Finite Group Databasebeta

← C2×#22562098 #31009262

Subgroups — C2×#22562098 #31009262

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863837C211yesmaximal, characteristic
2C2×#16863519C211yesmaximal, characteristic
3#22562098C218yesmaximal, direct product factor
4C2×#16863218C211yesmaximal, characteristic, complemented
5C2×#16863212C211yesmaximal, characteristic, complemented
6C2×#16863205C211yesmaximal, characteristic
7C2×#16863174C211yesmaximal, characteristic, complemented
8C2×#441×C67C211yesmaximal, characteristic, complemented
9C2×#14524065C2211yescharacteristic
10C2×#14524049—21no—
11#16863837C414yescomplemented
12#16863837C2214yes—
13C2×#14523989C412yescomplemented
14C2×#14523986C412yescomplemented
15C2×#14523983C2211yescharacteristic
16C2×#14523983C2211yescharacteristic
17#16863519—22no—
18#16863519C2214yes—
19C2×#14523979C2211yescharacteristic
20C2×#14523971—21no—
21#16863218C2214yescomplemented
22#16863212C2214yescomplemented
23#16863205C2214yes—
24#16863174C2214yescomplemented
25C22×Q16×C67C2211yescharacteristic
26C2×C4×Q8×C67—21no—
27#441×C67—22no—
28#441×C67C2214yescomplemented
29C2×#106×C67C2211yescharacteristic
30C2×#103×C67—21no—
31C2×#97×C67C2211yescharacteristic
32C2×#11761628—21no—
33#14524065—21no—
34#14524065—21no—
35#14524065C2×C412yes—
36#14524065C2312yes—
37C2×Dic268C2×C411yescharacteristic
38C2×#11761617—21no—
39C2×D536C2×C411yescharacteristic
40#14524049—24no—
41#14524049—42no—
42C2×#11761612—21no—
43C2×#11761611—41no—
44C2×C8×D67—41no—
45#14523989C2×C418yescomplemented
46#14523986C2×C418yescomplemented
47#14523983—21no—
48#14523983C2×C412yes—
49#14523983C2312yes—
50#14523983C2312yes—
51C2×C67⋊C16—21no—
52C2×C67⋊C16C2×C412yes—
53#14523979—21no—
54#14523979C2312yes—
55#14523971—24no—
56C22×Q8×C67—21no—
57C2×Q16×C67—21no—
58C2×Q16×C67—21no—
59C2×Q16×C67—21no—
60C2×Q16×C67C2×C412yes—
61C2×Q16×C67C2×C412yes—
62C2×Q16×C67C2312yes—
63C22×C8×C67D411yescharacteristic
64C22×C8×C67C2311yescharacteristic
65C4×Q8×C67—24no—
66C4×Q8×C67—42no—
67C2×C4⋊C4×C67—21no—
68C2×C4⋊C4×C67—41no—
69C2×C42×C67—41no—
70C2×C42×C67D411yescharacteristic
71#106×C67—21no—
72#106×C67C2312yes—
73#103×C67—24no—
74#97×C67D412yes—
75#97×C67C2312yes—
76#11761628—22no—
77#11761628—22no—
78#11761628—41no—
79#11761628—41no—
80C2×Dic134—21no—
81C2×D268—21no—
82C2×C22⋊D67—41no—
83C2×C4×D67—41no—
84Dic268—21no—
85Dic268C22×C412yes—
86#11761617—22no—
87#11761617—22no—
88D536—21no—
89D536C22×C412yes—
90#11761612—22no—
91#11761612—41no—
92#11761611—44no—
93C2×C67⋊C8—41no—
94C8×D67—44no—
95C67⋊C16—22no—
96C67⋊C16C22×C414yes—
97C2×Q8×C67—22no—
98C2×Q8×C67—22no—
99C2×Q8×C67—22no—
100C2×Q8×C67—41no—
101C2×Q8×C67—41no—
102C2×Q8×C67—41no—
103C22×C4×C67—21no—
104C22×C4×C67—41no—
105C22×C4×C67—41no—
106C22×C4×C67C2×D411yescharacteristic
107Q16×C67—22no—
108Q16×C67—22no—
109Q16×C67—22no—
110Q16×C67C22×C414yes—
111C2×C8×C67—21no—
112C2×C8×C67—21no—
113C2×C8×C67C22×C411yescharacteristic
114C2×C8×C67C22×C411yescharacteristic
115C2×C8×C67C2×D412yes—
