Finite Group Databasebeta

← C4×#17989655 #34030528

Subgroups — C4×#17989655 #34030528

Order 18688.

3402 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989655C212yesmaximal
2#22626371C218yesmaximal, complemented
3#22626369C216yesmaximal
4C4×Q8×#64408C218yesmaximal, complemented
5C4×#14556670C216yesmaximal
6C2×C4×Q8×D73C211yesmaximal, characteristic
7#17989655C4132yesdirect product factor
8C2×Q8×#64408C22116yes—
9C2×#14556670C22112yes—
10C4⋊C4×#64408C22148yes—
11C4×C4⋊#64408C22112yes—
12C4×C4⋊#64408C22112yes—
13C4×C22⋊#64408C2214yes—
14C4×C22⋊#64408C22112yes—
15#17989523C22112yes—
16C42×#64408C22112yes—
17C22×Q8×D73C2211yescharacteristic
18C2×Q8×Dic73C411yescharacteristic, complemented
19C2×C4×Dic146C413yescomplemented
20C2×#14556556C413yescomplemented
21C4×Q8×D73C414yescomplemented
22C4×Q8×D73C414yescomplemented
23C4×Q8×D73C2214yes—
24C4×Q8×D73C2214yes—
25C2×C4⋊C4×D73C2213yes—
26C2×C42×D73C2213yes—
27C2×C4×Q8×C73C411yescharacteristic, complemented
28Q8×#64408C2×C41128yescomplemented
29#14556670C2×C4196yes—
30C2×C4⋊#64408C23124yes—
31C2×C22⋊#64408C2318yes—
32C2×C4×#64408C2318yes—
33C2×C4×#64408C23124yes—
34C2×C4×#64408C23124yes—
35(C2×#64408)⋊C4C2318yes—
36(C2×#64408)⋊C4C23124yes—
37C22×Dic146C2×C413yes—
38C2×Q8×D73C2×C4116yes—
39C2×Q8×D73C2×C414yes—
40C2×Q8×D73C2314yes—
41C22×C4×D73C2311yescharacteristic
42C22×C4×D73C2313yes—
43C22×C4×D73C2313yes—
44Q8×Dic73C2×C414yes—
45Q8×Dic73C2×C414yes—
46C4×Dic146C2×C4112yes—
47C4×Dic146C2×C4112yes—
48#14556556C2×C4112yes—
49#14556556C2×C4112yes—
50C2×C4⋊Dic73C2×C413yes—
51C2×#11924007C2×C416yes—
52C4⋊C4×D73C2×C4112yes—
53C4⋊C4×D73C23112yes—
54C2×C4×Dic73C2×C413yes—
55C2×C4×Dic73C2×C413yes—
56C2×C4×Dic73C2×C413yes—
57C42×D73C2×C416yes—
58C42×D73C2×C416yes—
59C42×D73C2316yes—
60C42×D73C2316yes—
61C22×Q8×C73C2×C411yescharacteristic
62C4×Q8×C73C2×C414yes—
63C4×Q8×C73C2×C414yes—
64C2×C4⋊C4×C73C2×C413yes—
65C2×C42×C73C2×C413yes—
66C22×#64408C24116yes—
67C4⋊#64408C22×C4196yes—
68C22⋊#64408C22×C4132yes—
69C4×#64408—216no—
70C4×#64408—424no—
71C4×#64408C22×C4196yes—
72C23×D73C2411yescharacteristic
73C2×Dic146C42124yescomplemented
74C2×Dic146C22×C4112yes—
75Q8×D73C42132yescomplemented
76Q8×D73C22×C4132yes—
77C22×Dic73C22×C411yescharacteristic
78C22×Dic73C22×C413yes—
79C22×Dic73C22×C413yes—
80C2×C4×D73—24no—
81C2×C4×D73—46no—
82C2×C4×D73C22×C411yescharacteristic
83C2×C4×D73C22×C411yescharacteristic
84C2×C4×D73C22×C416yes—
85C2×C4×D73C22×C416yes—
86C2×C4×D73C22×C4124yes—
87C2×C4×D73C2411yescharacteristic
88C2×C4×D73C2416yes—
89C2×C4×D73C2416yes—
90C2×C4×D73C2411yescharacteristic
91C4⋊Dic73C22×C4112yes—
92#11924007C22×C4124yes—
93C4×Dic73C22×C416yes—
94C4×Dic73C22×C416yes—
95C4×Dic73C22×C416yes—
96C4×Dic73C22×C416yes—
97C4×Dic73C22×C416yes—
98C4×Dic73C22×C416yes—
99C2×Q8×C73C4218yescomplemented
100C2×Q8×C73C22×C414yes—
101C22×C4×C73C22×C411yescharacteristic
