Finite Group Databasebeta

← C2×#22611822 #33239417

Subgroups — C2×#22611822 #33239417

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×#17989027C211yesmaximal, characteristic
2C2×#17988695C211yesmaximal, characteristic, complemented
3#22611822C218yesmaximal, direct product factor
4C2×#17988411C211yesmaximal, characteristic, complemented
5C2×#17988402C211yesmaximal, characteristic, complemented
6C2×#17988397C211yesmaximal, characteristic
7C2×#17988365C211yesmaximal, characteristic, complemented
8C2×#444×C73C211yesmaximal, characteristic
9C22×Dic292C2211yescharacteristic
10C2×C4×Dic146—21no—
11#17989027C414yescomplemented
12#17989027C2214yes—
13C2×#14556537C412yescomplemented
14C2×#14556535C412yescomplemented
15C2×#14556531C2211yescharacteristic
16C2×#14556531C2211yescharacteristic
17#17988695—22no—
18#17988695C2214yescomplemented
19C2×#14556523C2211yescharacteristic
20C2×#14556516—21no—
21#17988411C2214yescomplemented
22#17988402C2214yescomplemented
23#17988397C2214yes—
24#17988365C2214yescomplemented
25C2×#135×C73C2211yescharacteristic
26C2×#131×C73—21no—
27#444×C73—22no—
28#444×C73C2214yes—
29C2×#108×C73C2211yescharacteristic
30C2×#104×C73—21no—
31C2×#97×C73C2211yescharacteristic
32C22×Dic146—21no—
33C2×Dic292—21no—
34C2×Dic292—21no—
35C2×Dic292—21no—
36C2×Dic292C2×C412yes—
37C2×Dic292C2×C412yes—
38C2×Dic292C2312yes—
39C4×Dic146—24no—
40C4×Dic146—42no—
41C2×C4⋊Dic73—21no—
42C2×#11924007—41no—
43C2×C4×Dic73—41no—
44#14556537C2×C418yescomplemented
45#14556535C2×C418yescomplemented
46#14556531—21no—
47#14556531C2×C412yes—
48#14556531C2312yes—
49#14556531C2312yes—
50C2×#11924003—21no—
51C2×#11924003C2×C412yes—
52#14556523—21no—
53#14556523C2312yes—
54#14556516—24no—
55C2×#41×C73—21no—
56#135×C73—21no—
57#135×C73—21no—
58#135×C73C2×C412yes—
59#135×C73C2312yes—
60C2×Q16×C73C2×C411yescharacteristic
61C2×SD16×C73—21no—
62C2×D8×C73C2×C411yescharacteristic
63#131×C73—24no—
64#131×C73—42no—
65C2×M16×C73—21no—
66C2×M16×C73—41no—
67C22×C8×C73—41no—
68C22×C8×C73D411yescharacteristic
69C22×C8×C73C2311yescharacteristic
70C2×C42×C73D411yescharacteristic
71#108×C73—21no—
72#108×C73C2312yes—
73#104×C73—24no—
74#97×C73D412yes—
75#97×C73C2312yes—
76C2×Dic146—22no—
77C2×Dic146—22no—
78C2×Dic146—22no—
79C2×Dic146—41no—
80C2×Dic146—41no—
81C2×Dic146—41no—
82C22×Dic73—41no—
83C22×Dic73—41no—
84Dic292—22no—
85Dic292—22no—
86Dic292—22no—
87Dic292C22×C414yes—
88C4⋊Dic73—22no—
89C4⋊Dic73—41no—
90#11924007—44no—
91C4×Dic73—44no—
92#11924003—22no—
93#11924003C22×C414yes—
94#41×C73—22no—
95#41×C73—22no—
96#41×C73—41no—
97#41×C73—41no—
98C2×Q8×C73—21no—
99C2×D4×C73—21no—
100C2×D4×C73—41no—
101C22×C4×C73—21no—
102C22×C4×C73—41no—
103C22×C4×C73C2×D411yescharacteristic
104Q16×C73—21no—
105Q16×C73C22×C412yes—
106SD16×C73—22no—
107SD16×C73—22no—
108D8×C73—21no—
109D8×C73C22×C412yes—
110M16×C73—22no—
111M16×C73—44no—
112M16×C73—41no—
