Finite Group Databasebeta

← #32986495

Subgroups — #32986495

Order 18688.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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#22620703C211yesmaximal, characteristic
2C2×#17988922C211yesmaximal, characteristic
3#22610916C211yesmaximal, characteristic
4#22610021C211yesmaximal, characteristic
5#22608503C211yesmaximal, characteristic
6#22607779C211yesmaximal, characteristic
7C73×#2502C211yesmaximal, characteristic
8C2×#14556570C2211yescharacteristic
9#17988922—22no—
10#17988922—21no—
11#17988922—22no—
12#17988920—21no—
13#17988826—22no—
14#17988743—21no—
15#17988742—21no—
16C2×#14556524—21no—
17C2×#14556514—21no—
18C2×C22⋊#10596831C2211yescharacteristic
19C2×C22⋊#10596831C2211yescharacteristic
20#17988447C2211yescharacteristic
21#17988353—22no—
22#17988349—22no—
23#17988324—22no—
24C2×C22⋊Q8×C73C2211yescharacteristic
25C22×C4⋊C4×C73C2211yescharacteristic
26#398×C73—22no—
27#391×C73—21no—
28#385×C73—22no—
29#384×C73—21no—
30C4×C22⋊C4×C73C2211yescharacteristic
31C22×Dic146D411yescharacteristic
32C2×#11924018—22no—
33C22×#10596831—21no—
34C22×#10596831D412yes—
35#14556570—21no—
36#14556570—21no—
37#14556570—21no—
38#14556570—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73D411yescharacteristic
41C2×#11924007—21no—
42C2×C22⋊Dic73—21no—
43C2×C22⋊Dic73—21no—
44#14556524—24no—
45#14556524—22no—
46#14556515—22no—
47#14556514—24no—
48#14556514—22no—
49C22⋊#10596831—21no—
50C22⋊#10596831—21no—
51C22⋊#10596831—21no—
52C22⋊#10596831D412yes—
53#14556498—21no—
54C22×Q8×C73D411yescharacteristic
55C23×C4×C73—21no—
56C23×C4×C73C2311yescharacteristic
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×C73—21no—
64C2×C4⋊C4×C73—21no—
65C2×C4⋊C4×C73—21no—
66C2×C4⋊C4×C73—21no—
67C2×C4⋊C4×C73—24no—
68C2×C4⋊C4×C73D411yescharacteristic
69C2×C4⋊C4×C73D412yes—
70C2×C22⋊C4×C73—21no—
71C2×C22⋊C4×C73D412yes—
72C2×C42×C73—21no—
73#95×C73—24no—
74#95×C73—21no—
75C2×Dic146—21no—
76C2×Dic146—21no—
77C2×Dic146—42no—
78C22×Dic73—21no—
79C22×Dic73—21no—
80C22×Dic73—41no—
81#11924018—82no—
82C2×#10596831—22no—
83C2×#10596831—22no—
84C2×#10596831—42no—
85C2×#10596831—42no—
86C4⋊Dic73—21no—
87C4⋊Dic73—21no—
88C4⋊Dic73—42no—
89#11924007—41no—
90#11924007—41no—
91C22⋊Dic73—41no—
92C22⋊Dic73—41no—
93C22⋊Dic73—42no—
94C24×C73C2×D411yescharacteristic
95C2×Q8×C73—21no—
96C2×Q8×C73—21no—
97C2×Q8×C73—42no—
98C22×C4×C73—22no—
99C22×C4×C73—21no—
100C22×C4×C73—22no—
101C22×C4×C73—21no—
102C22×C4×C73—42no—
103C22×C4×C73—41no—
104C22×C4×C73—41no—
105C22×C4×C73—82no—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73#4111yescharacteristic
112C4⋊C4×C73—22no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—22no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—84no—
121C22⋊C4×C73—22no—
122C22⋊C4×C73—41no—
123C22⋊C4×C73—41no—
124C22⋊C4×C73Q1614yescomplemented
125C42×C73—42no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73#12011yescharacteristic
129C23×C73#12111yescharacteristic
130C23×C73#12711yescharacteristic
131C23×C73C2×Q1612yes—
132C23×C73#13712yes—
133Q8×C73—42no—
134Q8×C73—81no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—22no—
138C2×C4×C73—24no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73—42no—
145C2×C4×C73—44no—
146C2×C4×C73—42no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—42no—
150C2×C4×C73—82no—
151C2×C4×C73—82no—
152C2×C4×C73#12011yescharacteristic
153C2×C4×C73#12011yescharacteristic
154C2×C4×C73#12111yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73—81no—
159C2×Dic73—81no—
160C2×Dic73—81no—
161C2×Dic73—81no—
162C2×Dic73—81no—
163Dic146—42no—
164Dic146—42no—
165#10596831—42no—
166#10596831—42no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—42no—
174C22×C73—42no—
175C22×C73#39211yescharacteristic
176C22×C73#45112yes—
177C22×C73#46212yes—
178C22×C73#49411yescharacteristic
179C22×C73#49711yescharacteristic
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—44no—
184C4×C73—81no—
185C4×C73—82no—