102C22×C4×C73C22×C413yes—
103C22×C4×C73C22×C413yes—
104C4⋊C4×C73C22×C4112yes—
105C42×C73C22×C416yes—
106C42×C73C22×C416yes—
107C23×C73C23×C411yescharacteristic
108C22×D73—22no—
109C22×D73—22no—
110C22×D73C23×C414yes—
111C22×D73C23×C411yescharacteristic
112C22×D73C2511yescharacteristic
113C2×#64408—216no—
114C2×#64408—216no—
115C2×#64408C23×C4132yes—
116Q8×C73C2×C42116yes—
117C2×C4×C73—22no—
118C2×C4×C73—43no—
119C2×C4×C73C2×C42112yes—
120C2×C4×C73C23×C411yescharacteristic
121C2×C4×C73C23×C416yes—
122C2×C4×C73C23×C416yes—
123C2×C4×C73C23×C411yescharacteristic
124C2×Dic73—22no—
125C2×Dic73—43no—
126C2×Dic73C2×C42112yes—
127C2×Dic73C23×C411yescharacteristic
128C2×Dic73C23×C416yes—
129C2×Dic73C23×C411yescharacteristic
130C2×Dic73C23×C416yes—
131C4×D73—22no—
132C4×D73—212no—
133C4×D73—22no—
134C4×D73—22no—
135C4×D73—46no—
136C4×D73C2×C42124yes—
137C4×D73C23×C4124yes—
138C4×D732−1+414yes—
139Dic146C2×C42148yes—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73C22×C4212yes—
143C22×C73C24×C411yescharacteristic
144C2×D73—21no—
145C2×D73—21no—
146C2×D73—28no—
147C2×D73C22×C4212yes—
148C2×D73C24×C412yes—
149C2×D73C2×2−1+412yes—
150C2×D73#58512yes—
151#64408—232no—
152C4×C73—21no—
153C4×C73—26no—
154C4×C73C22×C42112yes—
155C4×C73#51912yescomplemented
156Dic73—21no—
157Dic73—26no—
158Dic73C22×C42112yes—
159Dic73#51912yescomplemented
160C4×#519—731nomaximal, Sylow
161C2×C73—22no—
162C2×C73C23×C4211yescharacteristic
163C2×C73C2×#51911yescharacteristic
164C2×C73#342811yescharacteristic
165D73—22no—
166D73C4×2−1+414yes—
167C2×#519—732no—
168C22×C4×Q8—731no—
169#2276—738no—
170#2274—736no—
171C42×Q8—738no—
172C4×#117—736no—
173C73C4×#51911yescharacteristic, complemented, Sylow
174C23×Q8—731no—
175#519—7332no—
176C2×C4×Q8—7316no—
177C2×C4×Q8—731no—
178C2×C4×Q8—734no—
179C2×C4×Q8—734no—
180C2×C4×Q8—734no—
181C2×C4×Q8—734no—
182C2×C4×Q8—733no—
183C2×C4×Q8—733no—
184C2×C4×Q8#6440811yescharacteristic, complemented
185C2×#117—7312no—
186C22×C4⋊C4—733no—
187C22×C42—733no—
188C4×C4⋊C4—7312no—
189C4×C4⋊C4—7312no—
190C4×C4⋊C4—7348no—
191C4×C22⋊C4—734no—
192C4×C22⋊C4—7312no—
193#376—7312no—
194C43—7312no—
195C22×Q8—7316no—
196C22×Q8—734no—
197C22×Q8—734no—
198C22×Q8—733no—
199C22×Q8C2×#6440811yescharacteristic
200C23×C4—733no—
201C23×C4—733no—
202C23×C4—731no—
203C4×Q8—73128no—
204C4×Q8—734no—
205C4×Q8—734no—
206C4×Q8—7312no—
207C4×Q8—7312no—
208C4×Q8—7312no—
209C4×Q8—7312no—
210C4×Q8C2×#6440814yes—
211C4×Q8C2×#6440814yes—
212#117—7396no—
213C2×C4⋊C4—7324no—
214C2×C4⋊C4—736no—
215C2×C4⋊C4—733no—
216C2×C4⋊C4—7312no—
217C2×C4⋊C4—7312no—
218C2×C4⋊C4C2×#6440813yes—
219C2×C22⋊C4—738no—
220C2×C42—7324no—
221C2×C42—7324no—
222C2×C42—733no—
223C2×C42—733no—
224C2×C42—733no—
225C2×C42—736no—
226C2×C42—736no—
227C2×C42—736no—
228C2×C42—736no—
229C2×C42—738no—
230C2×C42C2×#6440813yes—
231#95—738no—
