Finite Group Databasebeta

← C2×#22611821 #33239416

Subgroups — C2×#22611821 #33239416

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×#17989030C211yesmaximal, characteristic
2C2×#17988708C211yesmaximal, characteristic
3C2×#17988693C211yesmaximal, characteristic, complemented
4#22611821C218yesmaximal, direct product factor
5C2×#17988402C211yesmaximal, characteristic, complemented
6C2×#17988396C211yesmaximal, characteristic
7C2×#17988364C211yesmaximal, characteristic, complemented
8C2×#365×C73C211yesmaximal, characteristic, complemented
9C2×#14556613C2211yescharacteristic
10C22×Dic292C2211yescharacteristic
11C2×#14556599—21no—
12C2×#14556556—21no—
13#17989030C414yescomplemented
14#17989030C2214yes—
15C2×Dic584C412yescomplemented
16C2×#14556536C412yescomplemented
17C2×#14556531C2211yescharacteristic
18#17988708—22no—
19#17988708C2214yes—
20#17988693—22no—
21#17988693C2214yescomplemented
22C2×#14556517—21no—
23C2×#14556514—21no—
24C2×#14556501C2211yescharacteristic
25#17988402C2214yescomplemented
26#17988396C2214yes—
27#17988364C2214yescomplemented
28C2×M32×C73C2211yescharacteristic
29C2×#108×C73C2211yescharacteristic
30C2×#106×C73C2211yescharacteristic
31#365×C73C2214yescomplemented
32C2×#11924028—21no—
33C22×Dic146—21no—
34#14556613—21no—
35#14556613—21no—
36#14556613C2×C412yes—
37#14556613C2312yes—
38C2×Dic292—21no—
39C2×Dic292—21no—
40C2×Dic292—21no—
41C2×Dic292C2×C412yes—
42C2×Dic292C2×C411yescharacteristic
43C2×Dic292C2×C412yes—
44C2×Dic292C2312yes—
45C2×#11924017—21no—
46C2×D584C2×C411yescharacteristic
47#14556599—24no—
48#14556599—42no—
49C2×#11924011—41no—
50C22×#10596831D411yescharacteristic
51C2×C8×D73—41no—
52#14556556—24no—
53#14556556—42no—
54C2×#11924007—41no—
55C2×C4×Dic73—41no—
56C2×C4×Dic73D411yescharacteristic
57Dic584C2×C418yescomplemented
58#14556536C2×C418yescomplemented
59#14556531—21no—
60#14556531C2312yes—
61C2×#11924003—21no—
62#14556517—24no—
63#14556514—24no—
64#14556501D412yes—
65#14556501C2312yes—
66C2×M16×C73—21no—
67C22×C8×C73C2311yescharacteristic
68C2×C4⋊C4×C73—21no—
69M32×C73C2×C412yes—
70M32×C73C2312yes—
71C2×C16×C73C2×C412yes—
72#108×C73—21no—
73#108×C73C2312yes—
74#106×C73—21no—
75#106×C73C2312yes—
76#11924028—22no—
77#11924028—22no—
78#11924028—41no—
79#11924028—41no—
80C2×Dic146—21no—
81C2×Dic146—22no—
82C2×Dic146—22no—
83C2×Dic146—22no—
84C2×Dic146—41no—
85C2×Dic146—41no—
86C2×Dic146—41no—
87C2×D292—21no—
88C2×C22⋊D73—41no—
89C22×Dic73—21no—
90C22×Dic73—41no—
91C2×C4×D73—41no—
92Dic292—22no—
93Dic292—22no—
94Dic292—21no—
95Dic292—22no—
96Dic292C22×C414yes—
97Dic292C22×C412yes—
98#11924017—22no—
99#11924017—22no—
100D584—21no—
101D584C22×C412yes—
102#11924011—44no—
103C2×#10596831—21no—
104C2×#10596831—21no—
105C2×#10596831C2×D412yes—
106C8×D73—44no—
107#11924007—44no—
108C4×Dic73—21no—
109C4×Dic73—44no—
110C4×Dic73C2×D412yes—
111#11924003—22no—
112C22×C4×C73—41no—
113C22×C4×C73C2×D411yescharacteristic
114M16×C73—22no—
115M16×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73C22×C411yescharacteristic
