Finite Group Databasebeta

← C2×#22553496 #30675304

Subgroups — C2×#22553496 #30675304

Order 16768.

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×#16862293C211yesmaximal, characteristic
2C2×#16862279C211yesmaximal, characteristic, complemented
3C2×#16862276C211yesmaximal, characteristic, complemented
4C2×#16862246C211yesmaximal, characteristic, complemented
5#22553496C218yesmaximal, direct product factor
6C2×#16862174C211yesmaximal, characteristic, complemented
7C2×#16862164C211yesmaximal, characteristic, complemented
8C2×#104×C131C211yesmaximal, characteristic, complemented
9#16862293—22no—
10#16862293—22no—
11#16862293—22no—
12#16862293C2214yes—
13C2×#14523513—21no—
14C2×#14523513C2211yescharacteristic
15C2×#14523512—22no—
16C2×#14523512—21no—
17C22×Dic262—21no—
18C22×Dic262C2211yescharacteristic
19C22×Dic262C2211yescharacteristic
20C2×Q8×D131—21no—
21#16862279—22no—
22#16862279C2214yescomplemented
23#16862276—22no—
24#16862276C2214yescomplemented
25C2×Dic524—21no—
26C2×#14523503—21no—
27C2×#14523502—21no—
28C2×#14523501—21no—
29C2×#14523497C2211yescharacteristic
30#16862246—22no—
31#16862246C2214yescomplemented
32C2×C4⋊Dic131—22no—
33#16862174C2214yescomplemented
34#16862164C2214yescomplemented
35C2×#41×C131C2211yescharacteristic
36C2×M16×C131C2211yescharacteristic
37C2×C42×C131C2211yescharacteristic
38#104×C131C2214yescomplemented
39#14523513—24no—
40#14523513—24no—
41#14523513—21no—
42#14523513—21no—
43#14523513D412yes—
44#14523513C2312yes—
45#14523512—28no—
46#14523512—24no—
47#14523512—28no—
48#14523512—24no—
49C2×Dic262—22no—
50C2×Dic262—24no—
51C2×Dic262—24no—
52C2×Dic262—21no—
53C2×Dic262—21no—
54C2×Dic262—21no—
55C2×Dic262—21no—
56C2×Dic262—21no—
57C2×Dic262—21no—
58C2×Dic262—21no—
59C2×Dic262—21no—
60C2×Dic262—21no—
61C2×Dic262D411yescharacteristic
62C2×Dic262D412yes—
63C2×Dic262C2312yes—
64C2×Dic262C2312yes—
65Q8×D131—24no—
66Q8×D131—24no—
67C2×D524D411yescharacteristic
68C2×C22⋊D131—22no—
69C2×C22⋊D131—21no—
70C22×Dic131—22no—
71C22×Dic131—21no—
72C22×Dic131D412yes—
73C2×C4×D131—22no—
74C2×C4×D131—21no—
75Dic524—24no—
76#14523503—24no—
77#14523502—24no—
78#14523501—24no—
79#14523497—21no—
80#14523497C2312yes—
81C2×C131⋊C8—21no—
82C4⋊Dic131—28no—
83#41×C131—21no—
84#41×C131—21no—
85#41×C131D412yes—
86#41×C131C2312yes—
87C2×Q8×C131D411yescharacteristic
88C2×D4×C131—21no—
89C2×D4×C131D411yescharacteristic
90C22×C4×C131—21no—
91C22×C4×C131—21no—
92C22×C4×C131C2311yescharacteristic
93M16×C131—21no—
94M16×C131C2312yes—
95C2×C8×C131—21no—
96C42×C131—21no—
97C42×C131C2312yes—
98C22×D131—21no—
99Dic262—28no—
100Dic262—22no—
101Dic262—21no—
102Dic262—22no—
103Dic262—22no—
104Dic262—22no—
105Dic262—22no—
106Dic262—22no—
107Dic262—41no—
108Dic262—41no—
109Dic262C2×D412yes—
