Finite Group Databasebeta

← #32983856

Subgroups — #32983856

Order 18688.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#22619016C211yesmaximal, characteristic, complemented
2#22614836C211yesmaximal, characteristic
3#22608355C212yesmaximal
4#22607791C212yesmaximal, complemented
5C73×#1801C211yesmaximal, characteristic, complemented
6C2×#14556567—21no—
7C2×C4×D292C2211yescharacteristic
8#17988718—24no—
9#17988644—22no—
10#17988640C2212yes—
11#17988517C412yes—
12#17988504C412yes—
13#17988478—22no—
14#17988453C2211yescharacteristic
15#17988444—21no—
16#17988314—24no—
17C2×C4×Q8×C73C2211yescharacteristic
18#423×C73C2212yes—
19C2×#101×C73—21no—
20#343×C73—21no—
21C22×D292C2×C411yescharacteristic
22C22×C4×D73—21no—
23#14556567—44no—
24C4×D292—22no—
25C4×D292—22no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—22no—
29C2×#11924005—22no—
30C2×#11924005—22no—
31C2×#11924005—21no—
32C4⋊#10596831—24no—
33C4⋊#10596831—22no—
34C22⋊#10596831—22no—
35#14556502—24no—
36C4×#10596831D414yes—
37#14556498—22no—
38#14556498—21no—
39#14556498C2×C412yes—
40C22×Q8×C73D411yescharacteristic
41C2×M16×C73—22no—
42C4×Q8×C73—22no—
43C4×Q8×C73—22no—
44C2×C4⋊C4×C73C2×C412yes—
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C42×C73D412yes—
47C2×C42×C73C2311yescharacteristic
48C4⋊C8×C73—22no—
49C4⋊C8×C73—22no—
50#101×C73—42no—
51C23×D73—21no—
52C2×D292—22no—
53C2×D292—42no—
54C22×Dic73—22no—
55C22×Dic73—21no—
56C2×C4×D73—42no—
57C2×C4×D73—42no—
58C2×C4×D73—41no—
59C2×C4×D73—41no—
60C2×#10596831—24no—
61C4⋊Dic73—22no—
62C4⋊Dic73—42no—
63#11924007—44no—
64#11924005—44no—
65#11924005—44no—
66#11924005—41no—
67#11924005—41no—
68C2×Q8×C73—22no—
69C2×Q8×C73—21no—
70C2×Q8×C73—22no—
71C2×Q8×C73—21no—
72C22×C4×C73C22⋊C412yes—
73C22×C4×C73C22⋊C412yes—
74C22×C4×C73C22×C411yescharacteristic
75C22×C4×C73C2×D411yescharacteristic
76C22×C4×C73C2×D411yescharacteristic
77M16×C73—42no—
78M16×C73—42no—
79C2×C8×C73—42no—
80C4⋊C4×C73—24no—
81C4⋊C4×C73—22no—
82C42×C73—22no—
83C42×C73—22no—
84C42×C73C2×D412yes—
85C42×C73#4112yes—
86C23×C73C2×C22⋊C411yescharacteristic
87C22×D73—22no—
88C22×D73—41no—
89C22×D73—41no—
90C22×D73—41no—
91C22×D73—41no—
92C22×D73—41no—
93C22×D73—41no—
94Q8×C73—42no—
95Q8×C73—42no—
96C2×C4×C73—22no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—41no—
101C2×C4×C73—41no—
102C2×C4×C73—42no—
103C2×C4×C73C4×D412yes—
104C2×C4×C73#12011yescharacteristic
105C2×C4×C73#12112yes—
106C2×C4×C73#12311yescharacteristic
107C2×C4×C73AGL(1,ℤ/8)12yes—
108C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C73#13711yescharacteristic
110C2×Dic73—42no—
111C2×Dic73—42no—
112C2×Dic73—41no—
113C2×Dic73—41no—
114C2×Dic73—42no—
115C2×Dic73—41no—
116D292—24no—
117D292—42no—
118C4×D73—82no—
119C4×D73—82no—
120C8×C73—82no—
121#10596831—44no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73#38611yescharacteristic
125C22×C73#41711yescharacteristic
126C22×C73#41811yescharacteristic
127C2×D73—42no—
128C2×D73—42no—
129C2×D73—42no—
130C2×D73—81no—
131C2×D73—81no—
132C2×D73—81no—
133C2×D73—81no—
134C4×C73—22no—
135C4×C73—21no—
136C4×C73—21no—
137C4×C73—42no—
138C4×C73—42no—
139C4×C73#46812yes—
140C4×C73#46912yes—
141Dic73—82no—
142Dic73—81no—
143#9490—731nomaximal, Sylow
144C2×C73—21no—
145C2×C73—21no—
146C2×C73#189111yescharacteristic
147C2×C73#189511yescharacteristic
