Finite Group Databasebeta

← #33004653

Subgroups — #33004653

Order 18688.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C4×#14556506C211yesmaximal, characteristic
3#22608055C214yesmaximal
4C73×#1799C211yesmaximal, characteristic
5C2×#14556591C411yescharacteristic, complemented
6C2×#14556570—21no—
7C2×#14556570—21no—
8C2×#14556506C2212yes—
9#17988523—21no—
10#17988523—21no—
11#17988490C2212yes—
12#17988456C412yescomplemented
13C4×C4⋊Dic73C411yescharacteristic, complemented
14C4×C22⋊Dic73—21no—
15#17988354—21no—
16C2×C4×D4×C73C2211yescharacteristic
17C2×#99×C73C2212yes—
18#380×C73—21no—
19#342×C73—21no—
20C22×Dic146—21no—
21#14556591—41no—
22#14556591—41no—
23#14556570—44no—
24#14556570—44no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73C2×C412yes—
28C2×#11924007—22no—
29C2×#11924007—22no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33C2×C4×Dic73—41no—
34C2×C4×Dic73C2×C412yes—
35#14556506—22no—
36#14556506Q814yescomplemented
37#14556498—21no—
38#14556498—41no—
39C22×D4×C73C2311yescharacteristic
40C23×C4×C73D411yescharacteristic
41C23×C4×C73D411yescharacteristic
42C4×D4×C73—41no—
43C4×D4×C73—41no—
44C2×C4⋊C4×C73—41no—
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C22⋊C4×C73—22no—
47C2×C22⋊C4×C73D411yescharacteristic
48C2×C22⋊C4×C73D411yescharacteristic
49C2×C42×C73C2×C411yescharacteristic
50#99×C73—24no—
51#95×C73—41no—
52C2×Dic146—42no—
53C2×Dic146—42no—
54C2×Dic146—41no—
55C2×Dic146—41no—
56C22×Dic73—22no—
57C22×Dic73—22no—
58C22×Dic73—42no—
59C4⋊Dic73—24no—
60C4⋊Dic73—42no—
61C4⋊Dic73—42no—
62#11924007—44no—
63#11924007—44no—
64C22⋊Dic73—24no—
65C22⋊Dic73—44no—
66C22⋊Dic73—44no—
67C22⋊Dic73—42no—
68C4×Dic73—42no—
69C4×Dic73—82no—
70C24×C73C2×D411yescharacteristic
71C24×C73C2×D411yescharacteristic
72C2×D4×C73—41no—
73C2×D4×C73—41no—
74C2×D4×C73C2×Q812yes—
75C2×D4×C73#4112yes—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—42no—
81C22×C4×C73—42no—
82C22×C4×C73—42no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22×C411yescharacteristic
88C4⋊C4×C73—41no—
89C4⋊C4×C73—82no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—41no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—42no—
94C42×C73—21no—
95C42×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73—41no—
104C23×C73C2×C22⋊C411yescharacteristic
105C23×C73C22⋊Q812yes—
106C23×C73#12411yescharacteristic
107C23×C73C4⋊Q811yescharacteristic
108D4×C73—42no—
109D4×C73—42no—
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—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73—82no—
124C2×C4×C73—82no—
125C2×C4×C73—82no—
126C2×C4×C73#9911yescharacteristic
127C2×C4×C73#9911yescharacteristic
128C2×C4×C73C4×Q812yes—
129C2×Dic73—24no—
130C2×Dic73—42no—
131C2×Dic73—42no—
132C2×Dic73—44no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—82no—
137C2×Dic73—82no—
138Dic146—82no—
139Dic146—82no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—22no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—42no—
146C22×C73—41no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—81no—
153C22×C73—81no—
154C22×C73#39111yescharacteristic
155C22×C73C2×#9911yescharacteristic
156C22×C73#41011yescharacteristic
157C4×C73—21no—
158C4×C73—22no—
159C4×C73—41no—
160C4×C73—41no—
161C4×C73—41no—
162C4×C73—162no—
163C4×C73#35312yes—
164Dic73—44no—
165Dic73—84no—
166Dic73—82no—
167#13298—731nomaximal, Sylow
168C2×C73—21no—
169C2×C73—21no—
170C2×C73—42no—
171C2×C73—41no—
172C2×C73—41no—
173C2×C73#199611yescharacteristic
