Finite Group Databasebeta

← #32975007

Subgroups — #32975007

Order 18688.

682 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 95 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#22611459C211yesmaximal, characteristic, complemented
2#22611334C212yesmaximal, complemented
3#22607752C211yesmaximal, characteristic, complemented
4#22607752C212yesmaximal, complemented
5C73×#2679C211yesmaximal, characteristic
6C2×#14556524—21no—
7C2×#14556512—21no—
8C2×#14556512—22no—
9#17988457C2211yescharacteristic, complemented
10#17988349—22no—
11#17988342C2211yescharacteristic, complemented
12#17988342C2212yescomplemented
13C2×#120×C73—22no—
14C2×#120×C73—21no—
15#394×C73—21no—
16#394×C73—22no—
17#390×C73C2211yescharacteristic
18#390×C73C2212yes—
19C22×#10596831—21no—
20C22×#10596831—22no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73—22no—
23#14556524—42no—
24#14556512—42no—
25#14556512—44no—
26C25×C73—21no—
27C22×D4×C73—21no—
28C22×D4×C73—22no—
29C22×D4×C73D412yes—
30C22×D4×C73D411yescharacteristic
31#120×C73—44no—
32#120×C73—42no—
33#120×C73—44no—
34#120×C73—42no—
35C2×C4⋊C4×C73D411yescharacteristic
36C2×C4⋊C4×C73D412yes—
37C2×C22⋊C4×C73—22no—
38C2×C22⋊C4×C73—21no—
39C2×C22⋊C4×C73—22no—
40C2×C22⋊C4×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C42×C73C2311yescharacteristic
43#95×C73—21no—
44C22×Dic73—41no—
45C2×#10596831—22no—
46C2×#10596831—41no—
47C2×#10596831—41no—
48C2×#10596831—42no—
49C2×#10596831—42no—
50C2×#10596831—42no—
51C4⋊Dic73—41no—
52C4⋊Dic73—41no—
53C4⋊Dic73—42no—
54C4⋊Dic73—42no—
55C24×C73—22no—
56C24×C73—21no—
57C24×C73—44no—
58C24×C73—42no—
59C24×C73—81no—
60C24×C73—82no—
61C24×C73—81no—
62C2×D4×C73—22no—
63C2×D4×C73—21no—
64C2×D4×C73—42no—
65C2×D4×C73—42no—
66C2×D4×C73—41no—
67C2×D4×C73—41no—
68C2×D4×C73—44no—
69C2×D4×C73—44no—
70C2×D4×C73—44no—
71C2×D4×C73—81no—
72C2×D4×C73—82no—
73C2×D4×C73D814yes—
74C2×D4×C73SD1612yes—
75C22×C4×C73—21no—
76C22×C4×C73—22no—
77C22×C4×C73C2×D412yes—
78C22×C4×C73C2×D411yescharacteristic
79C4⋊C4×C73—42no—
80C4⋊C4×C73—41no—
81C22⋊C4×C73—44no—
82C22⋊C4×C73—42no—
83C22⋊C4×C73—42no—
84C22⋊C4×C73—42no—
85C22⋊C4×C73—42no—
86C22⋊C4×C73—42no—
87C22⋊C4×C73—42no—
88C22⋊C4×C73—42no—
89C42×C73—41no—
90C23×C73—24no—
91C23×C73—22no—
92C23×C73—42no—
93C23×C73—42no—
94C23×C73—42no—
95C23×C73—41no—
96C23×C73—42no—
97C23×C73—42no—
98C23×C73—42no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73—41no—
102C23×C73—84no—
103C23×C73—88no—
104C23×C73—88no—
105C23×C73—84no—
106C23×C73—84no—
107C23×C73#12011yescharacteristic
108D4×C73—44no—
109D4×C73—42no—
110D4×C73—82no—
111D4×C73—81no—
112D4×C73—84no—
113D4×C73—88no—
114C2×C4×C73—21no—
115C2×C4×C73—22no—
116C2×C4×C73—22no—
117C2×C4×C73—21no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—41no—
122C2×C4×C73—42no—
123C2×C4×C73C2×D812yes—
124C2×C4×C73C2×SD1611yescharacteristic
125C2×C4×C73AGL(1,ℤ/8)12yes—
126C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
127C2×Dic73—82no—
128C2×Dic73—81no—
129#10596831—42no—
130#10596831—82no—
131C22×C73—44no—
132C22×C73—42no—
133C22×C73—44no—
134C22×C73—44no—
135C22×C73—44no—
136C22×C73—816no—
137C22×C73—82no—
138C22×C73—82no—
139C22×C73—82no—
140C22×C73—81no—
141C22×C73—82no—
142C22×C73—81no—
143C22×C73—82no—
144C22×C73#44712yes—
145C22×C73#45011yescharacteristic
146C22×C73#45311yescharacteristic
147C22×C73#45512yes—
148C22×C73#45811yescharacteristic