148C2×C73#191211yescharacteristic
149D73—82no—
150#2983—731no—
151#2536—731no—
152#1801D7311yescharacteristic, complemented
153#1486—732no—
154#1286—732no—
155C73#949011yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—1461no—
157C2×C4×Q8C2×D7311yescharacteristic
158C2×C4×D4—731no—
159#446—1464no—
160#432—1462no—
161#431—732no—
162#423C2×D7312yes—
163C2×#101—21no—
164#398—732no—
165#394—732no—
166#388—1462no—
167#382—731no—
168#380—1461no—
169#343—21no—
170D4⋊C8—1464no—
171C22×Q8C22⋊D7311yescharacteristic
172C22×D4—731no—
173C23×C4—1461no—
174C2×M16—22no—
175C22⋊Q8—2924no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1462no—
179C4×D4—1462no—
180C2×C4⋊C4—731no—
181C2×C4⋊C4—1462no—
182C2×C4⋊C4—1461no—
183C2×C4⋊C4C4×D7312yes—
184C2×C4⋊C4C22⋊D7311yescharacteristic
185C2×C22⋊C4—1462no—
186C2×C22⋊C4—1462no—
187C2×C22⋊C4—1461no—
188C2×C42D29212yes—
189C2×C42C22×D7311yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—1464no—
193C4⋊C8—1462no—
194#101—42no—
195C22⋊C8—1462no—
196#97—1464no—
197C4×C8—734no—
198#95—732no—
199#95—1462no—
200#95—1461no—
201C24—1461no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×D4—2922no—
207C2×D4—1462no—
208C22×C4—2922no—
209C22×C4—2922no—
210C22×C4—2921no—
211C22×C4—2921no—
212C22×C4—1462no—
213C22×C4—1461no—
214C22×C4#1192400512yes—
215C22×C4#1192400512yes—
216C22×C4C2×C4×D7311yescharacteristic
217C22×C4C2×C22⋊D7311yescharacteristic
218C22×C4C2×D29211yescharacteristic
219M16—42no—
220M16—42no—
221C2×C8—42no—
222C2×C8—1464no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—2924no—
226C4⋊C4—2922no—
227C4⋊C4—1462no—
228C22⋊C4—2924no—
229C22⋊C4—2924no—
230C22⋊C4—2921no—
231C22⋊C4—2921no—
232C42—22no—
233C42—22no—
234C42D4×D7312yes—
235C42#1192402712yes—
236C23—2921no—
237C23—2921no—
238C23—2921no—
239C23—2921no—
240C23—2921no—
241C23—2921no—
242C23—1462no—
243C23C2×#1192400511yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—2922no—
247D4—1464no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—2922no—
256C2×C4—2922no—
257C2×C4—2921no—
258C2×C4—2921no—
259C2×C4—2922no—
260C2×C4—2921no—
261C2×C4—5842no—
262C2×C4—5842no—
263C2×C4#1455655112yes—
264C2×C4#1455656111yescharacteristic
265C2×C4#1455656312yes—
266C2×C4#1455657511yescharacteristic
267C2×C4#1455661612yes—
268C2×C4#1455661711yescharacteristic
269C2×C4#1455662211yescharacteristic
270C8—82no—
271C8—2924no—
272C22—21no—
273C22—21no—
274C22—2922no—
275C22—2922no—
276C22—2922no—
277C22—5841no—
278C22—5841no—
279C22—5841no—
280C22—5841no—
281C22#1798846411yescharacteristic
282C22#1798858611yescharacteristic
283C22#1798859111yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—5842no—
290C4—5841no—
291C4#1798883512yes—
292C4#1798883812yes—
293C2—21no—
294C2—21no—
295C2—5842no—
296C2#2261034711yescharacteristic
297C2#2261036711yescharacteristic
298C2#2261046611yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1801C2
socleC22×C73#386
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1801D73
hypocenterC73#9490
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #22619016C2semidirect
4. #22607791C2semidirect
5. C73×#1801C2semidirect
152. #1801D73semidirect
155. C73#9490semidirect

Maximal subgroups

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

subgroupindex
#226190162
#226148362
#226083552
#226077912
C73×#18012
#949073