174C2×C73C2×#35311yescharacteristic
175C2×C73#212911yescharacteristic
176#2542—731no—
177#1799D7311yescharacteristic
178C4×#99—731no—
179#1393—734no—
180C73#1329811yescharacteristic, complemented, Sylow
181C2×C4⋊Q8—731no—
182C2×C22⋊Q8—1461no—
183C2×C22⋊Q8—1461no—
184C2×C4×D4C2×D7311yescharacteristic
185C2×#99—732no—
186C2×#99C2×D7312yes—
187#399—1461no—
188#399—1461no—
189#390—732no—
190#384—732no—
191#380—21no—
192C4×C4⋊C4—731no—
193C4×C22⋊C4—1461no—
194#342—21no—
195#342—1461no—
196C22×Q8—1461no—
197C22×D4C22×D7311yescharacteristic
198C23×C4D29211yescharacteristic
199C23×C4C22⋊D7311yescharacteristic
200C4⋊Q8—2921no—
201C4⋊Q8—2921no—
202C22⋊Q8—2924no—
203C22⋊Q8—2924no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—732no—
208C2×C4⋊C4—1462no—
209C2×C4⋊C4—1462no—
210C2×C4⋊C4—1461no—
211C2×C4⋊C4—1461no—
212C2×C4⋊C4C4×D7311yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—1462no—
215C2×C22⋊C4—1462no—
216C2×C22⋊C4—1462no—
217C2×C22⋊C4D29211yescharacteristic
218C2×C22⋊C4C22⋊D7311yescharacteristic
219C2×C42—2921no—
220C2×C42—732no—
221C2×C42C4×D7311yescharacteristic
222#99—24no—
223#99—734no—
224#99—1462no—
225#95—41no—
226#95—2921no—
227#95—1461no—
228C24C2×C22⋊D7311yescharacteristic
229C24C2×D29211yescharacteristic
230C2×Q8—2922no—
231C2×Q8—2922no—
232C2×Q8—2921no—
233C2×Q8—2921no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4Q8×D7312yes—
237C2×D4#1192402712yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—2922no—
246C22×C4—1462no—
247C22×C4—1462no—
248C22×C4#1192400511yescharacteristic
249C22×C4#1192400511yescharacteristic
250C22×C4#1192400511yescharacteristic
251C22×C4#1192400511yescharacteristic
252C22×C4C2×C4×D7311yescharacteristic
253C4⋊C4—41no—
254C4⋊C4—82no—
255C4⋊C4—2922no—
256C4⋊C4—2922no—
257C4⋊C4—2924no—
258C4⋊C4—2924no—
259C4⋊C4—1464no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—2922no—
265C22⋊C4—2924no—
266C22⋊C4—2924no—
267C22⋊C4—1464no—
268C42—21no—
269C42—21no—
270C42—2922no—
271C42—5842no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×#1192400511yescharacteristic
281C23#1455657112yes—
282C23#1455657811yescharacteristic
283C23#1455658911yescharacteristic
284Q8—5842no—
285Q8—5842no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—2922no—
305C2×C4—2922no—
306C2×C4—2922no—
307C2×C4—2922no—
308C2×C4—2924no—
309C2×C4—2922no—
310C2×C4—5842no—
311C2×C4—5842no—
312C2×C4—1464no—
313C2×C4#1455650511yescharacteristic
314C2×C4#1455650511yescharacteristic
315C2×C4#1455655612yes—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—21no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22#1798849211yescharacteristic
331C22C2×#1455650511yescharacteristic
332C22#1798856011yescharacteristic
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—162no—
339C4—2924no—
340C4—5844no—
341C4—5842no—
342C4#1798837912yes—
343C2—21no—
344C2—21no—
345C2—41no—
346C2—42no—
347C2—41no—
348C2#2261091511yescharacteristic
349C2C2×#1798837911yescharacteristic
350C2#2261160511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#1799C2
socleC22×C73C2×#99
prime radical, cosocleC22×D4×C73C23
hypercenter#1799D73
hypocenterC73#13298
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
5. C2×#14556591C4semidirect
12. #17988456C4semidirect
13. C4×C4⋊Dic73C4semidirect
36. #14556506Q8semidirect
180. C73#13298semidirect

Maximal subgroups

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

subgroupindex
#226148842
C4×#145565062
#226080552
C73×#17992
#1329873