110D524—21no—
111D524C2×D412yes—
112C22⋊D131—28no—
113C22⋊D131—22no—
114C22⋊D131—22no—
115C2×Dic131—28no—
116C2×Dic131—21no—
117C2×Dic131—22no—
118C2×Dic131—22no—
119C2×Dic131—22no—
120C2×Dic131—41no—
121C2×Dic131—41no—
122C2×Dic131C2×D414yes—
123C4×D131—28no—
124C4×D131—22no—
125C4×D131—22no—
126C131⋊C8—22no—
127C23×C131—21no—
128C23×C131C2×D411yescharacteristic
129Q8×C131—21no—
130Q8×C131C2×D412yes—
131D4×C131—22no—
132D4×C131—22no—
133D4×C131—21no—
134D4×C131C2×D412yes—
135C2×C4×C131—24no—
136C2×C4×C131—21no—
137C2×C4×C131—22no—
138C2×C4×C131—22no—
139C2×C4×C131—22no—
140C2×C4×C131C2×D411yescharacteristic
141C2×C4×C131C2×D411yescharacteristic
142C2×C4×C131C2×D411yescharacteristic
143C2×C4×C131C2×D411yescharacteristic
144C2×C4×C131C2×D411yescharacteristic
145C2×C4×C131C2411yescharacteristic
146C8×C131—22no—
147C22×C131—22no—
148C22×C131—22no—
149C22×C131—42no—
150C22×C131#12011yescharacteristic
151C22×C131#12011yescharacteristic
152C22×C131C22×D411yescharacteristic
153C2×D131—22no—
154C2×D131—42no—
155C4×C131—24no—
156C4×C131—22no—
157C4×C131#12011yescharacteristic
158C4×C131#12011yescharacteristic
159C4×C131C22×D411yescharacteristic
160C4×C131C22×D411yescharacteristic
161Dic131—22no—
162Dic131—24no—
163Dic131—42no—
164C2×C131—21no—
165C2×C131—21no—
166C2×C131—42no—
167C2×C131#45612yescomplemented
168C2×C131C2×#12011yescharacteristic
169D131—42no—
170C131C2×#45611yescharacteristic, complemented, Sylow
171C2×#456—1311nomaximal, Sylow
172C2×2−1+4—1311no—
173C2×#137—1311no—
174C2×#137—1311no—
175C2×C4⋊Q8—1311no—
176#456—1318no—
177C2×#104—1311no—
178C2×#104D13111yescharacteristic, complemented
179C2×#101—1311no—
1802−1+4—2622no—
1812−1+4—2622no—
1822−1+4—2622no—
1832−1+4—1314no—
184C2×#41—2621no—
185C2×#41—2621no—
186C2×#41—2622no—
187C2×#41—1311no—
188C2×#41C2×D13111yescharacteristic
189C22×Q8—2621no—
190C22×Q8—2621no—
191C22×Q8—1311no—
192C22×Q8—1311no—
193#137—2622no—
194#137—2622no—
195#137—1314no—
196#137—1314no—
197C2×Q16—2621no—
198C2×Q16—2621no—
199C2×SD16—2621no—
200C2×SD16—2621no—
201C2×M16—1311no—
202C2×M16C2×D13111yescharacteristic
203C4⋊Q8—2622no—
204C4⋊Q8—1314no—
205C2×C4⋊C4—2622no—
206C2×C42C2×D13111yescharacteristic
207#104—1314no—
208#104C2×D13114yescomplemented
209#101—1314no—
210#41—21no—
211#41—21no—
212#41—2624no—
213#41—2624no—
214#41—2628no—
215#41—2624no—
216#41—2624no—
217#41—2628no—
218#41—2621no—
219#41—2621no—
220#41—1312no—
221#41—1312no—
222#41D52412yes—
223#41C22×D13112yes—
224C2×Q8—2622no—
225C2×Q8—2624no—
226C2×Q8—2624no—
227C2×Q8—2624no—
228C2×Q8—2624no—
229C2×Q8—2621no—
230C2×Q8—2621no—
231C2×Q8—2621no—
232C2×Q8—2621no—
233C2×Q8—2621no—
234C2×Q8—2621no—
235C2×Q8—2621no—
236C2×Q8—2621no—
237C2×Q8—2621no—
238C2×Q8—1311no—
239C2×Q8—1312no—
