Finite Group Databasebeta

← #34005255

Subgroups — #34005255

Order 18688.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#22625788C211yesmaximal, characteristic
3#22624809C214yesmaximal, complemented
4#384×D73C211yesmaximal, characteristic
5C2×#14556658C2212yes—
6C2×#14556658C2212yes—
7C2×#14556648—21no—
8C2×#14556648—21no—
9#17989453—21no—
10#17989453—21no—
11C2×C4⋊C4×D73—21no—
12C2×C4⋊C4×D73—21no—
13C2×C4⋊C4×D73C412yes—
14C2×C42×D73C2211yescharacteristic
15#17988457C411yescharacteristic
16#17988456—21no—
17#17988456—21no—
18#17988456C2212yes—
19#384×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22#14556658—24no—
23#14556658—24no—
24#14556648—42no—
25#14556648—42no—
26C22×C4×D73—22no—
27C22×C4×D73C2×C411yescharacteristic
28C22×C4×D73D411yescharacteristic
29C22×C4×D73D411yescharacteristic
30C2×C4⋊Dic73—21no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73C2×C412yes—
33C2×#11924007—22no—
34C2×#11924007—22no—
35C2×#11924007—22no—
36C4⋊C4×D73—44no—
37C4⋊C4×D73—42no—
38C4⋊C4×D73—42no—
39C4⋊C4×D73—44no—
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73C2311yescharacteristic
43C42×D73—24no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C412yes—
47C2×C42×C73C2×C411yescharacteristic
48C2×#10596833—42no—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×#10596833—42no—
54C23×D73C22⋊C411yescharacteristic
55C22×Dic73—22no—
56C22×Dic73C22⋊C411yescharacteristic
57C22×Dic73C2×D411yescharacteristic
58C22×Dic73C2×D411yescharacteristic
59C2×C4×D73—22no—
60C2×C4×D73—22no—
61C2×C4×D73—22no—
62C2×C4×D73—22no—
63C2×C4×D73—22no—
64C2×C4×D73—22no—
65C2×C4×D73—22no—
66C2×C4×D73—44no—
67C2×C4×D73—44no—
68C2×C4×D73—44no—
69C2×C4×D73D812yes—
70C2×C4×D73SD1612yes—
71C2×C4×D73SD1612yes—
72C2×C4×D73Q1612yes—
73C4⋊Dic73—42no—
74C4⋊Dic73—42no—
75C4⋊Dic73—42no—
76#11924007—44no—
77#11924007—42no—
78#11924007—42no—
79#11924007—44no—
80C4×Dic73—22no—
81C4×Dic73—22no—
82C4×Dic73#4114yes—
83C22×C4×C73—22no—
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22×C411yescharacteristic
87C4⋊C4×C73—42no—
88C4⋊C4×C73—42no—
89C4⋊C4×C73—42no—
90C42×C73—22no—
91C23×C73C2×C22⋊C411yescharacteristic
92C22×D73—21no—
93C22×D73—21no—
94C22×D73—21no—
95C22×D73—21no—
96C22×D73#10112yes—
97C22×D73#10212yes—
98C22×D73#10312yes—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73#10212yes—
109C2×C4×C73#10312yes—
110C2×C4×C73#11712yes—
111C2×Dic73—21no—
112C2×Dic73—21no—
113C2×Dic73—42no—
114C2×Dic73—42no—
115C2×Dic73—42no—
116C2×Dic73#10012yes—
117C2×Dic73#12312yes—
118C2×Dic73#12312yes—
119C2×Dic73C2×D811yescharacteristic
120C2×Dic73C2×SD1611yescharacteristic
121C2×Dic73C2×SD1611yescharacteristic
122C2×Dic73C2×Q1611yescharacteristic
123C2×Dic73AGL(1,ℤ/8)12yes—
124C2×Dic73#13712yes—
125C4×D73—28no—
126C4×D73—24no—
127C4×D73—24no—
128C4×D73—24no—
129C4×D73—24no—
130C4×D73—88no—
131#10596833—82no—
132#10596833—82no—
133C22×C73#38111yescharacteristic
134C22×C73C2×#10011yescharacteristic
135C22×C73C2×#10111yescharacteristic
136C22×C73C2×#10211yescharacteristic
137C22×C73C2×#10311yescharacteristic
138C22×C73#41811yescharacteristic
139C22×C73#41911yescharacteristic
140C2×D73—22no—
141C2×D73—22no—
142C2×D73—22no—
143C2×D73—22no—
144C2×D73—22no—
145C2×D73—22no—
146C2×D73#33214yes—
147C4×C73—22no—
148C4×C73—22no—
149C4×C73—22no—
150C4×C73—82no—
151Dic73—22no—
152Dic73—82no—
153Dic73#48012yes—
154Dic73#48112yes—
155Dic73#48312yes—
156Dic73#48412yes—
157#8724—731nomaximal, Sylow
158C2×C73C2×#33211yescharacteristic
159C2×C73#154511yescharacteristic
160C2×C73#154811yescharacteristic
