Finite Group Databasebeta

← C2×#22611823 #33239418

Subgroups — C2×#22611823 #33239418

Order 18688.

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×#17988709C211yesmaximal, characteristic
2C2×#17988694C211yesmaximal, characteristic, complemented
3#22611823C218yesmaximal, direct product factor
4C2×#17988410C211yesmaximal, characteristic, complemented
5C2×#17988403C211yesmaximal, characteristic, complemented
6C2×#17988397C211yesmaximal, characteristic
7C2×#17988364C211yesmaximal, characteristic, complemented
8C2×#510×C73C211yesmaximal, characteristic
9C2×#14556598—21no—
10C2×Q8×Dic73—21no—
11C2×#14556531C2211yescharacteristic
12#17988709—22no—
13#17988709C2214yes—
14#17988694—22no—
15#17988694C2214yescomplemented
16C2×#14556527C2211yescharacteristic
17C2×#14556523C2211yescharacteristic
18C2×#14556518—21no—
19C2×#14556515—21no—
20C2×#14556501C2211yescharacteristic
21#17988410C2214yescomplemented
22#17988403C2214yescomplemented
23#17988397C2214yes—
24#17988364C2214yescomplemented
25C2×#135×C73C2211yescharacteristic
26C22×Q16×C73C2211yescharacteristic
27#510×C73C414yescomplemented
28#510×C73C2214yes—
29C2×Q32×C73C412yescomplemented
30C2×SD32×C73C412yescomplemented
31C2×M32×C73C2211yescharacteristic
32#14556598—24no—
33#14556598—42no—
34C2×#11924012—21no—
35C2×#11924012—41no—
36C22×#10596831—41no—
37C22×#10596831D411yescharacteristic
38Q8×Dic73—24no—
39Q8×Dic73—42no—
40C2×C4⋊Dic73—21no—
41C2×C4⋊Dic73—41no—
42C2×C4×Dic73—41no—
43C2×C4×Dic73D411yescharacteristic
44#14556531—21no—
45#14556531C2312yes—
46C2×#11924003—21no—
47#14556527—21no—
48#14556527C2312yes—
49#14556523—21no—
50#14556523C2312yes—
51#14556518—24no—
52#14556515—24no—
53#14556501D412yes—
54#14556501C2312yes—
55C2×#41×C73—21no—
56C22×Q8×C73—21no—
57#135×C73—21no—
58#135×C73—21no—
59#135×C73C2×C412yes—
60#135×C73C2312yes—
61C2×Q16×C73—21no—
62C2×Q16×C73—21no—
63C2×Q16×C73—21no—
64C2×Q16×C73C2×C412yes—
65C2×Q16×C73C2×C411yescharacteristic
66C2×Q16×C73C2×C412yes—
67C2×Q16×C73C2312yes—
68C2×SD16×C73—21no—
69C2×D8×C73C2×C411yescharacteristic
70C22×C8×C73C2311yescharacteristic
71Q32×C73C2×C418yescomplemented
72SD32×C73C2×C418yescomplemented
73M32×C73C2×C412yes—
74M32×C73C2312yes—
75C2×C16×C73C2×C412yes—
76C22×Dic73—21no—
77C22×Dic73—41no—
78#11924012—22no—
79#11924012—44no—
80#11924012—41no—
81C2×#10596831—21no—
82C2×#10596831—21no—
83C2×#10596831—44no—
84C2×#10596831—41no—
85C2×#10596831C2×D412yes—
86C4⋊Dic73—22no—
87C4⋊Dic73—44no—
88C4⋊Dic73—41no—
89C4×Dic73—21no—
90C4×Dic73—44no—
91C4×Dic73C2×D412yes—
92#11924003—22no—
93#41×C73—22no—
94#41×C73—22no—
95#41×C73—41no—
96#41×C73—41no—
97C2×Q8×C73—21no—
98C2×Q8×C73—22no—
99C2×Q8×C73—22no—
100C2×Q8×C73—22no—
101C2×Q8×C73—41no—
102C2×Q8×C73—41no—
103C2×Q8×C73—41no—
104C2×D4×C73—21no—
105C2×D4×C73—41no—
106C22×C4×C73—41no—
107C22×C4×C73—41no—
108C22×C4×C73C2×D411yescharacteristic
109Q16×C73—22no—
110Q16×C73—22no—
111Q16×C73—21no—
112Q16×C73—22no—
113Q16×C73C22×C414yes—
114Q16×C73C22×C412yes—
