Finite Group Databasebeta

← #32989167

Subgroups — #32989167

Order 18688.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#22615574C211yesmaximal, characteristic
2#22608482C211yesmaximal, characteristic
3#22608443C211yesmaximal, characteristic
4#22607825C211yesmaximal, characteristic
5#22607796C211yesmaximal, characteristic
6#22607787C211yesmaximal, characteristic
7C73×#1986C211yesmaximal, characteristic
8#17988644C2211yescharacteristic
9#17988644C2211yescharacteristic
10#17988526—21no—
11#17988507—21no—
12#17988493—21no—
13#17988491—21no—
14#17988471—21no—
15#17988470—21no—
16#17988455C2211yescharacteristic
17#17988433C2211yescharacteristic
18#17988324—21no—
19#17988324—21no—
20#17988324—21no—
21#17988324—21no—
22#533×C73C2211yescharacteristic
23C4×M16×C73C2211yescharacteristic
24C4×C4⋊C4×C73C2211yescharacteristic
25#339×C73—21no—
26#339×C73—21no—
27C22×Dic146—21no—
28C2×C4⋊Dic73—21no—
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×C4×Dic73—21no—
32C4⋊#10596831—21no—
33C4⋊#10596831—21no—
34C22⋊#10596831—21no—
35C22⋊#10596831—21no—
36#14556502—21no—
37#14556502—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498—21no—
45#14556498D411yescharacteristic
46#14556498D411yescharacteristic
47#14556498D411yescharacteristic
48#14556498D411yescharacteristic
49C2×M16×C73—21no—
50C4⋊Q8×C73Q811yescharacteristic, complemented
51C4⋊Q8×C73Q811yescharacteristic, complemented
52#125×C73—21no—
53C22⋊Q8×C73—21no—
54C22⋊Q8×C73—21no—
55C4×Q8×C73—21no—
56C4×Q8×C73—21no—
57#117×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C42×C73D411yescharacteristic
60C2×C42×C73D411yescharacteristic
61C2×C42×C73C2311yescharacteristic
62#97×C73—21no—
63C4×C8×C73—21no—
64#95×C73—21no—
65C2×Dic146—42no—
66C2×Dic146—42no—
67C2×Dic146—81no—
68C22×Dic73—21no—
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73—21no—
72C2×#10596831—41no—
73C2×#10596831—41no—
74C4⋊Dic73—42no—
75#11924007—42no—
76#11924007—81no—
77C4×Dic73—42no—
78C2×Q8×C73—21no—
79C2×Q8×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73C2×D411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85M16×C73—22no—
86M16×C73—41no—
87C2×C8×C73—41no—
88C4⋊C4×C73—21no—
89C4⋊C4×C73—21no—
90C4⋊C4×C73—21no—
91C4⋊C4×C73—21no—
92C4⋊C4×C73—42no—
93C4⋊C4×C73—41no—
94C4⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C42×C73—41no—
97C42×C73—41no—
98C42×C73—41no—
99C42×C73C2×Q811yescharacteristic
100C42×C73#4111yescharacteristic
101C42×C73#4111yescharacteristic
102C42×C73#4111yescharacteristic
103C23×C73#12011yescharacteristic
104Q8×C73—81no—
105Q8×C73—81no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—22no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73C22⋊Q811yescharacteristic
119C2×C4×C73C22⋊Q811yescharacteristic
120C2×C4×C73C22⋊Q811yescharacteristic
121C2×C4×C73#12411yescharacteristic
122C2×C4×C73#12411yescharacteristic
123C2×C4×C73#12411yescharacteristic
124C2×Dic73—22no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136Dic146—84no—
137C8×C73—42no—
138#10596831—81no—
139#10596831—81no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73#39311yescharacteristic
143C22×C73#45511yescharacteristic
144C22×C73#45611yescharacteristic
145C4×C73—21no—
146C4×C73—21no—
147C4×C73—21no—
148C4×C73—21no—
149C4×C73—21no—
150C4×C73—21no—
151C4×C73—42no—
152C4×C73—81no—
153C4×C73—81no—
154C4×C73—81no—
155Dic73—42no—
156Dic73—81no—
157Dic73—81no—
158Dic73—81no—