149C4×C73—22no—
150C4×C73—22no—
151C4×C73—21no—
152C4×C73—21no—
153C4×C73—81no—
154C4×C73—82no—
155Dic73—161no—
156#7786—731nomaximal, Sylow
157C2×C73—84no—
158C2×C73—82no—
159C2×C73#160712yes—
160C2×C73#161311yescharacteristic
161C2×C73#161911yescharacteristic
162C2×C73#163912yes—
163C2×C73#165911yescharacteristic
164#2679D7311yescharacteristic
165#2093—731no—
166#2071—732no—
167#1270—731no—
168#1270—732no—
169C73#778611yescharacteristic, complemented, Sylow
170C2×#120—22no—
171C2×#120—21no—
172C2×#107—1461no—
173C2×#102—1462no—
174C2×#102—1461no—
175#394—22no—
176#394—21no—
177#390C2×D7312yes—
178#390C2×D7311yescharacteristic
179#384—731no—
180#340—1462no—
181#336—732no—
182#336—731no—
183C25—21no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4C22⋊D7312yes—
187C22×D4C22⋊D7311yescharacteristic
188C22×C8—1462no—
189C22×C8—1461no—
190#120—44no—
191#120—42no—
192#120—44no—
193#120—42no—
194C2×C4⋊C4—1462no—
195C2×C4⋊C4—1461no—
196C2×C4⋊C4C22⋊D7312yes—
197C2×C4⋊C4C22⋊D7311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C42C22×D7311yescharacteristic
204#107—2922no—
205#102—2924no—
206#102—2922no—
207#95—21no—
208C24—22no—
209C24—21no—
210C24—44no—
211C24—42no—
212C24—81no—
213C24—82no—
214C24—81no—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—81no—
225C2×D4—82no—
226C2×D4#1192401314yes—
227C2×D4#1192401612yes—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—2921no—
231C22×C4C2×C22⋊D7312yes—
232C22×C4C2×C22⋊D7311yescharacteristic
233C2×C8—2921no—
234C2×C8—2921no—
235C2×C8—2922no—
236C2×C8—2922no—
237C2×C8—2922no—
238C2×C8—1462no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—2921no—
242C4⋊C4—2921no—
243C4⋊C4—2922no—
244C4⋊C4—2922no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—84no—
267C23—88no—
268C23—88no—
269C23—84no—
270C23—84no—
271C23#1455656011yescharacteristic
272D4—44no—
273D4—42no—
274D4—82no—
275D4—81no—
276D4—84no—
277D4—88no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—5842no—
288C2×C4—5841no—
289C2×C4C2×#1192401312yes—
290C2×C4C2×#1192401611yescharacteristic
291C2×C4#1455661712yes—
292C2×C4#1455661711yescharacteristic
293C8—2922no—
294C8—5842no—
295C22—44no—
296C22—42no—
297C22—44no—
298C22—44no—
299C22—44no—
300C22—816no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—81no—
305C22—82no—
306C22—81no—
307C22—82no—
308C22#1798872112yes—
309C22#1798873811yescharacteristic
310C22#1798875511yescharacteristic
311C22#1798876512yes—
312C22#1798877711yescharacteristic
313C4—22no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—81no—
318C4—82no—
319C4—11681no—
320C2—84no—
321C2—82no—
322C2#2260891712yes—
323C2#2260895111yescharacteristic
324C2#2260898511yescharacteristic
325C2#2260911012yes—
326C2#2260923711yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
Fitting, prime radicalC73×#2679C2
socleC23×C73#120
hypercenter#2679D73
hypocenterC73#7786
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2679

Complements

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

NH ≅ G/Ntype
1. #22611459C2semidirect
2. #22611334C2semidirect
3. #22607752C2semidirect
4. #22607752C2semidirect
9. #17988457C22semidirect
11. #17988342C22semidirect
12. #17988342C22semidirect
169. C73#7786semidirect

Maximal subgroups

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

subgroupindex
#226114592
#226113342
#226077522
C73×#26792
#778673