Finite Group Databasebeta

← #34009955

Subgroups — #34009955

Order 18688.

460 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 86 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#22625170C211yesmaximal, characteristic
2#22625166C211yesmaximal, characteristic
3#22624876C212yesmaximal, complemented
4#22624857C212yesmaximal, complemented
5#22612600C211yesmaximal, characteristic
6#17989604—21no—
7#17989555—21no—
8#17989433—21no—
9#17989431—21no—
10#17989430C2212yes—
11#17989430C2212yes—
12C2×#14556574C411yescharacteristic
13C2×C4×C22⋊D73C2211yescharacteristic
14#17988482—21no—
15#17988477—21no—
16#17988468—21no—
17#17988466C411yescharacteristic
18#17988460C2211yescharacteristic
19#17988458—21no—
20#17988450C2211yescharacteristic
21C4×C22⋊Dic73C411yescharacteristic
22C22×C4⋊C4×C73C411yescharacteristic
23C2×#11924036—41no—
24#14556658—41no—
25#14556648—41no—
26C22⋊#10596833—22no—
27C22⋊#10596833—22no—
28C4×#10596833—41no—
29C22×C22⋊D73C2×C411yescharacteristic
30C22×C4×D73D411yescharacteristic
31#14556574—22no—
32#14556574C814yescomplemented
33C4×C22⋊D73—41no—
34C4×C22⋊D73—41no—
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×#11924007D411yescharacteristic
38C2×C22⋊Dic73C2×C411yescharacteristic
39C2×C22⋊Dic73C2×C411yescharacteristic
40C2×C22⋊Dic73C2311yescharacteristic
41C2×#11924005—21no—
42C2×#11924005—21no—
43C2×#11924005D411yescharacteristic
44C2×C4×Dic73—21no—
45C2×C4×Dic73D411yescharacteristic
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50#14556498—21no—
51C23×C4×C73C2×C411yescharacteristic
52C23×C4×C73C2×C411yescharacteristic
53C23×C4×C73C2×C411yescharacteristic
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73—41no—
57C2×C4⋊C4×C73—41no—
58#11924036—82no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C23×D73C22⋊C411yescharacteristic
62C2×C22⋊D73—41no—
63C2×C22⋊D73—41no—
64C2×C22⋊D73C2×C812yes—
65C2×C22⋊D73M1612yes—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C2×D411yescharacteristic
70C22×Dic73C2×D411yescharacteristic
71C2×C4×D73—21no—
72C2×C4×D73—21no—
73C2×C4×D73—81no—
74#11924007—24no—
75#11924007—41no—
76#11924007—81no—
77C22⋊Dic73—21no—
78C22⋊Dic73—41no—
79C22⋊Dic73—41no—
80C22⋊Dic73C2×C812yescomplemented
81#11924005—24no—
82#11924005—41no—
83#11924005—81no—
84C4×Dic73—21no—
85C4×Dic73—21no—
86C4×Dic73—81no—
87C24×C73C22×C411yescharacteristic
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73—41no—
96C22×C4×C73—81no—
97C22×C4×C73—81no—
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22⋊C411yescharacteristic
100C22×C4×C73C2×C812yescomplemented
101C4⋊C4×C73—42no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—41no—
104C23×C73—21no—
105C23×C73—21no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73C2×C22⋊C411yescharacteristic
109C23×C73C22×C811yescharacteristic
110C23×C73C2×M1611yescharacteristic
111C22×D73—21no—
112C22×D73—21no—
113C22×D73—41no—
114C22×D73—41no—
115C22×D73C22⋊C812yes—
116C2×C4×C73—22no—
117C2×C4×C73—21no—
118C2×C4×C73—41no—
119C2×C4×C73—42no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—42no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—81no—
128C2×C4×C73—81no—
129C2×C4×C73—82no—
130C2×C4×C73#9911yescharacteristic
131C2×C4×C73#9911yescharacteristic
132C22⋊D73—42no—
133C22⋊D73—42no—
134C2×Dic73—21no—
135C2×Dic73—21no—
136C2×Dic73—22no—
137C2×Dic73—21no—
138C2×Dic73—21no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73—41no—
144C2×Dic73—41no—
145C2×Dic73—41no—
146C2×Dic73C22⋊C812yes—
147C4×D73—42no—
148C4×D73—81no—
149#10596833—82no—
150#10596833—82no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73—21no—
155C22×C73—41no—
156C22×C73—41no—
157C22×C73—42no—
158C22×C73—41no—
159C22×C73—41no—
160C22×C73C2×C22⋊C811yescharacteristic
161C22×C73C2×#9911yescharacteristic
162C22×C73#41311yescharacteristic
163C2×D73—22no—
164C2×D73—41no—
165C2×D73—41no—
166C2×D73—42no—
167C2×D73—81no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—41no—