116C2×C8×C67C2×D411yescharacteristic
117C2×C8×C67#4111yescharacteristic
118C2×C8×C67#4111yescharacteristic
119C2×C8×C67C2411yescharacteristic
120C4⋊C4×C67—22no—
121C4⋊C4×C67—44no—
122C4⋊C4×C67—41no—
123C42×C67—21no—
124C42×C67—44no—
125C42×C67C2×D412yes—
126C23×C67AGL(1,ℤ/8)11yescharacteristic
127C22×D67—41no—
128Q8×C67—24no—
129Q8×C67—42no—
130Q8×C67—42no—
131Q8×C67—42no—
132C2×C4×C67—22no—
133C2×C4×C67—42no—
134C2×C4×C67—42no—
135C2×C4×C67—42no—
136C2×C4×C67—42no—
137C2×C4×C67—42no—
138C2×C4×C67—82no—
139C2×C4×C67C4×D411yescharacteristic
140C2×C4×C67C4×D411yescharacteristic
141C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C67C22×D411yescharacteristic
145C22⋊D67—42no—
146C22⋊D67—42no—
147C2×Dic67—41no—
148D268—22no—
149D268—41no—
150C4×D67—42no—
151C4×D67—42no—
152Dic134—22no—
153Dic134—41no—
154C8×C67—22no—
155C8×C67C4×D411yescharacteristic
156C8×C67C4×D411yescharacteristic
157C8×C67C23×C411yescharacteristic
158C8×C67C2×#4111yescharacteristic
159C67⋊C8—42no—
160C22×C67—22no—
161C22×C67#44211yescharacteristic
162C22×C67#44211yescharacteristic
163C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
164C2×D67—42no—
165C2×D67—82no—
166C4×C67—42no—
167C4×C67—44no—
168C4×C67—82no—
169C4×C67#44211yescharacteristic
170C4×C67#44211yescharacteristic
171C4×C67C2×C4×D411yescharacteristic
172C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
173Dic67—42no—
174C2×#2180—671nomaximal, Sylow
175C2×C67—21no—
176C2×C67—21no—
177C2×C67#218012yescomplemented
178C2×C67C2×#44211yescharacteristic
179D67—82no—
180C2×#510—671no—
181C2×#444—671no—
182C2×#441D6711yescharacteristic, complemented
183#2180—678no—
184C2×#365—671no—
185C2×#362—671no—
186C2×#360—671no—
187C2×#347—671no—
188C67C2×#218011yescharacteristic, complemented, Sylow
189C2×#135—671no—
190C22×Q16C2×D6711yescharacteristic
191C2×#131—1341no—
192C2×C4×Q8—21no—
193#510—674no—
194#510—674no—
195C2×Q32—672no—
196C2×SD32—672no—
197C2×M32—671no—
198C2×M32—671no—
199#444—674no—
200#444—1342no—
201#441—22no—
202#441C2×D6714yescomplemented
203C2×#108—671no—
204C2×#106C2×D6711yescharacteristic
205C2×#104—1341no—
206C2×#103—21no—
207C2×#97C2×D6711yescharacteristic
208#365—674no—
209#362—674no—
210#360—674no—
211#347—674no—
212C2×#41—1341no—
213C22×Q8—21no—
214#135—672no—
215#135—672no—
216#135—1341no—
217#135—1341no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—671no—
222C2×Q16C4×D6712yes—
223C2×Q16C4×D6712yes—
224C2×Q16C22×D6712yes—
225C2×SD16—1341no—
226C2×D8—671no—
227#131—2682no—
228#131—1344no—
229C2×M16—2681no—
230C2×M16—1341no—
231C22×C8—2681no—
232C22×C8C22⋊D6711yescharacteristic
233C22×C8C22×D6711yescharacteristic
234C4×Q8—24no—
235C4×Q8—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C42—41no—
239C2×C42C22⋊D6711yescharacteristic
240Q32—678no—
241SD32—678no—
242M32—672no—
243M32—672no—
244M32—672no—
245M32—1341no—
246C2×C16—672no—
247C2×C16—1341no—
248#108—672no—
249#108—1341no—
250#106—21no—
251#106C22×D6712yes—
252#104—1344no—
253#103—24no—
254#97C22⋊D6712yes—
