Finite Group Databasebeta

← #33002782

Subgroups — #33002782

Order 18688.

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

normal subgroups only

This group has 536 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#22613043C211yesmaximal, characteristic
2C2×#17988382C211yesmaximal, characteristic
3#22610212C212yesmaximal
4#22608050C212yesmaximal
5C73×#1885C211yesmaximal, characteristic
6C2×#14556506C2212yes—
7#17988494—21no—
8#17988480—22no—
9#17988467C2212yes—
10#17988459—22no—
11C22⋊C4×Dic73—22no—
12#17988433—21no—
13#17988382C414yescomplemented
14#17988354—22no—
15C2×#124×C73C2211yescharacteristic
16#518×C73C2211yescharacteristic
17#419×C73—21no—
18C2×#100×C73C2211yescharacteristic
19#339×C73—21no—
20C23×Dic73D412yes—
21C2×C4⋊Dic73—21no—
22C2×#11924007—42no—
23C2×C22⋊Dic73—22no—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73D412yes—
26C2×C4×Dic73—22no—
27C2×C4×Dic73—21no—
28#14556506—22no—
29#14556506C2×C414yes—
30#14556498—21no—
31#14556498—21no—
32#14556498—22no—
33#14556498—22no—
34#14556498—42no—
35C22×Q8×C73D411yescharacteristic
36C22×D4×C73C2311yescharacteristic
37C2×M16×C73—21no—
38#124×C73—22no—
39#124×C73C2×C412yes—
40#124×C73Q812yes—
41C4×D4×C73—42no—
42#117×C73—21no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73—22no—
45C2×C42×C73D411yescharacteristic
46#103×C73—42no—
47#100×C73—22no—
48C22×Dic73—22no—
49C22×Dic73—22no—
50C22×Dic73—44no—
51C22×Dic73—42no—
52C22×Dic73—42no—
53C4⋊Dic73—81no—
54#11924007—84no—
55C22⋊Dic73—24no—
56C22⋊Dic73—44no—
57C22⋊Dic73—42no—
58C22⋊Dic73—42no—
59C4×Dic73—44no—
60C4×Dic73—81no—
61C24×C73C2×D412yes—
62C2×Q8×C73—21no—
63C2×Q8×C73—41no—
64C2×Q8×C73—41no—
65C2×Q8×C73C4⋊C412yescomplemented
66C2×D4×C73—41no—
67C2×D4×C73—41no—
68C2×D4×C73C22×C411yescharacteristic
69C2×D4×C73C2×Q811yescharacteristic
70C2×D4×C73#4112yes—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—22no—
74C22×C4×C73—22no—
75C22×C4×C73C2×D411yescharacteristic
76M16×C73—41no—
77M16×C73—41no—
78C2×C8×C73—41no—
79C4⋊C4×C73—22no—
80C22⋊C4×C73—24no—
81C22⋊C4×C73—24no—
82C22⋊C4×C73—42no—
83C22⋊C4×C73—41no—
84C42×C73—21no—
85C42×C73—41no—
86C42×C73C4⋊C412yescomplemented
87C23×C73—22no—
88C23×C73—22no—
89C23×C73—42no—
90C23×C73—42no—
91C23×C73C4×D412yes—
92C23×C73#12011yescharacteristic
93C23×C73C22⋊Q812yes—
94Q8×C73—42no—
95Q8×C73—81no—
96D4×C73—41no—
97D4×C73—41no—
98D4×C73—42no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—22no—
102C2×C4×C73—22no—
103C2×C4×C73—42no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73—42no—
107C2×C4×C73—82no—
108C2×C4×C73—82no—
109C2×C4×C73C2×C4⋊C411yescharacteristic
110C2×C4×C73#12311yescharacteristic
111C2×Dic73—24no—
112C2×Dic73—42no—
113C2×Dic73—42no—
114C2×Dic73—44no—
115C2×Dic73—42no—
116C2×Dic73—44no—
117C2×Dic73—42no—
118C2×Dic73—42no—
119C2×Dic73—82no—
120C2×Dic73—82no—
121C2×Dic73—82no—
122C8×C73—81no—
123C22×C73—22no—
124C22×C73—22no—
125C22×C73—22no—
126C22×C73—42no—
127C22×C73—42no—
128C22×C73—42no—
129C22×C73—42no—
130C22×C73—82no—
131C22×C73#38511yescharacteristic
132C22×C73B(4,𝔽2)11yescharacteristic
133C22×C73#45811yescharacteristic
134C4×C73—21no—
135C4×C73—21no—
136C4×C73—42no—
137C4×C73—42no—
138C4×C73—82no—
139Dic73—44no—
140Dic73—82no—
141Dic73—84no—
142#12965—731nomaximal, Sylow
143C2×C73—21no—
144C2×C73—21no—
145C2×C73—42no—
