Finite Group Databasebeta

← #32978377

Subgroups — #32978377

Order 18688.

692 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 131 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#22610752C211yesmaximal, characteristic, complemented
2#22608657C211yesmaximal, characteristic, complemented
3#22608551C212yesmaximal, complemented
4#22608469C212yesmaximal, complemented
5C73×#3544C211yesmaximal, characteristic
6#17988644C2212yes—
7C2×#14556515—22no—
8C2×#14556502C2211yescharacteristic
9#17988453C2211yescharacteristic
10#17988336—22no—
11#17988329—22no—
12#17988327—22no—
13#17988324—22no—
14#17988321—22no—
15#563×C73—22no—
16Q82×C73C411yescharacteristic, complemented
17#555×C73C411yescharacteristic, complemented
18#554×C73C412yescomplemented
19#552×C73—22no—
20#547×C73—22no—
21#544×C73—22no—
22#542×C73—22no—
23#536×C73—22no—
24C2×C4⋊Q8×C73C2211yescharacteristic
25#529×C73C2212yes—
26C22×#10596831—21no—
27C2×C4⋊Dic73—21no—
28C4⋊#10596831—22no—
29#14556515—44no—
30C22⋊#10596831—22no—
31#14556502—22no—
32#14556502—22no—
33#14556498—22no—
34C2×#41×C73—22no—
35C22×Q8×C73D412yes—
36C4⋊Q8×C73—22no—
37C4⋊Q8×C73—22no—
38C4⋊Q8×C73—22no—
39C4⋊Q8×C73—22no—
40C4⋊Q8×C73C2×C411yescharacteristic
41C4⋊Q8×C73C2×C411yescharacteristic
42C4⋊Q8×C73D412yes—
43#126×C73—24no—
44#126×C73—22no—
45#125×C73—24no—
46#125×C73—22no—
47#125×C73D412yes—
48#124×C73—22no—
49#124×C73—22no—
50#124×C73D412yes—
51#123×C73—24no—
52#123×C73—24no—
53#123×C73—22no—
54C22⋊Q8×C73—24no—
55C22⋊Q8×C73—24no—
56C22⋊Q8×C73—24no—
57C22⋊Q8×C73—22no—
58C22⋊Q8×C73D412yes—
59#121×C73—22no—
60#121×C73D412yes—
61C4×Q8×C73—22no—
62C4×Q8×C73C2×C412yes—
63C4×D4×C73—22no—
64C4×D4×C73—22no—
65C4×D4×C73C2×C412yes—
66#117×C73—22no—
67#117×C73—22no—
68#117×C73—22no—
69C2×C4⋊C4×C73—22no—
70C2×C4⋊C4×C73—22no—
71C2×C42×C73C2311yescharacteristic
72C22×Dic73—41no—
73C2×#10596831—22no—
74C2×#10596831—41no—
75C2×#10596831—41no—
76C2×#10596831—42no—
77C4⋊Dic73—42no—
78#41×C73—82no—
79#41×C73—82no—
80C2×Q8×C73—22no—
81C2×Q8×C73—22no—
82C2×Q8×C73—42no—
83C2×Q8×C73—42no—
84C2×Q8×C73C22⋊C412yes—
85C2×Q8×C73C22⋊C412yes—
86C2×Q8×C73C22⋊C412yes—
87C2×D4×C73—22no—
88C2×D4×C73—22no—
89C2×D4×C73C22⋊C412yes—
90C22×C4×C73—22no—
91C22×C4×C73—24no—
92C22×C4×C73—22no—
93C22×C4×C73C2×D411yescharacteristic
94C22×C4×C73C2×D412yes—
95C4⋊C4×C73—24no—
96C4⋊C4×C73—24no—
97C4⋊C4×C73—24no—
98C4⋊C4×C73—22no—
99C4⋊C4×C73—22no—
100C4⋊C4×C73—22no—
101C4⋊C4×C73—24no—
102C4⋊C4×C73—22no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—22no—
105C4⋊C4×C73C22⋊C412yes—
106C4⋊C4×C73C22⋊C412yes—
107C22⋊C4×C73—24no—
108C22⋊C4×C73—24no—
109C22⋊C4×C73—22no—
110C22⋊C4×C73—22no—
111C22⋊C4×C73—22no—
112C22⋊C4×C73—22no—
113C22⋊C4×C73—22no—
114C42×C73—22no—
115C42×C73—22no—
116C42×C73—22no—
117C42×C73C22×C411yescharacteristic
118C42×C73C2×D412yes—
119C42×C73C2×D411yescharacteristic
120C23×C73—22no—
121C23×C73#12011yescharacteristic
122Q8×C73—42no—
123Q8×C73—42no—
124Q8×C73—42no—
125Q8×C73—82no—
126Q8×C73—82no—
127D4×C73—42no—
128D4×C73—42no—
129D4×C73—42no—
130D4×C73—82no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—22no—
134C2×C4×C73—22no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—22no—
138C2×C4×C73—42no—
139C2×C4×C73—44no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—84no—
143C2×C4×C73C2×C22⋊C412yes—
144C2×C4×C73C2×C22⋊C411yescharacteristic
145C2×C4×C73#12012yes—
146C2×C4×C73#12011yescharacteristic
147C2×Dic73—82no—
148C2×Dic73—81no—
149#10596831—42no—
150#10596831—82no—
151C22×C73—22no—
152C22×C73—42no—
153C22×C73—42no—
154C22×C73—42no—
155C22×C73#37911yescharacteristic
156C22×C73#45311yescharacteristic
157C22×C73#45511yescharacteristic
