Finite Group Databasebeta

← #32982029

Subgroups — #32982029

Order 18688.

560 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 113 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#22615746C211yesmaximal, characteristic
2#22612548C211yesmaximal, characteristic
3#22608503C211yesmaximal, characteristic
4C2×#17988312C211yesmaximal, characteristic
5#22607799C211yesmaximal, characteristic
6#22607798C211yesmaximal, characteristic
7C73×#1520C211yesmaximal, characteristic
8C2×#14556547—21no—
9C22×#11924007C2211yescharacteristic
10C2×C22⋊#10596831C2211yescharacteristic
11#17988525C412yescomplemented
12#17988520—22no—
13#17988506C412yescomplemented
14#17988484—22no—
15#17988482—22no—
16#17988459—21no—
17#17988448—21no—
18#17988447C2211yescharacteristic
19#17988447C2211yescharacteristic
20#17988445—21no—
21#17988324—22no—
22#17988312—22no—
23C2×C22⋊Q8×C73C2211yescharacteristic
24C2×C22⋊C8×C73C2211yescharacteristic
25#380×C73C2211yescharacteristic
26#328×C73—22no—
27C23×Dic73D411yescharacteristic
28C23×Dic73D411yescharacteristic
29C22×#10596831—21no—
30#14556547—42no—
31#14556547—42no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73—21no—
34C2×#11924007—22no—
35C2×#11924007—22no—
36C2×#11924007—22no—
37C2×#11924007—21no—
38C2×#11924007—21no—
39C2×#11924007—21no—
40C2×#11924007—21no—
41C2×#11924007C2×C412yes—
42C2×#11924007D412yes—
43C2×C22⋊Dic73—21no—
44C2×C22⋊Dic73—21no—
45C2×C4×Dic73—22no—
46C22⋊#10596831—21no—
47C22⋊#10596831—21no—
48#14556498—22no—
49#14556498—22no—
50#14556498—22no—
51#14556498—22no—
52#14556498C2×C412yes—
53#14556498D412yes—
54C22×Q8×C73C2×C411yescharacteristic
55C23×C4×C73C2311yescharacteristic
56C22×C8×C73—21no—
57C22⋊Q8×C73—21no—
58C22⋊Q8×C73—21no—
59C22⋊Q8×C73—21no—
60C22⋊Q8×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73C2×C411yescharacteristic
64C2×C22⋊C4×C73—21no—
65C2×C22⋊C4×C73—21no—
66C22⋊C8×C73—21no—
67C22⋊C8×C73—21no—
68#95×C73D412yes—
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73—21no—
72C22×Dic73—21no—
73C22×Dic73—21no—
74C22×Dic73—21no—
75C22×Dic73—22no—
76C22×Dic73—22no—
77C22×Dic73—81no—
78C22×Dic73C22⋊C412yes—
79C22×Dic73C22⋊C412yes—
80C22×Dic73SD1612yes—
81C22×Dic73Q1612yes—
82C2×#10596831—41no—
83C2×#10596831—81no—
84C4⋊Dic73—81no—
85C4⋊Dic73—81no—
86#11924007—22no—
87#11924007—22no—
88#11924007—22no—
89#11924007—22no—
90#11924007—44no—
91#11924007—82no—
92#11924007—82no—
93C22⋊Dic73—41no—
94C22⋊Dic73—41no—
95C22⋊Dic73—41no—
96C22⋊Dic73—41no—
97C4×Dic73—44no—
98C24×C73C2×D411yescharacteristic
99C2×Q8×C73—21no—
100C2×Q8×C73—21no—
101C2×Q8×C73—81no—
102C22×C4×C73—21no—
103C22×C4×C73—21no—
104C22×C4×C73—22no—
105C22×C4×C73—81no—
106C22×C4×C73C22×C411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73#4111yescharacteristic
111C22×C4×C73#4111yescharacteristic
112C2×C8×C73—41no—
113C2×C8×C73—81no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—81no—
119C22⋊C4×C73—41no—
120C22⋊C4×C73—41no—
121C22⋊C4×C73—41no—
122C22⋊C4×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73C2×C22⋊C411yescharacteristic
126C23×C73#12011yescharacteristic
127C23×C73#12311yescharacteristic
128C23×C73C2×SD1611yescharacteristic
129C23×C73C2×Q1611yescharacteristic
130C23×C73AGL(1,ℤ/8)11yescharacteristic
131C23×C73#13711yescharacteristic
132Q8×C73—81no—
133Q8×C73—81no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—42no—
147C2×C4×C73C4×D411yescharacteristic
148C2×C4×C73C4×D411yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12111yescharacteristic
151C2×Dic73—22no—
152C2×Dic73—22no—
153C2×Dic73—22no—
154C2×Dic73—22no—
155C2×Dic73—22no—
156C2×Dic73—22no—
157C2×Dic73—22no—
158C2×Dic73—42no—
159C2×Dic73—42no—
160C2×Dic73—42no—
161C2×Dic73—42no—
162C2×Dic73—42no—
163C2×Dic73—42no—
164C2×Dic73—42no—
165C2×Dic73—42no—
166C2×Dic73—81no—
167C2×Dic73#10314yes—
168C8×C73—81no—
169#10596831—81no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73—41no—
177C22×C73—41no—
178C22×C73—41no—
179C22×C73—41no—
180C22×C73#38611yescharacteristic
181C22×C73C2×#10311yescharacteristic
182C22×C73#41711yescharacteristic
183C22×C73#45011yescharacteristic
184C22×C73#45111yescharacteristic
185C22×C73#48311yescharacteristic
186C22×C73#48411yescharacteristic
187C4×C73—21no—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—82no—
192C4×C73—81no—
193C4×C73—81no—
194Dic73—24no—
