Finite Group Databasebeta

← #32996604

Subgroups — #32996604

Order 18688.

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

normal subgroups only

This group has 516 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#22614884C211yesmaximal, characteristic
2#22608269C211yesmaximal, characteristic
3#22607954C214yesmaximal, complemented
4C4×#99×C73C211yesmaximal, characteristic
5C2×#14556591C411yescharacteristic, complemented
6C2×#14556570—21no—
7C2×#14556570—21no—
8#17988551—21no—
9#17988523C412yescomplemented
10#17988523C2212yes—
11#17988490—21no—
12#17988456—21no—
13C4×C4⋊Dic73C411yescharacteristic, complemented
14#17988313—21no—
15#17988311C2212yes—
16C2×C4×D4×C73C2211yescharacteristic
17C2×#99×C73—21no—
18C4×C22⋊C4×C73—21no—
19#342×C73C2212yes—
20C22×Dic146—21no—
21C2×#11924012—41no—
22#14556591—41no—
23#14556591—41no—
24#14556570—44no—
25#14556570—44no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73C2×C412yes—
29C2×#11924007—22no—
30C2×#11924007—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33C2×C4×Dic73—21no—
34C22⋊#10596831—22no—
35C22⋊#10596831—41no—
36#14556498C2×C412yes—
37C22×D4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C4×D4×C73—41no—
41C4×D4×C73—41no—
42C2×C4⋊C4×C73D411yescharacteristic
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73C2×C411yescharacteristic
45C2×C22⋊C4×C73C2311yescharacteristic
46C2×C42×C73—41no—
47C2×C42×C73C2×C411yescharacteristic
48#99×C73—24no—
49#95×C73—41no—
50C2×Dic146—42no—
51C2×Dic146—42no—
52C2×Dic146—41no—
53C2×Dic146—41no—
54C22×Dic73—22no—
55C22×Dic73—22no—
56#11924012—82no—
57C2×#10596831—42no—
58C4⋊Dic73—24no—
59C4⋊Dic73—42no—
60C4⋊Dic73—42no—
61#11924007—44no—
62#11924007—44no—
63C22⋊Dic73—44no—
64C22⋊Dic73—44no—
65C4×Dic73—42no—
66C24×C73C22⋊C411yescharacteristic
67C24×C73C2×D411yescharacteristic
68C2×D4×C73—22no—
69C2×D4×C73—22no—
70C2×D4×C73—41no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—42no—
76C22×C4×C73—42no—
77C22×C4×C73—42no—
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22×C411yescharacteristic
82C22×C4×C73C2×D411yescharacteristic
83C4⋊C4×C73—41no—
84C22⋊C4×C73—24no—
85C22⋊C4×C73—41no—
86C22⋊C4×C73—41no—
87C22⋊C4×C73—42no—
88C42×C73—21no—
89C42×C73—21no—
90C42×C73—82no—
91C23×C73—21no—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73C2×C22⋊C411yescharacteristic
101C23×C73#13712yes—
102D4×C73—44no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—22no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—42no—
110C2×C4×C73—42no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—44no—
115C2×C4×C73—81no—
116C2×C4×C73—81no—
117C2×C4×C73—82no—
118C2×C4×C73—82no—
119C2×C4×C73#9911yescharacteristic
120C2×C4×C73#9911yescharacteristic
121C2×C4×C73#10412yes—
122C2×Dic73—24no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73—42no—
128Dic146—82no—
129Dic146—82no—
130#10596831—82no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—22no—
134C22×C73—42no—
135C22×C73—42no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73—41no—
141C22×C73—81no—
142C22×C73—81no—
143C22×C73C2×#9911yescharacteristic
144C22×C73#41911yescharacteristic
145C22×C73C2×#10411yescharacteristic
146C4×C73—21no—
147C4×C73—22no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73—41no—
151C4×C73—84no—
152C4×C73#35312yes—
