Finite Group Databasebeta

← #32996319

Subgroups — #32996319

Order 18688.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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#22613190C211yesmaximal, characteristic
2C4×#14556506C211yesmaximal, characteristic
3#22607956C212yesmaximal, complemented
4#22607932C212yesmaximal, complemented
5C73×#1460C211yesmaximal, characteristic
6C2×#14556550C411yescharacteristic, complemented
7C2×#14556506—21no—
8#17988526C411yescharacteristic, complemented
9#17988523C2211yescharacteristic
10#17988517C411yescharacteristic, complemented
11#17988515—21no—
12#17988512C2211yescharacteristic
13#17988501—21no—
14C4×#11924007—21no—
15C4×C22⋊Dic73—21no—
16C4×C22⋊Dic73—21no—
17C4×C22⋊Dic73C411yescharacteristic, complemented
18#17988354C2212yes—
19C2×C4×D4×C73C2211yescharacteristic
20#431×C73—21no—
21#323×C73—21no—
22#323×C73C2212yes—
23#14556550—41no—
24#14556550—41no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—21no—
28C2×#11924007—21no—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—21no—
33C2×C22⋊Dic73—21no—
34C2×C4×Dic73—21no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73—41no—
37#14556506—24no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498—41no—
43#14556498C2×C411yescharacteristic
44#14556498C2×C411yescharacteristic
45C22×D4×C73D411yescharacteristic
46C23×C4×C73C2×C411yescharacteristic
47C23×C4×C73D411yescharacteristic
48C4×D4×C73—41no—
49C4×D4×C73—41no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73C2311yescharacteristic
53C2×C42×C73D411yescharacteristic
54C4⋊C8×C73—41no—
55C22⋊C8×C73—22no—
56C4×C8×C73—41no—
57C22×Dic73—21no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—42no—
62C4⋊Dic73—41no—
63C4⋊Dic73—41no—
64#11924007—42no—
65#11924007—42no—
66C22⋊Dic73—24no—
67C22⋊Dic73—42no—
68C22⋊Dic73—42no—
69C22⋊Dic73—42no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C4×Dic73—42no—
73C4×Dic73—41no—
74C4×Dic73—41no—
75C4×Dic73—82no—
76C24×C73C22⋊C411yescharacteristic
77C24×C73C2×D411yescharacteristic
78C2×D4×C73—22no—
79C2×D4×C73—22no—
80C2×D4×C73—41no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—42no—
86C22×C4×C73—42no—
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22×C411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C2×C8×C73—42no—
93C4⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C42×C73—21no—
97C42×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×C73C2×C22⋊C411yescharacteristic
108C23×C73#13512yes—
109D4×C73—44no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—22no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—81no—
123C2×C4×C73#9911yescharacteristic
124C2×C4×C73#9911yescharacteristic
125C2×C4×C73#10412yes—
126C2×Dic73—22no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—44no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—82no—
140C2×Dic73—82no—
141C8×C73—82no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—22no—
145C22×C73—42no—
146C22×C73—41no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—81no—
152C22×C73—81no—
153C22×C73—81no—
154C22×C73C2×#9911yescharacteristic
155C22×C73#41611yescharacteristic
156C22×C73C2×#10411yescharacteristic
157C4×C73—21no—
158C4×C73—22no—
159C4×C73—41no—
160C4×C73—41no—
161C4×C73—41no—
162C4×C73#35212yes—
163Dic73—42no—
164Dic73—81no—
165Dic73—84no—
166Dic73—81no—
167Dic73—81no—
168#11727—731nomaximal, Sylow
169C2×C73—21no—
170C2×C73—21no—
171C2×C73—42no—
172C2×C73—81no—
