Finite Group Databasebeta

← #34001446

Subgroups — #34001446

Order 18688.

504 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 106 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#22625823C211yesmaximal, characteristic
2#22625788C211yesmaximal, characteristic
3#22624751C212yesmaximal, complemented
4#22624750C212yesmaximal, complemented
5C4×#14556567C211yesmaximal, characteristic
6C2×#14556658—21no—
7C2×#14556648—21no—
8C2×#14556648—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×#14556567C411yescharacteristic
13C2×C42×D73C2211yescharacteristic
14#17988494C2211yescharacteristic
15#17988483C2211yescharacteristic
16#17988468—21no—
17#17988458—21no—
18#17988444C411yescharacteristic
19C4×#11924007—21no—
20C4×C4⋊Dic73C411yescharacteristic
21C4×#11924005—21no—
22C2×C4×Q8×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25#14556658—42no—
26#14556648—42no—
27#14556648—42no—
28C22×C4×D73C2×C411yescharacteristic
29C22×C4×D73D411yescharacteristic
30C22×C4×D73D411yescharacteristic
31#14556567—22no—
32#14556567C814yescomplemented
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×#11924005—21no—
38C2×#11924005—21no—
39C2×C4×Dic73—21no—
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73C2311yescharacteristic
43C42×D73—42no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C22×Q8×C73C2×C411yescharacteristic
50C4×Q8×C73—42no—
51C2×C4⋊C4×C73—21no—
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C42×C73—21no—
54C2×C42×C73C2×C411yescharacteristic
55C2×#10596833—42no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C23×D73C22⋊C411yescharacteristic
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73C22⋊C411yescharacteristic
65C22×Dic73C2×D411yescharacteristic
66C22×Dic73C2×D411yescharacteristic
67C2×C4×D73—22no—
68C2×C4×D73—21no—
69C2×C4×D73—22no—
70C2×C4×D73—21no—
71C2×C4×D73—41no—
72C2×C4×D73—41no—
73C2×C4×D73—41no—
74C2×C4×D73—41no—
75C2×C4×D73—41no—
76C2×C4×D73C2×C812yes—
77C2×C4×D73M1612yes—
78C4⋊Dic73—21no—
79C4⋊Dic73—41no—
80C4⋊Dic73C2×C812yes—
81#11924007—24no—
82#11924007—41no—
83#11924007—41no—
84#11924005—24no—
85#11924005—42no—
86C4×Dic73—22no—
87C4×Dic73—41no—
88C4×Dic73—41no—
89C4×Dic73—42no—
90C2×Q8×C73—21no—
91C2×Q8×C73—41no—
92C2×Q8×C73—41no—
93C2×Q8×C73C2×C812yes—
94C22×C4×C73—21no—
95C22×C4×C73—21no—
96C22×C4×C73C22⋊C411yescharacteristic
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22×C411yescharacteristic
99C4⋊C4×C73—41no—
100C4⋊C4×C73—42no—
101C42×C73—22no—
102C42×C73—42no—
103C23×C73C2×C22⋊C411yescharacteristic
104C22×D73—21no—
105C22×D73—21no—
106C22×D73—21no—
107C22×D73—21no—
108C22×D73—21no—
109C22×D73—21no—
110C22×D73C22⋊C812yes—
111Q8×C73—42no—
112Q8×C73—42no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—22no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73#9912yes—
127C2×C4×C73C22×C811yescharacteristic
128C2×C4×C73C2×M1611yescharacteristic
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—21no—
134C2×Dic73—21no—
135C2×Dic73—22no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73C22⋊C812yes—
142C2×Dic73AGL(1,ℤ/8)11yescharacteristic
143C2×Dic73#13711yescharacteristic
144C2×Dic73#13712yes—
145C4×D73—42no—
146C4×D73—42no—
147C4×D73—44no—
148C4×D73—42no—
149C4×D73—42no—
150#10596833—82no—
151#10596833—82no—
152C22×C73C2×C22⋊C811yescharacteristic
153C22×C73C2×#9911yescharacteristic
154C22×C73#41311yescharacteristic
155C22×C73#41711yescharacteristic
156C22×C73#41911yescharacteristic
157C22×C73#42111yescharacteristic
158C22×C73#42111yescharacteristic
159C2×D73—22no—
160C2×D73—22no—
161C2×D73—22no—
162C2×D73—41no—
163C2×D73—41no—
164C2×D73—41no—
165C2×D73—41no—
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—22no—
169C4×C73—42no—
170C4×C73—41no—
171C4×C73—81no—
172Dic73—22no—
173Dic73—22no—
174Dic73—21no—
175Dic73—21no—
176Dic73—42no—
177Dic73—81no—
178#7600—731nomaximal, Sylow
179C2×C73#145911yescharacteristic
180C2×C73#147711yescharacteristic
181C2×C73#147911yescharacteristic