153Dic73—44no—
154Dic73—82no—
155#11788—731nomaximal, Sylow
156C2×C73—21no—
157C2×C73—21no—
158C2×C73—42no—
159C2×C73—81no—
160C2×C73#150811yescharacteristic
161C2×C73#212311yescharacteristic
162C2×C73C2×#35311yescharacteristic
163#2542—731no—
164C4×#99D7311yescharacteristic
165#1465—731no—
166#1352—734no—
167C73#1178811yescharacteristic, complemented, Sylow
168C2×C4⋊Q8—731no—
169C2×C22⋊Q8—1461no—
170C2×C22⋊Q8—1461no—
171C2×C4×D4C2×D7311yescharacteristic
172C2×#99—21no—
173#407—1461no—
174#399—732no—
175#399—732no—
176#390—1461no—
177#384—1461no—
178C4×C4⋊C4—731no—
179C4×C22⋊C4—21no—
180#342C2×D7312yes—
181#324—1461no—
182#323—732no—
183C22×Q8—1461no—
184C22×D4C22⋊D7311yescharacteristic
185C23×C4D29211yescharacteristic
186C23×C4C22⋊D7311yescharacteristic
187C2×M16—2921no—
188C4⋊Q8—2921no—
189C4⋊Q8—2921no—
190C22⋊Q8—2924no—
191C22⋊Q8—2924no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—732no—
195C2×C4⋊C4—1462no—
196C2×C4⋊C4—1461no—
197C2×C4⋊C4—1462no—
198C2×C4⋊C4—1461no—
199C2×C4⋊C4D29211yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—1462no—
202C2×C22⋊C4—1462no—
203C2×C22⋊C4C4×D7311yescharacteristic
204C2×C22⋊C4C22×D7311yescharacteristic
205C2×C42—41no—
206C2×C42—1461no—
207C2×C42C4×D7311yescharacteristic
208#99—24no—
209C22⋊C8—2921no—
210C22⋊C8—1462no—
211#95—41no—
212#95—732no—
213C24#1192400511yescharacteristic
214C24C2×C22⋊D7311yescharacteristic
215C2×Q8—2922no—
216C2×Q8—2922no—
217C2×Q8—2921no—
218C2×Q8—2921no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—1462no—
230C22×C4—1462no—
231C22×C4#1192400511yescharacteristic
232C22×C4#1192400511yescharacteristic
233C22×C4#1192400511yescharacteristic
234C22×C4C2×C4×D7311yescharacteristic
235C22×C4C2×D29211yescharacteristic
236M16—5842no—
237C2×C8—2922no—
238C4⋊C4—41no—
239C4⋊C4—2922no—
240C4⋊C4—2922no—
241C4⋊C4—2924no—
242C4⋊C4—2924no—
243C4⋊C4—1464no—
244C22⋊C4—24no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—2924no—
249C22⋊C4—2924no—
250C42—21no—
251C42—21no—
252C42—82no—
253C42—2922no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×#1192400511yescharacteristic
264C23#1455662112yes—
265Q8—5842no—
266Q8—5842no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—2922no—
285C2×C4—2922no—
286C2×C4—2922no—
287C2×C4—2922no—
288C2×C4—2922no—
289C2×C4—1464no—
290C2×C4#1455650511yescharacteristic
291C2×C4#1455650511yescharacteristic
292C2×C4#1455651912yes—
293C8—5842no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×#1455650511yescharacteristic
307C22#1798859511yescharacteristic
308C22C2×#1455651911yescharacteristic
309C4—22no—
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—84no—
315C4—2924no—
316C4—5842no—
317C4#1798837912yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#2260845011yescharacteristic
323C2#2261158411yescharacteristic
324C2C2×#1798837911yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC4×#99×C73C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenterC4×#99D73
hypocenterC73#11788
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #22607954C2semidirect
5. C2×#14556591C4semidirect
9. #17988523C4semidirect
13. C4×C4⋊Dic73C4semidirect
167. C73#11788semidirect

Maximal subgroups

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

subgroupindex
#226148842
#226082692
#226079542
C4×#99×C732
#1178873