Finite Group Databasebeta

← #33002235

Subgroups — #33002235

Order 18688.

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

normal subgroups only

This group has 628 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#22612523C211yesmaximal, characteristic
2#22609849C211yesmaximal, characteristic
3#22608050C214yesmaximal, complemented
4C73×#1785C211yesmaximal, characteristic
5C2×#14556550C411yescharacteristic, complemented
6C22×C4⋊Dic73C411yescharacteristic, complemented
7C2×#14556506—21no—
8C2×#14556506C2212yes—
9#17988494—21no—
10#17988490—21no—
11#17988467C412yescomplemented
12#17988467C2212yes—
13#17988450—21no—
14#17988447—21no—
15#17988447—21no—
16C22×#41×C73C2211yescharacteristic
17C2×#101×C73—21no—
18C2×#100×C73C2212yes—
19#407×C73—21no—
20C23×Dic73C2×C412yes—
21#14556550—42no—
22C2×C4⋊Dic73—21no—
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—42no—
26C2×C22⋊Dic73—22no—
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73—22no—
29C2×C22⋊Dic73C2×C412yes—
30C2×C4×Dic73—21no—
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×C73D411yescharacteristic
40C22×D4×C73C2×C411yescharacteristic
41C22×D4×C73D411yescharacteristic
42C22×D4×C73C2311yescharacteristic
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73D411yescharacteristic
45C23×C4×C73D411yescharacteristic
46C2×M16×C73—22no—
47#101×C73—42no—
48#100×C73—24no—
49C22⋊C8×C73—42no—
50C22×Dic73—22no—
51C22×Dic73—22no—
52C22×Dic73—42no—
53C22×Dic73—42no—
54C22×Dic73—82no—
55C4⋊Dic73—42no—
56C4⋊Dic73—42no—
57C4⋊Dic73—42no—
58C22⋊Dic73—42no—
59C22⋊Dic73—42no—
60C22⋊Dic73—42no—
61C22⋊Dic73—82no—
62C22⋊Dic73—82no—
63C4×Dic73—42no—
64C24×C73C22⋊C411yescharacteristic
65C24×C73C22×C411yescharacteristic
66C24×C73C2×D411yescharacteristic
67#41×C73—416no—
68C2×Q8×C73—22no—
69C2×Q8×C73—22no—
70C2×Q8×C73—41no—
71C2×D4×C73—22no—
72C2×D4×C73—22no—
73C2×D4×C73—22no—
74C2×D4×C73—41no—
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×C73C22⋊C411yescharacteristic
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22⋊C411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93M16×C73—42no—
94M16×C73—42no—
95C2×C8×C73—42no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73C2×C22⋊C411yescharacteristic
110C23×C73#11712yes—
111C23×C73#12312yes—
112Q8×C73—44no—
113D4×C73—44no—
114D4×C73—44no—
115D4×C73—44no—
116C2×C4×C73—28no—
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—21no—
125C2×C4×C73—21no—
126C2×C4×C73—44no—
127C2×C4×C73—44no—
128C2×C4×C73#9912yes—
129C2×C4×C73#9912yes—
130C2×C4×C73#12312yes—
131C2×Dic73—42no—
132C2×Dic73—42no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—84no—
138C2×Dic73—82no—
139C2×Dic73—82no—
140C2×Dic73—82no—
141C8×C73—82no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—22no—
145C22×C73—22no—
146C22×C73—42no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73—42no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73—41no—
154C22×C73—41no—
155C22×C73—41no—
156C22×C73—41no—
157C22×C73—81no—
158C22×C73—81no—
159C22×C73#38111yescharacteristic
160C22×C73C2×#9911yescharacteristic
161C22×C73C2×#9911yescharacteristic
162C4×C73—22no—
163C4×C73—22no—
164C4×C73—22no—
165C4×C73—41no—
166Dic73—82no—
167Dic73—82no—
168Dic73—162no—
169#12855—731nomaximal, Sylow
170C2×C73—21no—
