Finite Group Databasebeta

← #34008153

Subgroups — #34008153

Order 18688.

600 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 122 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#22625787C211yesmaximal, characteristic
2#22625657C211yesmaximal, characteristic
3#22624769C214yesmaximal
4C2×#17988483C211yesmaximal, characteristic
5C2×#14556658—21no—
6C2×C22⋊#10596833C2212yes—
7C2×C22⋊#10596833C2212yes—
8C2×#14556646—21no—
9#17989453—21no—
10#17989453—21no—
11C22×C4⋊Dic73C411yescharacteristic
12C22×#11924007C412yes—
13C22×C4×Dic73C2211yescharacteristic
14#17988483—21no—
15#17988483—21no—
16#17988483—21no—
17#17988483—21no—
18C2×#14556498C2212yes—
19C22×C4⋊C4×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22#14556658—42no—
23C22⋊#10596833—22no—
24C22⋊#10596833—24no—
25C22⋊#10596833Q814yes—
26#14556646—42no—
27C23×Dic73C2×C411yescharacteristic
28C23×Dic73C2×C412yes—
29C23×Dic73C2311yescharacteristic
30C2×C4⋊Dic73—24no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73—21no—
33C2×#11924007—22no—
34C2×#11924007—22no—
35C2×#11924007—44no—
36C2×C4×Dic73—42no—
37C2×C4×Dic73D411yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41#14556498—22no—
42#14556498—22no—
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73C2×C412yes—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73—42no—
48C2×#10596833—24no—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×#10596833—42no—
54C22×Dic73—22no—
55C22×Dic73—22no—
56C22×Dic73—22no—
57C22×Dic73—22no—
58C22×Dic73—41no—
59C22×Dic73—41no—
60C22×Dic73—41no—
61C22×Dic73—42no—
62C22×Dic73—42no—
63C22×Dic73C22⋊C411yescharacteristic
64C22×Dic73C22⋊C411yescharacteristic
65C22×Dic73M1614yes—
66C22×Dic73C2×D411yescharacteristic
67C22×Dic73C2×D411yescharacteristic
68C22×Dic73C2×Q812yes—
69C22×Dic73#4112yes—
70C4⋊Dic73—24no—
71C4⋊Dic73—24no—
72#11924007—44no—
73#11924007—44no—
74C4×Dic73—22no—
75C4×Dic73—22no—
76C4×Dic73—41no—
77C4×Dic73—41no—
78C24×C73C22×C411yescharacteristic
79C22×C4×C73—22no—
80C22×C4×C73—22no—
81C22×C4×C73—22no—
82C22×C4×C73—22no—
83C22×C4×C73—24no—
84C22×C4×C73—42no—
85C22×C4×C73—42no—
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C4⋊C4×C73—42no—
89C4⋊C4×C73—42no—
90C23×C73—21no—
91C23×C73—21no—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73C2×C22⋊C411yescharacteristic
95C23×C73C4×Q812yes—
96C23×C73C2×M1612yes—
97C23×C73#13112yes—
98C2×C4×C73—28no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—84no—
107C2×C4×C73#9912yes—
108C2×C4×C73#10112yes—
109C2×Dic73—21no—
110C2×Dic73—21no—
111C2×Dic73—21no—
112C2×Dic73—21no—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—24no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—42no—
120C2×Dic73—42no—
121C2×Dic73—44no—
122C2×Dic73—44no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—44no—
126C2×Dic73#10012yes—
127C2×Dic73C22⋊Q812yes—
128C2×Dic73#12411yescharacteristic
129C2×Dic73C4⋊Q811yescharacteristic
130#10596833—44no—
131#10596833—82no—
132C22×C73—22no—
133C22×C73—22no—
134C22×C73—22no—
135C22×C73—22no—
136C22×C73—22no—
137C22×C73—22no—
138C22×C73#39111yescharacteristic
139C22×C73#40711yescharacteristic
140C22×C73#40811yescharacteristic
141C22×C73C2×#9911yescharacteristic
142C22×C73#41011yescharacteristic
143C22×C73C2×#10011yescharacteristic
144C22×C73C2×#10111yescharacteristic
145C22×C73#44614yes—
146C4×C73—22no—
147C4×C73—22no—
148C4×C73—82no—
149Dic73—22no—
150Dic73—22no—
151Dic73—21no—
152Dic73—21no—
153Dic73—44no—
154Dic73—41no—