240C2×Q8—1312no—
241C2×Q8—1312no—
242C2×Q8D52411yescharacteristic
243C2×D4—21no—
244C2×D4—2622no—
245C2×D4—2621no—
246C2×D4—1311no—
247C2×D4D52411yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—2622no—
251C22×C4—2622no—
252C22×C4—2621no—
253C22×C4—2621no—
254C22×C4—1312no—
255C22×C4C22×D13111yescharacteristic
256Q16—2624no—
257Q16—2624no—
258SD16—2624no—
259SD16—2624no—
260M16—21no—
261M16—2621no—
262M16—1312no—
263M16C22×D13112yes—
264C2×C8—21no—
265C2×C8—2621no—
266C4⋊C4—2628no—
267C42—21no—
268C42C22×D13112yes—
269C23—21no—
270C23—2621no—
271C23D4×D13111yescharacteristic
272Q8—21no—
273Q8—2628no—
274Q8—2621no—
275Q8—2622no—
276Q8—2622no—
277Q8—2622no—
278Q8—2622no—
279Q8—2622no—
280Q8—2622no—
281Q8—5241no—
282Q8—5241no—
283Q8—1312no—
284Q8C2×D52412yes—
285D4—22no—
286D4—22no—
287D4—21no—
288D4—2628no—
289D4—2622no—
290D4—2622no—
291D4—2621no—
292D4—1312no—
293D4C2×D52412yes—
294C2×C4—24no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—2628no—
300C2×C4—2628no—
301C2×C4—2622no—
302C2×C4—2621no—
303C2×C4—2622no—
304C2×C4—2622no—
305C2×C4—2622no—
306C2×C4—2622no—
307C2×C4—5241no—
308C2×C4—5241no—
309C2×C4—1314no—
310C2×C4D4×D13111yescharacteristic
311C2×C4D4×D13111yescharacteristic
312C2×C4D4×D13111yescharacteristic
313C2×C4C2×D52411yescharacteristic
314C2×C4C2×D52411yescharacteristic
315C2×C4C23×D13111yescharacteristic
316C8—22no—
317C8—2622no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—2622no—
322C22—5242no—
323C22#1686221611yescharacteristic
324C22#1686221611yescharacteristic
325C22C2×D4×D13111yescharacteristic
326C4—24no—
327C4—22no—
328C4—2622no—
329C4—2624no—
330C4—5242no—
331C4#1686221611yescharacteristic
332C4#1686221611yescharacteristic
333C4C2×D4×D13111yescharacteristic
334C4C22×D52411yescharacteristic
335C2—21no—
336C2—21no—
337C2—42no—
338C2—5242no—
339C2#2255349612yesdirect product factor
340C2C2×#1686221611yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#104×C131C2
socleC22×C131#120
prime radicalC2×#16862293C2
hypercenterC2×#104D131
hypocenterC131C2×#456
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16862279C2semidirect
3. C2×#16862276C2semidirect
4. C2×#16862246C2semidirect
5. #22553496C2direct (N × H)
6. C2×#16862174C2semidirect
7. C2×#16862164C2semidirect
8. C2×#104×C131C2semidirect
22. #16862279C22semidirect
24. #16862276C22semidirect
31. #16862246C22semidirect
33. #16862174C22semidirect
34. #16862164C22semidirect
38. #104×C131C22semidirect
167. C2×C131#456semidirect
170. C131C2×#456semidirect
178. C2×#104D131semidirect
208. #104C2×D131semidirect
339. C2#22553496direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168622932
C2×#168622792
C2×#168622762
C2×#168622462
#225534962
C2×#168621742
C2×#168621642
C2×#104×C1312
C2×#456131