171C2×C73—21no—
172C2×C73—42no—
173C2×C73—42no—
174C2×C73—81no—
175C2×C73#181911yescharacteristic
176C2×C73#212211yescharacteristic
177C2×C73#212311yescharacteristic
178#2311—731no—
179#1785D7311yescharacteristic
180#1782—731no—
181#1390—734no—
182C73#1285511yescharacteristic, complemented, Sylow
183C22×#41C2×D7311yescharacteristic
184C2×#117—731no—
185C22×C4⋊C4—731no—
186C2×#101—21no—
187C2×#100C2×D7312yes—
188C2×#99—732no—
189C2×#99—1461no—
190#407—21no—
191#391—1461no—
192#390—1461no—
193#386—732no—
194#386—732no—
195#381—1461no—
196#380—1461no—
197#380—1461no—
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—42no—
201C22×Q8C22⋊D7311yescharacteristic
202C22×D4C4×D7311yescharacteristic
203C22×D4D29211yescharacteristic
204C22×D4C22×D7311yescharacteristic
205C23×C4—732no—
206C23×C4C4×D7311yescharacteristic
207C23×C4D29211yescharacteristic
208C23×C4C22⋊D7311yescharacteristic
209C2×M16—22no—
210#117—2922no—
211C2×C4⋊C4—2922no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4—1461no—
214C2×C4⋊C4—1461no—
215C2×C22⋊C4—732no—
216C2×C22⋊C4—1462no—
217C2×C22⋊C4—1462no—
218C2×C22⋊C4—1462no—
219C2×C42—1461no—
220#101—42no—
221#100—24no—
222#99—2922no—
223#99—1464no—
224C22⋊C8—42no—
225#95—2922no—
226#95—1462no—
227#95—1462no—
228C24#1192400511yescharacteristic
229C24C2×C4×D7311yescharacteristic
230C24C2×D29211yescharacteristic
231#41—416no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#1192402714yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—2922no—
254C22×C4—2922no—
255C22×C4—5842no—
256C22×C4—1462no—
257C22×C4—1462no—
258C22×C4#1192400511yescharacteristic
259C22×C4#1192400511yescharacteristic
260C22×C4#1192400511yescharacteristic
261C22×C4C2×C22⋊D7311yescharacteristic
262M16—42no—
263M16—42no—
264C2×C8—42no—
265C4⋊C4—2922no—
266C4⋊C4—2922no—
267C4⋊C4—2922no—
268C22⋊C4—2922no—
269C22⋊C4—2922no—
270C22⋊C4—2922no—
271C22⋊C4—5842no—
272C22⋊C4—5842no—
273C42—2922no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×#1192400511yescharacteristic
288C23#1455654712yes—
289C23#1455657512yes—
290Q8—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—2922no—
307C2×C4—2922no—
308C2×C4—2922no—
309C2×C4—2922no—
310C2×C4—2922no—
311C2×C4—2922no—
312C2×C4—5844no—
313C2×C4—5842no—
314C2×C4—5842no—
315C2×C4—5842no—
316C2×C4#1455650512yes—
317C2×C4#1455650512yes—
318C2×C4#1455657212yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1798844811yescharacteristic
338C22C2×#1455650511yescharacteristic
339C22C2×#1455650511yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—5842no—
345C4—5842no—
346C4—11682no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2#2261000311yescharacteristic
353C2#2261158211yescharacteristic
354C2#2261158611yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC24×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#1785C2
socleC22×C73C2×#99
prime radicalC22×#41×C73C22
cosocleC22×D4×C73C23
hypercenter#1785D73
hypocenterC73#12855
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #22608050C2semidirect
5. C2×#14556550C4semidirect
6. C22×C4⋊Dic73C4semidirect
11. #17988467C4semidirect
182. C73#12855semidirect

Maximal subgroups

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

subgroupindex
#226125232
#226098492
#226080502
C73×#17852
#1285573