232#95—7324no—
233C24—731no—
234C2×Q8—7332no—
235C2×Q8—7332no—
236C2×Q8—7312no—
237C2×Q8—7324no—
238C2×Q8C4×#6440818yescomplemented
239C2×Q8C22×#6440814yes—
240C22×C4—2926no—
241C22×C4—733no—
242C22×C4—736no—
243C22×C4—733no—
244C22×C4—736no—
245C22×C4—736no—
246C22×C4—731no—
247C22×C4—7324no—
248C22×C4—7316no—
249C22×C4—731no—
250C22×C4—731no—
251C22×C4—731no—
252C22×C4—736no—
253C22×C4—731no—
254C22×C4—1464no—
255C22×C4C22×#6440811yescharacteristic
256C22×C4C22×#6440813yes—
257C22×C4C22×#6440813yes—
258C4⋊C4—7396no—
259C4⋊C4—7324no—
260C4⋊C4—7312no—
261C4⋊C4C22×#64408112yes—
262C22⋊C4—7332no—
263C42—29224no—
264C42—7396no—
265C42—736no—
266C42—736no—
267C42—736no—
268C42—736no—
269C42—736no—
270C42—736no—
271C42—14616no—
272C42C22×#6440816yes—
273C42C22×#6440816yes—
274C23—731no—
275C23—734no—
276C23—731no—
277C23—1462no—
278C23—1462no—
279C23C23×#6440811yescharacteristic
280Q8—7348no—
281Q8C2×C4×#64408116yes—
282C2×C4—22no—
283C2×C4—43no—
284C2×C4—2926no—
285C2×C4—2923no—
286C2×C4—7312no—
287C2×C4—731no—
288C2×C4—736no—
289C2×C4—7324no—
290C2×C4—7324no—
291C2×C4—734no—
292C2×C4—7332no—
293C2×C4—731no—
294C2×C4—736no—
295C2×C4—1462no—
296C2×C4—14612no—
297C2×C4—1462no—
298C2×C4—1462no—
299C2×C4—14616no—
300C2×C4—14616no—
301C2×C4—1462no—
302C2×C4C2×C4×#64408112yes—
303C2×C4C23×#6440811yescharacteristic
304C2×C4C23×#6440816yes—
305C2×C4C23×#6440816yes—
306C2×C4C23×#6440811yescharacteristic
307C22—21no—
308C22—21no—
309C22—732no—
310C22—732no—
311C22—732no—
312C22—732no—
313C22—1468no—
314C22—1461no—
315C22—1461no—
316C22C22×C4×#6440812yes—
317C22C24×#6440811yescharacteristic
318C4—21no—
319C4—26no—
320C4—7312no—
321C4—732no—
322C4—14632no—
323C4—1461no—
324C4—1466no—
325C4C22×C4×#64408112yes—
326C4#1798965512yesdirect product factor
327C2—22no—
328C2—734no—
329C2—1462no—
330C2C23×C4×#6440811yescharacteristic
331C2C2×#1798965511yescharacteristic
332C2#2262691111yescharacteristic
subgroupG/H
centerC2×C4C23×#64408
commutatorC2×C73C23×C42
FrattiniC22C24×#64408
FittingC2×C4×Q8×C73C4
socleC22×C73C24×C4
prime radicalC23×D73C24
cosocleC22×D73C25
hypercenterC2×C4×Q8#64408
hypocenterC73C4×#519
LCSG ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22626371C2semidirect
4. C4×Q8×#64408C2semidirect
7. #17989655C4direct (N × H)
18. C2×Q8×Dic73C4semidirect
19. C2×C4×Dic146C4semidirect
20. C2×#14556556C4semidirect
21. C4×Q8×D73C4semidirect
22. C4×Q8×D73C4semidirect
27. C2×C4×Q8×C73C4semidirect
28. Q8×#64408C2×C4semidirect
73. C2×Dic146C42semidirect
75. Q8×D73C42semidirect
99. C2×Q8×C73C42semidirect
155. C4×C73#519semidirect
159. Dic73#519semidirect
173. C73C4×#519semidirect
184. C2×C4×Q8#64408semidirect
238. C2×Q8C4×#64408semidirect
326. C4#17989655direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179896552
#226263712
#226263692
C4×Q8×#644082
C4×#145566702
C2×C4×Q8×D732
C4×#51973