Finite Group Databasebeta

← #34001406

Subgroups — #34001406

Order 18688.

578 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 122 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#22625788C211yesmaximal, characteristic
2#22625592C211yesmaximal, characteristic
3#22624751C212yesmaximal, complemented
4#22624750C212yesmaximal, complemented
5C4×#14556567C211yesmaximal, characteristic
6C22×#11924036—21no—
7C2×#14556658—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
23C2×#11924036—24no—
24C2×#11924036—42no—
25C2×#11924036—42no—
26C22×#10596833—22no—
27C22×#10596833—22no—
28#14556658—42no—
29#14556648—42no—
30C22×C4×D73C2×C411yescharacteristic
31C22×C4×D73D411yescharacteristic
32C22×C4×D73D411yescharacteristic
33#14556567—22no—
34#14556567C814yescomplemented
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×C4⋊Dic73C2×C411yescharacteristic
37C2×#11924007—21no—
38C2×#11924007—21no—
39C2×#11924005—21no—
40C2×#11924005—21no—
41C2×C4×Dic73—21no—
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73C2311yescharacteristic
45C42×D73—42no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50#14556498—21no—
51C22×Q8×C73C2×C411yescharacteristic
52C4×Q8×C73—42no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73C2×C411yescharacteristic
55C2×C42×C73—21no—
56C2×C42×C73C2×C411yescharacteristic
57#11924036—48no—
58C2×#10596833—24no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C2×#10596833—42no—
63C2×#10596833—42no—
64C23×D73C22⋊C411yescharacteristic
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73C22⋊C411yescharacteristic
68C22×Dic73C2×D411yescharacteristic
69C22×Dic73C2×D411yescharacteristic
70C2×C4×D73—22no—
71C2×C4×D73—22no—
72C2×C4×D73—21no—
73C2×C4×D73—21no—
74C2×C4×D73—41no—
75C2×C4×D73—41no—
76C2×C4×D73—41no—
77C2×C4×D73—41no—
78C2×C4×D73C2×C812yes—
79C2×C4×D73M1612yes—
80C2×C4×D73SD1612yes—
81C2×C4×D73Q1612yes—
82C4⋊Dic73—21no—
83C4⋊Dic73—41no—
84C4⋊Dic73C2×C812yescomplemented
85#11924007—24no—
86#11924007—41no—
87#11924007—41no—
88#11924005—24no—
89#11924005—42no—
90C4×Dic73—22no—
91C4×Dic73—41no—
92C4×Dic73—41no—
93C4×Dic73—42no—
94C2×Q8×C73—21no—
95C2×Q8×C73—41no—
96C2×Q8×C73—41no—
97C2×Q8×C73C2×C812yescomplemented
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73C22⋊C411yescharacteristic
101C22×C4×C73C22⋊C411yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C4⋊C4×C73—41no—
104C4⋊C4×C73—42no—
105C42×C73—22no—
106C42×C73—42no—
107C23×C73C2×C22⋊C411yescharacteristic
108C22×D73—21no—
109C22×D73—21no—
110C22×D73—21no—
111C22×D73—21no—
112C22×D73C22⋊C812yes—
113C22×D73#10312yes—
114C22×D73#10412yes—
115Q8×C73—42no—
116Q8×C73—42no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—22no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73#9912yes—
130C2×C4×C73#10312yes—
131C2×C4×C73C22×C811yescharacteristic
132C2×C4×C73C2×M1611yescharacteristic
133C2×Dic73—21no—
134C2×Dic73—21no—
135C2×Dic73—21no—
136C2×Dic73—21no—
137C2×Dic73—21no—
138C2×Dic73—21no—
139C2×Dic73—22no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73—41no—
144C2×Dic73—41no—
145C2×Dic73C22⋊C812yes—
146C2×Dic73C2×SD1611yescharacteristic
147C2×Dic73C2×Q1611yescharacteristic
148C2×Dic73AGL(1,ℤ/8)11yescharacteristic
149C2×Dic73#13711yescharacteristic
150C4×D73—24no—
151C4×D73—24no—
152C4×D73—42no—
153C4×D73—42no—
154C4×D73—42no—
155C4×D73—42no—
156#10596833—44no—
157#10596833—82no—
158C22×C73C2×C22⋊C811yescharacteristic
159C22×C73C2×#9911yescharacteristic
160C22×C73#41311yescharacteristic
161C22×C73C2×#10311yescharacteristic
162C22×C73#41711yescharacteristic
163C22×C73C2×#10411yescharacteristic
164C22×C73#42111yescharacteristic
165C2×D73—22no—
166C2×D73—22no—
167C2×D73—22no—
168C2×D73—22no—
169C2×D73—22no—
170C2×D73—22no—
171C2×D73Q8⋊C814yes—
172C4×C73—21no—
173C4×C73—21no—
174C4×C73—22no—
175C4×C73—22no—
176C4×C73—42no—
177C4×C73—81no—
178Dic73—22no—
179Dic73—21no—
180Dic73—21no—
181Dic73—21no—
182Dic73—21no—
183Dic73—42no—
184Dic73—81no—
185#7589—731nomaximal, Sylow
186C2×C73#145911yescharacteristic
187C2×C73C2×Q8⋊C811yescharacteristic
188C2×C73#147711yescharacteristic
189C2×C73#150411yescharacteristic
