Finite Group Databasebeta

← #32983835

Subgroups — #32983835

Order 18688.

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

normal subgroups only

This group has 542 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#22619011C211yesmaximal, characteristic, complemented
2#22614840C211yesmaximal, characteristic
3#22609924C211yesmaximal, characteristic, complemented
4#22608356C211yesmaximal, characteristic
5#22607791C211yesmaximal, characteristic, complemented
6#22607790C211yesmaximal, characteristic, complemented
7C73×#1486C211yesmaximal, characteristic
8C2×#14556570—21no—
9C2×C4×Dic146C2211yescharacteristic
10C8×Dic146—22no—
11#17988711—22no—
12#17988640C2211yescharacteristic
13#17988637—22no—
14#17988611C2211yescharacteristic
15C2×#14556509—21no—
16#17988518C412yes—
17#17988501C412yes—
18#17988475—22no—
19#17988453C2211yescharacteristic
20#17988447—21no—
21#17988355—21no—
22D4⋊#10596831—22no—
23C2×C4×D4×C73C2211yescharacteristic
24#431×C73C2211yescharacteristic
25#423×C73C2211yescharacteristic
26D4⋊C8×C73—22no—
27C22×Dic146D411yescharacteristic
28C2×#11924012—21no—
29#14556570—42no—
30#14556570—42no—
31C4×Dic146—21no—
32C4×Dic146—21no—
33C4×Dic146—21no—
34C4×Dic146—21no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73D411yescharacteristic
37C2×#11924007—22no—
38C2×#11924007C2×C412yes—
39C2×C22⋊Dic73—22no—
40C2×C22⋊Dic73—22no—
41C2×C4×Dic73D412yes—
42C4⋊#10596831—21no—
43C4⋊#10596831—21no—
44C4⋊#10596831—21no—
45#14556520—24no—
46#14556509—42no—
47C22⋊#10596831—21no—
48C4×#10596831D412yes—
49C8×Dic73—24no—
50#14556498—22no—
51#14556498—21no—
52#14556498C2×C412yes—
53C22×D4×C73C2×C411yescharacteristic
54C23×C4×C73—21no—
55C2×M16×C73—21no—
56C4×D4×C73—21no—
57C4×D4×C73—21no—
58C4×D4×C73—21no—
59C4×D4×C73—21no—
60C2×C4⋊C4×C73C2×C411yescharacteristic
61C2×C22⋊C4×C73—21no—
62C2×C42×C73C2311yescharacteristic
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4⋊C8×C73—21no—
66C22⋊C8×C73—21no—
67C4×C8×C73D412yes—
68C2×Dic146—21no—
69C2×Dic146—21no—
70C2×Dic146—21no—
71C2×Dic146—21no—
72C2×Dic146D812yes—
73C2×Dic146SD1612yes—
74C22×Dic73—22no—
75C22×Dic73C22⋊C412yes—
76C22×Dic73C22⋊C412yes—
77#11924012—41no—
78#11924012—41no—
79C2×#10596831—22no—
80C2×#10596831—41no—
81C4⋊Dic73—21no—
82C4⋊Dic73—21no—
83C4⋊Dic73—41no—
84C4⋊Dic73—41no—
85#11924007—22no—
86#11924007—22no—
87#11924007—44no—
88C22⋊Dic73—44no—
89C22⋊Dic73—44no—
90C4×Dic73—22no—
91C4×Dic73—22no—
92C24×C73—21no—
93C2×D4×C73—21no—
94C2×D4×C73—21no—
95C2×D4×C73—42no—
96C22×C4×C73—21no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73—41no—
103C22×C4×C73C22×C411yescharacteristic
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106M16×C73—41no—
107M16×C73—41no—
108C2×C8×C73—22no—
109C2×C8×C73—41no—
110C4⋊C4×C73—21no—
111C4⋊C4×C73—21no—
112C22⋊C4×C73—41no—
113C22⋊C4×C73—41no—
114C42×C73C2×D411yescharacteristic
115C42×C73C2×D411yescharacteristic
116C42×C73#4111yescharacteristic
117C42×C73#4111yescharacteristic
118C23×C73—22no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73C2×C22⋊C411yescharacteristic
126D4×C73—22no—
127D4×C73—22no—
128D4×C73—41no—
129D4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—81no—
136C2×C4×C73—81no—
137C2×C4×C73—81no—
138C2×C4×C73—81no—
139C2×C4×C73C4×D411yescharacteristic
140C2×C4×C73C4×D411yescharacteristic
141C2×C4×C73#12011yescharacteristic
142C2×C4×C73#12111yescharacteristic
143C2×C4×C73#12111yescharacteristic
144C2×C4×C73#12311yescharacteristic
145C2×C4×C73C2×D811yescharacteristic
146C2×C4×C73C2×SD1611yescharacteristic
147C2×C4×C73#13511yescharacteristic
148C2×C4×C73#13511yescharacteristic
149C2×Dic73—22no—
150C2×Dic73—22no—
151C2×Dic73—22no—
152C2×Dic73—42no—
153C2×Dic73—42no—
154C2×Dic73—42no—
155C2×Dic73—42no—
156C2×Dic73#10214yes—
157Dic146—22no—
158Dic146—22no—
159Dic146—22no—
160Dic146—22no—
161C8×C73—42no—
162C8×C73—81no—
163#10596831—42no—
164#10596831—81no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—42no—
168C22×C73—42no—
169C22×C73—42no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73#38611yescharacteristic
175C22×C73C2×#10211yescharacteristic
176C22×C73#41611yescharacteristic
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—81no—
182C4×C73#46511yescharacteristic
183C4×C73#46511yescharacteristic
184C4×C73#46611yescharacteristic
