Finite Group Databasebeta

← C2×#22614214 #33255801

Subgroups — C2×#22614214 #33255801

Order 18688.

2918 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11924007C211yesmaximal, characteristic
2#22614214C2116yesmaximal, direct product factor
3C2×#17988525C212yesmaximal, complemented
4C2×#17988518C212yesmaximal, complemented
5C2×#17988471C214yesmaximal, complemented
6C2×#17988459C212yesmaximal
7C2×#17988450C211yesmaximal, characteristic
8C2×#17988448C212yesmaximal
9C22×#117×C73C211yesmaximal, characteristic
10C24×Dic73C2211yescharacteristic
11C24×Dic73C2211yescharacteristic
12C22×#11924007—28no—
13C22×#11924007C2212yes—
14C22×#11924007C2212yes—
15C22×#11924007C2218yes—
16C22×#11924007C2214yes—
17C22×C22⋊Dic73C2212yes—
18#17988525C22116yescomplemented
19#17988518C22116yescomplemented
20#17988471C22132yescomplemented
21#17988459C22116yes—
22#17988450C2218yes—
23#17988448C22116yes—
24C2×#14556498C2214yes—
25C2×#14556498C2214yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2214yes—
29C24×C4×C73C2211yescharacteristic
30C2×#117×C73—28no—
31C2×#117×C73C2218yes—
32C22×C4⋊C4×C73C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34C22×C22⋊C4×C73C2212yes—
35C22×C42×C73C2212yes—
36C23×Dic73—28no—
37C23×Dic73—24no—
38C23×Dic73—24no—
39C23×Dic73—28no—
40C23×Dic73C2312yes—
41C23×Dic73C2314yes—
42C23×Dic73C2314yes—
43C23×Dic73C2314yes—
44C23×Dic73C2312yes—
45C2×#11924007—216no—
46C2×#11924007—28no—
47C2×#11924007—28no—
48C2×#11924007—232no—
49C2×#11924007C23116yes—
50C2×#11924007C2318yes—
51C2×#11924007C2318yes—
52C2×C22⋊Dic73—28no—
53C2×C22⋊Dic73C2318yes—
54#14556498C23116yes—
55#14556498C23116yes—
56#14556498C2318yes—
57#14556498C2318yes—
58#14556498C23116yes—
59C25×C73C2311yescharacteristic
60C23×C4×C73—28no—
61C23×C4×C73D418yes—
62C23×C4×C73C2312yes—
63C23×C4×C73C2312yes—
64C23×C4×C73C2311yescharacteristic
65C23×C4×C73C2311yescharacteristic
66C23×C4×C73C2314yes—
67#117×C73—232no—
68C2×C4⋊C4×C73—24no—
69C2×C4⋊C4×C73—24no—
70C2×C4⋊C4×C73C2314yes—
71C2×C4⋊C4×C73C2314yes—
72C2×C22⋊C4×C73—28no—
73C2×C22⋊C4×C73C2318yes—
74C2×C42×C73—28no—
75C2×C42×C73C2318yes—
76C22×Dic73—232no—
77C22×Dic73—28no—
78C22×Dic73—24no—
79C22×Dic73—24no—
80C22×Dic73—216no—
81C22×Dic73—28no—
82C22×Dic73—24no—
83C22×Dic73—28no—
84C22×Dic73—28no—
85C22×Dic73—216no—
86C22×Dic73—28no—
87C22×Dic73—28no—
88C22×Dic73—416no—
89C22×Dic73#4118yes—
90C22×Dic73C2418yes—
91C22×Dic73C2414yes—
92C22×Dic73C2414yes—
93#11924007—216no—
94#11924007—216no—
95#11924007—232no—
96C22⋊Dic73—216no—
97C24×C73—24no—
98C24×C73—24no—
99C24×C73C2×D414yes—
100C24×C73#4118yes—
101C24×C73C2412yes—
102C24×C73C2411yescharacteristic
103C22×C4×C73—232no—
104C22×C4×C73—216no—
105C22×C4×C73—24no—
106C22×C4×C73—24no—
107C22×C4×C73—24no—
108C22×C4×C73—24no—
109C22×C4×C73—28no—
110C22×C4×C73—28no—
111C22×C4×C73C2×D414yes—
112C22×C4×C73C2×D414yes—
113C22×C4×C73C2×D4116yes—
114C22×C4×C73C2414yes—
115C22×C4×C73C2414yes—
