Finite Group Databasebeta

← #32996351

Subgroups — #32996351

Order 18688.

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

normal subgroups only

This group has 434 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#22613214C211yesmaximal, characteristic
2#22608255C211yesmaximal, characteristic, complemented
3#22607957C212yesmaximal, complemented
4#22607933C212yesmaximal
5C4×#101×C73C211yesmaximal, characteristic, complemented
6C2×C4×D292C411yescharacteristic
7#17988640—21no—
8#17988526C411yescharacteristic
9#17988526C2211yescharacteristic
10#17988513—21no—
11#17988504C411yescharacteristic
12#17988504C2211yescharacteristic
13#17988498—21no—
14C4×#11924005—21no—
15C4×#11924005—21no—
16#17988433C411yescharacteristic
17#17988313—21no—
18#17988313C2212yes—
19C2×C4×Q8×C73C2211yescharacteristic
20C2×#101×C73—21no—
21C4×M16×C73—21no—
22#343×C73C2212yes—
23C22×D292C2×C411yescharacteristic
24C22×C4×D73—21no—
25C4×D292—41no—
26C4×D292—41no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924005—21no—
29C2×#11924005—21no—
30C2×#11924005—21no—
31C2×#11924005—21no—
32C2×#11924005—21no—
33C2×C4×Dic73—21no—
34C4⋊#10596831—41no—
35C22⋊#10596831—22no—
36C4×#10596831—41no—
37#14556498—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498C2×C411yescharacteristic
43#14556498C2×C411yescharacteristic
44C22×Q8×C73D411yescharacteristic
45C2×M16×C73—22no—
46C4×Q8×C73—41no—
47C4×Q8×C73—41no—
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C4⋊C4×C73C2311yescharacteristic
51C2×C42×C73C2×C411yescharacteristic
52C2×C42×C73D411yescharacteristic
53C2×C42×C73D411yescharacteristic
54#101×C73—24no—
55#97×C73—41no—
56C4×C8×C73—41no—
57C23×D73—21no—
58C2×D292—41no—
59C2×D292—42no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C2×C4×D73—42no—
64C2×C4×D73—42no—
65C2×C4×D73—41no—
66C2×C4×D73—41no—
67C2×#10596831—42no—
68C4⋊Dic73—41no—
69#11924005—42no—
70#11924005—42no—
71#11924005—42no—
72#11924005—42no—
73#11924005—42no—
74C4×Dic73—42no—
75C2×Q8×C73—22no—
76C2×Q8×C73—22no—
77C2×Q8×C73—41no—
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85M16×C73—24no—
86M16×C73—42no—
87C2×C8×C73—42no—
88C4⋊C4×C73—41no—
89C4⋊C4×C73—41no—
90C4⋊C4×C73—41no—
91C42×C73—21no—
92C42×C73—21no—
93C42×C73—21no—
94C42×C73—21no—
95C42×C73—21no—
96C42×C73—21no—
97C23×C73C2×C22⋊C411yescharacteristic
98C22×D73—22no—
99C22×D73—41no—
100C22×D73—41no—
101C22×D73—41no—
102C22×D73—41no—
103C22×D73—41no—
104C22×D73—41no—
105Q8×C73—44no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—22no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73#9911yescharacteristic
121C2×C4×C73#9911yescharacteristic
122C2×C4×C73#10412yes—
123C2×C4×C73#13512yes—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133D292—42no—
134D292—42no—
135C4×D73—82no—
136C4×D73—82no—
137C8×C73—44no—
138#10596831—82no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73C2×#9911yescharacteristic
142C22×C73#41611yescharacteristic
143C22×C73C2×#10411yescharacteristic
144C2×D73—42no—
145C2×D73—42no—
146C2×D73—42no—
147C2×D73—81no—
148C2×D73—81no—
149C2×D73—81no—
150C2×D73—81no—
151C4×C73—21no—
152C4×C73—22no—
153C4×C73—22no—
154C4×C73—41no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73#35512yes—
159Dic73—81no—
160Dic73—81no—
161Dic73—81no—
162#11734—731nomaximal, Sylow
163C2×C73—21no—
164C2×C73—21no—
165C2×C73#150111yescharacteristic
166C2×C73#212011yescharacteristic
167C2×C73C2×#35511yescharacteristic
168D73—82no—
169#2398—731no—