185C4×C73#46611yescharacteristic
186Dic73—24no—
187Dic73—42no—
188Dic73—82no—
189#9486—731nomaximal, Sylow
190C2×C73—21no—
191C2×C73—21no—
192C2×C73—82no—
193C2×C73#189111yescharacteristic
194C2×C73#189311yescharacteristic
195C2×C73#191211yescharacteristic
196#2982—731no—
197#2536—731no—
198#1801—731no—
199#1486—731no—
200#1486D7311yescharacteristic
201#1286—731no—
202#1286—731no—
203C73#948611yescharacteristic, complemented, Sylow
204C2×C22⋊Q8—1461no—
205C2×C4×Q8—731no—
206C2×C4×D4C2×D7311yescharacteristic
207C8×Q8—1462no—
208C8×Q8—1462no—
209#431—731no—
210#431—1462no—
211#431C2×D7311yescharacteristic
212#423—731no—
213#423C2×D7311yescharacteristic
214C2×#101—1461no—
215#398—732no—
216#394—732no—
217#388—1462no—
218#382—731no—
219#380—1461no—
220#343—1461no—
221D4⋊C8—22no—
222D4⋊C8—1462no—
223C22×Q8—731no—
224C22×D4C4×D7311yescharacteristic
225C23×C4—21no—
226C2×M16—21no—
227C2×M16—1461no—
228C22⋊Q8—2922no—
229C22⋊Q8—2922no—
230C4×Q8—1461no—
231C4×Q8—1461no—
232C4×Q8—1461no—
233C4×Q8—1461no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C2×C4⋊C4—732no—
239C2×C4⋊C4—731no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1461no—
242C2×C4⋊C4C4×D7311yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—1462no—
245C2×C22⋊C4—1462no—
246C2×C42—732no—
247C2×C42C22×D7311yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—1464no—
252C4⋊C8—1461no—
253C4⋊C8—1461no—
254C4⋊C8—1461no—
255#101—2922no—
256C22⋊C8—21no—
257C22⋊C8—1461no—
258C4×C8—732no—
259C4×C8—1464no—
260C4×C8C22⋊D7312yes—
261#95—732no—
262#95—1462no—
263#95—1461no—
264C24—21no—
265C2×Q8—732no—
266C2×Q8—732no—
267C2×Q8—1461no—
268C2×Q8—1461no—
269C2×Q8—1461no—
270C2×Q8—1461no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—42no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—732no—
282C22×C4—732no—
283C22×C4—1462no—
284C22×C4C2×C4×D7311yescharacteristic
285C22×C4D4×D7311yescharacteristic
286C22×C4D4×D7311yescharacteristic
287M16—41no—
288M16—41no—
289M16—2921no—
290M16—2921no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—2921no—
294C2×C8—1462no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—2924no—
298C4⋊C4—2921no—
299C4⋊C4—2921no—
300C4⋊C4—1462no—
301C4⋊C4—1462no—
302C4⋊C4—1461no—
303C4⋊C4—1461no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—2924no—
307C22⋊C4—2924no—
308C42—1462no—
309C42—1462no—
310C42D4×D7311yescharacteristic
311C42C2×C22⋊D7311yescharacteristic
312C42#1192402711yescharacteristic
313C42#1192402811yescharacteristic
314C23—22no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C22⋊C4×D7311yescharacteristic
322Q8—1462no—
323Q8—1462no—
324Q8—1462no—
325Q8—1462no—
326D4—22no—
327D4—22no—
328D4—41no—
329D4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—2922no—
340C2×C4—2922no—
341C2×C4—2922no—
342C2×C4—2922no—
343C2×C4—734no—
344C2×C4—1462no—
345C2×C4—1462no—
346C2×C4—1462no—
347C2×C4#1455655111yescharacteristic
348C2×C4C4×C22⋊D7311yescharacteristic
349C2×C4#1455655911yescharacteristic
350C2×C4#1455656211yescharacteristic
351C2×C4#1455656411yescharacteristic
352C2×C4#1455657311yescharacteristic
353C2×C4D8×D7311yescharacteristic
354C2×C4SD16×D7311yescharacteristic
355C2×C4#1455661011yescharacteristic
356C2×C4#1455661211yescharacteristic
357C8—42no—
358C8—81no—
359C8—2922no—
360C8—5841no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1798846311yescharacteristic
371C22#102×D7311yescharacteristic
372C22#1798858011yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—2922no—
379C4—5842no—
380C4—1464no—
381C4#1798881311yescharacteristic
382C4#1798881511yescharacteristic
383C4#1798882011yescharacteristic
384C4#1798882211yescharacteristic
385C2—21no—
386C2—21no—
387C2—82no—
388C2#2261034611yescharacteristic
389C2#2261035611yescharacteristic
390C2#2261046511yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1486C2
socleC22×C73#386
prime radicalC22×D4×C73C2×C4
cosocleC2×C42×C73C23
hypercenter#1486D73
hypocenterC73#9486
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #22619011C2semidirect
3. #22609924C2semidirect
5. #22607791C2semidirect
6. #22607790C2semidirect
203. C73#9486semidirect

Maximal subgroups

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

subgroupindex
#226190112
#226148402
#226099242
#226083562
#226077912
#226077902
C73×#14862
#948673