Finite Group Databasebeta

← #32989226

Subgroups — #32989226

Order 18688.

594 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 75 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#22615601C211yesmaximal, characteristic, complemented
2#22609875C212yesmaximal
3#22607839C211yesmaximal, characteristic, complemented
4#22607783C212yesmaximal, complemented
5C73×#1782C211yesmaximal, characteristic
6C2×#14556518—22no—
7C2×#14556518—22no—
8#17988551C2212yes—
9#17988510—22no—
10#17988484—22no—
11#17988470—22no—
12C4×C22⋊Dic73C2212yes—
13C22×#41×C73C2211yescharacteristic
14C2×#99×C73—22no—
15#381×C73C2211yescharacteristic
16#381×C73C2211yescharacteristic
17C2×#11924012—22no—
18C2×C4⋊Dic73—22no—
19C2×#11924007—22no—
20C2×#11924007—22no—
21C2×C22⋊Dic73—22no—
22C2×C4×Dic73D412yes—
23C2×C4×Dic73D412yes—
24#14556518—44no—
25#14556518—44no—
26C22⋊#10596831—22no—
27C22⋊#10596831—22no—
28#14556498—22no—
29#14556498—22no—
30C2×#41×C73—22no—
31C2×#41×C73—22no—
32C2×#41×C73—44no—
33C22×Q8×C73Q811yescharacteristic
34C22×D4×C73—21no—
35C22×D4×C73Q811yescharacteristic
36C23×C4×C73—21no—
37C23×C4×C73C2311yescharacteristic
38C2×C22⋊C4×C73—21no—
39C2×C22⋊C4×C73D412yes—
40#99×C73—44no—
41#95×C73—22no—
42#95×C73—22no—
43C22×Dic73—22no—
44C22×Dic73—22no—
45#11924012—42no—
46#11924012—42no—
47C2×#10596831—42no—
48C4⋊Dic73—44no—
49#11924007—44no—
50#11924007—82no—
51C22⋊Dic73—44no—
52C4×Dic73—42no—
53C4×Dic73—42no—
54C24×C73—21no—
55C24×C73C2×D411yescharacteristic
56#41×C73—24no—
57#41×C73—24no—
58#41×C73—42no—
59#41×C73—42no—
60#41×C73—44no—
61#41×C73—44no—
62C2×Q8×C73—22no—
63C2×Q8×C73—42no—
64C2×D4×C73—24no—
65C2×D4×C73—22no—
66C2×D4×C73—42no—
67C2×D4×C73—42no—
68C2×D4×C73—41no—
69C2×D4×C73—41no—
70C22×C4×C73—21no—
71C22×C4×C73—22no—
72C22×C4×C73—42no—
73C22×C4×C73—42no—
74C22×C4×C73—42no—
75C22×C4×C73—42no—
76C22×C4×C73—41no—
77C22×C4×C73C2×D412yes—
78C22×C4×C73C2×Q811yescharacteristic
79C22×C4×C73#4112yes—
80C22×C4×C73#4111yescharacteristic
81C22⋊C4×C73—42no—
82C22⋊C4×C73—42no—
83C23×C73—22no—
84C23×C73—22no—
85C23×C73—42no—
86C23×C73—41no—
87C23×C73—41no—
88C23×C73—82no—
89C23×C73#12011yescharacteristic
90C23×C73C22⋊Q811yescharacteristic
91C23×C73#12411yescharacteristic
92Q8×C73—24no—
93Q8×C73—42no—
94D4×C73—24no—
95D4×C73—42no—
96D4×C73—44no—
97D4×C73—44no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—22no—
101C2×C4×C73—41no—
102C2×C4×C73—41no—
103C2×C4×C73—42no—
104C2×C4×C73—44no—
105C2×C4×C73—44no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73—42no—
111C2×C4×C73—81no—
112C2×C4×C73—82no—
113C2×C4×C73C22⋊Q812yes—
114C2×C4×C73#12412yes—
115C2×Dic73—24no—
116C2×Dic73—42no—
117C2×Dic73—42no—
118C2×Dic73—42no—
119C2×Dic73—42no—
120C2×Dic73—42no—
121#10596831—82no—
122C22×C73—22no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—22no—
126C22×C73—22no—
127C22×C73—42no—
128C22×C73—82no—
129C22×C73—82no—
130C22×C73—82no—
131C22×C73—82no—
132C22×C73#39311yescharacteristic
