Finite Group Databasebeta

← #32974237

Subgroups — #32974237

Order 18688.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612461C211yesmaximal, characteristic
2#22609692C211yesmaximal, characteristic
3#22609623C211yesmaximal, characteristic
4#22607756C211yesmaximal, characteristic, complemented
5#22607756C211yesmaximal, characteristic, complemented
6#22607754C211yesmaximal, characteristic, complemented
7C73×#1271C211yesmaximal, characteristic, complemented
8C22×#11924011—21no—
9C2×#14556571C411yescharacteristic
10C2×#14556556C411yescharacteristic
11C2×C42×D73C2211yescharacteristic
12C2×#14556503—21no—
13C2×#14556501—21no—
14#17988493C2211yescharacteristic
15#17988470—21no—
16#17988458—21no—
17#17988444C411yescharacteristic
18C4⋊C4×Dic73—21no—
19C4⋊C4×Dic73C411yescharacteristic
20#17988434—21no—
21#17988342C2211yescharacteristic
22#17988342C2211yescharacteristic
23#17988341—21no—
24#389×C73C2211yescharacteristic
25#336×C73C2211yescharacteristic
26#336×C73C2211yescharacteristic
27C22×Dic146C2×C411yescharacteristic
28C22×C4×D73C2×C411yescharacteristic
29C22×C4×D73D411yescharacteristic
30C22×C4×D73D411yescharacteristic
31C2×#11924011—24no—
32C2×#11924011—42no—
33C2×#11924011—42no—
34C22×#10596831—21no—
35C22×#10596831—21no—
36#14556571—22no—
37#14556571—22no—
38#14556556—42no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73C2×C411yescharacteristic
41C2×#11924007—21no—
42C2×#11924005—21no—
43C2×#11924005—21no—
44C2×C4×Dic73—21no—
45C2×C4×Dic73—21no—
46C2×C4×Dic73C2×C411yescharacteristic
47C2×C4×Dic73D411yescharacteristic
48C2×C4×Dic73D411yescharacteristic
49C42×D73—42no—
50#14556503—24no—
51#14556501—42no—
52#14556498—21no—
53#14556498—21no—
54#14556498—21no—
55#14556498—21no—
56C22×C8×C73—21no—
57C22×C8×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73C2×C411yescharacteristic
60C2×C4⋊C4×C73C2×C411yescharacteristic
61C2×C42×C73C2311yescharacteristic
62#95×C73—21no—
63C23×D73C22⋊C411yescharacteristic
64C2×Dic146—21no—
65C2×Dic146—41no—
66C2×Dic146—41no—
67C2×Dic146M1612yescomplemented
68C22×Dic73—21no—
69C22×Dic73—21no—
70C22×Dic73C22⋊C411yescharacteristic
71C22×Dic73C22⋊C411yescharacteristic
72C22×Dic73C22⋊C411yescharacteristic
73C2×C4×D73—22no—
74C2×C4×D73—21no—
75C2×C4×D73—21no—
76C2×C4×D73—22no—
77C2×C4×D73—21no—
78C2×C4×D73—21no—
79C2×C4×D73—21no—
80C2×C4×D73—21no—
81C2×C4×D73—22no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×C4×D73—41no—
85#11924011—48no—
86C2×#10596831—22no—
87C2×#10596831—22no—
88C2×#10596831—41no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C2×#10596831—41no—
92C4⋊Dic73—24no—
93C4⋊Dic73—41no—
94#11924007—42no—
95#11924005—24no—
96#11924005—42no—
97C4×Dic73—22no—
98C4×Dic73—22no—
99C4×Dic73—41no—
100C4×Dic73—42no—
101C4×Dic73—42no—
102C22×C4×C73—21no—
103C22×C4×C73—21no—
104C22×C4×C73C22×C411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73C2×D411yescharacteristic
107C2×C8×C73—22no—
108C2×C8×C73—22no—
109C2×C8×C73—41no—
110C2×C8×C73—41no—
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—41no—
115C4⋊C4×C73—41no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73M1612yescomplemented