116C22×C4×C73C2412yes—
117C22×C4×C73C2412yes—
118C4⋊C4×C73—28no—
119C4⋊C4×C73—28no—
120C22⋊C4×C73—216no—
121C42×C73—216no—
122C23×C73—28no—
123C23×C73—28no—
124C23×C73—232no—
125C23×C73—24no—
126C23×C73—24no—
127C23×C73#123116yes—
128C23×C73C22×D414yes—
129C23×C73C22×D412yes—
130C23×C73C2×#4114yes—
131C23×C73C2×#4118yes—
132C23×C73C2×#4114yes—
133C23×C732−1+414yes—
134C23×C73C2511yescharacteristic
135C2×C4×C73—232no—
136C2×C4×C73—28no—
137C2×C4×C73—28no—
138C2×C4×C73—24no—
139C2×C4×C73—24no—
140C2×C4×C73—24no—
141C2×C4×C73—24no—
142C2×C4×C73—28no—
143C2×C4×C73—28no—
144C2×C4×C73—48no—
145C2×C4×C73—48no—
146C2×C4×C73C22×D414yes—
147C2×C4×C73C22×D414yes—
148C2×Dic73—232no—
149C2×Dic73—216no—
150C2×Dic73—24no—
151C2×Dic73—24no—
152C2×Dic73—24no—
153C2×Dic73—28no—
154C2×Dic73—28no—
155C2×Dic73—28no—
156C2×Dic73—28no—
157C2×Dic73—416no—
158C2×Dic73—48no—
159C2×Dic73—416no—
160C2×Dic73C2×#4118yes—
161C22×C73—24no—
162C22×C73—24no—
163C22×C73—216no—
164C22×C73—232no—
165C22×C73C2×#12318yes—
166C22×C73C2×#12314yes—
167C22×C73#52514yes—
168C22×C73#53614yes—
169C22×C73#552116yes—
170C22×C73C23×D411yescharacteristic
171C22×C73C22×#4112yes—
172C22×C73C22×#4112yes—
173C22×C73C2×2−1+412yes—
174C4×C73—24no—
175C4×C73—24no—
176C4×C73—44no—
177C4×C73—44no—
178Dic73—28no—
179Dic73—44no—
180Dic73—48no—
181C2×#2484—731nomaximal, Sylow
182C2×C73—28no—
183C2×C73#248418yescomplemented
184C2×C73C22×#12311yescharacteristic
185C2×C73C2×#52511yescharacteristic
186C2×C73C2×#53611yescharacteristic
187C2×C73C2×#55214yes—
188C22×#117D7311yescharacteristic
189C23×C4⋊C4—731no—
190#2484—7316no—
191C2×#400—732no—
192C2×#398—732no—
193C2×#387—734no—
194C2×#385—732no—
195C2×#381—731no—
196C2×#381—732no—
197C73C2×#248411yescharacteristic, complemented, Sylow
198C24×C4—731no—
199C24×C4—731no—
200C24×C4C2×D7311yescharacteristic
201C2×#117—28no—
202C2×#117C2×D7318yes—
203C22×C4⋊C4—732no—
204C22×C4⋊C4—734no—
205C22×C4⋊C4—738no—
206C22×C4⋊C4—732no—
207C22×C4⋊C4—1468no—
208C22×C4⋊C4C2×D7311yescharacteristic
209C22×C4⋊C4C2×D7311yescharacteristic
210C22×C22⋊C4—732no—
211C22×C22⋊C4C2×D7312yes—
212C22×C42C2×D7312yes—
213#400—7316no—
214#398—7316no—
215#387—7332no—
216#385—7316no—
217#381—738no—
218#381—7316no—
219C2×#95—732no—
220C2×#95—734no—
221C2×#95—734no—
222C2×#95—734no—
223C2×#95—732no—
224C25C22×D7311yescharacteristic
225C23×C4—28no—
226C23×C4—732no—
227C23×C4—734no—
228C23×C4—734no—
229C23×C4—734no—
230C23×C4—732no—
231C23×C4—1464no—
232C23×C4—1468no—
233C23×C4—1468no—
234C23×C4—1464no—
235C23×C4C22⋊D7318yes—
236C23×C4C22×D7312yes—
237C23×C4C22×D7312yes—
238C23×C4C22×D7314yes—
239C23×C4C22×D7311yescharacteristic
240C23×C4C22×D7311yescharacteristic
241#117—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—738no—
245C2×C4⋊C4—738no—
246C2×C4⋊C4—7316no—
247C2×C4⋊C4—14616no—
248C2×C4⋊C4—1468no—
249C2×C4⋊C4—1468no—
250C2×C4⋊C4—14632no—
251C2×C4⋊C4C22×D7314yes—
252C2×C4⋊C4C22×D7314yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—738no—
