Finite Group Databasebeta

← C2×#22618197 #33292575

Subgroups — C2×#22618197 #33292575

Order 18688.

1806 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618197C2116yesmaximal, direct product factor
2C2×#17988528C212yesmaximal
3C2×#17988526C211yesmaximal, characteristic
4C2×#17988521C212yesmaximal
5C2×#17988520C212yesmaximal
6C2×#17988518C212yesmaximal
7C2×#17988506C212yesmaximal
8C2×#17988456C211yesmaximal, characteristic
9C2×#17988454C212yesmaximal
10C2×#395×C73C211yesmaximal, characteristic
11C22×C4⋊Dic73C2211yescharacteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×#11924007C2212yes—
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×#11924007C2212yes—
17C22×#11924007C2212yes—
18C22×C4×Dic73C2211yescharacteristic
19#17988528C22116yes—
20#17988526C2218yes—
21#17988521C22116yes—
22#17988520C22116yes—
23#17988518C22116yes—
24#17988506C22116yes—
25#17988456C2218yes—
26#17988454C22116yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2211yescharacteristic
34C2×#14556498C2212yes—
35C23×Q8×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2211yescharacteristic
37C22×C4⋊C4×C73C2212yes—
38#395×C73C2218yes—
39C2×#95×C73C2212yes—
40C2×#95×C73C2211yescharacteristic
41C23×Dic73C2312yes—
42C23×Dic73C2312yes—
43C23×Dic73C2312yes—
44C23×Dic73C2312yes—
45C2×C4⋊Dic73—42no—
46C2×C4⋊Dic73—42no—
47C2×C4⋊Dic73C2314yes—
48C2×C4⋊Dic73C2314yes—
49C2×#11924007—44no—
50C2×#11924007—44no—
51C2×#11924007—44no—
52C2×#11924007—44no—
53C2×#11924007—44no—
54C2×#11924007C2318yes—
55C2×#11924007C2318yes—
56C2×#11924007C2318yes—
57C2×#11924007C2318yes—
58C2×#11924007C2318yes—
59C2×C4×Dic73Q818yescomplemented
60C2×C4×Dic73C2314yes—
61#14556498C2318yes—
62#14556498C2318yes—
63#14556498C2318yes—
64#14556498C2318yes—
65#14556498C2318yes—
66#14556498C2318yes—
67#14556498C2314yes—
68#14556498C2318yes—
69C22×Q8×C73—44no—
70C22×Q8×C73—42no—
71C22×Q8×C73C2314yes—
72C23×C4×C73C2312yes—
73C23×C4×C73C2312yes—
74C23×C4×C73C2311yescharacteristic
75C23×C4×C73C2311yescharacteristic
76C23×C4×C73C2311yescharacteristic
77C2×C4⋊C4×C73—42no—
78C2×C4⋊C4×C73—44no—
79C2×C4⋊C4×C73C2314yes—
80C2×C4⋊C4×C73C2318yes—
81#95×C73C2318yes—
82#95×C73C2314yes—
83C22×Dic73—24no—
84C22×Dic73—24no—
85C22×Dic73—24no—
86C22×Dic73—24no—
87C22×Dic73—24no—
88C22×Dic73—24no—
89C22×Dic73—24no—
90C22×Dic73—24no—
91C22×Dic73—24no—
92C22×Dic73—28no—
93C22×Dic73C2×Q818yes—
94C22×Dic73C2414yes—
95C22×Dic73C2414yes—
96C22×Dic73C2414yes—
97C22×Dic73C2414yes—
98C4⋊Dic73—44no—
99C4⋊Dic73—44no—
100#11924007—48no—
101#11924007—48no—
102#11924007—48no—
103#11924007—48no—
104#11924007—48no—
105C4×Dic73C2×Q8116yescomplemented
106C24×C73C2411yescharacteristic
107C2×Q8×C73—44no—
108C2×Q8×C73—48no—
109C2×Q8×C73—416no—
110C22×C4×C73—22no—
111C22×C4×C73—22no—
112C22×C4×C73—24no—
113C22×C4×C73—22no—
114C22×C4×C73—24no—
115C22×C4×C73—24no—
116C22×C4×C73—22no—
117C22×C4×C73—24no—
118C22×C4×C73—24no—
