Finite Group Databasebeta

← #33103579

Subgroups — #33103579

Order 18688.

704 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618589C212yesmaximal
2#22610680C212yesmaximal, complemented
3#22610532C214yesmaximal, complemented
4#22610030C212yesmaximal
5#22609969C211yesmaximal, characteristic
6#22609812C212yesmaximal, complemented
7C8×C22⋊Dic73C211yesmaximal, characteristic
8C2×#427×C73C211yesmaximal, characteristic
9#17988648—24no—
10#17988644—22no—
11#17988640—22no—
12C2×#14556527C2212yes—
13#17988614C2214yes—
14#17988551C2212yes—
15C2×C22⋊#10596831C2211yescharacteristic
16#17988543C2214yes—
17C2×C8×Dic73C2212yes—
18C4×C22⋊Dic73C2211yescharacteristic
19#17988352C2212yes—
20#17988346C2214yes—
21#17988346C2214yes—
22#17988341C2212yes—
23C23×C8×C73C2211yescharacteristic
24C2×#117×C73C2212yes—
25#427×C73C411yescharacteristic, complemented
26#427×C73C411yescharacteristic, complemented
27#427×C73C412yescomplemented
28#427×C73C2212yes—
29#427×C73C2211yescharacteristic
30#427×C73C2211yescharacteristic
31C2×#107×C73C411yescharacteristic, complemented
32C2×#107×C73C411yescharacteristic, complemented
33C2×#106×C73C412yescomplemented
34C2×#11924012C2314yes—
35C22×#10596831C2312yes—
36C2×C22⋊Dic73—21no—
37C2×C4×Dic73C2312yes—
38#14556527—24no—
39C4⋊#10596831—24no—
40C4⋊#10596831—24no—
41C22⋊#10596831—22no—
42C22⋊#10596831—22no—
43C22⋊#10596831D411yescharacteristic
44C22⋊#10596831D411yescharacteristic
45C22⋊#10596831D412yes—
46#14556502—24no—
47C4×#10596831—24no—
48C8×Dic73—24no—
49#14556498—21no—
50C23×C4×C73C2311yescharacteristic
51C22×C8×C73—24no—
52C22×C8×C73C2×C411yescharacteristic
53C22×C8×C73C2×C411yescharacteristic
54C22×C8×C73C2×C411yescharacteristic
55C22×C8×C73C2×C411yescharacteristic
56C22×C8×C73C2311yescharacteristic
57C22×C8×C73C2311yescharacteristic
58#117×C73—22no—
59#117×C73—22no—
60#117×C73C2×C412yes—
61#117×C73C2×C412yes—
62#117×C73C2312yes—
63#117×C73C2312yes—
64C2×C4⋊C4×C73C2×C412yes—
65C2×C4⋊C4×C73C2×C412yes—
66C2×C22⋊C4×C73—22no—
67C2×C42×C73—22no—
68#107×C73C2×C412yes—
69#107×C73C2×C412yes—
70#107×C73C2×C411yescharacteristic
71#107×C73C2×C411yescharacteristic
72#107×C73C2×C411yescharacteristic
73#107×C73C2×C411yescharacteristic
74#106×C73C2×C412yes—
75#106×C73C2×C412yes—
76#106×C73C2×C412yes—
77#106×C73C2×C412yes—
78C22×Dic73—22no—
79#11924012—24no—
80#11924012—24no—
81C2×#10596831—22no—
82C2×#10596831—22no—
83C2×#10596831—24no—
84C2×#10596831C2×D412yes—
85C2×#10596831C2×D412yes—
86C22⋊Dic73—24no—
87C4×Dic73—22no—
88C4×Dic73—22no—
89C24×C73#4111yescharacteristic
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—22no—
93C22×C4×C73—22no—
94C22×C4×C73C22×C411yescharacteristic
95C22×C4×C73C22×C411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73#4111yescharacteristic
99C22×C4×C73C2411yescharacteristic
100C2×C8×C73—22no—
101C2×C8×C73—22no—
102C2×C8×C73—22no—
103C2×C8×C73—22no—
104C2×C8×C73—22no—
105C2×C8×C73C22×C411yescharacteristic
106C2×C8×C73C22×C411yescharacteristic
107C2×C8×C73C22×C411yescharacteristic
108C2×C8×C73C22×C411yescharacteristic
109C2×C8×C73#4114yes—
