Finite Group Databasebeta

← #34009813

Subgroups — #34009813

Order 18688.

594 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×#17989432C212yesmaximal, complemented
2#22624862C212yesmaximal, complemented
3#22624860C212yesmaximal, complemented
4#22612519C211yesmaximal, characteristic
5C2×C22⋊#10596833—22no—
6#17989432—22no—
7#17989432C2214yes—
8C22×#11924028C411yescharacteristic
9C22×C4⋊Dic73C2211yescharacteristic
10#17988493C411yescharacteristic
11#17988493C2211yescharacteristic
12#17988489C411yescharacteristic
13#17988489C2211yescharacteristic
14#17988464—22no—
15#17988444—22no—
16C2×#117×C73C411yescharacteristic
17C22×#10596833—42no—
18C22⋊#10596833—24no—
19C22⋊#10596833—42no—
20C2×#11924028—22no—
21C2×#11924028—42no—
22C2×#11924028—42no—
23C2×#11924028C814yescomplemented
24C22×Dic146C2×C411yescharacteristic
25C22×D292C2×C411yescharacteristic
26C22×C22⋊D73—21no—
27C23×Dic73D412yes—
28C22×C4×D73—21no—
29C2×C4⋊Dic73—42no—
30C2×C4⋊Dic73C2×C411yescharacteristic
31C2×C4⋊Dic73D412yes—
32C2×C4⋊Dic73C2311yescharacteristic
33C2×#11924005—22no—
34C2×#11924005—22no—
35#14556498—22no—
36#14556498—22no—
37C23×C4×C73C2×C411yescharacteristic
38#117×C73—42no—
39C2×C4⋊C4×C73—21no—
40C2×C22⋊C4×C73—21no—
41C2×C42×C73C2×C411yescharacteristic
42C2×C42×C73C2×C411yescharacteristic
43C2×#10596833—44no—
44C2×#10596833—84no—
45C23×D73—21no—
46#11924028—48no—
47#11924028—48no—
48C2×Dic146—21no—
49C2×Dic146—41no—
50C2×Dic146—41no—
51C2×Dic146C2×C812yescomplemented
52C2×D292—21no—
53C2×D292—41no—
54C2×D292—41no—
55C2×D292C2×C812yescomplemented
56C2×C22⋊D73—24no—
57C2×C22⋊D73—42no—
58C2×C22⋊D73—42no—
59C22×Dic73—22no—
60C22×Dic73—21no—
61C22×Dic73—22no—
62C22×Dic73—82no—
63C22×Dic73C22⋊C412yes—
64C22×Dic73C2×D412yes—
65C2×C4×D73—24no—
66C2×C4×D73—42no—
67C2×C4×D73—42no—
68C4⋊Dic73—41no—
69C4⋊Dic73—42no—
70C4⋊Dic73—41no—
71#11924005—82no—
72#11924005—82no—
73C24×C73C22⋊C411yescharacteristic
74C22×C4×C73—22no—
75C22×C4×C73—22no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C2×C812yes—
80C22×C4×C73M1612yes—
81C22×C4×C73C22×C411yescharacteristic
82C4⋊C4×C73—42no—
83C22⋊C4×C73—42no—
84C42×C73—41no—
85C42×C73—41no—
86C23×C73—41no—
87C23×C73—41no—
88C23×C73C22⋊C812yes—
89C23×C73#9912yes—
90C23×C73#10012yes—
91C23×C73C2×C22⋊C411yescharacteristic
92C22×D73—22no—
93C22×D73—42no—
94C22×D73—42no—
95C22×D73—41no—
96C22×D73—41no—
97C2×C4×C73—24no—
98C2×C4×C73—24no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—41no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73C22⋊C812yes—
110C2×C4×C73C22×C811yescharacteristic
111C2×C4×C73C2×M1611yescharacteristic
112C22⋊D73—48no—
113C2×Dic73—22no—
114C2×Dic73—22no—
115C2×Dic73—22no—
116C2×Dic73—44no—
117C2×Dic73—41no—
118C2×Dic73—41no—
119C2×Dic73—42no—
120C2×Dic73—42no—
121C2×Dic73—82no—
122D292—42no—
123D292—42no—
124C4×D73—48no—
125Dic146—42no—
126Dic146—42no—
127#10596833—84no—
128C22×C73—21no—
129C22×C73—22no—
130C22×C73—21no—
131C22×C73—22no—
132C22×C73—42no—
133C22×C73—42no—
134C22×C73#32314yes—
135C22×C73C2×C22⋊C811yescharacteristic
136C22×C73C2×#9911yescharacteristic
137C22×C73C2×#10011yescharacteristic
138C2×D73—44no—
139C2×D73—42no—
140C2×D73—82no—
141C2×D73—81no—
142C2×D73—81no—
143C4×C73—21no—
144C4×C73—21no—
145C4×C73—22no—
146C4×C73—81no—
147C4×C73—81no—
148Dic73—42no—
149Dic73—42no—
150Dic73—42no—
151#11432—731nomaximal, Sylow
152C2×C73—21no—
153C2×C73—21no—
154C2×C73—24no—
155C2×C73C2×#32311yescharacteristic