119C22×C4×C73—24no—
120C22×C4×C73—24no—
121C22×C4×C73—24no—
122C22×C4×C73C2×Q814yes—
123C22×C4×C73C2414yes—
124C22×C4×C73C2414yes—
125C22×C4×C73C2412yes—
126C22×C4×C73C2412yes—
127C22×C4×C73C2412yes—
128C4⋊C4×C73—44no—
129C4⋊C4×C73—48no—
130C23×C73C22×Q812yes—
131C23×C73H(5,𝔽2)14yes—
132C23×C73H(5,𝔽2)12yes—
133C23×C732−1+412yes—
134C23×C732−1+414yes—
135C23×C73C2511yescharacteristic
136Q8×C73—416no—
137C2×C4×C73—28no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—22no—
141C2×C4×C73—24no—
142C2×C4×C73—24no—
143C2×C4×C73—24no—
144C2×C4×C73—22no—
145C2×C4×C73—24no—
146C2×C4×C73—24no—
147C2×C4×C73—24no—
148C2×C4×C73—24no—
149C2×C4×C73—24no—
150C2×C4×C73—44no—
151C2×C4×C73—48no—
152C2×C4×C73—44no—
153C2×C4×C73—48no—
154C2×C4×C73C22×Q814yes—
155C2×Dic73—216no—
156C2×Dic73—24no—
157C2×Dic73—24no—
158C2×Dic73—24no—
159C2×Dic73—24no—
160C2×Dic73—24no—
161C2×Dic73—24no—
162C2×Dic73—24no—
163C2×Dic73—24no—
164C2×Dic73—24no—
165C2×Dic73—28no—
166C2×Dic73—48no—
167C2×Dic73—48no—
168C2×Dic73—48no—
169C2×Dic73C22×Q818yes—
170C22×C73#54418yes—
171C22×C73#54414yes—
172C22×C73#56014yes—
173C22×C73#56314yes—
174C22×C73#56318yes—
175C22×C73C23×Q811yescharacteristic
176C22×C73C2×H(5,𝔽2)11yescharacteristic
177C22×C73C2×H(5,𝔽2)12yes—
178C22×C73C2×2−1+411yescharacteristic
179C22×C73C2×2−1+412yes—
180C4×C73—24no—
181C4×C73—42no—
182C4×C73—42no—
183C4×C73—44no—
184C4×C73—44no—
185Dic73—28no—
186Dic73—44no—
187Dic73—44no—
188Dic73—44no—
189C2×#2843—731nomaximal, Sylow
190C2×C73#284318yescomplemented
191C2×C73C2×#54412yes—
192C2×C73C2×#54411yescharacteristic
193C2×C73C2×#56011yescharacteristic
194C2×C73C2×#56311yescharacteristic
195C2×C73C2×#56312yes—
196#2843—7316no—
197C2×#400—732no—
198C2×#400—731no—
199C2×#398—732no—
200C2×#398—732no—
201C2×#398—732no—
202C2×#395—732no—
203C2×#395D7311yescharacteristic
204C2×#384—731no—
205C2×#383—732no—
206C73C2×#284311yescharacteristic, complemented, Sylow
207C23×Q8C2×D7311yescharacteristic
208C22×C4⋊C4—732no—
209C22×C4⋊C4—732no—
210C22×C4⋊C4—732no—
211C22×C4⋊C4—732no—
212C22×C4⋊C4—731no—
213C22×C4⋊C4—732no—
214C22×C4⋊C4—731no—
215C22×C4⋊C4C2×D7312yes—
216C22×C4⋊C4C2×D7311yescharacteristic
217C22×C42—731no—
218#400—738no—
219#400—7316no—
220#398—7316no—
221#398—7316no—
222#398—7316no—
223#395—7316no—
224#395C2×D7318yes—
225#384—738no—
226#383—7316no—
227C2×#95—732no—
228C2×#95—732no—
229C2×#95—732no—
230C2×#95—732no—
231C2×#95—731no—
232C2×#95—732no—
233C2×#95—732no—
234C2×#95—732no—
235C2×#95C2×D7312yes—
236C2×#95C2×D7311yescharacteristic
237C22×Q8—44no—
238C22×Q8—42no—
239C22×Q8C22×D7314yes—
240C23×C4—732no—
241C23×C4—732no—
242C23×C4—732no—
243C23×C4—732no—
244C23×C4C22×D7311yescharacteristic
245C23×C4C22×D7312yes—
246C23×C4C22×D7311yescharacteristic
247C23×C4C22×D7312yes—
248C23×C4C22×D7311yescharacteristic
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—2924no—
252C2×C4⋊C4—2924no—
