Finite Group Databasebeta

← #34009927

Subgroups — #34009927

Order 18688.

418 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 92 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#22626159C211yesmaximal, characteristic, complemented
2#22626156C211yesmaximal, characteristic, complemented
3#22624868C212yesmaximal, complemented
4#22624851C212yesmaximal, complemented
5C2×#17988656C211yesmaximal, characteristic
6C2×#14556666—21no—
7C2×#14556664—21no—
8#17989482—21no—
9#17989482—21no—
10C22⋊#11924031C2212yes—
11C22⋊#11924031C2212yes—
12C22×C8×D73C2211yescharacteristic
13C2×C4×C22⋊D73C411yescharacteristic
14#17988656—21no—
15#17988656—21no—
16#17988656—21no—
17#17988656—21no—
18C2×#14556520C2211yescharacteristic
19C2×C22⋊#10596831C2211yescharacteristic
20C2×#14556503C411yescharacteristic
21C2×#14556501C411yescharacteristic
22C22×M16×C73C411yescharacteristic
23#14556666—42no—
24#14556664—42no—
25C2×#11924031—22no—
26C2×#11924031—22no—
27C22×C22⋊D73C811yescharacteristic
28C22×C4×D73C2×C411yescharacteristic
29C22×#10596831C2×C411yescharacteristic
30C22×#10596831C2311yescharacteristic
31C2×C8×D73—42no—
32C2×C8×D73D411yescharacteristic
33C2×C8×D73D411yescharacteristic
34C2×C8×D73D411yescharacteristic
35C2×C8×D73D411yescharacteristic
36C4×C22⋊D73—21no—
37C4×C22⋊D73—21no—
38C4×C22⋊D73—21no—
39C4×C22⋊D73—21no—
40C2×#11924007C811yescharacteristic
41C2×C22⋊Dic73C811yescharacteristic
42C2×#11924005C811yescharacteristic
43C2×C4×Dic73C2×C411yescharacteristic
44#14556520—21no—
45#14556520—21no—
46C22⋊#10596831—21no—
47C22⋊#10596831—21no—
48#14556503—21no—
49#14556503—21no—
50#14556501—21no—
51#14556501—21no—
52C23×C4×C73C2×C411yescharacteristic
53C2×M16×C73—21no—
54C2×M16×C73—21no—
55C2×M16×C73—42no—
56C22×C8×C73C2×C411yescharacteristic
57C22×C8×C73C2×C411yescharacteristic
58#11924031—42no—
59#11924031—42no—
60C23×D73C2×C811yescharacteristic
61C2×C22⋊D73—21no—
62C2×C22⋊D73—21no—
63C2×C22⋊D73—42no—
64C22×Dic73C2×C811yescharacteristic
65C22×Dic73C2×C811yescharacteristic
66C22×Dic73C2×C811yescharacteristic
67C2×C4×D73—41no—
68C2×C4×D73—41no—
69C2×C4×D73C22⋊C411yescharacteristic
70C2×C4×D73C22⋊C411yescharacteristic
71C2×C4×D73M1611yescharacteristic
72C2×C4×D73M1611yescharacteristic
73C2×#10596831—21no—
74C2×#10596831—41no—
75C2×#10596831—41no—
76C2×#10596831C2×D411yescharacteristic
77C2×#10596831C2×D411yescharacteristic
78C8×D73—22no—
79C8×D73—22no—
80C8×D73—41no—
81C8×D73—41no—
82#11924007—21no—
83#11924007—21no—
84C22⋊Dic73—21no—
85C22⋊Dic73—21no—
86#11924005—21no—
87#11924005—21no—
88C4×Dic73—21no—
89C4×Dic73—21no—
90C24×C73C2×C811yescharacteristic
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73C2×C811yescharacteristic
96C22×C4×C73C22×C411yescharacteristic
97M16×C73—42no—
98M16×C73—42no—
99C2×C8×C73—22no—
100C2×C8×C73—21no—
101C2×C8×C73—41no—
102C2×C8×C73C22⋊C411yescharacteristic
103C2×C8×C73C22⋊C411yescharacteristic
104C23×C73—21no—
105C23×C73—22no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73C22×C811yescharacteristic
109C22×D73—22no—
110C22×D73—22no—
111C22×D73—41no—
112C22×D73C22⋊C811yescharacteristic
113C22×D73C22⋊C811yescharacteristic
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—42no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73C2×C22⋊C411yescharacteristic
122C2×C4×C73C2×M1611yescharacteristic
123C22⋊D73—42no—
124C22⋊D73—42no—
125C2×Dic73—21no—
126C2×Dic73—21no—
127C2×Dic73—22no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73C22⋊C811yescharacteristic
131C2×Dic73C22⋊C811yescharacteristic
132C4×D73—21no—
133C4×D73—21no—
134C4×D73—21no—
135C4×D73—21no—
136C4×D73—41no—
137C4×D73—41no—
138C8×C73—21no—
139C8×C73—41no—
140C8×C73#9912yes—
141#10596831—21no—
142#10596831—21no—
143#10596831—41no—
144C22×C73—21no—
145C22×C73—21no—
146C22×C73—41no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—82no—
150C22×C73C2×C22⋊C811yescharacteristic
151C22×C73#50412yes—
152C2×D73—21no—
153C2×D73—21no—
154C2×D73—21no—
155C2×D73—21no—
156C2×D73—21no—
157C2×D73—44no—
158C2×D73#34912yes—
