Finite Group Databasebeta

← #33002239

Subgroups — #33002239

Order 18688.

670 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 99 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#22612422C211yesmaximal, characteristic
2#22609849C211yesmaximal, characteristic
3#22608055C214yesmaximal, complemented
4C73×#1782C211yesmaximal, characteristic
5C2×#14556550C411yescharacteristic, complemented
6C22×C4×Dic73C411yescharacteristic, complemented
7C2×#14556506—21no—
8C2×#14556506C2212yes—
9#17988494C412yescomplemented
10#17988490C2212yes—
11#17988467—21no—
12#17988467—21no—
13#17988450—21no—
14C4×C22⋊Dic73—21no—
15C4×C22⋊Dic73—21no—
16C22×#41×C73C2211yescharacteristic
17C2×#99×C73—21no—
18C2×#99×C73C2212yes—
19#381×C73—21no—
20C23×Dic73—21no—
21#14556550—42no—
22C2×C4⋊Dic73C2×C412yes—
23C2×C22⋊Dic73—21no—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26C2×C22⋊Dic73—22no—
27C2×C4×Dic73—21no—
28C2×C4×Dic73—21no—
29C2×C4×Dic73—42no—
30C2×C4×Dic73C2×C412yes—
31#14556506—24no—
32#14556506—42no—
33#14556498—22no—
34#14556498—22no—
35#14556498—42no—
36C2×#41×C73—24no—
37C2×#41×C73—42no—
38C2×#41×C73—42no—
39C22×Q8×C73C2×C411yescharacteristic
40C22×D4×C73D411yescharacteristic
41C22×D4×C73D411yescharacteristic
42C22×D4×C73C2311yescharacteristic
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73D411yescharacteristic
45C23×C4×C73D411yescharacteristic
46C2×C22⋊C4×C73—22no—
47#99×C73—24no—
48#99×C73—42no—
49#95×C73—42no—
50C22×Dic73—22no—
51C22×Dic73—22no—
52C22×Dic73—44no—
53C22×Dic73—42no—
54C22×Dic73—42no—
55C4⋊Dic73—42no—
56C22⋊Dic73—44no—
57C22⋊Dic73—44no—
58C22⋊Dic73—42no—
59C22⋊Dic73—42no—
60C22⋊Dic73—42no—
61C4×Dic73—42no—
62C4×Dic73—42no—
63C4×Dic73—42no—
64C24×C73C22⋊C411yescharacteristic
65C24×C73C2×D411yescharacteristic
66C24×C73C2×D411yescharacteristic
67#41×C73—416no—
68C2×Q8×C73—22no—
69C2×Q8×C73—41no—
70C2×Q8×C73—41no—
71C2×D4×C73—22no—
72C2×D4×C73—22no—
73C2×D4×C73—22no—
74C2×D4×C73—22no—
75C2×D4×C73—41no—
76C2×D4×C73—41no—
77C2×D4×C73—41no—
78C2×D4×C73—41no—
79C2×D4×C73#4114yes—
80C22×C4×C73—22no—
81C22×C4×C73—22no—
82C22×C4×C73—22no—
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—41no—
88C22×C4×C73—41no—
89C22×C4×C73—42no—
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22⋊C411yescharacteristic
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22×C411yescharacteristic
94C22⋊C4×C73—42no—
95C22⋊C4×C73—42no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—42no—
103C23×C73—42no—
104C23×C73—42no—
105C23×C73#9912yes—
106C23×C73C2×C22⋊C411yescharacteristic
107C23×C73#12312yes—
108C23×C73#12312yes—
109Q8×C73—44no—
110D4×C73—44no—
111D4×C73—44no—
112D4×C73—44no—
113C2×C4×C73—28no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—44no—
125C2×C4×C73—44no—
126C2×C4×C73—82no—
127C2×C4×C73—84no—
128C2×C4×C73#9912yes—
129C2×C4×C73#11712yes—
130C2×Dic73—24no—
131C2×Dic73—42no—
132C2×Dic73—48no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—82no—
138C2×Dic73—84no—
139C22×C73—22no—
140C22×C73—22no—
141C22×C73—22no—
142C22×C73—22no—
143C22×C73—42no—
144C22×C73—42no—
145C22×C73—42no—
146C22×C73—42no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73—42no—
150C22×C73—81no—
151C22×C73—81no—
152C22×C73#38111yescharacteristic
153C22×C73C2×#9911yescharacteristic
154C22×C73C2×#9911yescharacteristic
155C4×C73—22no—
156C4×C73—22no—
157C4×C73—41no—
