Finite Group Databasebeta

← #32997024

Subgroups — #32997024

Order 18688.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#22614836C211yesmaximal, characteristic
2#22608255C211yesmaximal, characteristic, complemented
3#22607936C212yesmaximal, complemented
4#22607933C212yesmaximal
5C73×#1801C211yesmaximal, characteristic, complemented
6C2×#14556567—21no—
7C2×C4×D292C411yescharacteristic
8#17988640—21no—
9#17988517C411yescharacteristic
10#17988517C2211yescharacteristic
11#17988504C411yescharacteristic
12#17988504C2211yescharacteristic
13#17988478—21no—
14#17988478—21no—
15#17988453C411yescharacteristic
16#17988444—21no—
17#17988313—21no—
18#17988313C2212yes—
19C2×C4×Q8×C73C2211yescharacteristic
20#423×C73—21no—
21C2×#101×C73—21no—
22#343×C73C2212yes—
23C22×D292C2×C411yescharacteristic
24C22×C4×D73—21no—
25#14556567—42no—
26#14556567—42no—
27C4×D292—41no—
28C4×D292—41no—
29C2×C4⋊Dic73—21no—
30C2×C4⋊Dic73C2×C411yescharacteristic
31C2×#11924007—21no—
32C2×#11924007—21no—
33C2×#11924005—21no—
34C2×#11924005—21no—
35C2×#11924005—21no—
36C2×#11924005—21no—
37C2×#11924005—21no—
38C4⋊#10596831—41no—
39C22⋊#10596831—22no—
40C4×#10596831—41no—
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498C2×C411yescharacteristic
45#14556498C2×C411yescharacteristic
46C22×Q8×C73D411yescharacteristic
47C2×M16×C73—22no—
48C4×Q8×C73—41no—
49C4×Q8×C73—41no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73D411yescharacteristic
52C2×C4⋊C4×C73C2311yescharacteristic
53C2×C42×C73C2×C411yescharacteristic
54C2×C42×C73D411yescharacteristic
55C2×C42×C73D411yescharacteristic
56C4⋊C8×C73—41no—
57C4⋊C8×C73—41no—
58#101×C73—42no—
59C23×D73—21no—
60C2×D292—41no—
61C2×D292—42no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73—21no—
65C2×C4×D73—42no—
66C2×C4×D73—42no—
67C2×C4×D73—42no—
68C2×#10596831—42no—
69C4⋊Dic73—42no—
70C4⋊Dic73—41no—
71#11924007—42no—
72#11924007—41no—
73#11924007—41no—
74#11924005—42no—
75#11924005—42no—
76#11924005—41no—
77#11924005—41no—
78#11924005—42no—
79#11924005—41no—
80#11924005—41no—
81C2×Q8×C73—41no—
82C2×Q8×C73—41no—
83C2×Q8×C73D812yes—
84C2×Q8×C73SD1612yes—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22×C411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92M16×C73—42no—
93M16×C73—42no—
94C2×C8×C73—42no—
95C4⋊C4×C73—41no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C42×C73—21no—
99C42×C73—21no—
100C42×C73—21no—
101C42×C73—21no—
102C42×C73—21no—
103C42×C73—21no—
104C23×C73C2×C22⋊C411yescharacteristic
105C22×D73—22no—
106C22×D73—41no—
107C22×D73—41no—
108C22×D73—41no—
109C22×D73—41no—
110C22×D73—41no—
111C22×D73—41no—
112Q8×C73—42no—
113Q8×C73—42no—
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—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73#9911yescharacteristic
128C2×C4×C73#9911yescharacteristic
129C2×C4×C73#10212yes—
130C2×C4×C73#10212yes—
131C2×C4×C73C2×D811yescharacteristic
132C2×C4×C73C2×SD1611yescharacteristic
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142D292—42no—
143D292—42no—
144C4×D73—84no—
145C8×C73—82no—
146#10596831—82no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73C2×#9911yescharacteristic
150C22×C73C2×#10211yescharacteristic
151C22×C73#42111yescharacteristic
152C2×D73—42no—
153C2×D73—42no—
154C2×D73—42no—
155C2×D73—81no—
156C2×D73—81no—
157C2×D73—81no—
158C2×D73—81no—
159C4×C73—21no—
160C4×C73—21no—
161C4×C73—21no—
162C4×C73—22no—
163C4×C73—22no—
164C4×C73—21no—
165C4×C73—41no—
166C4×C73—41no—
167C4×C73—41no—
168Dic73—81no—
169Dic73—81no—
170Dic73—81no—
171#11879—731nomaximal, Sylow
172C2×C73—21no—
173C2×C73—21no—
174C2×C73#151911yescharacteristic
175C2×C73#212011yescharacteristic
176C2×C73#213311yescharacteristic
177D73—82no—
178#2536—731no—
