Finite Group Databasebeta

← #34001351

Subgroups — #34001351

Order 18688.

520 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 118 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#22625847C211yesmaximal, characteristic, complemented
2C4×#14556658C211yesmaximal, characteristic, complemented
3#22624750C212yesmaximal, complemented
4#22624747C212yesmaximal
5#22612393C211yesmaximal, characteristic
6C2×#14556658—21no—
7C2×#14556658—21no—
8C2×C4×#10596833—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×D4×Dic73C411yescharacteristic
13C2×#14556547—21no—
14#17988490C2211yescharacteristic
15#17988485C411yescharacteristic
16#17988483C2211yescharacteristic
17#17988481—21no—
18#17988475—21no—
19C4⋊C4×Dic73C411yescharacteristic
20C22⋊C4×Dic73—21no—
21C42×Dic73C2211yescharacteristic
22#390×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25#14556658—24no—
26#14556658—42no—
27C4×#10596833—42no—
28C23×Dic73—21no—
29D4×Dic73—22no—
30D4×Dic73C814yescomplemented
31#14556547—42no—
32#14556547—42no—
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×C22⋊Dic73—21no—
38C2×C22⋊Dic73—21no—
39C2×C22⋊Dic73—21no—
40C2×C4×Dic73—21no—
41C2×C4×Dic73C2×C411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73D411yescharacteristic
46C2×C4×Dic73C2311yescharacteristic
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50C22×D4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C42×C73C2×C411yescharacteristic
55C2×#10596833—24no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C22×Dic73—21no—
62C22×Dic73—24no—
63C22×Dic73—21no—
64C22×Dic73—42no—
65C22×Dic73—42no—
66C22×Dic73C22⋊C411yescharacteristic
67C22×Dic73C22⋊C411yescharacteristic
68C22×Dic73C2×D411yescharacteristic
69C22×Dic73C2×D411yescharacteristic
70C4⋊Dic73—21no—
71C4⋊Dic73—41no—
72C4⋊Dic73C2×C812yes—
73#11924007—42no—
74#11924007—41no—
75#11924007—41no—
76C22⋊Dic73—24no—
77C22⋊Dic73—41no—
78C22⋊Dic73—41no—
79C22⋊Dic73—41no—
80C22⋊Dic73—41no—
81C4×Dic73—22no—
82C4×Dic73—21no—
83C4×Dic73—21no—
84C4×Dic73—41no—
85C4×Dic73—41no—
86C4×Dic73—42no—
87C4×Dic73C2×C812yes—
88C4×Dic73M1612yes—
89C4×Dic73D812yes—
90C4×Dic73SD1612yes—
91C24×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73—41no—
94C2×D4×C73—41no—
95C2×D4×C73C2×C812yes—
96C22×C4×C73—21no—
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22×C411yescharacteristic
100C4⋊C4×C73—41no—
101C22⋊C4×C73—42no—
102C22⋊C4×C73—42no—
103C42×C73—41no—
104C23×C73—22no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73—41no—
111C23×C73C2×C22⋊C411yescharacteristic
112D4×C73—42no—
113D4×C73—42no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
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#10212yes—
125C2×C4×C73C22×C811yescharacteristic
126C2×C4×C73C2×M1611yescharacteristic
127C2×Dic73—21no—
128C2×Dic73—21no—
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—22no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—48no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73C22⋊C812yes—
141C2×Dic73C22⋊C812yes—
142C2×Dic73#10212yes—
143C2×Dic73#10412yes—
144C2×Dic73C2×D811yescharacteristic
145C2×Dic73C2×SD1611yescharacteristic
146C2×Dic73AGL(1,ℤ/8)11yescharacteristic
147C2×Dic73#13711yescharacteristic
148#10596833—44no—
149#10596833—82no—
150C22×C73—42no—
151C22×C73—42no—
152C22×C73—42no—
153C22×C73—81no—
154C22×C73—81no—
155C22×C73—81no—
156C22×C73—81no—
157C22×C73C2×C22⋊C811yescharacteristic
158C22×C73#41011yescharacteristic
159C22×C73#41311yescharacteristic
160C22×C73C2×#10211yescharacteristic
161C22×C73#41711yescharacteristic
162C22×C73C2×#10411yescharacteristic
163C22×C73#42111yescharacteristic
164C4×C73—22no—
165C4×C73—21no—
166C4×C73—21no—
167C4×C73—41no—
168C4×C73—81no—
169Dic73—22no—
170Dic73—21no—
171Dic73—21no—
172Dic73—21no—
173Dic73—21no—
174Dic73—42no—
175Dic73—81no—
176Dic73D4⋊C814yes—
177#7574—731nomaximal, Sylow
178C2×C73—82no—
179C2×C73#145811yescharacteristic
180C2×C73C2×D4⋊C811yescharacteristic
181C2×C73#147711yescharacteristic
