Finite Group Databasebeta

← #33058478

Subgroups — #33058478

Order 18688.

930 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 183 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#22619875C212yesmaximal
2#22614218C211yesmaximal, characteristic
3#22608973C214yesmaximal, complemented
4#22608939C214yesmaximal, complemented
5#22608464C212yesmaximal
6#22608261C211yesmaximal, characteristic
7C2×#544×C73C211yesmaximal, characteristic
8C2×#14556550C2211yescharacteristic
9#17988925—24no—
10#17988920C2214yes—
11#17988915C2214yes—
12#17988627—22no—
13#17988596—22no—
14#17988551C2212yes—
15#17988526C2212yes—
16#17988517C2212yes—
17#17988468—22no—
18#17988460—21no—
19#17988450—21no—
20#17988450C2211yescharacteristic
21#17988324C2218yes—
22#17988313C2214yes—
23#544×C73—24no—
24#544×C73—22no—
25#544×C73C2214yes—
26C2×C4⋊Q8×C73—22no—
27C2×#125×C73—22no—
28C2×C22⋊Q8×C73—21no—
29C2×C22⋊Q8×C73C2212yes—
30C2×#117×C73—21no—
31C22×C4⋊C4×C73C2211yescharacteristic
32C2×#11924012—22no—
33#14556550—22no—
34#14556550—22no—
35C2×C4⋊Dic73C2311yescharacteristic
36C2×C4⋊Dic73C2311yescharacteristic
37C2×#11924007—22no—
38C2×C22⋊Dic73—21no—
39C2×C22⋊Dic73—21no—
40C2×C4×Dic73—21no—
41#14556524—24no—
42#14556522—24no—
43#14556515—24no—
44#14556515—24no—
45C22⋊#10596831—22no—
46C22⋊#10596831C2314yes—
47#14556498—22no—
48#14556498—22no—
49#14556498—21no—
50#14556498—22no—
51#14556498C2312yes—
52C22×Q8×C73—21no—
53C22×Q8×C73D412yes—
54C23×C4×C73D411yescharacteristic
55C23×C4×C73D411yescharacteristic
56C23×C4×C73C2311yescharacteristic
57C4⋊Q8×C73—28no—
58C4⋊Q8×C73—28no—
59#125×C73—28no—
60#125×C73—28no—
61C22⋊Q8×C73—24no—
62C22⋊Q8×C73—24no—
63C22⋊Q8×C73—22no—
64C22⋊Q8×C73—22no—
65C22⋊Q8×C73D414yes—
66C22⋊Q8×C73C2314yes—
67#117×C73—24no—
68#117×C73—24no—
69C2×C4⋊C4×C73—21no—
70C2×C4⋊C4×C73—21no—
71C2×C4⋊C4×C73—21no—
72C2×C4⋊C4×C73—22no—
73C2×C4⋊C4×C73—22no—
74C2×C4⋊C4×C73—22no—
75C2×C4⋊C4×C73—22no—
76C2×C4⋊C4×C73D412yes—
77C2×C4⋊C4×C73D412yes—
78C2×C4⋊C4×C73C2312yes—
79C2×C22⋊C4×C73—22no—
80C2×C42×C73—22no—
81C22×Dic73—22no—
82C22×Dic73—21no—
83C22×Dic73—21no—
84#11924012—44no—
85C2×#10596831—24no—
86C4⋊Dic73—22no—
87C4⋊Dic73—22no—
88#11924007—44no—
89C22⋊Dic73—42no—
90C22⋊Dic73—41no—
91C22⋊Dic73—41no—
92C4×Dic73—42no—
93C24×C73C2×D411yescharacteristic
94C2×Q8×C73—24no—
95C2×Q8×C73—22no—
96C2×Q8×C73—44no—
97C2×Q8×C73—82no—
98C2×Q8×C73C2×D414yes—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73—24no—
102C22×C4×C73—22no—
103C22×C4×C73—22no—
104C22×C4×C73—21no—
105C22×C4×C73—21no—
106C22×C4×C73—42no—
107C22×C4×C73—42no—
108C22×C4×C73C2×D412yes—
109C22×C4×C73C2×D412yes—
110C22×C4×C73C2×D412yes—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D412yes—
113C22×C4×C73C2411yescharacteristic
114C4⋊C4×C73—24no—
115C4⋊C4×C73—24no—
116C4⋊C4×C73—24no—
117C4⋊C4×C73—28no—
118C4⋊C4×C73—24no—
119C4⋊C4×C73—24no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—24no—
122C4⋊C4×C73—24no—
123C4⋊C4×C73C2×D414yes—
124C22⋊C4×C73—24no—
125C22⋊C4×C73—24no—
126C42×C73—28no—
127C23×C73—22no—
128C23×C73—22no—
129C23×C73—41no—
130C23×C73#12011yescharacteristic
131C23×C73#12011yescharacteristic
132C23×C73C22×D411yescharacteristic
133Q8×C73—44no—
134Q8×C73—82no—
135Q8×C73—84no—
136C2×C4×C73—24no—
137C2×C4×C73—24no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—24no—
141C2×C4×C73—24no—
142C2×C4×C73—24no—
143C2×C4×C73—22no—
144C2×C4×C73—42no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—44no—
148C2×C4×C73—42no—
149C2×C4×C73—44no—
150C2×C4×C73—42no—
151C2×C4×C73—81no—
152C2×C4×C73#12012yes—
153C2×C4×C73#13714yes—
154C2×C4×C73C22×D412yes—
155C2×Dic73—42no—
156C2×Dic73—42no—
157C2×Dic73—42no—
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—42no—
161C2×Dic73—42no—
162#10596831—44no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—44no—
171C22×C73#45812yes—
