Finite Group Databasebeta

← #34001435

Subgroups — #34001435

Order 18688.

654 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 124 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#22625785C212yesmaximal, complemented
2#22624752C212yesmaximal, complemented
3#22624748C212yesmaximal, complemented
4#22612419C211yesmaximal, characteristic
5C2×C4⋊#10596833—22no—
6C2×C22⋊#10596833—22no—
7#17989453C2214yes—
8C22×#11924028C411yescharacteristic
9C22×C4×Dic73C2211yescharacteristic
10#17988492C411yescharacteristic
11#17988492C2211yescharacteristic
12#17988488C411yescharacteristic
13#17988488C2211yescharacteristic
14#17988464—22no—
15C4×#11924005—22no—
16C2×#117×C73C411yescharacteristic
17C22×#10596833—24no—
18C4⋊#10596833—44no—
19C22⋊#10596833—44no—
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×#11924005—22no—
30C2×#11924005—22no—
31C2×C4×Dic73—42no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73D412yes—
34C2×C4×Dic73C2311yescharacteristic
35#14556498—22no—
36#14556498—22no—
37C23×C4×C73C2×C411yescharacteristic
38#117×C73—42no—
39C2×C4⋊C4×C73C2×C411yescharacteristic
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C22⋊C4×C73—21no—
42C2×C42×C73—21no—
43C2×#10596833—44no—
44C2×#10596833—44no—
45C2×#10596833—44no—
46C23×D73—21no—
47#11924028—48no—
48#11924028—48no—
49C2×Dic146—21no—
50C2×Dic146—41no—
51C2×Dic146—41no—
52C2×Dic146C2×C812yes—
53C2×D292—21no—
54C2×D292—41no—
55C2×D292—41no—
56C2×D292C2×C812yes—
57C2×C22⋊D73—24no—
58C2×C22⋊D73—42no—
59C2×C22⋊D73—42no—
60C22×Dic73—21no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—42no—
64C22×Dic73—42no—
65C22×Dic73C22⋊C412yes—
66C22×Dic73C2×D412yes—
67C2×C4×D73—24no—
68C2×C4×D73—42no—
69C2×C4×D73—42no—
70#11924005—44no—
71#11924005—42no—
72#11924005—42no—
73C4×Dic73—24no—
74C4×Dic73—41no—
75C4×Dic73—41no—
76C24×C73C22⋊C411yescharacteristic
77C22×C4×C73—22no—
78C22×C4×C73—22no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C2×C812yes—
83C22×C4×C73M1612yes—
84C22×C4×C73C22×C411yescharacteristic
85C4⋊C4×C73—41no—
86C4⋊C4×C73—41no—
87C22⋊C4×C73—42no—
88C42×C73—42no—
89C23×C73—21no—
90C23×C73—21no—
91C23×C73—22no—
92C23×C73—22no—
93C23×C73C22⋊C812yes—
94C23×C73C2×C22⋊C411yescharacteristic
95C22×D73—22no—
96C22×D73—41no—
97C22×D73—41no—
98C22×D73—42no—
99C22×D73—42no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—24no—
103C2×C4×C73—24no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73C22⋊C812yes—
109C2×C4×C73#9912yes—
110C2×C4×C73#10012yes—
111C2×C4×C73C22×C811yescharacteristic
112C2×C4×C73C2×M1611yescharacteristic
113C22⋊D73—48no—
114C2×Dic73—22no—
115C2×Dic73—22no—
116C2×Dic73—22no—
117C2×Dic73—22no—
118C2×Dic73—22no—
119C2×Dic73—22no—
120C2×Dic73—44no—
121C2×Dic73—44no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73AGL(1,ℤ/8)12yes—
125C2×Dic73#13712yes—
126D292—42no—
127D292—42no—
128C4×D73—48no—
129Dic146—42no—
130Dic146—42no—
131#10596833—84no—
132C22×C73—22no—
133C22×C73—24no—
134C22×C73—41no—
135C22×C73—41no—
136C22×C73—42no—
137C22×C73C2×C22⋊C811yescharacteristic
138C22×C73C2×#9911yescharacteristic
139C22×C73C2×#10011yescharacteristic
140C22×C73#41712yes—
141C22×C73#42112yes—
142C2×D73—42no—
143C2×D73—44no—
144C2×D73—81no—
145C2×D73—81no—
146C2×D73—82no—
147C4×C73—21no—
148C4×C73—21no—
149C4×C73—81no—
150C4×C73—81no—
151C4×C73#32314yes—
152Dic73—22no—
153Dic73—22no—
154Dic73—42no—
155Dic73—42no—
156#7597—731nomaximal, Sylow
157C2×C73—42no—
158C2×C73C2×#32311yescharacteristic
159C2×C73#147712yes—
160C2×C73#151312yes—