118C42×C73—41no—
119C23×C73C2×C22⋊C411yescharacteristic
120C22×D73—21no—
121C22×D73—21no—
122C22×D73—21no—
123C22×D73—21no—
124C22×D73—21no—
125C22×D73—21no—
126C22×D73#10412yes—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—22no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73C2×M1611yescharacteristic
140C2×C4×C73C2×M1611yescharacteristic
141C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C73#13711yescharacteristic
143C2×C4×C73#13711yescharacteristic
144C2×C4×C73#13711yescharacteristic
145C2×Dic73—21no—
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—21no—
149C2×Dic73—22no—
150C2×Dic73—21no—
151C2×Dic73—21no—
152C2×Dic73—21no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73#9912yes—
159C2×Dic73#10412yes—
160C4×D73—24no—
161C4×D73—24no—
162C4×D73—42no—
163C4×D73—42no—
164C4×D73—42no—
165C4×D73—42no—
166Dic146—42no—
167Dic146—42no—
168C8×C73—42no—
169C8×C73—42no—
170#10596831—42no—
171#10596831—42no—
172C22×C73#40711yescharacteristic
173C22×C73C2×#9911yescharacteristic
174C22×C73#41311yescharacteristic
175C22×C73#41711yescharacteristic
176C22×C73#41911yescharacteristic
177C22×C73C2×#10411yescharacteristic
178C22×C73C2×#10411yescharacteristic
179C2×D73—22no—
180C2×D73—22no—
181C2×D73—22no—
182C2×D73—41no—
183C2×D73—41no—
184C2×D73—41no—
185C2×D73—41no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—21no—
192C4×C73—42no—
193C4×C73—81no—
194Dic73—22no—
195Dic73—22no—
196Dic73—42no—
197Dic73—41no—
198Dic73—81no—
199#7634—731nomaximal, Sylow
200C2×C73#146511yescharacteristic
201C2×C73#148911yescharacteristic
202C2×C73#149111yescharacteristic
203C2×C73#150411yescharacteristic
204C2×C73#150811yescharacteristic
205C2×C73#152411yescharacteristic
206C2×C73#155911yescharacteristic
207D73—42no—
208C4×C22⋊Q8—731no—
209#1749—731no—
210#1728—731no—
211#1272—731no—
212#1272—731no—
213#1271—731no—
214#1271D7311yescharacteristic, complemented
215C73#763411yescharacteristic, complemented, Sylow
216C22×M16—1461no—
217C2×C22⋊Q8—731no—
218C2×C4×Q8—731no—
219C22×C42—731no—
220C2×C22⋊C8—1461no—
221C2×#97—1461no—
222#391—731no—
223#389C2×D7311yescharacteristic
224#387—1461no—
225#385—1461no—
226#380—731no—
227C4×C4⋊C4—731no—
228C4×C4⋊C4—1461no—
229C4×C22⋊C4—1461no—
230#336—731no—
231#336—731no—
232#336—1461no—
233#336C2×D7311yescharacteristic
234#336C2×D7311yescharacteristic
235C22×Q8—731no—
236C23×C4—731no—
237C23×C4—731no—
238C23×C4—731no—
239C2×M16—2922no—
240C2×M16—2922no—
241C2×M16—1464no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—1461no—
245C22×C8—1461no—
246C22⋊Q8—1462no—
247C22⋊Q8—1462no—
248C4×Q8—2922no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—731no—
251C2×C4⋊C4—1461no—
252C2×C4⋊C4—1461no—
253C2×C4⋊C4C4×D7311yescharacteristic
254C2×C4⋊C4C4×D7311yescharacteristic
255C2×C22⋊C4—1461no—
256C2×C22⋊C4—1461no—
257C2×C42—2922no—
258C2×C42—731no—
259C2×C42—731no—
260C2×C42—731no—
261C2×C42—1461no—
262C2×C42—1461no—
263C2×C42C22×D7311yescharacteristic
264C22⋊C8—1464no—
265#97—2922no—
266#95—21no—
267#95—1461no—
268#95—1461no—
269#95—1461no—
270#95—1461no—
271C24—731no—