115SD16×C73—22no—
116SD16×C73—22no—
117D8×C73—21no—
118D8×C73C22×C412yes—
119C2×C8×C73C22×C411yescharacteristic
120C2×C8×C73C22×C411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73#4111yescharacteristic
123C2×C8×C73#4111yescharacteristic
124C2×C8×C73C2411yescharacteristic
125C16×C73C22×C414yes—
126C23×C73—41no—
127C23×C73AGL(1,ℤ/8)11yescharacteristic
128Q8×C73—22no—
129Q8×C73—24no—
130Q8×C73—42no—
131Q8×C73—42no—
132Q8×C73—42no—
133Q8×C73—41no—
134D4×C73—22no—
135D4×C73—42no—
136D4×C73—42no—
137D4×C73—41no—
138C2×C4×C73—42no—
139C2×C4×C73—41no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73C4×D411yescharacteristic
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C73C22×D411yescharacteristic
150C2×Dic73—22no—
151C2×Dic73—42no—
152C2×Dic73—42no—
153C2×Dic73—82no—
154C8×C73C4×D411yescharacteristic
155C8×C73C4×D411yescharacteristic
156C8×C73C23×C411yescharacteristic
157C8×C73C2×#4111yescharacteristic
158#10596831—22no—
159#10596831—42no—
160C22×C73—22no—
161C22×C73—42no—
162C22×C73—82no—
163C22×C73#44211yescharacteristic
164C22×C73#44211yescharacteristic
165C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
166C4×C73—42no—
167C4×C73—44no—
168C4×C73#44211yescharacteristic
169C4×C73#44211yescharacteristic
170C4×C73C2×C4×D411yescharacteristic
171C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
172Dic73—42no—
173Dic73—82no—
174C2×#2180—731nomaximal, Sylow
175C2×C73—21no—
176C2×C73—21no—
177C2×C73—82no—
178C2×C73#218012yescomplemented
179C2×C73C2×#44211yescharacteristic
180C2×#510D7311yescharacteristic
181C2×#444—731no—
182C2×#441—731no—
183#2180—738no—
184C2×#365—731no—
185C2×#362—731no—
186C2×#360—731no—
187C2×#347—731no—
188C73C2×#218011yescharacteristic, complemented, Sylow
189C2×#135C2×D7311yescharacteristic
190C22×Q16C2×D7311yescharacteristic
191C2×#131—1461no—
192C2×C4×Q8—1461no—
193#510Dic7314yescomplemented
194#510C2×D7314yes—
195C2×Q32Dic7312yescomplemented
196C2×SD32Dic7312yescomplemented
197C2×M32—731no—
198C2×M32C2×D7311yescharacteristic
199#444—734no—
200#444—1462no—
201#441—734no—
202#441—1462no—
203C2×#108—731no—
204C2×#106—731no—
205C2×#104—1461no—
206C2×#103—1461no—
207C2×#97—731no—
208#365—734no—
209#362—734no—
210#360—734no—
211#347—734no—
212C2×#41—21no—
213C22×Q8—21no—
214#135—21no—
215#135—21no—
216#135C2×Dic7312yes—
217#135C22×D7312yes—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16C2×Dic7312yes—
222C2×Q16C2×Dic7312yes—
223C2×Q16C2×Dic7311yescharacteristic
224C2×Q16C22×D7312yes—
225C2×SD16—21no—
226C2×D8C2×Dic7311yescharacteristic
227#131—2922no—
228#131—1464no—
229C2×M16—2921no—
230C2×M16—1461no—
231C22×C8—2921no—
232C22×C8—731no—
233C22×C8C22×D7311yescharacteristic
234C4×Q8—2922no—
235C4×Q8—1464no—
236C2×C4⋊C4—2921no—
237C2×C4⋊C4—1461no—
238C2×C42—2921no—
239C2×C42—731no—
240Q32C2×Dic7318yescomplemented
241SD32C2×Dic7318yescomplemented
242M32—732no—
243M32—1461no—
244M32C2×Dic7312yes—
245M32C22×D7312yes—
246C2×C16—1461no—
247C2×C16C2×Dic7312yes—
248#108—732no—