190C2×C73#152011yescharacteristic
191C2×C73#153211yescharacteristic
192C2×C73#155011yescharacteristic
193D73—24no—
194C4×C22⋊Q8—731no—
195#1797—731no—
196C2×#336—731no—
197#1272—732no—
198#1271—732no—
199C73#758911yescharacteristic, complemented, Sylow
200C23×C8—1461no—
201C2×C22⋊Q8—731no—
202C2×C4×Q8#6440811yescharacteristic
203C22×C42—731no—
204C2×C4⋊C8—1461no—
205C2×C22⋊C8—1461no—
206#391—731no—
207#389—731no—
208#387—1461no—
209#385—1461no—
210#380—731no—
211C4×C4⋊C4—731no—
212C4×C4⋊C4—1461no—
213C4×C22⋊C4—1461no—
214#336—732no—
215#336—732no—
216#336—1461no—
217C22×Q8C2×#6440811yescharacteristic
218C23×C4—731no—
219C23×C4—731no—
220C23×C4—731no—
221C22×C8—2922no—
222C22×C8—2922no—
223C22×C8—1462no—
224C22×C8—1462no—
225C22×C8—1464no—
226C22⋊Q8—734no—
227C22⋊Q8—1462no—
228C4×Q8—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—731no—
231C2×C4⋊C4—731no—
232C2×C4⋊C4—1461no—
233C2×C4⋊C4—1461no—
234C2×C4⋊C4C2×#6440811yescharacteristic
235C2×C22⋊C4—1461no—
236C2×C22⋊C4—1461no—
237C2×C42—21no—
238C2×C42—2922no—
239C2×C42—731no—
240C2×C42—731no—
241C2×C42—731no—
242C2×C42—1461no—
243C2×C42C2×#6440811yescharacteristic
244C4⋊C8—2922no—
245C22⋊C8—2922no—
246#95—1461no—
247#95—1461no—
248#95—1461no—
249#95—1461no—
250#95—1461no—
251C24—731no—
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8#1192403612yescomplemented
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—2921no—
259C22×C4—2921no—
260C22×C4—2921no—
261C22×C4—2921no—
262C22×C4—732no—
263C22×C4—732no—
264C22×C4—732no—
265C22×C4—732no—
266C22×C4—731no—
267C22×C4—731no—
268C22×C4—731no—
269C22×C4—1461no—
270C22×C4—1461no—
271C22×C4—1462no—
272C22×C4—1461no—
273C22×C4—1461no—
274C22×C4—1462no—
275C22×C4C22⋊#6440811yescharacteristic
276C22×C4C22⋊#6440811yescharacteristic
277C22×C4C22×#6440811yescharacteristic
278C2×C8—2922no—
279C2×C8—2922no—
280C2×C8—2928no—
281C2×C8—2922no—
282C2×C8—2922no—
283C2×C8—2922no—
284C2×C8—1464no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—2921no—
288C4⋊C4—2921no—
289C4⋊C4—2921no—
290C4⋊C4—732no—
291C4⋊C4—1464no—
292C4⋊C4—1461no—
293C22⋊C4—2922no—
294C22⋊C4—1464no—
295C42—22no—
296C42—42no—
297C42—2922no—
298C42—2921no—
299C42—2921no—
300C42—1462no—
301C23—732no—
302C23—732no—
303C23—732no—
304C23—1461no—
305C23—1461no—
306C23—1461no—
307C23—1461no—
308C23C2×C22⋊#6440811yescharacteristic
309Q8—42no—
310Q8—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—2922no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—2921no—
328C2×C4—2921no—
329C2×C4—2921no—
330C2×C4—2921no—
331C2×C4—2921no—
332C2×C4—732no—
333C2×C4—731no—
334C2×C4—731no—
335C2×C4—731no—
336C2×C4—731no—
337C2×C4—1464no—
338C2×C4—1464no—
339C2×C4—1462no—
340C2×C4—1461no—
341C2×C4—1461no—
342C2×C4—1461no—
343C2×C4—1461no—
344C2×C4—1461no—
345C2×C4—1461no—
346C2×C4#1455664912yes—
347C2×C4#1455665412yes—
348C2×C4C2×#1192403611yescharacteristic
349C2×C4D73⋊M1611yescharacteristic
350C8—2924no—
351C8—5842no—
352C22—734no—
353C22—1462no—
354C22—1462no—
355C22—1462no—
356C22—1462no—
357C22—1462no—
358C22—1462no—
359C22#1798955511yescharacteristic
360C22C2×#1455664911yescharacteristic
361C22#1798957311yescharacteristic
362C22C2×#1455665411yescharacteristic
363C22#1798957811yescharacteristic
364C22D73⋊#10411yescharacteristic
365C22#1798958611yescharacteristic
366C4—22no—
367C4—22no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—81no—
372C4—2922no—
373C4—5841no—
374C4—1461no—
375C4—1461no—
376C4—1461no—
377C4—1461no—
378C4—1462no—
379C2—1464no—
380C2#2262516611yescharacteristic
381C2#2262522011yescharacteristic
382C2#2262522911yescharacteristic
383C2#2262531711yescharacteristic
384C2#2262537111yescharacteristic
385C2#2262540811yescharacteristic
386C2#2262544811yescharacteristic
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#7589
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
34. #14556567C8semidirect
84. C4⋊Dic73C2×C8semidirect
97. C2×Q8×C73C2×C8semidirect
199. C73#7589semidirect
255. C2×Q8#11924036semidirect

Maximal subgroups

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

subgroupindex
#226257882
#226255922
#226247512
#226247502
C4×#145565672
#758973