173C2×C73#150111yescharacteristic
174C2×C73C2×#35211yescharacteristic
175C2×C73#212611yescharacteristic
176#2395—731no—
177C4×#99—731no—
178#1460D7311yescharacteristic
179#1352—732no—
180#1344—732no—
181C73#1172711yescharacteristic, complemented, Sylow
182C2×C4×D4C2×D7311yescharacteristic
183C2×#117—731no—
184#431—21no—
185C2×#99—1461no—
186#400—731no—
187#399—731no—
188#398—731no—
189#397—731no—
190#397—1461no—
191#394—1461no—
192C4×C4⋊C4—1461no—
193C4×C22⋊C4—731no—
194C4×C22⋊C4—1461no—
195C4×C22⋊C4—1461no—
196#342—732no—
197#323—21no—
198#323C2×D7312yes—
199C22×D4D29211yescharacteristic
200C23×C4C4×D7311yescharacteristic
201C23×C4D29211yescharacteristic
202C4×D4—41no—
203C4×D4—41no—
204#117—2921no—
205#117—2921no—
206C2×C4⋊C4—731no—
207C2×C4⋊C4—731no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4C4×D7311yescharacteristic
211C2×C22⋊C4—1461no—
212C2×C22⋊C4—1461no—
213C2×C22⋊C4—1461no—
214C2×C22⋊C4—1461no—
215C2×C22⋊C4—1462no—
216C2×C22⋊C4C22⋊D7311yescharacteristic
217C2×C22⋊C4C22×D7311yescharacteristic
218C2×C42—2921no—
219C2×C42—1461no—
220C2×C42—1461no—
221C2×C42C22⋊D7311yescharacteristic
222C4⋊C8—41no—
223#99—1464no—
224C22⋊C8—22no—
225C4×C8—41no—
226#95—2921no—
227#95—731no—
228#95—731no—
229#95—1461no—
230#95—1461no—
231#95—1461no—
232#95—1461no—
233C24#1192400511yescharacteristic
234C24C2×D29211yescharacteristic
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—2922no—
245C22×C4—1461no—
246C22×C4—1461no—
247C22×C4—1461no—
248C22×C4—1461no—
249C22×C4#1192400511yescharacteristic
250C22×C4#1192400511yescharacteristic
251C22×C4#1192400511yescharacteristic
252C22×C4C2×C4×D7311yescharacteristic
253C22×C4C2×C22⋊D7311yescharacteristic
254C2×C8—42no—
255C4⋊C4—41no—
256C4⋊C4—2921no—
257C4⋊C4—2921no—
258C4⋊C4—2922no—
259C4⋊C4—2922no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2922no—
263C22⋊C4—2922no—
264C22⋊C4—2922no—
265C22⋊C4—2922no—
266C22⋊C4—2922no—
267C22⋊C4—1464no—
268C42—21no—
269C42—21no—
270C42—2922no—
271C42—2921no—
272C42—2921no—
273C42—5842no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×#1192400511yescharacteristic
284C23#1455661312yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2924no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—2921no—
309C2×C4—2921no—
310C2×C4—2921no—
311C2×C4—5842no—
312C2×C4—5842no—
313C2×C4—1462no—
314C2×C4#1455650511yescharacteristic
315C2×C4#1455650511yescharacteristic
316C2×C4#1455651712yes—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×#1455650511yescharacteristic
331C22#1798858411yescharacteristic
332C22C2×#1455651711yescharacteristic
333C4—22no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2922no—
339C4—5841no—
340C4—5844no—
341C4—5841no—
342C4—5841no—
343C4#1798837812yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#2260842311yescharacteristic
349C2C2×#1798837811yescharacteristic
350C2#2261159411yescharacteristic
subgroupG/H
centerC2×C4#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1460C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1460D73
hypocenterC73#11727
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #22607956C2semidirect
4. #22607932C2semidirect
6. C2×#14556550C4semidirect
8. #17988526C4semidirect
10. #17988517C4semidirect
17. C4×C22⋊Dic73C4semidirect
181. C73#11727semidirect

Maximal subgroups

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

subgroupindex
#226131902
C4×#145565062
#226079562
#226079322
C73×#14602
#1172773