186C4×C73—81no—
187C4×C73#45812yes—
188Dic73—81no—
189Dic73—81no—
190Dic73—161no—
191#9959—731nomaximal, Sylow
192C2×C73—21no—
193C2×C73—22no—
194C2×C73—22no—
195C2×C73—21no—
196C2×C73#200811yescharacteristic
197C2×C73#201611yescharacteristic
198C2×C73#202211yescharacteristic
199#3189—731no—
200C2×#484—731no—
201#2502D7311yescharacteristic
202#1996—731no—
203#1824—731no—
204#1520—731no—
205#1282—731no—
206C73#995911yescharacteristic, complemented, Sylow
207C2×C22⋊Q8—731no—
208C2×C22⋊Q8C2×D7311yescharacteristic
209C22×C4⋊C4C2×D7311yescharacteristic
210#484—1462no—
211#484—1461no—
212#484—1462no—
213#484—1461no—
214#467—1462no—
215#451—1461no—
216#451—1461no—
217C2×#107—1461no—
218C2×#103—1461no—
219C2×C22⋊C8—731no—
220C2×C22⋊C8—731no—
221#398—22no—
222#391—21no—
223#385—22no—
224#384—21no—
225#380—731no—
226C4×C22⋊C4C2×D7311yescharacteristic
227#342—1462no—
228#340—1462no—
229#328—1462no—
230C22×Q8—731no—
231C22×Q8C22⋊D7311yescharacteristic
232C23×C4—21no—
233C23×C4C22×D7311yescharacteristic
234C2×Q16—1462no—
235C22×C8—732no—
236C22×C8—1461no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—1461no—
242C22⋊Q8—1461no—
243C22⋊Q8—1461no—
244C22⋊Q8—1461no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—731no—
253C2×C4⋊C4—1461no—
254C2×C4⋊C4—1461no—
255C2×C4⋊C4C22⋊D7312yes—
256C2×C4⋊C4C22⋊D7311yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1461no—
259C2×C22⋊C4—1461no—
260C2×C22⋊C4C22⋊D7312yes—
261C2×C42—21no—
262#107—1464no—
263#107—1462no—
264#103—1464no—
265#103—1462no—
266#103—1462no—
267C22⋊C8—732no—
268C22⋊C8—1461no—
269C22⋊C8—1461no—
270C22⋊C8—1461no—
271#95—24no—
272#95—21no—
273#95—1461no—
274C24C2×C22⋊D7311yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—2922no—
279C2×Q8—1461no—
280C2×Q8—1461no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—2921no—
290C22×C4—1461no—
291C22×C4—1461no—
292C22×C4D4×D7311yescharacteristic
293C22×C4D4×D7311yescharacteristic
294C22×C4D4×D7311yescharacteristic
295C22×C4C2×C22⋊D7311yescharacteristic
296C22×C4C2×C22⋊D7311yescharacteristic
297C22×C4#1192402711yescharacteristic
298Q16—5842no—
299C2×C8—2922no—
300C2×C8—2922no—
301C2×C8—1462no—
302C2×C8—1462no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—84no—
312C4⋊C4—2922no—
313C4⋊C4—2921no—
314C4⋊C4—2921no—
315C4⋊C4—1461no—
316C4⋊C4—1461no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—2921no—
321C22⋊C4—2921no—
322C22⋊C4—2922no—
323C22⋊C4#1192401814yescomplemented
324C42—42no—
325C23—41no—
326C23—41no—
327C23#1455656011yescharacteristic
328C23#1455656211yescharacteristic
329C23#1455658711yescharacteristic
330C23C2×#1192401812yes—
331C23#1455662212yes—
332Q8—42no—
333Q8—81no—
334Q8—2922no—
335Q8—2922no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—2921no—
354C2×C4—2921no—
355C2×C4—5841no—
356C2×C4—5841no—
357C2×C4—5841no—
358C2×C4—5841no—
359C2×C4—5841no—
360C2×C4#1455655911yescharacteristic
361C2×C4#1455655911yescharacteristic
362C2×C4#1455656311yescharacteristic
363C2×C4#1455656311yescharacteristic
364C8—2922no—
365C8—2922no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1798849511yescharacteristic
375C22#1798874312yes—
376C22#1798879412yes—
377C22#1798896611yescharacteristic
378C22#1798897911yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—44no—
383C4—82no—
384C4—81no—
385C4—81no—
386C4—5841no—
387C4—5841no—
388C4—11681no—
389C4#1798877612yes—
390C2—22no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2#2261098411yescharacteristic
395C2#2261103111yescharacteristic
396C2#2261105911yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#2502C2
socleC22×C73#392
prime radicalC24×C73C2×D4
hypercenter#2502D73
hypocenterC73#9959
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
124. C22⋊C4×C73Q16semidirect
206. C73#9959semidirect
323. C22⋊C4#11924018semidirect

Maximal subgroups

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

subgroupindex
#226207032
C2×#179889222
#226109162
#226100212
#226085032
#226077792
C73×#25022
#995973