161C2×C73#155611yescharacteristic
162C2×C73#182111yescharacteristic
163C2×C73#182311yescharacteristic
164C2×C73#182411yescharacteristic
165D73—24no—
166C2×#384—731no—
167#1797—731no—
168#1797—731no—
169#1296—734no—
170C73#872411yescharacteristic, complemented, Sylow
171C22×C4⋊C4—732no—
172C22×C4⋊C4—1461no—
173C22×C4⋊C4—1461no—
174C22×C42—731no—
175C2×C4⋊C8—732no—
176C2×C4⋊C8—732no—
177C2×C22⋊C8—1461no—
178C2×C22⋊C8—1461no—
179#384—732no—
180#384—731no—
181#384—1461no—
182#384—1461no—
183#384#6440811yescharacteristic
184#336—1461no—
185#336—1461no—
186C23×C4—731no—
187C23×C4—731no—
188C23×C4—731no—
189C23×C4—1462no—
190C22×C8—1462no—
191C22×C8—1462no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—2922no—
195C2×C4⋊C4—2922no—
196C2×C4⋊C4—2924no—
197C2×C4⋊C4—2924no—
198C2×C4⋊C4—732no—
199C2×C4⋊C4—1462no—
200C2×C4⋊C4—1462no—
201C2×C4⋊C4—1462no—
202C2×C4⋊C4—1461no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4C2×#6440812yes—
205C2×C42—731no—
206C2×C42—731no—
207C2×C42—731no—
208C2×C42—1464no—
209C2×C42C2×#6440811yescharacteristic
210C4⋊C8—1464no—
211C4⋊C8—1464no—
212C22⋊C8—2922no—
213C22⋊C8—2922no—
214C24—731no—
215C22×C4—22no—
216C22×C4—2924no—
217C22×C4—2924no—
218C22×C4—2924no—
219C22×C4—731no—
220C22×C4—731no—
221C22×C4—731no—
222C22×C4—732no—
223C22×C4—732no—
224C22×C4—732no—
225C22×C4—732no—
226C22×C4—1462no—
227C22×C4—1462no—
228C22×C4—1462no—
229C22×C4—1462no—
230C22×C4—1462no—
231C22×C4—1462no—
232C22×C4—1462no—
233C22×C4—1462no—
234C22×C4C22⋊#6440811yescharacteristic
235C22×C4C22⋊#6440811yescharacteristic
236C22×C4C22×#6440811yescharacteristic
237C2×C8—2922no—
238C2×C8—2922no—
239C2×C8—2922no—
240C2×C8—2922no—
241C2×C8—2922no—
242C2×C8—2922no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—2922no—
247C4⋊C4—2922no—
248C4⋊C4—2922no—
249C4⋊C4—2922no—
250C4⋊C4—2924no—
251C4⋊C4—2924no—
252C4⋊C4—2922no—
253C42—22no—
254C42—734no—
255C42—1462no—
256C42—1462no—
257C23—732no—
258C23—732no—
259C23—732no—
260C23—1461no—
261C23—1461no—
262C23—1461no—
263C23—1461no—
264C23C2×C22⋊#6440811yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—2922no—
275C2×C4—2922no—
276C2×C4—2922no—
277C2×C4—5848no—
278C2×C4—732no—
279C2×C4—732no—
280C2×C4—731no—
281C2×C4—731no—
282C2×C4—731no—
283C2×C4—731no—
284C2×C4—732no—
285C2×C4—732no—
286C2×C4—732no—
287C2×C4—1464no—
288C2×C4—1464no—
289C2×C4—1464no—
290C2×C4—1464no—
291C2×C4—1468no—
292C2×C4—1461no—
293C2×C4—1461no—
294C2×C4#1455665312yes—
295C2×C4#1455665412yes—
296C2×C4#1455667012yes—
297C8—5842no—
298C8—5842no—
299C22—734no—
300C22—1462no—
301C22—1462no—
302C22—1462no—
303C22—1462no—
304C22—1462no—
305C22—1462no—
306C22#1798953011yescharacteristic
307C22D73⋊#10011yescharacteristic
308C22D73⋊#10111yescharacteristic
309C22C2×#1455665311yescharacteristic
310C22C2×#1455665411yescharacteristic
311C22#1798957911yescharacteristic
312C22#1798958011yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—82no—
317C4—5842no—
318C4—732no—
319C4—732no—
320C4—732no—
321C4—732no—
322C4—1462no—
323C2—1464no—
324C2D73⋊#33211yescharacteristic
325C2#2262543911yescharacteristic
326C2#2262544511yescharacteristic
327C2#2262546111yescharacteristic
328C2#2262583111yescharacteristic
329C2#2262583411yescharacteristic
330C2#2262583511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#384#64408
hypocenterC73#8724
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #22624809C2semidirect
170. C73#8724semidirect

Maximal subgroups

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

subgroupindex
#226257882
#226248092
#384×D732
#872473