133C22×C73B(4,𝔽2)12yes—
134C4×C73—22no—
135C4×C73—22no—
136C4×C73—42no—
137C4×C73—82no—
138C4×C73—161no—
139Dic73—44no—
140Dic73—82no—
141#10388—731nomaximal, Sylow
142C2×C73—22no—
143C2×C73—41no—
144C2×C73—41no—
145C2×C73—82no—
146C2×C73#202911yescharacteristic
147C2×C73#203112yes—
148#2608—731no—
149#1789—732no—
150#1782D7311yescharacteristic
151#1306—731no—
152#1284—732no—
153C73#1038811yescharacteristic, complemented, Sylow
154C22×#41C2×D7311yescharacteristic
155C2×#104—1462no—
156C2×#104—1462no—
157C2×#99—22no—
158#407—732no—
159#396—1462no—
160#389—1462no—
161#387—1462no—
162#381C2×D7311yescharacteristic
163#381C2×D7311yescharacteristic
164C4×C22⋊C4—732no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—44no—
168C22×Q8Dic14611yescharacteristic
169C22×D4—21no—
170C22×D4Dic14611yescharacteristic
171C23×C4—21no—
172C23×C4C22×D7311yescharacteristic
173C2×M16—1462no—
174C2×C4⋊C4—1462no—
175C2×C4⋊C4—1462no—
176C2×C4⋊C4—1462no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—1462no—
179C2×C22⋊C4C22⋊D7312yes—
180C2×C42—732no—
181C2×C42—732no—
182#104—2924no—
183#104—2924no—
184#99—44no—
185C22⋊C8—1462no—
186C22⋊C8—1462no—
187#95—22no—
188#95—22no—
189#95—1462no—
190#95—1462no—
191C24—21no—
192C24C2×C22⋊D7311yescharacteristic
193#41—24no—
194#41—24no—
195#41—42no—
196#41—42no—
197#41—44no—
198#41—44no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—1462no—
215C22×C4—1462no—
216C22×C4D4×D7312yes—
217C22×C4C2×Dic14611yescharacteristic
218C22×C4#1192402712yes—
219C22×C4#1192402811yescharacteristic
220M16—2922no—
221M16—2922no—
222C2×C8—2922no—
223C4⋊C4—2924no—
224C4⋊C4—2924no—
225C4⋊C4—5842no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—2924no—
229C42—2922no—
230C42—2922no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—82no—
237C23#1455655911yescharacteristic
238C23#1455657011yescharacteristic
239C23#1455657811yescharacteristic
240Q8—24no—
241Q8—42no—
242D4—24no—
243D4—42no—
244D4—44no—
245D4—44no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—2922no—
262C2×C4—2922no—
263C2×C4—2922no—
264C2×C4—2922no—
265C2×C4—2922no—
266C2×C4—1464no—
267C2×C4#1455656812yes—
268C2×C4#1455657712yes—
269C8—5842no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22#1798849911yescharacteristic
281C22#1798877312yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—82no—
286C4—161no—
287C4—2924no—
288C4—5842no—
289C2—22no—
290C2—41no—
291C2—41no—
292C2—82no—
293C2#2261109511yescharacteristic
294C2#2261110512yes—
subgroupG/H
centerC22#17988499
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#1782C2
socleC22×C73#393
prime radicalC22×#41×C73C22
hypercenter#1782D73
hypocenterC73#10388
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #22615601C2semidirect
3. #22607839C2semidirect
4. #22607783C2semidirect
153. C73#10388semidirect

Maximal subgroups

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

subgroupindex
#226156012
#226098752
#226078392
#226077832
C73×#17822
#1038873