Finite Group Databasebeta

← #34001696

Subgroups — #34001696

Order 18688.

438 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 96 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#22625859C211yesmaximal, characteristic, complemented
2C4×#14556658C211yesmaximal, characteristic, complemented
3#22624749C212yesmaximal, complemented
4#22624747C212yesmaximal
5#22618295C211yesmaximal, characteristic
6C2×#14556658—21no—
7C2×C4×#10596833—21no—
8#17989453—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×#14556584—21no—
13#17988490C411yescharacteristic
14#17988490C2211yescharacteristic
15#17988486C2211yescharacteristic
16#17988485C411yescharacteristic
17#17988483C411yescharacteristic
18#17988481—21no—
19#17988480—21no—
20#17988475—21no—
21C42×Dic73C2211yescharacteristic
22#390×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25#14556658—24no—
26C4×#10596833—42no—
27#14556584—42no—
28#14556584—42no—
29C2×C4⋊Dic73—21no—
30C2×C4⋊Dic73C2×C411yescharacteristic
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×#11924007—21no—
33C2×#11924007—21no—
34C2×C22⋊Dic73—21no—
35C2×C22⋊Dic73—21no—
36C2×C22⋊Dic73—21no—
37C2×C22⋊Dic73—21no—
38C2×C4×Dic73C2×C411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73C2311yescharacteristic
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47C22×D4×C73C2×C411yescharacteristic
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C42×C73C2×C411yescharacteristic
52C2×#10596833—24no—
53C2×#10596833—42no—
54C2×#10596833—42no—
55C2×#10596833—42no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C22⋊C411yescharacteristic
62C22×Dic73C2×D411yescharacteristic
63C22×Dic73C2×D411yescharacteristic
64C4⋊Dic73—41no—
65C4⋊Dic73—41no—
66C4⋊Dic73—41no—
67C4⋊Dic73—41no—
68#11924007—42no—
69#11924007—42no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C22⋊Dic73—41no—
73C22⋊Dic73—41no—
74C22⋊Dic73—42no—
75C4×Dic73—22no—
76C4×Dic73—22no—
77C4×Dic73—22no—
78C4×Dic73—41no—
79C4×Dic73—41no—
80C4×Dic73D812yes—
81C4×Dic73SD1612yes—
82C24×C73—21no—
83C2×D4×C73—22no—
84C2×D4×C73—41no—
85C2×D4×C73—41no—
86C22×C4×C73—21no—
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22×C411yescharacteristic
90C4⋊C4×C73—22no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C42×C73—41no—
94C23×C73—41no—
95C23×C73—41no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73C2×C22⋊C411yescharacteristic
102D4×C73—82no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73#10212yes—
114C2×C4×C73#13112yes—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73#10212yes—
130C2×Dic73#10412yes—
131C2×Dic73C2×D811yescharacteristic
132C2×Dic73C2×SD1611yescharacteristic
133C2×Dic73#13512yes—
134#10596833—44no—
135#10596833—82no—
136C22×C73—81no—
137C22×C73—81no—
138C22×C73—81no—
139C22×C73—81no—
140C22×C73—81no—
141C22×C73—81no—
142C22×C73—81no—
143C22×C73#40811yescharacteristic
144C22×C73#41011yescharacteristic
145C22×C73#41311yescharacteristic
146C22×C73C2×#10211yescharacteristic
147C22×C73#41611yescharacteristic
148C22×C73C2×#10411yescharacteristic
149C22×C73#42111yescharacteristic
150C4×C73—22no—
151C4×C73—22no—
152C4×C73—41no—
153C4×C73—81no—
154Dic73—22no—
155Dic73—22no—
156Dic73—21no—
157Dic73—21no—
158Dic73—22no—
159Dic73—81no—
160Dic73—81no—
161#7672—731nomaximal, Sylow
162C2×C73—161no—
163C2×C73#147311yescharacteristic
164C2×C73#148011yescharacteristic
165C2×C73#149411yescharacteristic
166C2×C73#150211yescharacteristic
167C2×C73#152111yescharacteristic
168C2×C73#152911yescharacteristic