113C2×C8×C73—21no—
114C2×C8×C73—21no—
115C2×C8×C73—44no—
116C2×C8×C73—41no—
117C2×C8×C73C22×C411yescharacteristic
118C2×C8×C73C22×C411yescharacteristic
119C2×C8×C73C2×D412yes—
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C2×C8×C73#4111yescharacteristic
123C2×C8×C73C2411yescharacteristic
124C42×C73—21no—
125C42×C73C2×D412yes—
126C23×C73—41no—
127C23×C73AGL(1,ℤ/8)11yescharacteristic
128Q8×C73—22no—
129Q8×C73—41no—
130D4×C73—22no—
131D4×C73—42no—
132D4×C73—42no—
133D4×C73—41no—
134C2×C4×C73—22no—
135C2×C4×C73—42no—
136C2×C4×C73—41no—
137C2×C4×C73—42no—
138C2×C4×C73—42no—
139C2×C4×C73C4×D411yescharacteristic
140C2×C4×C73C4×D411yescharacteristic
141C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C73C22×D411yescharacteristic
145C2×Dic73—42no—
146C2×Dic73—42no—
147C2×Dic73—42no—
148C2×Dic73—42no—
149C2×Dic73—82no—
150Dic146—24no—
151Dic146—42no—
152Dic146—42no—
153Dic146—42no—
154C8×C73—22no—
155C8×C73—42no—
156C8×C73C4×D411yescharacteristic
157C8×C73C4×D411yescharacteristic
158C8×C73C23×C411yescharacteristic
159C8×C73C2×#4111yescharacteristic
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—42no—
168C4×C73#44211yescharacteristic
169C4×C73#44211yescharacteristic
170C4×C73C2×C4×D411yescharacteristic
171C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
172Dic73—44no—
173Dic73—82no—
174C2×#2180—731nomaximal, Sylow
175C2×C73—21no—
176C2×C73—21no—
177C2×C73—82no—
178C2×C73#218012yescomplemented
179C2×C73C2×#44211yescharacteristic
180C2×#510—731no—
181C2×#444D7311yescharacteristic
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×Q16—731no—
191C2×#131—21no—
192C2×C4×Q8—1461no—
193#510—734no—
194#510—734no—
195C2×Q32—732no—
196C2×SD32—732no—
197C2×M32—731no—
198C2×M32—731no—
199#444—22no—
200#444C2×D7314yes—
201#441—734no—
202#441—1462no—
203C2×#108C2×D7311yescharacteristic
204C2×#106—731no—
205C2×#104—21no—
206C2×#103—1461no—
207C2×#97C2×D7311yescharacteristic
208#365—734no—
209#362—734no—
210#360—734no—
211#347—734no—
212C2×#41—21no—
213C22×Q8—1461no—
214#135—21no—
215#135—21no—
216#135C4×D7312yes—
217#135C22×D7312yes—
218C2×Q16—732no—
219C2×Q16—732no—
220C2×Q16—732no—
221C2×Q16—1461no—
222C2×Q16—1461no—
223C2×Q16—1461no—
224C2×Q16C4×D7311yescharacteristic
225C2×SD16—21no—
226C2×D8C4×D7311yescharacteristic
227#131—24no—
228#131—42no—
229C2×M16—21no—
230C2×M16—41no—
231C22×C8—41no—
232C22×C8C22⋊D7311yescharacteristic
233C22×C8C22×D7311yescharacteristic
234C4×Q8—2922no—
235C4×Q8—1464no—
236C2×C4⋊C4—2921no—
237C2×C4⋊C4—1461no—
238C2×C42—2921no—
239C2×C42C22⋊D7311yescharacteristic
240Q32—738no—
241SD32—738no—
242M32—732no—
243M32—732no—
244M32—732no—
245M32—1461no—
246C2×C16—732no—
247C2×C16—1461no—
248#108—21no—
249#108C22×D7312yes—
250#106—732no—
251#106—1461no—
252#104—24no—
253#103—1464no—
254#97C22⋊D7312yes—
255#97C22×D7312yes—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×Q8—2921no—