146C2×C73—42no—
147C2×C73#186711yescharacteristic
148C2×C73#219712yes—
149#2372—731no—
150C2×#354—731no—
151#1885D7311yescharacteristic
152#1867—732no—
153#1390—732no—
154C73#1296511yescharacteristic, complemented, Sylow
155C2×#124C2×D7311yescharacteristic
156#518C2×D7311yescharacteristic
157#419—21no—
158C2×#100C2×D7311yescharacteristic
159C2×#99—732no—
160#391—1461no—
161#388—1462no—
162#386—732no—
163#385—1462no—
164C4×C22⋊C4—1462no—
165#376—1461no—
166#354—734no—
167#342—1462no—
168#339—21no—
169C22×Q8C22⋊D7311yescharacteristic
170C22×D4C22×D7311yescharacteristic
171C23×C4—732no—
172C2×M16—21no—
173#124—22no—
174#124Dic14612yes—
175#124C4×D7312yes—
176C4×D4—42no—
177#117—21no—
178C2×C4⋊C4—2922no—
179C2×C4⋊C4—1461no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—732no—
183C2×C22⋊C4—1462no—
184C2×C22⋊C4—1462no—
185C2×C42—1462no—
186C2×C42—1461no—
187C2×C42C22⋊D7311yescharacteristic
188#103—42no—
189#100—22no—
190#99—734no—
191#99—1462no—
192#95—2922no—
193#95—1461no—
194#95—1461no—
195#95—1462no—
196#95—1462no—
197C24D4×D7312yes—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8#1192400712yescomplemented
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×C4×D7311yescharacteristic
205C2×D4C2×Dic14611yescharacteristic
206C2×D4#1192402712yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—2922no—
212C22×C4—2922no—
213C22×C4—2924no—
214C22×C4—1462no—
215C22×C4—1462no—
216C22×C4C2×C22⋊D7311yescharacteristic
217M16—41no—
218M16—41no—
219C2×C8—41no—
220C4⋊C4—22no—
221C4⋊C4—5841no—
222C4⋊C4—5844no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—41no—
227C22⋊C4—2922no—
228C22⋊C4—2924no—
229C22⋊C4—2922no—
230C22⋊C4—1464no—
231C42—21no—
232C42—41no—
233C42—2924no—
234C42—5841no—
235C42#1192400712yescomplemented
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23#1455655112yes—
241C23#1455655911yescharacteristic
242C23#1455656812yes—
243Q8—42no—
244Q8—81no—
245D4—41no—
246D4—41no—
247D4—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—2922no—
259C2×C4—2922no—
260C2×C4—2924no—
261C2×C4—2922no—
262C2×C4—2922no—
263C2×C4—2924no—
264C2×C4—2922no—
265C2×C4—5842no—
266C2×C4—5842no—
267C2×C4—5842no—
268C2×C4—1464no—
269C2×C4C2×#1192400711yescharacteristic
270C2×C4#1455657211yescharacteristic
271C8—81no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22#1798845911yescharacteristic
281C22#1798877311yescharacteristic
282C22#1798877611yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—82no—
288C4—2924no—
289C4—5842no—
290C4—5844no—
291C2—21no—
292C2—21no—
293C2—42no—
294C2—42no—
295C2#2261021111yescharacteristic
296C2#2261192012yes—
subgroupG/H
centerC22#17988773
commutatorC2×D4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#1885C2
socleC22×C73B(4,𝔽2)
prime radical, cosocleC22×D4×C73C23
hypercenter#1885D73
hypocenterC73#12965
LCSG ▷ C2×D4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1885

Complements

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

NH ≅ G/Ntype
13. #17988382C4semidirect
65. C2×Q8×C73C4⋊C4semidirect
86. C42×C73C4⋊C4semidirect
154. C73#12965semidirect
201. C2×Q8#11924007semidirect
235. C42#11924007semidirect

Maximal subgroups

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

subgroupindex
#226130432
C2×#179883822
#226102122
#226080502
C73×#18852
#1296573