159#10379—731nomaximal, Sylow
160C2×C73—21no—
161C2×C73—21no—
162C2×C73#202811yescharacteristic
163C2×C73#203111yescharacteristic
164C2×C73#203111yescharacteristic
165#2605—731no—
166#1986D7311yescharacteristic
167#1515—731no—
168#1506—731no—
169#1300—731no—
170#1288—731no—
171#1285—731no—
172C73#1037911yescharacteristic, complemented, Sylow
173#533C2×D7311yescharacteristic
174#432—731no—
175#432—731no—
176C4×M16C2×D7311yescharacteristic
177#400—1461no—
178#395—1461no—
179#391—1461no—
180#391—1461no—
181#387—1461no—
182#387—1461no—
183#383—731no—
184C4×C4⋊C4C2×D7311yescharacteristic
185#376—731no—
186#339—21no—
187#339—21no—
188#328—1461no—
189#328—1461no—
190#328—1461no—
191#328—1461no—
192C22×Q8—1461no—
193C2×M16—21no—
194C4⋊Q8Dic14611yescharacteristic, complemented
195C4⋊Q8Dic14611yescharacteristic, complemented
196#125—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201#117—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4—1461no—
205C2×C4⋊C4—1461no—
206C2×C42—1461no—
207C2×C42C22⋊D7311yescharacteristic
208C2×C42C22⋊D7311yescharacteristic
209C2×C42C22×D7311yescharacteristic
210C4⋊C8—1461no—
211C4⋊C8—1461no—
212C22⋊C8—1461no—
213C22⋊C8—1461no—
214#97—21no—
215#97—1461no—
216#97—1461no—
217C4×C8—21no—
218#95—21no—
219#95—731no—
220#95—731no—
221#95—731no—
222#95—731no—
223#95—1461no—
224#95—1461no—
225#95—1461no—
226#95—1461no—
227#95—1461no—
228#95—1461no—
229#95—1461no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—2922no—
233C2×Q8—2922no—
234C2×Q8—5841no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1461no—
238C22×C4—1461no—
239C22×C4—1461no—
240C22×C4—1461no—
241C22×C4D4×D7311yescharacteristic
242C22×C4D4×D7311yescharacteristic
243C22×C4C2×C22⋊D7311yescharacteristic
244M16—22no—
245M16—41no—
246C2×C8—41no—
247C2×C8—2921no—
248C2×C8—2921no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2922no—
257C4⋊C4—2922no—
258C4⋊C4—5841no—
259C22⋊C4—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42—2922no—
264C42C2×Dic14611yescharacteristic
265C42#1192402711yescharacteristic
266C42#1192402711yescharacteristic
267C42#1192402811yescharacteristic
268C23#1455655911yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—5844no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—2921no—
295C2×C4—1462no—
296C2×C4#1455656811yescharacteristic
297C2×C4#1455656811yescharacteristic
298C2×C4#1455657011yescharacteristic
299C2×C4#1455657711yescharacteristic
300C2×C4#1455657711yescharacteristic
301C2×C4#1455657811yescharacteristic
302C8—42no—
303C8—5841no—
304C8—5841no—
305C22—21no—
306C22—21no—
307C22#1798849911yescharacteristic
308C22#1798876411yescharacteristic
309C22#1798876911yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—2922no—
321C4—5841no—
322C4—5841no—
323C4—5841no—
324C2—21no—
325C2—21no—
326C2#2261109211yescharacteristic
327C2#2261110111yescharacteristic
328C2#2261110311yescharacteristic
subgroupG/H
centerC22#17988499
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1986C2
socleC22×C73#393
prime radicalC23×C73#120
hypercenter#1986D73
hypocenterC73#10379
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
50. C4⋊Q8×C73Q8semidirect
51. C4⋊Q8×C73Q8semidirect
172. C73#10379semidirect
194. C4⋊Q8Dic146semidirect
195. C4⋊Q8Dic146semidirect

Maximal subgroups

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

subgroupindex
#226155742
#226084822
#226084432
#226078252
#226077962
#226077872
C73×#19862
#1037973