182C2×C73#151111yescharacteristic
183C2×C73#151311yescharacteristic
184C2×C73#153511yescharacteristic
185C2×C73#155911yescharacteristic
186D73—42no—
187C4×C22⋊Q8—731no—
188#1817—731no—
189#1797—731no—
190#1272—732no—
191#1271—732no—
192C73#760011yescharacteristic, complemented, Sylow
193C2×C22⋊Q8—731no—
194C2×C4×Q8#6440811yescharacteristic
195C22×C42—731no—
196C2×C4⋊C8—1461no—
197C2×C22⋊C8—1461no—
198C2×C22⋊C8—1461no—
199#391—731no—
200#389—731no—
201#387—1461no—
202#385—1461no—
203#380—731no—
204C4×C4⋊C4—731no—
205C4×C4⋊C4—1461no—
206C4×C22⋊C4—1461no—
207#336—732no—
208#336—732no—
209#336—1461no—
210C22×Q8C2×#6440811yescharacteristic
211C23×C4—731no—
212C23×C4—731no—
213C23×C4—731no—
214C22×C8—1462no—
215C22×C8—1462no—
216C22⋊Q8—734no—
217C22⋊Q8—1462no—
218C4×Q8—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—731no—
221C2×C4⋊C4—731no—
222C2×C4⋊C4—1461no—
223C2×C4⋊C4—1461no—
224C2×C4⋊C4C2×#6440811yescharacteristic
225C2×C22⋊C4—1461no—
226C2×C22⋊C4—1461no—
227C2×C42—21no—
228C2×C42—2922no—
229C2×C42—731no—
230C2×C42—731no—
231C2×C42—731no—
232C2×C42—1461no—
233C2×C42C2×#6440811yescharacteristic
234C4⋊C8—2922no—
235C22⋊C8—2922no—
236C22⋊C8—2922no—
237#95—1461no—
238#95—1461no—
239#95—1461no—
240#95—1461no—
241#95—1461no—
242C24—731no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8#1192403612yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—2921no—
250C22×C4—2921no—
251C22×C4—2921no—
252C22×C4—2921no—
253C22×C4—2921no—
254C22×C4—732no—
255C22×C4—732no—
256C22×C4—731no—
257C22×C4—731no—
258C22×C4—731no—
259C22×C4—1461no—
260C22×C4—1462no—
261C22×C4—1461no—
262C22×C4—1461no—
263C22×C4—1461no—
264C22×C4—1462no—
265C22×C4C22⋊#6440811yescharacteristic
266C22×C4C22⋊#6440811yescharacteristic
267C22×C4C22×#6440811yescharacteristic
268C2×C8—2922no—
269C2×C8—2922no—
270C2×C8—2922no—
271C2×C8—2922no—
272C2×C8—2922no—
273C2×C8—2922no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—2921no—
277C4⋊C4—2921no—
278C4⋊C4—2921no—
279C4⋊C4—732no—
280C4⋊C4—1464no—
281C4⋊C4—1461no—
282C22⋊C4—2922no—
283C22⋊C4—1464no—
284C42—22no—
285C42—42no—
286C42—2922no—
287C42—2921no—
288C42—2921no—
289C42—1462no—
290C23—732no—
291C23—1461no—
292C23—1461no—
293C23—1461no—
294C23—1461no—
295C23—1461no—
296C23—1461no—
297C23C2×C22⋊#6440811yescharacteristic
298Q8—42no—
299Q8—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—2924no—
314C2×C4—2922no—
315C2×C4—2922no—
316C2×C4—2922no—
317C2×C4—2922no—
318C2×C4—2921no—
319C2×C4—2921no—
320C2×C4—2921no—
321C2×C4—2921no—
322C2×C4—2921no—
323C2×C4—732no—
324C2×C4—731no—
325C2×C4—731no—
326C2×C4—732no—
327C2×C4—1462no—
328C2×C4—1461no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4—1461no—
333C2×C4—1461no—
334C2×C4#1455664912yes—
335C2×C4C2×#1192403611yescharacteristic
336C2×C4D73⋊M1611yescharacteristic
337C8—5842no—
338C8—5842no—
339C22—2921no—
340C22—2921no—
341C22—2921no—
342C22—2921no—
343C22—1462no—
344C22—1462no—
345C22—1462no—
346C22#1798955511yescharacteristic
347C22C2×#1455664911yescharacteristic
348C22#1798957311yescharacteristic
349C22#1798957811yescharacteristic
350C22#1798958011yescharacteristic
351C22#1798958411yescharacteristic
352C22#1798958611yescharacteristic
353C4—22no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—42no—
358C4—81no—
359C4—2922no—
360C4—5841no—
361C4—1462no—
362C4—1461no—
363C4—1461no—
364C4—1462no—
365C2—2922no—
366C2#2262516611yescharacteristic
367C2#2262522911yescharacteristic
368C2#2262523411yescharacteristic
369C2#2262534111yescharacteristic
370C2#2262534611yescharacteristic
371C2#2262541711yescharacteristic
372C2#2262546611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C4×Q8#64408
hypocenterC73#7600
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #22624751C2semidirect
4. #22624750C2semidirect
32. #14556567C8semidirect
192. C73#7600semidirect

Maximal subgroups

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

subgroupindex
#226258232
#226257882
#226247512
#226247502
C4×#145565672
#760073