272C2×Q8—2921no—
273C2×Q8—2921no—
274C2×Q8—732no—
275C2×Q8—1461no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—2921no—
279C22×C4—2921no—
280C22×C4—2921no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—731no—
284C22×C4—1462no—
285C22×C4—1462no—
286C22×C4—1462no—
287C22×C4—1461no—
288C22×C4—1461no—
289C22×C4—1461no—
290C22×C4—1461no—
291C22×C4—1461no—
292C22×C4—1461no—
293C22×C4—1461no—
294C22×C4—1461no—
295C22×C4C2×C4×D7311yescharacteristic
296C22×C4D4×D7311yescharacteristic
297C22×C4D4×D7311yescharacteristic
298M16—2928no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—2921no—
306C2×C8—2921no—
307C2×C8—2921no—
308C2×C8—2921no—
309C2×C8—1462no—
310C2×C8—1462no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—2921no—
316C4⋊C4—2922no—
317C4⋊C4—1464no—
318C4⋊C4#1192401112yescomplemented
319C22⋊C4—2922no—
320C22⋊C4—1464no—
321C42—41no—
322C42—2921no—
323C42—2922no—
324C42—2922no—
325C42—1462no—
326C42—1462no—
327C23—732no—
328C23—1461no—
329C23—1461no—
330C23—1461no—
331C23—1461no—
332C23—1461no—
333C23—1461no—
334C23C22⋊C4×D7311yescharacteristic
335Q8—2922no—
336Q8—2922no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—2922no—
350C2×C4—2922no—
351C2×C4—2922no—
352C2×C4—2922no—
353C2×C4—2921no—
354C2×C4—2921no—
355C2×C4—2921no—
356C2×C4—2921no—
357C2×C4—2921no—
358C2×C4—732no—
359C2×C4—732no—
360C2×C4—1464no—
361C2×C4—1464no—
362C2×C4—1462no—
363C2×C4—1461no—
364C2×C4—1461no—
365C2×C4—1461no—
366C2×C4—1461no—
367C2×C4—1461no—
368C2×C4—1461no—
369C2×C4—1461no—
370C2×C4M16×D7311yescharacteristic
371C2×C4C2×#1192401111yescharacteristic
372C2×C4#1455661511yescharacteristic
373C2×C4#1455662011yescharacteristic
374C2×C4#1455662011yescharacteristic
375C2×C4#1455662311yescharacteristic
376C8—42no—
377C8—42no—
378C8—2922no—
379C8—2922no—
380C22—2921no—
381C22—2921no—
382C22—2921no—
383C22—2921no—
384C22—1462no—
385C22—1462no—
386C22—1462no—
387C22#1798854911yescharacteristic
388C22#99×D7311yescharacteristic
389C22#1798857011yescharacteristic
390C22#1798858511yescharacteristic
391C22#1798859411yescharacteristic
392C22#104×D7311yescharacteristic
393C22#104×D7311yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—2921no—
403C4—2922no—
404C4—5841no—
405C4—1462no—
406C4—1462no—
407C2—2922no—
408C2#2260826811yescharacteristic
409C2#2260836711yescharacteristic
410C2#2260837611yescharacteristic
411C2#2260843211yescharacteristic
412C2#2260844911yescharacteristic
413C2#2260851811yescharacteristic
414C2#2260866011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1271C2
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C42×C73C23
hypercenter#1271D73
hypocenterC73#7634
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #22607756C2semidirect
5. #22607756C2semidirect
6. #22607754C2semidirect
7. C73×#1271C2semidirect
67. C2×Dic146M16semidirect
117. C4⋊C4×C73M16semidirect
214. #1271D73semidirect
215. C73#7634semidirect
318. C4⋊C4#11924011semidirect

Maximal subgroups

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

subgroupindex
#226124612
#226096922
#226096232
#226077562
#226077542
C73×#12712
#763473