156C2×C73#212711yescharacteristic
157C2×C73#213311yescharacteristic
158D73—82no—
159#2311—731no—
160C2×#324—732no—
161#1319—732no—
162#1318—732no—
163C73#1143211yescharacteristic, complemented, Sylow
164C22×#41—731no—
165C2×#117#6440811yescharacteristic
166C22×C4⋊C4—731no—
167C2×C22⋊C8—1462no—
168#391—731no—
169#391—731no—
170#390—731no—
171#390—731no—
172#386—1462no—
173#380—1462no—
174#324—734no—
175#324—1462no—
176C2×#41—2922no—
177C2×#41—2922no—
178C2×#41—734no—
179C2×#41—1462no—
180C22×Q8—731no—
181C22×D4—731no—
182C22×D4—1461no—
183C23×C4—732no—
184C23×C4—1461no—
185C23×C4C2×#6440811yescharacteristic
186C22×C8—2922no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—2922no—
190C2×C4⋊C4—732no—
191C2×C4⋊C4—731no—
192C2×C4⋊C4—731no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—1462no—
195C2×C22⋊C4—1462no—
196C2×C42C2×#6440811yescharacteristic
197C2×C42C2×#6440811yescharacteristic
198C22⋊C8—2922no—
199C22⋊C8—1464no—
200#95—1462no—
201#95—1462no—
202C24—1461no—
203C24C22⋊#6440811yescharacteristic
204#41—2928no—
205#41—2928no—
206C2×Q8—2921no—
207C2×Q8—2921no—
208C2×Q8—732no—
209C2×Q8—1461no—
210C2×D4—2921no—
211C2×D4—2921no—
212C2×D4—2922no—
213C2×D4—2922no—
214C2×D4—732no—
215C2×D4—1464no—
216C2×D4—1461no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—2922no—
222C22×C4—2922no—
223C22×C4—5842no—
224C22×C4—732no—
225C22×C4—732no—
226C22×C4—1462no—
227C22×C4—1461no—
228C22×C4—1464no—
229C22×C4—1462no—
230C22×C4C22⋊#6440811yescharacteristic
231C22×C4#1192403612yes—
232C22×C4#1192403812yes—
233C22×C4C22×#6440811yescharacteristic
234C2×C8—2924no—
235C2×C8—5844no—
236C4⋊C4—42no—
237C4⋊C4—2922no—
238C4⋊C4—2921no—
239C4⋊C4—2921no—
240C22⋊C4—42no—
241C22⋊C4—5842no—
242C22⋊C4—5842no—
243C42—41no—
244C42—41no—
245C23—41no—
246C23—41no—
247C23—2922no—
248C23—2922no—
249C23—2921no—
250C23—2921no—
251C23—1462no—
252C23#1455664812yes—
253C23#1455665012yes—
254C23#1455665112yes—
255C23C2×C22⋊#6440811yescharacteristic
256Q8—2922no—
257Q8—2922no—
258D4—2922no—
259D4—2922no—
260D4—2928no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—2928no—
274C2×C4—2924no—
275C2×C4—2921no—
276C2×C4—2921no—
277C2×C4—2922no—
278C2×C4—2922no—
279C2×C4—5842no—
280C2×C4—1462no—
281C2×C4—1462no—
282C2×C4—1462no—
283C2×C4#1455664812yes—
284C2×C4C2×#1192403611yescharacteristic
285C2×C4C2×#1192403811yescharacteristic
286C8—5844no—
287C22—21no—
288C22—22no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—2924no—
294C22—2922no—
295C22—5842no—
296C22—5841no—
297C22—5841no—
298C22#1798943114yes—
299C22C2×#1455664811yescharacteristic
300C22C2×#1455665011yescharacteristic
301C22C2×#1455665111yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—81no—
306C4—81no—
307C4—2922no—
308C4—2922no—
309C4—2922no—
310C2—21no—
311C2—21no—
312C2—24no—
313C2—5842no—
314C2C2×#1798943111yescharacteristic
315C2#2262620811yescharacteristic
316C2#2262622811yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#99
prime radicalC22×#11924028C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×#117#64408
hypocenterC73#11432
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#17989432C2semidirect
2. #22624862C2semidirect
3. #22624860C2semidirect
23. C2×#11924028C8semidirect
51. C2×Dic146C2×C8semidirect
55. C2×D292C2×C8semidirect
163. C73#11432semidirect

Maximal subgroups

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

subgroupindex
C2×#179894322
#226248622
#226248602
#226125192
#1143273