159C4×C73—21no—
160C4×C73—41no—
161C4×C73C2×#9911yescharacteristic
162C4×C73#41311yescharacteristic
163Dic73—21no—
164Dic73—21no—
165Dic73—41no—
166Dic73—41no—
167#11467—731nomaximal, Sylow
168C2×C73—22no—
169C2×C73—81no—
170C2×C73#145911yescharacteristic
171C2×C73#211311yescharacteristic
172C2×C73C2×#34911yescharacteristic
173D73—22no—
174D73—41no—
175C2×#435—731no—
176#2112—731no—
177C2×C22⋊C16—731no—
178#1322—732no—
179#1314—732no—
180C73#1146711yescharacteristic, complemented, Sylow
181C22×M16#6440811yescharacteristic
182C23×C8—731no—
183C2×C4×D4—731no—
184C2×M32—1461no—
185C22×C16—1461no—
186#435—1461no—
187#435—1461no—
188#435—1461no—
189#435—1461no—
190C2×C4⋊C8—731no—
191C2×C22⋊C8—731no—
192C2×C22⋊C8—731no—
193C2×#97—731no—
194C22⋊C16—732no—
195C22⋊C16—732no—
196C22⋊C16—1461no—
197C22⋊C16—1461no—
198C22×D4—731no—
199C23×C4—731no—
200C23×C4C2×#6440811yescharacteristic
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—42no—
204C22×C8—2922no—
205C22×C8—731no—
206C22×C8—731no—
207C22×C8—731no—
208C22×C8—731no—
209C22×C8—731no—
210C22×C8—731no—
211C22×C8C2×#6440811yescharacteristic
212C22×C8C2×#6440811yescharacteristic
213C4×D4—1461no—
214C4×D4—1461no—
215C4×D4—1461no—
216C4×D4—1461no—
217C2×C4⋊C4—731no—
218C2×C22⋊C4—731no—
219C2×C22⋊C4—731no—
220C2×C42—731no—
221M32—2922no—
222C2×C16—2922no—
223C2×C16—1462no—
224C2×C16—1462no—
225C4⋊C8—1461no—
226C4⋊C8—1461no—
227C22⋊C8—1461no—
228C22⋊C8—1461no—
229C22⋊C8—1461no—
230C22⋊C8—1461no—
231#97—1461no—
232#97—1461no—
233C24—731no—
234C24#1192403611yescharacteristic
235C2×D4—2922no—
236C2×D4—1461no—
237C2×D4—1461no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—2921no—
243C22×C4—2921no—
244C22×C4—731no—
245C22×C4—731no—
246C22×C4—731no—
247C22×C4—731no—
248C22×C4—731no—
249C22×C4—731no—
250C22×C4—731no—
251C22×C4#1192403611yescharacteristic
252C22×C4C22×#6440811yescharacteristic
253M16—42no—
254M16—42no—
255C2×C8—22no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—2921no—
259C2×C8—2921no—
260C2×C8—2921no—
261C2×C8—2921no—
262C2×C8—731no—
263C2×C8—731no—
264C2×C8—1461no—
265C2×C8—1462no—
266C2×C8—1462no—
267C2×C8C22⋊#6440811yescharacteristic
268C2×C8C22⋊#6440811yescharacteristic
269C4⋊C4—1461no—
270C4⋊C4—1461no—
271C22⋊C4—1461no—
272C22⋊C4—1461no—
273C22⋊C4—1461no—
274C22⋊C4—1461no—
275C42—1461no—
276C42—1461no—
277C16—2922no—
278C16—2922no—
279C23—21no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—2921no—
284C23—731no—
285C23—731no—
286C23—1462no—
287C23—1462no—
288C23C2×#1192403611yescharacteristic
289D4—2922no—
290D4—2922no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—731no—
303C2×C4—731no—
304C2×C4—1462no—
305C2×C4—1461no—
306C2×C4—1461no—
307C2×C4—1461no—
308C2×C4—1461no—
309C2×C4—1461no—
310C2×C4—1461no—
311C2×C4C2×C22⋊#6440811yescharacteristic
312C2×C4D73⋊M1611yescharacteristic
313C8—21no—
314C8—41no—
315C8—2921no—
316C8—1461no—
317C8—1461no—
318C8#1455664912yes—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—82no—
325C22—2924no—
326C22—732no—
327C22—1461no—
328C22—1461no—
329C22—1461no—
330C22—1461no—
331C22—1461no—
332C22#1798955511yescharacteristic
333C22#1798964612yes—
334C4—21no—
335C4—41no—
336C4—2921no—
337C4—2921no—
338C4—1461no—
339C4—1461no—
340C4C2×#1455664911yescharacteristic
341C4#1798957311yescharacteristic
342C2—22no—
343C2—81no—
344C2—2921no—
345C2—1462no—
346C2#2262516611yescharacteristic
347C2#2262616211yescharacteristic
348C2#2262616511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC23×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC22×M16×C73C4
socleC22×C73C2×C22⋊C8
prime radicalC22×C22⋊D73C8
cosocleC22×#10596831C23
hypercenterC22×M16#64408
hypocenterC73#11467
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #22626159C2semidirect
2. #22626156C2semidirect
3. #22624868C2semidirect
4. #22624851C2semidirect
180. C73#11467semidirect

Maximal subgroups

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

subgroupindex
#226261592
#226261562
#226248682
#226248512
C2×#179886562
#1146773