110C4⋊C4×C73—22no—
111C4⋊C4×C73—22no—
112C4⋊C4×C73C22×C412yes—
113C4⋊C4×C73C22×C412yes—
114C4⋊C4×C73C22×C412yes—
115C4⋊C4×C73C22×C412yes—
116C22⋊C4×C73—22no—
117C22⋊C4×C73—22no—
118C22⋊C4×C73—22no—
119C22⋊C4×C73—22no—
120C42×C73—22no—
121C42×C73—22no—
122C42×C73—22no—
123C42×C73—22no—
124C23×C73—22no—
125C23×C73—22no—
126C23×C73—41no—
127C23×C73C4×D411yescharacteristic
128C23×C73C4×D411yescharacteristic
129C23×C73C2×#4111yescharacteristic
130C2×C4×C73—22no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—22no—
141C2×C4×C73—44no—
142C2×C4×C73—82no—
143C2×C4×C73C4×D411yescharacteristic
144C2×C4×C73C4×D411yescharacteristic
145C2×C4×C73C23×C411yescharacteristic
146C2×C4×C73C22×D411yescharacteristic
147C2×C4×C73C2×#4112yes—
148C2×C4×C732−1+412yes—
149C2×Dic73—24no—
150C2×Dic73—42no—
151C2×Dic73—42no—
152C8×C73—21no—
153C8×C73—21no—
154C8×C73#11712yes—
155C8×C73#11712yes—
156#10596831—22no—
157#10596831—22no—
158#10596831—44no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—44no—
167C22×C73#44312yes—
168C22×C73C2×C4×D411yescharacteristic
169C22×C73#55212yes—
170C4×C73—21no—
171C4×C73—21no—
172C4×C73—42no—
173C4×C73—42no—
174C4×C73—42no—
175C4×C73—42no—
176C4×C73C2×#11711yescharacteristic
177C4×C73#51911yescharacteristic
178C4×C73#54712yes—
179Dic73—44no—
180#22606—731nomaximal, Sylow
181C2×C73—22no—
182C2×C73—22no—
183C2×C73—41no—
184C2×C73#234411yescharacteristic
185C2×C73C2×#44311yescharacteristic
186C2×C73#295711yescharacteristic
187#2924—732no—
188C2×#427D7311yescharacteristic
189#1945—732no—
190#1922—734no—
191#1826—732no—
192#1810—731no—
193#1772—732no—
194C8×C22⋊C4—731no—
195C73#2260611yescharacteristic, complemented, Sylow
196C23×C8C2×D7311yescharacteristic
197C2×#117C2×D7312yes—
198#433—1464no—
199#432—1462no—
200#431—1462no—
201C2×#108—732no—
202#427Dic7311yescharacteristic, complemented
203#427Dic7311yescharacteristic, complemented
204#427Dic7312yescomplemented
205#427C2×D7312yes—
206#427C2×D7311yescharacteristic
207#427C2×D7311yescharacteristic
208C2×#107Dic7311yescharacteristic, complemented
209C2×#107Dic7311yescharacteristic, complemented
210C2×#106Dic7312yescomplemented
211#424—734no—
212#407—732no—
213C2×C22⋊C8—731no—
214#405—734no—
215C2×C4×C8—732no—
216C4×C22⋊C4—731no—
217#341—732no—
218#339—734no—
219#339—734no—
220#336—732no—
221C23×C4C22×D7311yescharacteristic
222C2×M16—734no—
223C22×C8—24no—
224C22×C8—732no—
225C22×C8C2×Dic7311yescharacteristic
226C22×C8C2×Dic7311yescharacteristic
227C22×C8C2×Dic7311yescharacteristic
228C22×C8C2×Dic7311yescharacteristic
229C22×C8C22×D7311yescharacteristic
230C22×C8C22×D7311yescharacteristic
231#117—22no—
232#117—22no—
233#117C2×Dic7312yes—
234#117C2×Dic7312yes—
235#117C22×D7312yes—
236#117C22×D7312yes—
237C2×C4⋊C4C2×Dic7312yes—
238C2×C4⋊C4C2×Dic7312yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—1461no—
241C2×C42—22no—
242C2×C42—732no—
243#108—1464no—
244#107C2×Dic7312yes—
245#107C2×Dic7312yes—
246#107C2×Dic7311yescharacteristic
247#107C2×Dic7311yescharacteristic
248#107C2×Dic7311yescharacteristic
