Finite Group Databasebeta

← #32996815

Subgroups — #32996815

Order 18688.

400 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 87 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#22609923C211yesmaximal, characteristic
2#22608269C211yesmaximal, characteristic
3#22607961C212yesmaximal, complemented
4#22607937C212yesmaximal, complemented
5C73×#2638C211yesmaximal, characteristic
6#17988611—21no—
7C2×#14556508—21no—
8#17988551—21no—
9#17988356C2212yes—
10#17988313C2212yes—
11#17988311—21no—
12C2×C22⋊Q8×C73C411yescharacteristic
13C2×C4×D4×C73C2211yescharacteristic
14#398×C73C2211yescharacteristic
15#397×C73—21no—
16#397×C73C411yescharacteristic
17#395×C73C2211yescharacteristic
18#394×C73—21no—
19#393×C73C411yescharacteristic
20#387×C73—21no—
21#385×C73—21no—
22#381×C73C411yescharacteristic
23C2×#11924012—22no—
24C2×#11924012—41no—
25C4⋊#10596831—41no—
26C4⋊#10596831—41no—
27#14556508—42no—
28C22⋊#10596831—22no—
29C22⋊#10596831—41no—
30C22×Q8×C73C2×C411yescharacteristic
31C22×D4×C73D411yescharacteristic
32C23×C4×C73C2×C411yescharacteristic
33C23×C4×C73D411yescharacteristic
34C22⋊Q8×C73—41no—
35C22⋊Q8×C73—41no—
36C4×D4×C73—41no—
37C4×D4×C73—41no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73C2311yescharacteristic
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73D411yescharacteristic
49C2×C42×C73D411yescharacteristic
50#95×C73—21no—
51#95×C73—21no—
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73C2×C411yescharacteristic
56#95×C73C2×C411yescharacteristic
57#11924012—42no—
58#11924012—42no—
59#11924012—82no—
60C2×#10596831—42no—
61C2×#10596831—42no—
62C24×C73C22⋊C411yescharacteristic
63C24×C73C22⋊C411yescharacteristic
64C2×Q8×C73—41no—
65C2×Q8×C73—81no—
66C2×D4×C73—21no—
67C2×D4×C73—21no—
68C2×D4×C73—41no—
69C2×D4×C73—41no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—81no—
79C22×C4×C73—81no—
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C4⋊C4×C73—41no—
86C4⋊C4×C73—41no—
87C4⋊C4×C73—41no—
88C4⋊C4×C73—41no—
89C4⋊C4×C73—41no—
90C4⋊C4×C73—41no—
91C4⋊C4×C73—81no—
92C22⋊C4×C73—41no—
93C22⋊C4×C73—41no—
94C22⋊C4×C73—42no—
95C22⋊C4×C73—42no—
96C22⋊C4×C73—81no—
97C22⋊C4×C73—81no—
98C42×C73—21no—
99C42×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—21no—
105C23×C73—21no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73C2×C22⋊C411yescharacteristic
111Q8×C73—82no—
112D4×C73—42no—
113D4×C73—42no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—41no—
120C2×C4×C73—42no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—81no—
137C2×C4×C73—81no—
138C2×C4×C73—81no—
139C2×C4×C73#9911yescharacteristic
140C2×C4×C73#9911yescharacteristic
141C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C73#13711yescharacteristic
143#10596831—82no—
144#10596831—82no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—42no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73—41no—
154C22×C73—81no—
155C22×C73—81no—
156C22×C73—81no—
157C22×C73C2×#9911yescharacteristic
158C22×C73#41711yescharacteristic
159C22×C73#42111yescharacteristic
160C4×C73—21no—
161C4×C73—21no—
162C4×C73—21no—
163C4×C73—21no—
164C4×C73—41no—
165C4×C73—41no—
166C4×C73—41no—
167C4×C73—81no—
168C4×C73—81no—
169C4×C73—81no—
170C4×C73—81no—
171#11834—731nomaximal, Sylow
172C2×C73—21no—
173C2×C73—21no—
174C2×C73—81no—
175C2×C73—81no—
176C2×C73#151311yescharacteristic
177C2×C73#212111yescharacteristic
178C2×C73#213511yescharacteristic
179#2638D7311yescharacteristic
180#1800—731no—
181#1465—731no—
182#1354—732no—
183#1346—732no—
184C73#1183411yescharacteristic, complemented, Sylow
185C2×C22⋊Q8Dic7311yescharacteristic
186C2×C4×D4C2×D7311yescharacteristic
187#423—1461no—
188C2×#100—1461no—
189#407—1461no—
190#398C2×D7311yescharacteristic
191#397—21no—
192#397Dic7311yescharacteristic
193#395C2×D7311yescharacteristic
194#394—21no—
195#393Dic7311yescharacteristic
196#387—21no—
197#385—21no—
198#381Dic7311yescharacteristic
199#343—732no—
200#324—732no—
201#323—1461no—
202C22×Q8C2×Dic7311yescharacteristic
203C22×D4C22⋊D7311yescharacteristic
204C23×C4C2×Dic7311yescharacteristic
205C23×C4C22⋊D7311yescharacteristic
206C2×M16—2921no—
207C2×M16—1462no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×Dic7311yescharacteristic
216C2×C4⋊C4C22×D7311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4C2×Dic7311yescharacteristic
222C2×C22⋊C4C22⋊D7311yescharacteristic
223C2×C42C22⋊D7311yescharacteristic
224C4⋊C8—2921no—
225C4⋊C8—2921no—
226#100—2922no—
227C22⋊C8—2921no—
228C22⋊C8—1462no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95C2×Dic7311yescharacteristic
235#95C2×Dic7311yescharacteristic
236C24C22⋊Dic7311yescharacteristic
237C24C22⋊Dic7311yescharacteristic
238C2×Q8—41no—
239C2×Q8—81no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C22⋊Dic7311yescharacteristic
255C22×C4C22⋊Dic7311yescharacteristic
256C22×C4C22×Dic7311yescharacteristic
257C22×C4C2×C22⋊D7311yescharacteristic
258C22×C4C2×C22⋊D7311yescharacteristic
259M16—2922no—
260M16—2922no—
261M16—5842no—
262C2×C8—2922no—
263C2×C8—2922no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C42—21no—
278C42—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊Dic7311yescharacteristic
290Q8—82no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4#1455650611yescharacteristic
319C2×C4#1455650611yescharacteristic
320C2×C4#1455661611yescharacteristic
321C2×C4#1455662111yescharacteristic
322C8—5842no—
323C8—5842no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×#1455650611yescharacteristic
337C22#1798858811yescharacteristic
338C22#1798860411yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—81no—
354C2#2260847411yescharacteristic
355C2#2261157711yescharacteristic
356C2#2261162611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2638C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C4⋊C4×C73C23
hypercenter#2638D73
hypocenterC73#11834
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #22607961C2semidirect
4. #22607937C2semidirect
184. C73#11834semidirect

Maximal subgroups

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

subgroupindex
#226099232
#226082692
#226079612
#226079372
C73×#26382
#1183473