118C2×C8×C73C22×C411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73#4111yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C2×C8×C73C2411yescharacteristic
123C4⋊C4×C73—22no—
124C4⋊C4×C73—41no—
125C16×C73C22×C414yes—
126C23×C73AGL(1,ℤ/8)11yescharacteristic
127C22×D73—41no—
128C2×C4×C73—42no—
129C2×C4×C73—82no—
130C2×C4×C73C4×D411yescharacteristic
131C2×C4×C73C4×D411yescharacteristic
132C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C73C22×D411yescharacteristic
136C22⋊D73—42no—
137C22⋊D73—42no—
138C2×Dic73—22no—
139C2×Dic73—42no—
140C2×Dic73—42no—
141C2×Dic73—41no—
142C2×Dic73—42no—
143C2×Dic73—42no—
144D292—22no—
145D292—41no—
146C4×D73—42no—
147C4×D73—42no—
148Dic146—22no—
149Dic146—24no—
150Dic146—42no—
151Dic146—42no—
152Dic146—42no—
153Dic146—41no—
154C8×C73—42no—
155C8×C73C4×D411yescharacteristic
156C8×C73C4×D411yescharacteristic
157C8×C73C23×C411yescharacteristic
158C8×C73C2×#4111yescharacteristic
159#10596831—22no—
160C22×C73—22no—
161C22×C73#44211yescharacteristic
162C22×C73#44211yescharacteristic
163C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
164C2×D73—42no—
165C2×D73—82no—
166C4×C73—82no—
167C4×C73#44211yescharacteristic
168C4×C73#44211yescharacteristic
169C4×C73C2×C4×D411yescharacteristic
170C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
171Dic73—42no—
172Dic73—44no—
173Dic73—42no—
174C2×#2180—731nomaximal, Sylow
175C2×C73—21no—
176C2×C73—21no—
177C2×C73#218012yescomplemented
178C2×C73C2×#44211yescharacteristic
179D73—82no—
180C2×#510—731no—
181C2×#444—731no—
182C2×#441—731no—
183#2180—738no—
184C2×#365D7311yescharacteristic, complemented
185C2×#362—731no—
186C2×#360—731no—
187C2×#347—731no—
188C73C2×#218011yescharacteristic, complemented, Sylow
189C2×#135—731no—
190C22×Q16—731no—
191C2×#131—1461no—
192C2×C4×Q8—1461no—
193#510—734no—
194#510—734no—
195C2×Q32—732no—
196C2×SD32—732no—
197C2×M32—731no—
198C2×M32C2×D7311yescharacteristic
199#444—734no—
200#444—1462no—
201#441—734no—
202#441—1462no—
203C2×#108C2×D7311yescharacteristic
204C2×#106C2×D7311yescharacteristic
205C2×#104—1461no—
206C2×#103—1461no—
207C2×#97—731no—
208#365C2×D7314yescomplemented
209#362—734no—
210#360—734no—
211#347—734no—
212C2×#41—1461no—
213C22×Q8—1461no—
214#135—732no—
215#135—732no—
216#135—1461no—
217#135—1461no—
218C2×Q16—732no—
219C2×Q16—732no—
220C2×Q16—731no—
221C2×Q16—732no—
222C2×Q16—1461no—
223C2×Q16—1461no—
224C2×Q16—1461no—
225C2×SD16—1461no—
226C2×D8—731no—
227#131—2922no—
228#131—1464no—
229C2×M16—21no—
230C2×M16—2921no—
231C22×C8—2921no—
232C22×C8—731no—
233C22×C8C22×D7311yescharacteristic
234C4×Q8—2922no—
235C4×Q8—1464no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—2921no—
238C2×C42—2921no—
239C2×C42—731no—
240Q32—738no—
241SD32—738no—
242M32—732no—
243M32—1461no—
244M32C4×D7312yes—
245M32C22×D7312yes—
246C2×C16—1461no—
247C2×C16C4×D7312yes—
248#108—21no—
249#108C22×D7312yes—
250#106—21no—
251#106C22×D7312yes—
252#104—1464no—
253#103—1464no—
254#97—732no—
255#97—732no—
256#41—2921no—
257#41—2921no—
258#41—1462no—
259#41—1462no—
260C2×Q8—2921no—
261C2×Q8—2921no—
262C2×Q8—2921no—
263C2×Q8—1461no—