182C2×C73#150911yescharacteristic
183C2×C73#151711yescharacteristic
184C2×C73#152911yescharacteristic
185C2×C73#154711yescharacteristic
186#2293—731no—
187#1831—731no—
188C4×C4⋊C8—731no—
189#1271—732no—
190#1270—732no—
191C73#757411yescharacteristic, complemented, Sylow
192C2×C4×D4—731no—
193C2×#117—1461no—
194C2×C4⋊C8—1461no—
195C2×C4⋊C8—1461no—
196C2×C4×C8—1461no—
197#390—731no—
198#390#6440811yescharacteristic
199#389—731no—
200#389—731no—
201#388—1461no—
202#388—1461no—
203C4×C4⋊C4—731no—
204C4×C22⋊C4—1461no—
205C43—731no—
206#336—732no—
207#336—732no—
208#336—1461no—
209C22×D4C2×#6440811yescharacteristic
210C23×C4—1461no—
211C22×C8—1462no—
212C22×C8—1462no—
213C4×D4—734no—
214C4×D4—1462no—
215#117—2922no—
216#117—2922no—
217C2×C4⋊C4—731no—
218C2×C4⋊C4—731no—
219C2×C4⋊C4—1461no—
220C2×C4⋊C4—1461no—
221C2×C4⋊C4C2×#6440811yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—1461no—
225C2×C22⋊C4—1461no—
226C2×C22⋊C4—1461no—
227C2×C42—731no—
228C2×C42—731no—
229C2×C42—731no—
230C2×C42—731no—
231C2×C42—731no—
232C2×C42—731no—
233C2×C42—1461no—
234C2×C42C2×#6440811yescharacteristic
235C4⋊C8—2922no—
236C4⋊C8—1464no—
237C4×C8—2922no—
238#95—1461no—
239#95—1461no—
240#95—1461no—
241C24—21no—
242C2×D4—21no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4#1192403612yes—
246C22×C4—21no—
247C22×C4—2922no—
248C22×C4—2922no—
249C22×C4—731no—
250C22×C4—731no—
251C22×C4—731no—
252C22×C4—731no—
253C22×C4—1461no—
254C22×C4—1464no—
255C22×C4—1461no—
256C22×C4C22⋊#6440811yescharacteristic
257C22×C4C22⋊#6440811yescharacteristic
258C22×C4C22×#6440811yescharacteristic
259C2×C8—2922no—
260C2×C8—2922no—
261C2×C8—2922no—
262C2×C8—2922no—
263C2×C8—2922no—
264C2×C8—1464no—
265C4⋊C4—41no—
266C4⋊C4—2921no—
267C4⋊C4—2921no—
268C4⋊C4—2921no—
269C4⋊C4—2922no—
270C4⋊C4—732no—
271C4⋊C4—1461no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—2921no—
275C22⋊C4—2921no—
276C22⋊C4—2921no—
277C22⋊C4—2921no—
278C22⋊C4—1464no—
279C42—41no—
280C42—2921no—
281C42—2921no—
282C42—2922no—
283C42—732no—
284C42—732no—
285C42—732no—
286C42—732no—
287C42—1461no—
288C42—1462no—
289C42—1461no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊#6440811yescharacteristic
298D4—42no—
299D4—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—2921no—
311C2×C4—2921no—
312C2×C4—2928no—
313C2×C4—2921no—
314C2×C4—2921no—
315C2×C4—2921no—
316C2×C4—732no—
317C2×C4—732no—
318C2×C4—732no—
319C2×C4—732no—
320C2×C4—731no—
321C2×C4—731no—
322C2×C4—731no—
323C2×C4—731no—
324C2×C4—1462no—
325C2×C4—1461no—
326C2×C4—1461no—
327C2×C4—1461no—
328C2×C4—1461no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4#1455665312yes—
332C2×C4C2×#1192403611yescharacteristic
333C2×C4D73⋊M1611yescharacteristic
334C8—2924no—
335C8—5842no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#1798955511yescharacteristic
344C22#1798956711yescharacteristic
345C22#1798957411yescharacteristic
346C22C2×#1455665311yescharacteristic
347C22#1798957811yescharacteristic
348C22D73⋊#10411yescharacteristic
349C22#1798958511yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—41no—
354C4—81no—
355C4—2922no—
356C4—5841no—
357C4—734no—
358C4—1461no—
359C4—1461no—
360C4—1461no—
361C4—1461no—
362C4—1462no—
363C2—82no—
364C2#2262516311yescharacteristic
365C2#2262521711yescharacteristic
366C2#2262522711yescharacteristic
367C2#2262533211yescharacteristic
368C2#2262536011yescharacteristic
369C2#2262539711yescharacteristic
370C2#2262544311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#390×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC22×D4×C73C2×C4
cosocleC2×C4×Dic73C23
hypercenter#390#64408
hypocenterC73#7574
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #22625847C2semidirect
2. C4×#14556658C2semidirect
3. #22624750C2semidirect
30. D4×Dic73C8semidirect
191. C73#7574semidirect

Maximal subgroups

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

subgroupindex
#226258472
C4×#145566582
#226247502
#226247472
#226123932
#757473