249#108—1461no—
250#106—732no—
251#106—1461no—
252#104—1464no—
253#103—1464no—
254#97—732no—
255#97—732no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—2921no—
272C22×C4—1461no—
273C22×C4D4×D7311yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—21no—
277Q16—22no—
278Q16C22×Dic7314yes—
279Q16C22×Dic7312yes—
280SD16—22no—
281SD16—22no—
282D8—21no—
283D8C22×Dic7312yes—
284M16—2921no—
285M16—2924no—
286M16—1462no—
287C2×C8—2921no—
288C2×C8—2924no—
289C2×C8—732no—
290C2×C8—1461no—
291C2×C8—1461no—
292C2×C8C22×Dic7311yescharacteristic
293C2×C8C22×Dic7311yescharacteristic
294C2×C8D4×D7311yescharacteristic
295C2×C8#1192402711yescharacteristic
296C2×C8#1192402711yescharacteristic
297C2×C8C23×D7311yescharacteristic
298C4⋊C4—2924no—
299C4⋊C4—2921no—
300C4⋊C4—1462no—
301C42—2924no—
302C42—732no—
303C42—1461no—
304C16—1462no—
305C16C22×Dic7314yes—
306C23—41no—
307C23#1455661511yescharacteristic
308Q8—22no—
309Q8—24no—
310Q8—42no—
311Q8—42no—
312Q8—41no—
313Q8—42no—
314D4—22no—
315D4—42no—
316D4—42no—
317D4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—5842no—
327C2×C4—1462no—
328C2×C4D4×Dic7311yescharacteristic
329C2×C4D4×Dic7311yescharacteristic
330C2×C4#1455661511yescharacteristic
331C2×C4#1455661511yescharacteristic
332C2×C4#1455661511yescharacteristic
333C2×C4C2×D4×D7311yescharacteristic
334C8—2922no—
335C8—1462no—
336C8D4×Dic7311yescharacteristic
337C8D4×Dic7311yescharacteristic
338C8C23×Dic7311yescharacteristic
339C8C2×#1192402711yescharacteristic
340C22—22no—
341C22—42no—
342C22—82no—
343C22#1798869911yescharacteristic
344C22#1798869911yescharacteristic
345C22C2×#1455661511yescharacteristic
346C4—42no—
347C4—44no—
348C4—2922no—
349C4—5842no—
350C4#1798869911yescharacteristic
351C4#1798869911yescharacteristic
352C4C2×D4×Dic7311yescharacteristic
353C4C2×#1455661511yescharacteristic
354C2—21no—
355C2—21no—
356C2—82no—
357C2#2261182312yesdirect product factor
358C2C2×#1798869911yescharacteristic
subgroupG/H
centerC22#17988699
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC2×#510×C73C2
socleC22×C73#442
prime radicalC2×#135×C73C22
cosocleC2×C8×C73C24
hypercenterC2×#510D73
hypocenterC73C2×#2180
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
2. C2×#17988694C2semidirect
3. #22611823C2direct (N × H)
4. C2×#17988410C2semidirect
5. C2×#17988403C2semidirect
7. C2×#17988364C2semidirect
15. #17988694C22semidirect
21. #17988410C22semidirect
22. #17988403C22semidirect
24. #17988364C22semidirect
27. #510×C73C4semidirect
29. C2×Q32×C73C4semidirect
30. C2×SD32×C73C4semidirect
71. Q32×C73C2×C4semidirect
72. SD32×C73C2×C4semidirect
178. C2×C73#2180semidirect
188. C73C2×#2180semidirect
193. #510Dic73semidirect
195. C2×Q32Dic73semidirect
196. C2×SD32Dic73semidirect
240. Q32C2×Dic73semidirect
241. SD32C2×Dic73semidirect
357. C2#22611823direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179887092
C2×#179886942
#226118232
C2×#179884102
C2×#179884032
C2×#179883972
C2×#179883642
C2×#510×C732
C2×#218073