249#107C2×Dic7311yescharacteristic
250#106C2×Dic7312yes—
251#106C2×Dic7312yes—
252#106C2×Dic7312yes—
253#106C2×Dic7312yes—
254C4⋊C8—1464no—
255C4⋊C8—1464no—
256C22⋊C8—731no—
257C22⋊C8—731no—
258C22⋊C8—732no—
259C22⋊C8—1462no—
260C22⋊C8—1462no—
261#97—1464no—
262C4×C8—1464no—
263C4×C8—1464no—
264#95—1461no—
265C24#1192402711yescharacteristic
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—1462no—
271C22×C4C22×Dic7311yescharacteristic
272C22×C4C22×Dic7311yescharacteristic
273C22×C4D4×D7311yescharacteristic
274C22×C4D4×D7311yescharacteristic
275C22×C4#1192402711yescharacteristic
276C22×C4C23×D7311yescharacteristic
277M16—1464no—
278M16—1464no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—732no—
285C2×C8—732no—
286C2×C8—1464no—
287C2×C8—1462no—
288C2×C8—1462no—
289C2×C8C22×Dic7311yescharacteristic
290C2×C8C22×Dic7311yescharacteristic
291C2×C8C22×Dic7311yescharacteristic
292C2×C8C22×Dic7311yescharacteristic
293C2×C8#1192402814yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4C22×Dic7312yes—
297C4⋊C4C22×Dic7312yes—
298C4⋊C4C22×Dic7312yes—
299C4⋊C4C22×Dic7312yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—1464no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—1462no—
310C42—1462no—
311C23—22no—
312C23—22no—
313C23—41no—
314C23D4×Dic7311yescharacteristic
315C23D4×Dic7311yescharacteristic
316C23C2×#1192402711yescharacteristic
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—44no—
329C2×C4—82no—
330C2×C4—2922no—
331C2×C4—2922no—
332C2×C4—1464no—
333C2×C4D4×Dic7311yescharacteristic
334C2×C4D4×Dic7311yescharacteristic
335C2×C4C23×Dic7311yescharacteristic
336C2×C4C2×D4×D7311yescharacteristic
337C2×C4C2×#1192402812yes—
338C2×C4#1455663912yes—
339C8—21no—
340C8—21no—
341C8—2924no—
342C8—1462no—
343C8—1462no—
344C8#1455655012yes—
345C8#1455655012yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—44no—
354C22#1798870412yes—
355C22C2×D4×Dic7311yescharacteristic
356C22#1798927212yes—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—2924no—
364C4C2×#1455655011yescharacteristic
365C4#1798906511yescharacteristic
366C4#1798923812yes—
367C2—22no—
368C2—22no—
369C2—41no—
370C2#2261280011yescharacteristic
371C2C2×#1798870411yescharacteristic
372C2#2261884811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#427×C73C2
socleC22×C73C2×C4×D4
prime radicalC24×C73#41
cosocleC22×C4×C73C24
hypercenterC2×#427D73
hypocenterC73#22606
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #22610680C2semidirect
3. #22610532C2semidirect
6. #22609812C2semidirect
25. #427×C73C4semidirect
26. #427×C73C4semidirect
27. #427×C73C4semidirect
31. C2×#107×C73C4semidirect
32. C2×#107×C73C4semidirect
33. C2×#106×C73C4semidirect
195. C73#22606semidirect
202. #427Dic73semidirect
203. #427Dic73semidirect
204. #427Dic73semidirect
208. C2×#107Dic73semidirect
209. C2×#107Dic73semidirect
210. C2×#106Dic73semidirect

Maximal subgroups

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

subgroupindex
#226185892
#226106802
#226105322
#226100302
#226099692
#226098122
C8×C22⋊Dic732
C2×#427×C732
#2260673