171C4×C73—42no—
172C4×C73—81no—
173Dic73—22no—
174Dic73—42no—
175Dic73—41no—
176Dic73—41no—
177Dic73—81no—
178#11474—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—21no—
181C2×C73—41no—
182C2×C73—41no—
183C2×C73#145911yescharacteristic
184C2×C73#212111yescharacteristic
185C2×C73#212311yescharacteristic
186D73—42no—
187#2322—731no—
188#1460—731no—
189#1459—731no—
190#1326—732no—
191#1317—732no—
192C73#1147411yescharacteristic, complemented, Sylow
193C2×#123—731no—
194C2×C4×D4—731no—
195C22×C4⋊C4#6440811yescharacteristic
196#431—1461no—
197C2×C22⋊C8—1461no—
198#389—1461no—
199#388—1461no—
200#387—1461no—
201#386—731no—
202#385—731no—
203#385—1461no—
204#381—731no—
205C4×C22⋊C4—731no—
206#324—1461no—
207#323—732no—
208#323—732no—
209#323—1461no—
210C22×D4—731no—
211C23×C4—731no—
212C23×C4C2×#6440811yescharacteristic
213C23×C4C2×#6440811yescharacteristic
214C23×C4C2×#6440811yescharacteristic
215C22×C8—2921no—
216#123—734no—
217#123—1462no—
218C4×D4—2921no—
219C4×D4—2921no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—731no—
225C2×C4⋊C4—1461no—
226C2×C4⋊C4—1461no—
227C2×C22⋊C4—731no—
228C2×C22⋊C4—731no—
229C2×C22⋊C4—731no—
230C2×C22⋊C4—731no—
231C2×C22⋊C4—1461no—
232C2×C22⋊C4—1461no—
233C2×C42—731no—
234C2×C42—1461no—
235C4⋊C8—2921no—
236C22⋊C8—2921no—
237C22⋊C8—1462no—
238C22⋊C8—1462no—
239C4×C8—2921no—
240#95—1461no—
241#95—1461no—
242#95—1461no—
243#95—1461no—
244#95—1461no—
245C24—731no—
246C24C22×#6440811yescharacteristic
247C2×D4—2921no—
248C2×D4—2921no—
249C2×D4—732no—
250C2×D4—732no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—5841no—
262C22×C4—731no—
263C22×C4—731no—
264C22×C4—731no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4—1461no—
268C22×C4—1461no—
269C22×C4C22⋊#6440811yescharacteristic
270C22×C4C22⋊#6440811yescharacteristic
271C22×C4#1192403612yescomplemented
272C2×C8—2922no—
273C2×C8—2922no—
274C2×C8—5842no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—2921no—
279C4⋊C4—5841no—
280C4⋊C4—1464no—
281C22⋊C4—2921no—
282C22⋊C4—2921no—
283C22⋊C4—2921no—
284C22⋊C4—5841no—
285C22⋊C4—732no—
286C22⋊C4—1464no—
287C22⋊C4—1461no—
288C42—5841no—
289C42—1461no—
290C42—1461no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—2921no—
296C23—2921no—
297C23—732no—
298C23—1461no—
299C23—1461no—
300C23C2×C22⋊#6440811yescharacteristic
301C23C2×#1192403611yescharacteristic
302C23D73⋊M1611yescharacteristic
303D4—2922no—
304D4—2922no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—2922no—
320C2×C4—2921no—
321C2×C4—2921no—
322C2×C4—2921no—
323C2×C4—2921no—
324C2×C4—2921no—
325C2×C4—2921no—
326C2×C4—2921no—
327C2×C4—5841no—
328C2×C4—732no—
329C2×C4—1462no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4—1461no—
333C2×C4—1461no—
334C2×C4#1455664911yescharacteristic
335C2×C4#1455664911yescharacteristic
336C8—5842no—
337C8—5842no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22—2922no—
348C22—2921no—
349C22—2921no—
350C22—5841no—
351C22—1462no—
352C22#1798955511yescharacteristic
353C22C2×#1455664911yescharacteristic
354C22#1798957311yescharacteristic
355C4—21no—
356C4—21no—
357C4—41no—
358C4—42no—
359C4—81no—
360C4—2922no—
361C4—2921no—
362C4—2921no—
363C4—5841no—
364C4—1462no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—41no—
369C2—2922no—
370C2#2262516611yescharacteristic
371C2#2262618611yescharacteristic
372C2#2262619311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
prime radicalC22×C22⋊D73C2×C4
cosocleC2×C22⋊Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#11474
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22624876C2semidirect
4. #22624857C2semidirect
32. #14556574C8semidirect
80. C22⋊Dic73C2×C8semidirect
100. C22×C4×C73C2×C8semidirect
192. C73#11474semidirect
271. C22×C4#11924036semidirect

Maximal subgroups

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

subgroupindex
#226251702
#226251662
#226248762
#226248572
#226126002
#1147473