158C4×C73—41no—
159C4×C73—162no—
160Dic73—44no—
161Dic73—82no—
162Dic73—162no—
163#12857—731nomaximal, Sylow
164C2×C73—21no—
165C2×C73—21no—
166C2×C73—42no—
167C2×C73—42no—
168C2×C73—42no—
169C2×C73#181911yescharacteristic
170C2×C73#212612yes—
171#2297—731no—
172#1782—731no—
173#1782D7311yescharacteristic
174#1393—734no—
175C73#1285711yescharacteristic, complemented, Sylow
176C22×#41C2×D7311yescharacteristic
177C2×#117—731no—
178C22×C42—731no—
179C2×#99—21no—
180C2×#99—732no—
181C2×#99—1461no—
182C2×#99C2×D7312yes—
183#391—732no—
184#390—732no—
185#386—1461no—
186#386—1461no—
187#381—21no—
188#381—1461no—
189C4×C22⋊C4—1461no—
190C4×C22⋊C4—1461no—
191C2×#41—24no—
192C2×#41—42no—
193C2×#41—42no—
194C22×Q8C4×D7311yescharacteristic
195C22×D4D29211yescharacteristic
196C22×D4C22⋊D7311yescharacteristic
197C22×D4C22×D7311yescharacteristic
198C23×C4—1461no—
199C23×C4C4×D7311yescharacteristic
200C23×C4D29211yescharacteristic
201C23×C4C22⋊D7311yescharacteristic
202#117—2922no—
203C2×C4⋊C4—732no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—1462no—
206C2×C22⋊C4—1461no—
207C2×C22⋊C4—1462no—
208C2×C22⋊C4—1462no—
209C2×C42—2922no—
210C2×C42—732no—
211C2×C42—1461no—
212C2×C42—1461no—
213#99—24no—
214#99—42no—
215#99—2922no—
216#99—1464no—
217#95—42no—
218#95—2922no—
219#95—1462no—
220#95—1462no—
221C24#1192400511yescharacteristic
222C24C2×C22⋊D7311yescharacteristic
223C24C2×D29211yescharacteristic
224#41—416no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4#1192402714yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—2922no—
248C22×C4—2924no—
249C22×C4—2922no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4#1192400511yescharacteristic
253C22×C4#1192400511yescharacteristic
254C22×C4#1192400511yescharacteristic
255C22×C4C2×C4×D7311yescharacteristic
256C4⋊C4—2922no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—2922no—
260C22⋊C4—2924no—
261C22⋊C4—2924no—
262C22⋊C4—2922no—
263C22⋊C4—2922no—
264C42—2922no—
265C42—2922no—
266C42—2922no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23#1455650512yes—
277C23C2×#1192400511yescharacteristic
278C23#1455657212yes—
279C23#1455657512yes—
280Q8—44no—
281D4—44no—
282D4—44no—
283D4—44no—
284C2×C4—28no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—2922no—
300C2×C4—2928no—
301C2×C4—2922no—
302C2×C4—2922no—
303C2×C4—2922no—
304C2×C4—2922no—
305C2×C4—5842no—
306C2×C4—5844no—
307C2×C4—1464no—
308C2×C4#1455650512yes—
309C2×C4#1455654712yes—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—81no—
322C22—81no—
323C22#1798844811yescharacteristic
324C22C2×#1455650511yescharacteristic
325C22C2×#1455650511yescharacteristic
326C4—22no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—162no—
331C4—2924no—
332C4—5842no—
333C4—11682no—
334C2—21no—
335C2—21no—
336C2—42no—
337C2—42no—
338C2—42no—
339C2#2261000311yescharacteristic
340C2#2261159412yes—
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#1782C2
socleC22×C73C2×#99
prime radicalC22×#41×C73C22
cosocleC22×D4×C73C23
hypercenter#1782D73
hypocenterC73#12857
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #22608055C2semidirect
5. C2×#14556550C4semidirect
6. C22×C4×Dic73C4semidirect
9. #17988494C4semidirect
175. C73#12857semidirect

Maximal subgroups

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

subgroupindex
#226124222
#226098492
#226080552
C73×#17822
#1285773