155#9883—731nomaximal, Sylow
156C2×C73—24no—
157C2×C73#146311yescharacteristic
158C2×C73#146611yescharacteristic
159C2×C73#147011yescharacteristic
160C2×C73#147111yescharacteristic
161C2×C73#198511yescharacteristic
162C2×C73#199611yescharacteristic
163C2×C73#199711yescharacteristic
164C2×#389—731no—
165#1797—731no—
166#1749—731no—
167#1280—734no—
168C73#988311yescharacteristic, complemented, Sylow
169C22×C4⋊C4—732no—
170C22×C4⋊C4—731no—
171C22×C4⋊C4#6440811yescharacteristic
172C22×C42—731no—
173C2×C4⋊C8—1461no—
174C2×C22⋊C8—732no—
175C2×C22⋊C8—732no—
176C2×#97—1461no—
177#389—1461no—
178#389—1461no—
179#389—1461no—
180#389—1461no—
181C2×#95—732no—
182#336—1461no—
183#336—1461no—
184C23×C4—731no—
185C23×C4—732no—
186C23×C4—731no—
187C23×C4C2×#6440812yes—
188C23×C4C2×#6440811yescharacteristic
189C22×C8—1462no—
190C22×C8—1462no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—2924no—
195C2×C4⋊C4—1462no—
196C2×C4⋊C4—1462no—
197C2×C4⋊C4—1464no—
198C2×C4⋊C4—1461no—
199C2×C4⋊C4—1461no—
200C2×C42—2922no—
201C2×C42—731no—
202C2×C42—731no—
203C2×C42—731no—
204C2×C42—731no—
205C4⋊C8—2922no—
206C22⋊C8—734no—
207C22⋊C8—1464no—
208C22⋊C8—1462no—
209#97—2922no—
210#95—1462no—
211#95—1462no—
212C24C22×#6440811yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—2922no—
221C22×C4—2922no—
222C22×C4—2921no—
223C22×C4—2921no—
224C22×C4—2921no—
225C22×C4—732no—
226C22×C4—732no—
227C22×C4—734no—
228C22×C4—731no—
229C22×C4—731no—
230C22×C4—731no—
231C22×C4—731no—
232C22×C4—1462no—
233C22×C4—1462no—
234C22×C4—1462no—
235C22×C4—1462no—
236C22×C4C22⋊#6440811yescharacteristic
237C22×C4C22⋊#6440811yescharacteristic
238C2×C8—2922no—
239C2×C8—2922no—
240C2×C8—2922no—
241C2×C8—2922no—
242C2×C8—2922no—
243C2×C8—1464no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—2924no—
247C4⋊C4—2924no—
248C4⋊C4—1464no—
249C4⋊C4—1464no—
250C42—2921no—
251C42—2921no—
252C42—1462no—
253C42—1462no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23C2×C22⋊#6440811yescharacteristic
259C23Q8×#6440812yes—
260C23D73⋊M1612yes—
261C23#1455667812yes—
262C2×C4—28no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—84no—
271C2×C4—2924no—
272C2×C4—2922no—
273C2×C4—2922no—
274C2×C4—2924no—
275C2×C4—2924no—
276C2×C4—2922no—
277C2×C4—2922no—
278C2×C4—731no—
279C2×C4—732no—
280C2×C4—731no—
281C2×C4—732no—
282C2×C4—1464no—
283C2×C4—1461no—
284C2×C4—1461no—
285C2×C4—1461no—
286C2×C4—1461no—
287C2×C4—1461no—
288C2×C4—1461no—
289C2×C4—1461no—
290C2×C4—1461no—
291C2×C4—1461no—
292C2×C4#1455664912yes—
293C2×C4#1455665212yes—
294C8—2924no—
295C8—5842no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22(C2×#64408)⋊Q811yescharacteristic
303C22#1798955911yescharacteristic
304C22#1798956111yescharacteristic
305C22C2×#1455664911yescharacteristic
306C22#1798956811yescharacteristic
307C22D73⋊#10011yescharacteristic
308C22C2×#1455665211yescharacteristic
309C22#1798963914yes—
310C4—22no—
311C4—22no—
312C4—82no—
313C4—2924no—
314C4—2921no—
315C4—1462no—
316C4—1461no—
317C4—1461no—
318C4—1462no—
319C2—24no—
320C2#2262518111yescharacteristic
321C2#2262519211yescharacteristic
322C2#2262520411yescharacteristic
323C2#2262520611yescharacteristic
324C2#2262610311yescharacteristic
325C2#2262612011yescharacteristic
326C2#2262612411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#9883
LCSG ▷ C24×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
168. C73#9883semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226256572
#226247692
C2×#179884832
#988373