no—
282Q8—2542no—
283Q8—2541no—
284Q8C2×C22⋊D12712yes—
285D4—22no—
286D4—22no—
287D4—21no—
288D4—1272no—
289D4—2548no—
290D4—2542no—
291D4—2542no—
292D4—2541no—
293D4C2×C22⋊D12712yes—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—1274no—
301C2×C4—2548no—
302C2×C4—2548no—
303C2×C4—2544no—
304C2×C4—2542no—
305C2×C4—2542no—
306C2×C4—2542no—
307C2×C4—2541no—
308C2×C4—2542no—
309C2×C4—2542no—
310C2×C4D4×D12711yescharacteristic
311C2×C4D4×D12711yescharacteristic
312C2×C4D4×D12711yescharacteristic
313C2×C4C2×C22⋊D12711yescharacteristic
314C2×C4C2×C22⋊D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—22no—
317C8—2542no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—5082no—
322C22—2542no—
323C22#1686117711yescharacteristic
324C22#1686117711yescharacteristic
325C22C2×D4×D12711yescharacteristic
326C4—22no—
327C4—42no—
328C4—2544no—
329C4—2544no—
330C4—2542no—
331C4#1686117711yescharacteristic
332C4#1686117711yescharacteristic
333C4C2×D4×D12711yescharacteristic
334C4C22×C22⋊D12711yescharacteristic
335C2—21no—
336C2—21no—
337C2—42no—
338C2—5082no—
339C2#2255005712yesdirect product factor
340C2C2×#1686117711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127#120
prime radicalC2×#16861256C2
hypercenterC2×#137D127
hypocenterC127C2×#456
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. C2×#16861207C2semidirect
4. #22550057C2direct (N × H)
5. C2×#16861136C2semidirect
6. C2×#16861135C2semidirect
7. C2×#16861127C2semidirect
8. C2×#137×C127C2semidirect
21. #16861240C22semidirect
26. #16861207C22semidirect
29. #16861136C22semidirect
30. #16861135C22semidirect
31. #16861127C22semidirect
35. #137×C127C22semidirect
167. C2×C127#456semidirect
171. C127C2×#456semidirect
174. C2×#137D127semidirect
196. #137C2×D127semidirect
339. C2#22550057direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612402
C2×#168612072
#225500572
C2×#168611362
C2×#168611352
C2×#168611272
C2×#137×C1272
C2×#456127