253C2×C4⋊C4—2924no—
254C2×C4⋊C4—2924no—
255C2×C4⋊C4—2924no—
256C2×C4⋊C4—2922no—
257C2×C4⋊C4—2922no—
258C2×C4⋊C4—738no—
259C2×C4⋊C4—738no—
260C2×C4⋊C4—734no—
261C2×C4⋊C4—738no—
262C2×C4⋊C4—738no—
263C2×C4⋊C4—738no—
264C2×C4⋊C4—734no—
265C2×C4⋊C4C22×D7314yes—
266C2×C4⋊C4C22×D7318yes—
267C2×C42—738no—
268C2×C42—734no—
269#95—738no—
270#95—738no—
271#95—738no—
272#95—738no—
273#95—738no—
274#95—738no—
275#95—734no—
276#95—738no—
277#95C22×D7318yes—
278#95C22×D7314yes—
279C24C23×D7311yescharacteristic
280C2×Q8—44no—
281C2×Q8—48no—
282C2×Q8—416no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—738no—
296C22×C4—734no—
297C22×C4—734no—
298C22×C4—734no—
299C22×C4—734no—
300C22×C4—1464no—
301C22×C4—1464no—
302C22×C4—1464no—
303C22×C4—1464no—
304C22×C4—1464no—
305C22×C4—1464no—
306C22×C4—1468no—
307C22×C4—1464no—
308C22×C4—1464no—
309C22×C4—1464no—
310C22×C4Q8×D7314yes—
311C22×C4C23×D7314yes—
312C22×C4C23×D7314yes—
313C22×C4C23×D7312yes—
314C22×C4C23×D7312yes—
315C22×C4C23×D7312yes—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—2928no—
319C4⋊C4—2928no—
320C4⋊C4—2928no—
321C4⋊C4—2928no—
322C4⋊C4—2928no—
323C4⋊C4—2924no—
324C4⋊C4—2924no—
325C42—7316no—
326C23C2×Q8×D7312yes—
327C23#1455663614yes—
328C23#1455663612yes—
329C23#1455663812yes—
330C23#1455663914yes—
331C23C24×D7311yescharacteristic
332Q8—416no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—2928no—
351C2×C4—2928no—
352C2×C4—2928no—
353C2×C4—738no—
354C2×C4—1464no—
355C2×C4—1464no—
356C2×C4—1464no—
357C2×C4—1464no—
358C2×C4—1464no—
359C2×C4—1464no—
360C2×C4—1468no—
361C2×C4—1464no—
362C2×C4—1464no—
363C2×C4—1464no—
364C2×C4—14616no—
365C2×C4C2×Q8×D7314yes—
366C22#1798921418yes—
367C22#1798921614yes—
368C22#1798932614yes—
369C22#1798933614yes—
370C22#1798933818yes—
371C22C22×Q8×D7311yescharacteristic
372C22C2×#1455663611yescharacteristic
373C22C2×#1455663612yes—
374C22C2×#1455663811yescharacteristic
375C22C2×#1455663912yes—
376C4—24no—
377C4—42no—
378C4—44no—
379C4—42no—
380C4—44no—
381C4—2924no—
382C4—2924no—
383C4—2924no—
384C4—1468no—
385C2#2261819718yesdirect product factor
386C2C2×#1798921412yes—
387C2C2×#1798921611yescharacteristic
388C2C2×#1798932611yescharacteristic
389C2C2×#1798933611yescharacteristic
390C2C2×#1798933812yes—
subgroupG/H
centerC24C23×D73
commutator, cosocleC23×C73C25
FrattiniC23C24×D73
FittingC2×#395×C73C2
socle, prime radicalC24×C73C24
hypercenterC2×#395D73
hypocenterC73C2×#2843
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
1. #22618197C2direct (N × H)
59. C2×C4×Dic73Q8semidirect
105. C4×Dic73C2×Q8semidirect
190. C2×C73#2843semidirect
206. C73C2×#2843semidirect
385. C2#22618197direct (N × H)

Maximal subgroups

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

subgroupindex
#226181972
C2×#179885282
C2×#179885262
C2×#179885212
C2×#179885202
C2×#179885182
C2×#179885062
C2×#179884562
C2×#179884542
C2×#395×C732
C2×#284373