170C4×#101D7311yescharacteristic, complemented
171#1461—731no—
172#1353—732no—
173#1345—732no—
174C73#1173411yescharacteristic, complemented, Sylow
175C2×C4×Q8C2×D7311yescharacteristic
176C2×C4×D4—731no—
177#431—1461no—
178C2×#101—21no—
179C4×M16—21no—
180#400—731no—
181#400—731no—
182#397—1461no—
183#394—731no—
184#394—731no—
185#393—1461no—
186C4×C22⋊C4—1461no—
187C4×C22⋊C4—1461no—
188#376—731no—
189#343C2×D7312yes—
190#324—732no—
191#324—1461no—
192C22×Q8C22⋊D7311yescharacteristic
193C22×D4—731no—
194C23×C4—1461no—
195C2×M16—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—2921no—
199C4×D4—2921no—
200C2×C4⋊C4—731no—
201C2×C4⋊C4C4×D7311yescharacteristic
202C2×C4⋊C4D29211yescharacteristic
203C2×C4⋊C4C22×D7311yescharacteristic
204C2×C22⋊C4—1461no—
205C2×C22⋊C4—1461no—
206C2×C22⋊C4—1461no—
207C2×C22⋊C4—1461no—
208C2×C22⋊C4—1461no—
209C2×C42—1461no—
210C2×C42C4×D7311yescharacteristic
211C2×C42D29211yescharacteristic
212C2×C42C22⋊D7311yescharacteristic
213C4⋊C8—2921no—
214#101—24no—
215C22⋊C8—1462no—
216#97—41no—
217C4×C8—41no—
218C4×C8—2921no—
219#95—731no—
220#95—731no—
221#95—1461no—
222#95—1461no—
223#95—1461no—
224#95—1461no—
225#95—1461no—
226C24—1461no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×D4—2921no—
231C2×D4—2922no—
232C22×C4—2922no—
233C22×C4—2922no—
234C22×C4—2921no—
235C22×C4—2921no—
236C22×C4—1461no—
237C22×C4—1461no—
238C22×C4—1461no—
239C22×C4#1192400511yescharacteristic
240C22×C4#1192400511yescharacteristic
241C22×C4#1192400511yescharacteristic
242C22×C4#1192400511yescharacteristic
243C22×C4C2×C4×D7311yescharacteristic
244C22×C4C2×C22⋊D7311yescharacteristic
245C22×C4C2×D29211yescharacteristic
246M16—24no—
247M16—42no—
248C2×C8—42no—
249C2×C8—2922no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—2921no—
254C22⋊C4—2922no—
255C22⋊C4—2922no—
256C22⋊C4—2922no—
257C22⋊C4—2922no—
258C22⋊C4—2922no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—2922no—
266C23—2921no—
267C23—2921no—
268C23—2921no—
269C23—2921no—
270C23—2921no—
271C23—2921no—
272C23—1462no—
273C23C2×#1192400511yescharacteristic
274Q8—44no—
275D4—2922no—
276D4—2922no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—2921no—
295C2×C4—2921no—
296C2×C4—2921no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—5842no—
301C2×C4—5842no—
302C2×C4#1455650511yescharacteristic
303C2×C4#1455650511yescharacteristic
304C2×C4#1455651912yes—
305C2×C4#1455661112yes—
306C8—44no—
307C8—5842no—
308C22—21no—
309C22—21no—
310C22—2922no—
311C22—2922no—
312C22—2922no—
313C22—5841no—
314C22—5841no—
315C22—5841no—
316C22—5841no—
317C22C2×#1455650511yescharacteristic
318C22#1798858111yescharacteristic
319C22C2×#1455651911yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—5841no—
328C4—5841no—
329C4—5841no—
330C4#1798838312yes—
331C2—21no—
332C2—21no—
333C2—5842no—
334C2#2260842011yescharacteristic
335C2#2261157311yescharacteristic
336C2C2×#1798838311yescharacteristic
subgroupG/H
centerC2×C4#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC4×#101×C73C2
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊C4×C73C23
hypercenterC4×#101D73
hypocenterC73#11734
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #22608255C2semidirect
3. #22607957C2semidirect
5. C4×#101×C73C2semidirect
170. C4×#101D73semidirect
174. C73#11734semidirect

Maximal subgroups

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

subgroupindex
#226132142
#226082552
#226079572
#226079332
C4×#101×C732
#1173473