161C2×C73#153611yescharacteristic
162C2×C73#156011yescharacteristic
163D73—82no—
164#2297—731no—
165#1797—732no—
166#1272—732no—
167#1270—732no—
168C73#759711yescharacteristic, complemented, Sylow
169C22×#41—731no—
170C2×#117#6440811yescharacteristic
171C22×C42—731no—
172C2×C4⋊C8—1462no—
173C2×C22⋊C8—1462no—
174#391—731no—
175#391—731no—
176#390—731no—
177#390—731no—
178#386—1462no—
179C4×C22⋊C4—1462no—
180#336—734no—
181C2×#41—2922no—
182C2×#41—2922no—
183C2×#41—734no—
184C2×#41—1462no—
185C22×Q8—731no—
186C22×D4—731no—
187C22×D4—1461no—
188C23×C4—732no—
189C23×C4—1461no—
190C23×C4C2×#6440811yescharacteristic
191C22×C8—1464no—
192#117—42no—
193C2×C4⋊C4C2×#6440811yescharacteristic
194C2×C4⋊C4C2×#6440811yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—1462no—
197C2×C22⋊C4—1462no—
198C2×C42—21no—
199C2×C42—2922no—
200C2×C42—731no—
201C2×C42—731no—
202C2×C42—732no—
203C4⋊C8—2924no—
204C22⋊C8—2924no—
205#95—1462no—
206#95—1462no—
207C24—1461no—
208C24C22⋊#6440811yescharacteristic
209#41—2928no—
210#41—2928no—
211C2×Q8—2921no—
212C2×Q8—2921no—
213C2×Q8—732no—
214C2×Q8—1461no—
215C2×D4—2921no—
216C2×D4—2921no—
217C2×D4—2922no—
218C2×D4—2922no—
219C2×D4—732no—
220C2×D4—1464no—
221C2×D4—1461no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—2922no—
227C22×C4—2922no—
228C22×C4—2922no—
229C22×C4—2922no—
230C22×C4—732no—
231C22×C4—732no—
232C22×C4—1462no—
233C22×C4—1462no—
234C22×C4—1461no—
235C22×C4—1464no—
236C22×C4C22⋊#6440811yescharacteristic
237C22×C4#1192403612yes—
238C22×C4#1192403812yes—
239C22×C4C22×#6440811yescharacteristic
240C2×C8—2924no—
241C2×C8—2924no—
242C2×C8—2924no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—2922no—
247C22⋊C4—2922no—
248C22⋊C4—2924no—
249C42—42no—
250C42—2921no—
251C42—2921no—
252C42—1464no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—2921no—
258C23—2921no—
259C23—2922no—
260C23—2922no—
261C23—1462no—
262C23#1455664812yes—
263C23C2×C22⋊#6440811yescharacteristic
264Q8—2922no—
265Q8—2922no—
266D4—2922no—
267D4—2922no—
268D4—2928no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—2921no—
278C2×C4—2921no—
279C2×C4—2924no—
280C2×C4—2924no—
281C2×C4—2928no—
282C2×C4—732no—
283C2×C4—732no—
284C2×C4—1462no—
285C2×C4—1462no—
286C2×C4—1462no—
287C2×C4—1462no—
288C2×C4—1462no—
289C2×C4—1462no—
290C2×C4#1455664812yes—
291C2×C4#1455665012yes—
292C2×C4#1455665112yes—
293C2×C4C2×#1192403611yescharacteristic
294C2×C4C2×#1192403811yescharacteristic
295C8—5844no—
296C22—22no—
297C22—24no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—2922no—
302C22—2924no—
303C22—5842no—
304C22—5841no—
305C22—5841no—
306C22C2×#1455664811yescharacteristic
307C22C2×#1455665011yescharacteristic
308C22C2×#1455665111yescharacteristic
309C22#1798957812yes—
310C22#1798958412yes—
311C4—21no—
312C4—21no—
313C4—81no—
314C4—81no—
315C4—2922no—
316C4—2922no—
317C4—1462no—
318C4—1462no—
319C4#1798943114yes—
320C2—42no—
321C2—5842no—
322C2C2×#1798943111yescharacteristic
323C2#2262522812yes—
324C2#2262534812yes—
325C2#2262541911yescharacteristic
326C2#2262546811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC22×#11924028C4
cosocleC2×C4×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#7597
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #22625785C2semidirect
2. #22624752C2semidirect
3. #22624748C2semidirect
23. C2×#11924028C8semidirect
168. C73#7597semidirect

Maximal subgroups

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

subgroupindex
#226257852
#226247522
#226247482
#226124192
#759773