172C22×C73C2×#12011yescharacteristic
173C22×C73C2×#13712yes—
174C4×C73—24no—
175C4×C73—22no—
176C4×C73—22no—
177C4×C73—44no—
178C4×C73—42no—
179C4×C73—44no—
180Dic73—82no—
181Dic73—81no—
182Dic73—81no—
183#18148—731nomaximal, Sylow
184C2×C73—22no—
185C2×C73—22no—
186C2×C73—41no—
187C2×C73#300411yescharacteristic
188C2×C73#302011yescharacteristic
189C2×C73C2×#45811yescharacteristic
190C2×#544D7311yescharacteristic
191#3095—732no—
192#2484—731no—
193#1617—734no—
194#1611—734no—
195#1511—732no—
196#1463—731no—
197C73#1814811yescharacteristic, complemented, Sylow
198#544—24no—
199#544—22no—
200#544C2×D7314yes—
201C2×C4⋊Q8—22no—
202C2×#125—22no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8C2×D7312yes—
205C2×#117—21no—
206C2×#117—731no—
207C22×C4⋊C4C2×D7311yescharacteristic
208#485—1464no—
209#484—734no—
210#483—734no—
211#428—1462no—
212#419—1462no—
213#407—732no—
214#400—732no—
215#398—732no—
216#387—1462no—
217#385—1461no—
218#381—731no—
219#381—1461no—
220#328—738no—
221#324—734no—
222C22×Q8—21no—
223C22×Q8C22⋊D7312yes—
224C23×C4C22⋊D7311yescharacteristic
225C23×C4C22⋊D7311yescharacteristic
226C23×C4C22×D7311yescharacteristic
227C2×M16—1462no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230#125—28no—
231#125—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8C22⋊D7314yes—
237C22⋊Q8C22×D7314yes—
238#117—24no—
239#117—24no—
240#117—1462no—
241#117—1462no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—731no—
250C2×C4⋊C4—731no—
251C2×C4⋊C4—1462no—
252C2×C4⋊C4C22⋊D7312yes—
253C2×C4⋊C4C22⋊D7312yes—
254C2×C4⋊C4C22×D7312yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—1461no—
257C2×C22⋊C4—1461no—
258C2×C42—22no—
259C2×C42—1461no—
260#107—1464no—
261#106—1464no—
262#103—1464no—
263#103—1464no—
264C22⋊C8—734no—
265C22⋊C8—1462no—
266#95—732no—
267#95—1461no—
268#95—1462no—
269#95—1462no—
270#95—1462no—
271C24C2×C22⋊D7311yescharacteristic
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—82no—
276C2×Q8C2×C22⋊D7314yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—1462no—
287C22×C4—1461no—
288C22×C4—1461no—
289C22×C4C2×C22⋊D7312yes—
290C22×C4C2×C22⋊D7312yes—
291C22×C4C2×C22⋊D7312yes—
292C22×C4C2×C22⋊D7311yescharacteristic
293C22×C4C2×C22⋊D7312yes—
294C22×C4C23×D7311yescharacteristic
295M16—2924no—
296C2×C8—1464no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—2924no—
307C4⋊C4—1462no—
308C4⋊C4—1462no—
309C4⋊C4C2×C22⋊D7314yes—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—2922no—
313C22⋊C4—2921no—
314C22⋊C4—2921no—
315C42—28no—
316C42—2922no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23#1455656011yescharacteristic
321C23#1455656011yescharacteristic
322C23C22×C22⋊D7311yescharacteristic
323Q8—44no—
324Q8—82no—
325Q8—84no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—2922no—
343C2×C4—2922no—
344C2×C4—2922no—
345C2×C4—2922no—
346C2×C4—2922no—
347C2×C4—2921no—
348C2×C4—2921no—
349C2×C4#1455656012yes—
350C2×C4#1455662214yes—
351C2×C4C22×C22⋊D7312yes—
352C8—2924no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22#1798877712yes—
362C22C2×#1455656011yescharacteristic
363C22C2×#1455662212yes—
364C4—24no—
365C4—22no—
366C4—22no—
367C4—44no—
368C4—42no—
369C4—44no—
370C4—5842no—
371C4—5841no—
372C4—5841no—
373C2—22no—
374C2—22no—
375C2—41no—
376C2#2261916511yescharacteristic
377C2#2261929511yescharacteristic
378C2C2×#1798877711yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#544×C73C2
socleC22×C73C2×#120
prime radicalC24×C73C2×D4
hypercenterC2×#544D73
hypocenterC73#18148
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #22608973C2semidirect
4. #22608939C2semidirect
197. C73#18148semidirect

Maximal subgroups

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

subgroupindex
#226198752
#226142182
#226089732
#226089392
#226084642
#226082612
C2×#544×C732
#1814873