264C2×Q8—1462no—
265C2×Q8—1462no—
266C2×Q8—1462no—
267C2×D4—2921no—
268C2×D4—1461no—
269C22×C4—41no—
270C22×C4—2921no—
271C22×C4—2921no—
272C22×C4—1461no—
273C22×C4D4×D7311yescharacteristic
274Q16—734no—
275Q16—732no—
276Q16—1462no—
277Q16—1462no—
278Q16—1461no—
279Q16—1462no—
280SD16—1462no—
281SD16—1462no—
282D8—732no—
283D8—1461no—
284M16—22no—
285M16—41no—
286M16—2924no—
287C2×C8—41no—
288C2×C8—2924no—
289C2×C8—732no—
290C2×C8—1461no—
291C2×C8—1461no—
292C2×C8C2×C4×D7311yescharacteristic
293C2×C8C2×C4×D7311yescharacteristic
294C2×C8D4×D7311yescharacteristic
295C2×C8#1192402911yescharacteristic
296C2×C8#1192402911yescharacteristic
297C2×C8C23×D7311yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—2924no—
301C42—2924no—
302C42—732no—
303C42—1461no—
304C16—1462no—
305C16C2×C4×D7314yes—
306C23—2921no—
307C23#1455661811yescharacteristic
308Q8—2922no—
309Q8—2922no—
310Q8—2921no—
311Q8—2922no—
312Q8—1462no—
313Q8—1464no—
314D4—2922no—
315D4—2922no—
316D4—2921no—
317D4—1462no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—2922no—
321C2×C4—2922no—
322C2×C4—2921no—
323C2×C4—2922no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—1462no—
328C2×C4#1455655311yescharacteristic
329C2×C4#1455655311yescharacteristic
330C2×C4#1455661811yescharacteristic
331C2×C4#1455661811yescharacteristic
332C2×C4#1455661811yescharacteristic
333C2×C4C2×D4×D7311yescharacteristic
334C8—42no—
335C8—1462no—
336C8#1455655311yescharacteristic
337C8#1455655311yescharacteristic
338C8C22×C4×D7311yescharacteristic
339C8C2×#1192402911yescharacteristic
340C22—22no—
341C22—2922no—
342C22—5842no—
343C22#1798869811yescharacteristic
344C22#1798869811yescharacteristic
345C22C2×#1455661811yescharacteristic
346C4—82no—
347C4—2922no—
348C4—2922no—
349C4—2924no—
350C4#1798869811yescharacteristic
351C4#1798869811yescharacteristic
352C4C2×#1455655311yescharacteristic
353C4C2×#1455661811yescharacteristic
354C2—21no—
355C2—21no—
356C2—5842no—
357C2#2261182112yesdirect product factor
358C2C2×#1798869811yescharacteristic
subgroupG/H
centerC22#17988698
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC2×#365×C73C2
socleC22×C73#442
prime radicalC2×#14556613C22
cosocleC2×C8×C73C24
hypercenterC2×#365D73
hypocenterC73C2×#2180
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#365

Complements

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

NH ≅ G/Ntype
3. C2×#17988693C2semidirect
4. #22611821C2direct (N × H)
5. C2×#17988402C2semidirect
7. C2×#17988364C2semidirect
8. C2×#365×C73C2semidirect
13. #17989030C4semidirect
15. C2×Dic584C4semidirect
16. C2×#14556536C4semidirect
21. #17988693C22semidirect
25. #17988402C22semidirect
27. #17988364C22semidirect
31. #365×C73C22semidirect
57. Dic584C2×C4semidirect
58. #14556536C2×C4semidirect
177. C2×C73#2180semidirect
184. C2×#365D73semidirect
188. C73C2×#2180semidirect
208. #365C2×D73semidirect
357. C2#22611821direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179890302
C2×#179887082
C2×#179886932
#226118212
C2×#179884022
C2×#179883962
C2×#179883642
C2×#365×C732
C2×#218073