Finite Group Databasebeta

← #34005261

Subgroups — #34005261

Order 18688.

514 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 114 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#22625787C211yesmaximal, characteristic
2#22625787C211yesmaximal, characteristic
3#22624809C212yesmaximal, complemented
4#22624809C212yesmaximal, complemented
5#22612333C211yesmaximal, characteristic
6C2×#14556658C2212yes—
7C2×#14556658C2212yes—
8C2×C22⋊#10596833—21no—
9C2×C22⋊#10596833—21no—
10#17989453—21no—
11#17989453—21no—
12C2×#14556547—21no—
13C22×C4×Dic73C2211yescharacteristic
14#17988456C2211yescharacteristic
15#17988456C2211yescharacteristic
16#17988451—21no—
17#17988447C411yescharacteristic
18#17988445—21no—
19C4⋊C4×Dic73C411yescharacteristic
20C4⋊C4×Dic73C411yescharacteristic
21#17988438—21no—
22#380×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25#14556658—24no—
26#14556658—24no—
27C22⋊#10596833—42no—
28C22⋊#10596833—42no—
29C23×Dic73D411yescharacteristic
30C23×Dic73D411yescharacteristic
31#14556547—44no—
32C2×C4⋊Dic73C2×C411yescharacteristic
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×#11924007—21no—
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×#11924007—21no—
38C2×C22⋊Dic73—21no—
39C2×C4×Dic73—24no—
40C2×C4×Dic73—21no—
41C2×C4×Dic73—21no—
42C2×C4×Dic73C2×C411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73C2311yescharacteristic
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C23×C4×C73C2×C411yescharacteristic
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73—21no—
53#95×C73—21no—
54C2×#10596833—42no—
55C2×#10596833—42no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C22×Dic73—22no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—22no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C22⋊C411yescharacteristic
70C22×Dic73SD1612yes—
71C22×Dic73Q1612yes—
72C22×Dic73C2×D411yescharacteristic
73C22×Dic73C2×D411yescharacteristic
74C4⋊Dic73—41no—
75C4⋊Dic73—41no—
76#11924007—42no—
77#11924007—41no—
78#11924007—42no—
79#11924007—42no—
80#11924007—41no—
81C22⋊Dic73—42no—
82C4×Dic73—22no—
83C4×Dic73—22no—
84C4×Dic73—22no—
85C4×Dic73—42no—
86C4×Dic73—42no—
87C4×Dic73#4114yes—
88C24×C73C22⋊C411yescharacteristic
89C22×C4×C73—22no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—41no—
93C22×C4×C73—41no—
94C22×C4×C73C22⋊C411yescharacteristic
95C22×C4×C73C22×C411yescharacteristic
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C22⋊C4×C73—42no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—21no—
105C23×C73#10312yes—
106C23×C73C2×C22⋊C411yescharacteristic
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—44no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73#11712yes—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—24no—
123C2×Dic73—24no—
124C2×Dic73—21no—
125C2×Dic73—21no—
126C2×Dic73—21no—
127C2×Dic73—42no—
128C2×Dic73—42no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73#10312yes—
136C2×Dic73#12312yes—
137C2×Dic73#12312yes—
138C2×Dic73C2×SD1611yescharacteristic
139C2×Dic73C2×Q1611yescharacteristic
140C2×Dic73#13512yes—
141C2×Dic73AGL(1,ℤ/8)12yes—
142C2×Dic73AGL(1,ℤ/8)11yescharacteristic
143C2×Dic73#13711yescharacteristic
144#10596833—82no—
145#10596833—82no—
146C22×C73—22no—
147C22×C73—22no—
148C22×C73—22no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73#38111yescharacteristic
154C22×C73#41311yescharacteristic
155C22×C73#41311yescharacteristic
156C22×C73C2×#10311yescharacteristic
157C22×C73#41611yescharacteristic
158C22×C73#41711yescharacteristic
159C22×C73#41811yescharacteristic
160C4×C73—22no—
161C4×C73—41no—
162C4×C73—81no—
163C4×C73—81no—
164Dic73—22no—
165Dic73—41no—
166Dic73—81no—
167Dic73—81no—
168Dic73#48112yes—
169Dic73#48212yes—
170Dic73#48412yes—
171Dic73#48512yes—
172#8727—731nomaximal, Sylow