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—22no—
161C4×C73—22no—
162C4×C73—42no—
163C4×C73—42no—
164C4×C73—42no—
165C4×C73—42no—
166C4×C73—42no—
167Dic73—161no—
168#8498—731nomaximal, Sylow
169C2×C73—21no—
170C2×C73—21no—
171C2×C73—82no—
172C2×C73#179211yescharacteristic
173C2×C73#179612yes—
174#3544D7311yescharacteristic
175#1960—731no—
176#1558—731no—
177#1532—732no—
178#1512—732no—
179C73#849811yescharacteristic, complemented, Sylow
180#563—22no—
181Q82Dic7311yescharacteristic, complemented
182#555Dic7311yescharacteristic, complemented
183#554Dic7312yescomplemented
184#552—22no—
185#547—22no—
186#544—22no—
187#542—22no—
188#536—22no—
189C2×C4⋊Q8C2×D7311yescharacteristic
190#529C2×D7312yes—
191#432—732no—
192C2×#103—1462no—
193C2×#97—731no—
194#382—731no—
195#333—1462no—
196#330—1462no—
197#329—1462no—
198#328—1462no—
199#327—1462no—
200C2×#41—22no—
201C22×Q8C22⋊D7312yes—
202C22×C8—1461no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8C2×Dic7311yescharacteristic
208C4⋊Q8C2×Dic7311yescharacteristic
209C4⋊Q8C22⋊D7312yes—
210#126—24no—
211#126—22no—
212#125—24no—
213#125—22no—
214#125C22⋊D7312yes—
215#124—22no—
216#124—22no—
217#124C22⋊D7312yes—
218#123—24no—
219#123—24no—
220#123—22no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8C22⋊D7312yes—
226#121—22no—
227#121C22⋊D7312yes—
228C4×Q8—22no—
229C4×Q8C2×Dic7312yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C2×Dic7312yes—
233#117—22no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—1461no—
239C2×C42C22×D7311yescharacteristic
240C4⋊C8—1462no—
241#103—2924no—
242C22⋊C8—1462no—
243#97—1462no—
244#97—1462no—
245#95—1462no—
246#41—82no—
247#41—82no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊Dic7312yes—
253C2×Q8C22⋊Dic7312yes—
254C2×Q8C22⋊Dic7312yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊Dic7312yes—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—2921no—
262C22×C4C2×C22⋊D7311yescharacteristic
263C22×C4C2×C22⋊D7312yes—
264C2×C8—2921no—
265C2×C8—2921no—
266C2×C8—2922no—
267C2×C8—1462no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—2922no—
279C4⋊C4C22⋊Dic7312yes—
280C4⋊C4C22⋊Dic7312yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42C22×Dic7311yescharacteristic
292C42C2×C22⋊D7312yes—
293C42C2×C22⋊D7311yescharacteristic
294C23—22no—
295C23#1455656011yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—42no—
299Q8—82no—
300Q8—82no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—5842no—
318C2×C4—5841no—
319C2×C4C2×C22⋊Dic7312yes—
320C2×C4C2×C22⋊Dic7311yescharacteristic
321C2×C4#1455656012yes—
322C2×C4#1455656011yescharacteristic
323C8—2922no—
324C8—5842no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#1798844311yescharacteristic
330C22#1798875511yescharacteristic
331C22#1798876511yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—11681no—
342C2—21no—
343C2—21no—
344C2—82no—
345C2#2260988511yescharacteristic
346C2#2260990612yes—
subgroupG/H
centerC22#17988443
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
Fitting, prime radicalC73×#3544C2
socleC22×C73#379
cosocleC2×C42×C73C23
hypercenter#3544D73
hypocenterC73#8498
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3544

Complements

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

NH ≅ G/Ntype
1. #22610752C2semidirect
2. #22608657C2semidirect
3. #22608551C2semidirect
4. #22608469C2semidirect
16. Q82×C73C4semidirect
17. #555×C73C4semidirect
18. #554×C73C4semidirect
179. C73#8498semidirect
181. Q82Dic73semidirect
182. #555Dic73semidirect
183. #554Dic73semidirect

Maximal subgroups

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

subgroupindex
#226107522
#226086572
#226085512
#226084692
C73×#35442
#849873