195Dic73—42no—
196Dic73—82no—
197Dic73—82no—
198#9200—731nomaximal, Sylow
199C2×C73—21no—
200C2×C73—21no—
201C2×C73—21no—
202C2×C73—21no—
203C2×C73—21no—
204C2×C73—21no—
205C2×C73#187211yescharacteristic
206C2×C73#191211yescharacteristic
207C2×C73#191311yescharacteristic
208#2627—731no—
209#2316—731no—
210#1520—731no—
211#1520D7311yescharacteristic
212C2×#324—731no—
213#1289—731no—
214#1289—731no—
215C73#920011yescharacteristic, complemented, Sylow
216C2×C22⋊Q8C2×D7311yescharacteristic
217C2×#117—1461no—
218C22×C4⋊C4—731no—
219C2×C22⋊C8—731no—
220C2×C22⋊C8C2×D7311yescharacteristic
221#400—732no—
222#398—1462no—
223#395—732no—
224#389—1462no—
225#389—1462no—
226#385—1461no—
227#381—1461no—
228#380—731no—
229#380—731no—
230#380—1461no—
231#380C2×D7311yescharacteristic
232#328—22no—
233#328—1462no—
234#324—1462no—
235C22×Q8C4×D7311yescharacteristic
236C23×C4—731no—
237C23×C4—731no—
238C23×C4C22×D7311yescharacteristic
239C22×C8—21no—
240C22×C8—1461no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245#117—2922no—
246#117—2922no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—732no—
250C2×C4⋊C4—732no—
251C2×C4⋊C4—1462no—
252C2×C4⋊C4—1462no—
253C2×C4⋊C4—1462no—
254C2×C4⋊C4—1461no—
255C2×C4⋊C4—1461no—
256C2×C4⋊C4—1461no—
257C2×C4⋊C4—1461no—
258C2×C4⋊C4—1461no—
259C2×C4⋊C4—1461no—
260C2×C4⋊C4C4×D7311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1461no—
264C2×C22⋊C4—1461no—
265C2×C42—1462no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—1461no—
269C22⋊C8—1461no—
270#95—732no—
271#95—732no—
272#95—1462no—
273#95—1462no—
274#95—1462no—
275#95—1462no—
276#95C22⋊D7312yes—
277C24D4×D7311yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—81no—
285C22×C4—5841no—
286C22×C4—732no—
287C22×C4—732no—
288C22×C4—732no—
289C22×C4—732no—
290C22×C4—1462no—
291C22×C4—1462no—
292C22×C4—1461no—
293C22×C4—1461no—
294C22×C4—1461no—
295C22×C4—1461no—
296C22×C4—1461no—
297C22×C4—1461no—
298C22×C4C2×C4×D7311yescharacteristic
299C22×C4D4×D7311yescharacteristic
300C22×C4D4×D7311yescharacteristic
301C22×C4C2×C22⋊D7311yescharacteristic
302C22×C4#1192402711yescharacteristic
303C22×C4#1192402811yescharacteristic
304C2×C8—41no—
305C2×C8—81no—
306C2×C8—2921no—
307C2×C8—5841no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—2924no—
314C4⋊C4—5842no—
315C4⋊C4—5842no—
316C4⋊C4—5841no—
317C4⋊C4—5841no—
318C4⋊C4—1462no—
319C4⋊C4—1462no—
320C4⋊C4—1462no—
321C4⋊C4—1462no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—2921no—
327C22⋊C4—2921no—
328C22⋊C4—2921no—
329C22⋊C4—2921no—
330C42—2924no—
331C23—41no—
332C23—41no—
333C23C22⋊C4×D7311yescharacteristic
334C23#1455655911yescharacteristic
335C23#1455657311yescharacteristic
336C23SD16×D7311yescharacteristic
337C23Q16×D7311yescharacteristic
338C23#1455661811yescharacteristic
339C23#1455662311yescharacteristic
340Q8—81no—
341Q8—81no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—2922no—
356C2×C4—2922no—
357C2×C4—2922no—
358C2×C4—2922no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—2922no—
362C2×C4—2922no—
363C2×C4—5841no—
364C2×C4—734no—
365C2×C4—1462no—
366C2×C4—1462no—
367C2×C4—1462no—
368C2×C4—1462no—
369C2×C4—1462no—
370C2×C4—1462no—
371C2×C4—1462no—
372C2×C4#1455655111yescharacteristic
373C2×C4C4×C22⋊D7311yescharacteristic
374C2×C4#1455656211yescharacteristic
375C2×C4#1455656411yescharacteristic
376C8—81no—
377C8—5841no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#1798846311yescharacteristic
389C22#103×D7311yescharacteristic
390C22#1798858711yescharacteristic
391C22#1798873611yescharacteristic
392C22#1798874111yescharacteristic
393C22#1798891611yescharacteristic
394C22#1798892111yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—2922no—
403C4—5842no—
404C4—5842no—
405C4—1464no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#2261023511yescharacteristic
413C2#2261046511yescharacteristic
414C2#2261047011yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1520C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#1520D73
hypocenterC73#9200
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
11. #17988525C4semidirect
13. #17988506C4semidirect
215. C73#9200semidirect

Maximal subgroups

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

subgroupindex
#226157462
#226125482
#226085032
C2×#179883122
#226077992
#226077982
C73×#15202
#920073