173C2×C73—42no—
174C2×C73#154211yescharacteristic
175C2×C73#155011yescharacteristic
176C2×C73#155311yescharacteristic
177C2×C73#155711yescharacteristic
178C2×C73#182211yescharacteristic
179C2×C73#182411yescharacteristic
180C2×C73#182511yescharacteristic
181#2285—731no—
182#1797—731no—
183#1797—731no—
184#1296—732no—
185#1296—732no—
186C73#872711yescharacteristic, complemented, Sylow
187C2×#117—1461no—
188C22×C42—731no—
189C2×C4⋊C8—732no—
190C2×C4⋊C8—732no—
191C2×C22⋊C8—1461no—
192C2×C22⋊C8—1461no—
193#384—731no—
194#384—731no—
195#382—1461no—
196#380—731no—
197#380—1461no—
198#380#6440811yescharacteristic
199C4×C4⋊C4—731no—
200C4×C4⋊C4—731no—
201C4×C4⋊C4—1461no—
202#336—1461no—
203#336—1461no—
204C23×C4—731no—
205C23×C4—731no—
206C23×C4C2×#6440811yescharacteristic
207C22×C8—1462no—
208C22×C8—1462no—
209#117—2924no—
210C2×C4⋊C4—731no—
211C2×C4⋊C4—731no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4—1461no—
214C2×C4⋊C4—1461no—
215C2×C4⋊C4—1461no—
216C2×C4⋊C4C2×#6440811yescharacteristic
217C2×C4⋊C4C2×#6440811yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—1461no—
220C2×C42—731no—
221C2×C42—731no—
222C2×C42—731no—
223C2×C42—731no—
224C2×C42—1461no—
225C2×C42—1461no—
226C2×C42—1464no—
227C4⋊C8—1464no—
228C4⋊C8—1464no—
229C22⋊C8—2922no—
230C22⋊C8—2922no—
231#95—21no—
232#95—1461no—
233#95—1461no—
234#95—1461no—
235C24C22⋊#6440811yescharacteristic
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—731no—
242C22×C4—731no—
243C22×C4—732no—
244C22×C4—732no—
245C22×C4—731no—
246C22×C4—731no—
247C22×C4—1461no—
248C22×C4—1461no—
249C22×C4—1461no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4—1461no—
253C22×C4—1462no—
254C22×C4—1462no—
255C22×C4C22⋊#6440811yescharacteristic
256C22×C4C22×#6440811yescharacteristic
257C2×C8—2922no—
258C2×C8—2922no—
259C2×C8—2922no—
260C2×C8—2922no—
261C2×C8—2922no—
262C2×C8—2922no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—2921no—
266C4⋊C4—2921no—
267C4⋊C4—2922no—
268C4⋊C4—2921no—
269C4⋊C4—2921no—
270C4⋊C4—2922no—
271C4⋊C4—2922no—
272C22⋊C4—42no—
273C22⋊C4—2922no—
274C42—2922no—
275C42—2922no—
276C42—734no—
277C42—1462no—
278C42—1462no—
279C42—1462no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23#1455665412yes—
287C23C2×C22⋊#6440811yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—44no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—2922no—
301C2×C4—2922no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—732no—
309C2×C4—732no—
310C2×C4—731no—
311C2×C4—731no—
312C2×C4—732no—
313C2×C4—731no—
314C2×C4—731no—
315C2×C4—732no—
316C2×C4—732no—
317C2×C4—1461no—
318C2×C4—1461no—
319C2×C4—1461no—
320C2×C4—1464no—
321C2×C4—1464no—
322C2×C4—1461no—
323C2×C4—1461no—
324C2×C4#1455667012yes—
325C8—5842no—
326C8—5842no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22#1798953011yescharacteristic
335C22#1798957411yescharacteristic
336C22#1798957411yescharacteristic
337C22C2×#1455665411yescharacteristic
338C22#1798957711yescharacteristic
339C22#1798957811yescharacteristic
340C22#1798957911yescharacteristic
341C4—22no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—2921no—
346C4—5841no—
347C4—5841no—
348C4—732no—
349C4—732no—
350C4—732no—
351C4—732no—
352C4—1462no—
353C2—42no—
354C2#2262543311yescharacteristic
355C2#2262544911yescharacteristic
356C2#2262545511yescharacteristic
357C2#2262546311yescharacteristic
358C2#2262583311yescharacteristic
359C2#2262583511yescharacteristic
360C2#2262583611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#380×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#380#64408
hypocenterC73#8727
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #22624809C2semidirect
4. #22624809C2semidirect
186. C73#8727semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226248092
#226123332
#872773