169C2×C73#154011yescharacteristic
170#2860—731no—
171#1839—731no—
172C4×C4⋊C8—731no—
173#1271—732no—
174#1270—732no—
175C73#767211yescharacteristic, complemented, Sylow
176C2×#126—1461no—
177C2×C4⋊C8—1461no—
178C2×C4×C8—1461no—
179#390—731no—
180#390—731no—
181#390#6440811yescharacteristic
182#389—731no—
183#389—731no—
184#389—731no—
185#388—1461no—
186#388—1461no—
187#388—1461no—
188C43—731no—
189#336—732no—
190#336—732no—
191#336—1461no—
192#336—1461no—
193C22×D4C2×#6440811yescharacteristic
194C22×C8—1462no—
195C22×C8—1462no—
196#126—2922no—
197#126—2922no—
198C2×C4⋊C4—731no—
199C2×C4⋊C4—731no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4—1461no—
202C2×C4⋊C4—1461no—
203C2×C4⋊C4C2×#6440811yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—1461no—
207C2×C22⋊C4—1461no—
208C2×C22⋊C4—1461no—
209C2×C22⋊C4—1461no—
210C2×C42—731no—
211C2×C42—731no—
212C2×C42—731no—
213C2×C42—731no—
214C2×C42—731no—
215C2×C42—731no—
216C2×C42C2×#6440811yescharacteristic
217C4⋊C8—1464no—
218C4×C8—2922no—
219#95—1461no—
220#95—1461no—
221#95—1461no—
222C24—21no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—21no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—731no—
230C22×C4—731no—
231C22×C4—1461no—
232C22×C4—1461no—
233C22×C4C22⋊#6440811yescharacteristic
234C22×C4C22⋊#6440811yescharacteristic
235C22×C4C22×#6440811yescharacteristic
236C2×C8—2922no—
237C2×C8—2922no—
238C2×C8—2922no—
239C2×C8—2922no—
240C2×C8—2922no—
241C2×C8—1464no—
242C4⋊C4—22no—
243C4⋊C4—2921no—
244C4⋊C4—2921no—
245C4⋊C4—2922no—
246C4⋊C4—2921no—
247C4⋊C4—2921no—
248C4⋊C4—2922no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—2922no—
252C22⋊C4—2922no—
253C22⋊C4—2921no—
254C22⋊C4—2922no—
255C22⋊C4—2921no—
256C42—41no—
257C42—2921no—
258C42—2921no—
259C42—732no—
260C42—732no—
261C42—1462no—
262C42—1462no—
263C42—1462no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C2×C22⋊#6440811yescharacteristic
272D4—82no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—2921no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—731no—
290C2×C4—732no—
291C2×C4—731no—
292C2×C4—732no—
293C2×C4—732no—
294C2×C4—1461no—
295C2×C4—1461no—
296C2×C4—1461no—
297C2×C4—1461no—
298C2×C4—1461no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4#1455665312yes—
303C2×C4#1455667812yes—
304C8—2924no—
305C8—5842no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1798956111yescharacteristic
314C22#1798956611yescharacteristic
315C22#1798957411yescharacteristic
316C22C2×#1455665311yescharacteristic
317C22#1798957711yescharacteristic
318C22D73⋊#10411yescharacteristic
319C22#1798958511yescharacteristic
320C4—22no—
321C4—22no—
322C4—41no—
323C4—81no—
324C4—5841no—
325C4—5841no—
326C4—1462no—
327C4—1461no—
328C4—1461no—
329C4—1462no—
330C4—1462no—
331C2—161no—
332C2#2262521211yescharacteristic
333C2#2262523711yescharacteristic
334C2#2262528311yescharacteristic
335C2#2262531011yescharacteristic
336C2#2262537311yescharacteristic
337C2#2262539711yescharacteristic
338C2#2262542911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#390×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC22×D4×C73C2×C4
cosocleC2×C4×Dic73C23
hypercenter#390#64408
hypocenterC73#7672
LCSG ▷ C22×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. #22625859C2semidirect
2. C4×#14556658C2semidirect
3. #22624749C2semidirect
175. C73#7672semidirect

Maximal subgroups

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

subgroupindex
#226258592
C4×#145566582
#226247492
#226247472
#226182952
#767273