255#97C22×D6712yes—
256#41—2681no—
257#41—2681no—
258#41—1342no—
259#41—1342no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—1341no—
267C2×D4—2681no—
268C2×D4—1341no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—2681no—
273C22×C4C2×C22⋊D6711yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—22no—
277Q16—672no—
278Q16—1341no—
279Q16C2×C4×D6714yes—
280SD16—1342no—
281SD16—1342no—
282D8—672no—
283D8—1341no—
284M16—2684no—
285M16—2681no—
286M16—1342no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—2684no—
290C2×C8—2681no—
291C2×C8C2×C4×D6711yescharacteristic
292C2×C8C2×C4×D6711yescharacteristic
293C2×C8C2×C22⋊D6712yes—
294C2×C8C2×C22⋊D6711yescharacteristic
295C2×C8#1176162811yescharacteristic
296C2×C8#1176162811yescharacteristic
297C2×C8C23×D6711yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C42—21no—
302C42—44no—
303C42C2×C22⋊D6712yes—
304C16—674no—
305C16—1342no—
306C23—2681no—
307C23#1452406911yescharacteristic
308Q8—24no—
309Q8—42no—
310Q8—42no—
311Q8—42no—
312Q8—2681no—
313Q8—1342no—
314D4—2682no—
315D4—2682no—
316D4—2681no—
317D4—1342no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—2681no—
326C2×C4—2682no—
327C2×C4—2682no—
328C2×C4C4×C22⋊D6711yescharacteristic
329C2×C4C4×C22⋊D6711yescharacteristic
330C2×C4#1452406811yescharacteristic
331C2×C4#1452406811yescharacteristic
332C2×C4#1452406911yescharacteristic
333C2×C4C22×C22⋊D6711yescharacteristic
334C8—22no—
335C8—2682no—
336C8C4×C22⋊D6711yescharacteristic
337C8C4×C22⋊D6711yescharacteristic
338C8C22×C4×D6711yescharacteristic
339C8C2×#1176162811yescharacteristic
340C22—22no—
341C22—2682no—
342C22—5362no—
343C22#1686350611yescharacteristic
344C22#1686350611yescharacteristic
345C22C2×#1452406911yescharacteristic
346C4—42no—
347C4—44no—
348C4—82no—
349C4—2682no—
350C4#1686350611yescharacteristic
351C4#1686350611yescharacteristic
352C4C2×C4×C22⋊D6711yescharacteristic
353C4C2×#1452406811yescharacteristic
354C2—21no—
355C2—21no—
356C2—5362no—
357C2#2256209812yesdirect product factor
358C2C2×#1686350611yescharacteristic
subgroupG/H
centerC22#16863506
commutatorC8×C67C23×C4
FrattiniC2×C8C23×D67
FittingC2×#441×C67C2
socleC22×C67#442
prime radicalC2×#14524065C22
cosocleC2×C8×C67C24
hypercenterC2×#441D67
hypocenterC67C2×#2180
LCSG ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
3. #22562098C2direct (N × H)
4. C2×#16863218C2semidirect
5. C2×#16863212C2semidirect
7. C2×#16863174C2semidirect
8. C2×#441×C67C2semidirect
11. #16863837C4semidirect
13. C2×#14523989C4semidirect
14. C2×#14523986C4semidirect
21. #16863218C22semidirect
22. #16863212C22semidirect
24. #16863174C22semidirect
28. #441×C67C22semidirect
45. #14523989C2×C4semidirect
46. #14523986C2×C4semidirect
177. C2×C67#2180semidirect
182. C2×#441D67semidirect
188. C67C2×#2180semidirect
202. #441C2×D67semidirect
357. C2#22562098direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168638372
C2×#168635192
#225620982
C2×#168632182
C2×#168632122
C2×#168632052
C2×#168631742
C2×#441×C672
C2×#218067