262C2×Q8—2921no—
263C2×Q8—2921no—
264C2×Q8—1462no—
265C2×Q8—1462no—
266C2×Q8—1462no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—2921no—
272C22×C4—2921no—
273C22×C4C2×C22⋊D7311yescharacteristic
274Q16—21no—
275Q16—734no—
276Q16—1462no—
277Q16—1462no—
278Q16—1462no—
279Q16C2×C4×D7312yes—
280SD16—22no—
281SD16—22no—
282D8—21no—
283D8C2×C4×D7312yes—
284M16—22no—
285M16—44no—
286M16—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8C2×C4×D7311yescharacteristic
292C2×C8C2×C4×D7311yescharacteristic
293C2×C8C2×C22⋊D7312yes—
294C2×C8C2×C22⋊D7311yescharacteristic
295C2×C8#1192402811yescharacteristic
296C2×C8#1192402811yescharacteristic
297C2×C8C23×D7311yescharacteristic
298C4⋊C4—2924no—
299C4⋊C4—2921no—
300C4⋊C4—1462no—
301C42—21no—
302C42—2924no—
303C42C2×C22⋊D7312yes—
304C16—734no—
305C16—1462no—
306C23—41no—
307C23#1455661611yescharacteristic
308Q8—22no—
309Q8—41no—
310Q8—2922no—
311Q8—2922no—
312Q8—2922no—
313Q8—1464no—
314D4—22no—
315D4—42no—
316D4—42no—
317D4—41no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—2922no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—5842no—
328C2×C4C4×C22⋊D7311yescharacteristic
329C2×C4C4×C22⋊D7311yescharacteristic
330C2×C4#1455661611yescharacteristic
331C2×C4#1455661711yescharacteristic
332C2×C4#1455661711yescharacteristic
333C2×C4C22×C22⋊D7311yescharacteristic
334C8—22no—
335C8—42no—
336C8C4×C22⋊D7311yescharacteristic
337C8C4×C22⋊D7311yescharacteristic
338C8C22×C4×D7311yescharacteristic
339C8C2×#1192402811yescharacteristic
340C22—22no—
341C22—42no—
342C22—82no—
343C22#1798869711yescharacteristic
344C22#1798869711yescharacteristic
345C22C2×#1455661611yescharacteristic
346C4—42no—
347C4—42no—
348C4—2924no—
349C4—5842no—
350C4#1798869711yescharacteristic
351C4#1798869711yescharacteristic
352C4C2×C4×C22⋊D7311yescharacteristic
353C4C2×#1455661711yescharacteristic
354C2—21no—
355C2—21no—
356C2—82no—
357C2#2261182212yesdirect product factor
358C2C2×#1798869711yescharacteristic
subgroupG/H
centerC22#17988697
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC2×#444×C73C2
socleC22×C73#442
prime radicalC2×#135×C73C22
cosocleC2×C8×C73C24
hypercenterC2×#444D73
hypocenterC73C2×#2180
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. C2×#17988695C2semidirect
3. #22611822C2direct (N × H)
4. C2×#17988411C2semidirect
5. C2×#17988402C2semidirect
7. C2×#17988365C2semidirect
11. #17989027C4semidirect
13. C2×#14556537C4semidirect
14. C2×#14556535C4semidirect
18. #17988695C22semidirect
21. #17988411C22semidirect
22. #17988402C22semidirect
24. #17988365C22semidirect
44. #14556537C2×C4semidirect
45. #14556535C2×C4semidirect
178. C2×C73#2180semidirect
188. C73C2×#2180semidirect
357. C2#22611822direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179890272
C2×#179886952
#226118222
C2×#179884112
C2×#179884022
C2×#179883972
C2×#179883652
C2×#444×C732
C2×#218073