255C2×C22⋊C4—1468no—
256C2×C22⋊C4C22×D7318yes—
257C2×C42—28no—
258C2×C42C22×D7318yes—
259#95—738no—
260#95—7316no—
261#95—7316no—
262#95—738no—
263#95—7316no—
264C24—24no—
265C24—24no—
266C24C2×C22⋊D7314yes—
267C24#1192402818yes—
268C24C23×D7312yes—
269C24C23×D7311yescharacteristic
270C22×C4—232no—
271C22×C4—216no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—29216no—
279C22×C4—738no—
280C22×C4—734no—
281C22×C4—738no—
282C22×C4—734no—
283C22×C4—1468no—
284C22×C4—1464no—
285C22×C4—1468no—
286C22×C4—1468no—
287C22×C4—14616no—
288C22×C4—1468no—
289C22×C4—1468no—
290C22×C4—14616no—
291C22×C4—14632no—
292C22×C4—1468no—
293C22×C4—1464no—
294C22×C4—1464no—
295C22×C4C2×C22⋊D7314yes—
296C22×C4C2×C22⋊D7314yes—
297C22×C4C2×C22⋊D73116yes—
298C22×C4C23×D7314yes—
299C22×C4C23×D7314yes—
300C22×C4C23×D7312yes—
301C22×C4C23×D7312yes—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—14632no—
305C4⋊C4—14616no—
306C4⋊C4—14616no—
307C22⋊C4—216no—
308C22⋊C4—14616no—
309C42—216no—
310C23—24no—
311C23—28no—
312C23—24no—
313C23—232no—
314C23—28no—
315C23#14556574116yes—
316C23C22×C22⋊D7314yes—
317C23C22×C22⋊D7312yes—
318C23#41×D7314yes—
319C23C2×#1192402818yes—
320C23C2×#1192402814yes—
321C23#1455663814yes—
322C23C24×D7311yescharacteristic
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—232no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—29216no—
335C2×C4—29216no—
336C2×C4—2928no—
337C2×C4—738no—
338C2×C4—1464no—
339C2×C4—1468no—
340C2×C4—1468no—
341C2×C4—1468no—
342C2×C4—1468no—
343C2×C4—14632no—
344C2×C4—1464no—
345C2×C4—1464no—
346C2×C4—14616no—
347C2×C4C22×C22⋊D7314yes—
348C2×C4C22×C22⋊D7314yes—
349C22—24no—
350C22—24no—
351C22—232no—
352C22—216no—
353C22C2×#1455657418yes—
354C22C2×#1455657414yes—
355C22#1798909414yes—
356C22#1798915814yes—
357C22#17989270116yes—
358C22C23×C22⋊D7311yescharacteristic
359C22C2×#41×D7312yes—
360C22C22×#1192402812yes—
361C22C2×#1455663812yes—
362C4—24no—
363C4—24no—
364C4—44no—
365C4—44no—
366C4—2924no—
367C4—2928no—
368C4—1468no—
369C2—28no—
370C2#2261421418yesdirect product factor
371C2C22×#1455657411yescharacteristic
372C2C2×#1798909411yescharacteristic
373C2C2×#1798915811yescharacteristic
374C2C2×#1798927014yes—
subgroupG/H
centerC24C23×D73
commutator, cosocleC23×C73C25
FrattiniC23C24×D73
FittingC22×#117×C73C2
socleC24×C73C24
prime radicalC25×C73C23
hypercenterC22×#117D73
hypocenterC73C2×#2484
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. #22614214C2direct (N × H)
3. C2×#17988525C2semidirect
4. C2×#17988518C2semidirect
5. C2×#17988471C2semidirect
18. #17988525C22semidirect
19. #17988518C22semidirect
20. #17988471C22semidirect
183. C2×C73#2484semidirect
197. C73C2×#2484semidirect
370. C2#22614214direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119240072
#226142142
C2×#179885252
C2×#179885182
C2×#179884712
C2×#179884592
C2×#179884502
C2×#179884482
C22×#117×C732
C2×#248473