179#1801D7311yescharacteristic, complemented
180#1461—731no—
181#1346—732no—
182#1345—732no—
183C73#1187911yescharacteristic, complemented, Sylow
184C2×C22⋊Q8—1461no—
185C2×C4×Q8C2×D7311yescharacteristic
186C2×C4×D4—731no—
187#431—1461no—
188#423—21no—
189C2×#101—21no—
190#398—731no—
191#398—731no—
192#394—731no—
193#394—731no—
194#388—1461no—
195#388—1461no—
196#382—731no—
197#380—1461no—
198#343C2×D7312yes—
199#324—732no—
200#324—1461no—
201C22×Q8C22⋊D7311yescharacteristic
202C22×D4—731no—
203C23×C4—1461no—
204C2×M16—22no—
205C22⋊Q8—2922no—
206C22⋊Q8—2922no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—2921no—
210C4×D4—2921no—
211C2×C4⋊C4—731no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4—1461no—
214C2×C4⋊C4—1461no—
215C2×C4⋊C4C4×D7311yescharacteristic
216C2×C4⋊C4D29211yescharacteristic
217C2×C4⋊C4C22×D7311yescharacteristic
218C2×C22⋊C4—1461no—
219C2×C22⋊C4—1461no—
220C2×C22⋊C4—1461no—
221C2×C22⋊C4—1461no—
222C2×C22⋊C4—1461no—
223C2×C42C4×D7311yescharacteristic
224C2×C42D29211yescharacteristic
225C2×C42C22⋊D7311yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—41no—
228C4⋊C8—2921no—
229#101—42no—
230C22⋊C8—1462no—
231C4×C8—2921no—
232#95—731no—
233#95—731no—
234#95—1461no—
235#95—1461no—
236#95—1461no—
237C24—1461no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#1192401312yes—
241C2×Q8#1192401512yes—
242C2×D4—2921no—
243C2×D4—2922no—
244C22×C4—2922no—
245C22×C4—2922no—
246C22×C4—2922no—
247C22×C4—1461no—
248C22×C4—1461no—
249C22×C4—1461no—
250C22×C4#1192400511yescharacteristic
251C22×C4#1192400511yescharacteristic
252C22×C4#1192400511yescharacteristic
253C22×C4#1192400511yescharacteristic
254C22×C4C2×C4×D7311yescharacteristic
255C22×C4C2×C22⋊D7311yescharacteristic
256C22×C4C2×D29211yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—2922no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2921no—
265C4⋊C4—2922no—
266C4⋊C4—2922no—
267C4⋊C4—2921no—
268C4⋊C4—2921no—
269C22⋊C4—2922no—
270C22⋊C4—2922no—
271C22⋊C4—2921no—
272C22⋊C4—2921no—
273C22⋊C4—2922no—
274C22⋊C4—2921no—
275C22⋊C4—2921no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—2921no—
283C23—2921no—
284C23—2921no—
285C23—2921no—
286C23—2921no—
287C23—2921no—
288C23—1462no—
289C23C2×#1192400511yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—2922no—
293D4—2922no—
294C2×C4—21no—
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—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—2921no—
308C2×C4—2921no—
309C2×C4—2921no—
310C2×C4—2921no—
311C2×C4—2921no—
312C2×C4—2921no—
313C2×C4—2921no—
314C2×C4—2921no—
315C2×C4—2921no—
316C2×C4—5844no—
317C2×C4#1455650511yescharacteristic
318C2×C4#1455650511yescharacteristic
319C2×C4#1455651112yes—
320C2×C4#1455651112yes—
321C2×C4C2×#1192401311yescharacteristic
322C2×C4C2×#1192401511yescharacteristic
323C8—82no—
324C8—5842no—
325C22—21no—
326C22—21no—
327C22—2922no—
328C22—2922no—
329C22—2922no—
330C22—5841no—
331C22—5841no—
332C22—5841no—
333C22—5841no—
334C22C2×#1455650511yescharacteristic
335C22C2×#1455651111yescharacteristic
336C22#1798860511yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—5841no—
347C4—5841no—
348C4—5841no—
349C2—21no—
350C2—21no—
351C2—5842no—
352C2#2260849811yescharacteristic
353C2#2261157311yescharacteristic
354C2#2261161811yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1801C2
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1801D73
hypocenterC73#11879
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #22608255C2semidirect
3. #22607936C2semidirect
5. C73×#1801C2semidirect
179. #1801D73semidirect
183. C73#11879semidirect

Maximal subgroups

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

subgroupindex
#226148362
#226082552
#226079362
#226079332
C73×#18012
#1187973