Finite Group Databasebeta

← #32988723

Subgroups — #32988723

Order 18688.

516 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 105 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#22621072C211yesmaximal, characteristic
2#22618308C211yesmaximal, characteristic
3#22610909C211yesmaximal, characteristic
4#22608612C211yesmaximal, characteristic
5#22608602C211yesmaximal, characteristic
6#22607763C211yesmaximal, characteristic
7C73×#3209C211yesmaximal, characteristic
8C2×#14556591C2211yescharacteristic
9#17988983—21no—
10#17988937—21no—
11#17988915—22no—
12#17988872—22no—
13#17988640C2211yescharacteristic
14#17988611C2211yescharacteristic
15#17988494—22no—
16#17988492—22no—
17#17988452—22no—
18C42×Dic73C2211yescharacteristic
19#17988346—22no—
20#17988336—22no—
21#17988332—22no—
22#533×C73C2211yescharacteristic
23C2×C4⋊Q8×C73C2211yescharacteristic
24#494×C73—21no—
25#487×C73—21no—
26#462×C73—22no—
27#451×C73—22no—
28#431×C73C2211yescharacteristic
29C22×Dic146—21no—
30C2×#11924012—21no—
31#14556591—21no—
32#14556591—21no—
33#14556591—21no—
34#14556591—21no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73D412yes—
37C2×C4×Dic73—21no—
38C2×C4×Dic73—21no—
39C2×C4×Dic73D412yes—
40#14556522—24no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831D412yes—
44#14556515—24no—
45C22⋊#10596831—21no—
46C4×#10596831—21no—
47#14556498—28no—
48C22×Q8×C73D412yes—
49C2×Q16×C73—24no—
50C4⋊Q8×C73—21no—
51C4⋊Q8×C73—21no—
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C4⋊Q8×C73D411yescharacteristic
55C4⋊Q8×C73D411yescharacteristic
56#125×C73—21no—
57C22⋊Q8×C73—21no—
58C22⋊Q8×C73—21no—
59C4×Q8×C73—21no—
60C4×Q8×C73—21no—
61#117×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C42×C73C2311yescharacteristic
65C4⋊C8×C73D412yes—
66#103×C73—24no—
67C22⋊C8×C73—21no—
68C4×C8×C73—21no—
69C2×Dic146—41no—
70C2×Dic146—41no—
71C2×Dic146—42no—
72C2×Dic146—42no—
73C22×Dic73—22no—
74C22×Dic73—22no—
75#11924012—42no—
76C2×#10596831—22no—
77C2×#10596831—41no—
78C4⋊Dic73—22no—
79C4⋊Dic73—22no—
80C4⋊Dic73—41no—
81C4⋊Dic73—41no—
82C4×Dic73—24no—
83C4×Dic73—22no—
84C4×Dic73—22no—
85C4×Dic73—42no—
86C2×Q8×C73—22no—
87C2×Q8×C73—21no—
88C2×Q8×C73—21no—
89C2×Q8×C73—22no—
90C2×Q8×C73—44no—
91C22×C4×C73—22no—
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73C2×D411yescharacteristic
94C22×C4×C73C2×D411yescharacteristic
95Q16×C73—84no—
96C2×C8×C73—22no—
97C4⋊C4×C73—21no—
98C4⋊C4×C73—21no—
99C4⋊C4×C73—21no—
100C4⋊C4×C73—21no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—41no—
104C4⋊C4×C73—41no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73—41no—
107C22⋊C4×C73—41no—
108C42×C73—41no—
109C42×C73C2×D411yescharacteristic
110C42×C73C2×D411yescharacteristic
111C42×C73C2×D411yescharacteristic
112C42×C73#4111yescharacteristic
113C23×C73#12011yescharacteristic
114Q8×C73—42no—
115Q8×C73—44no—
116Q8×C73—42no—
117Q8×C73—82no—
118C2×C4×C73—22no—
119C2×C4×C73—22no—
120C2×C4×C73—44no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—42no—
124C2×C4×C73#12011yescharacteristic
125C2×C4×C73#12011yescharacteristic
126C2×C4×C73#12111yescharacteristic
127C2×C4×C73#12111yescharacteristic
128C2×C4×C73#12111yescharacteristic
129C2×C4×C73#12711yescharacteristic
130C2×C4×C73#13512yes—
131C2×C4×C73#13712yes—
132C2×Dic73—24no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—44no—
137Dic146—82no—
138Dic146—81no—
139Dic146—81no—
140C8×C73—42no—
141#10596831—42no—
142#10596831—81no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73#39211yescharacteristic
146C22×C73#45212yes—
147C4×C73—21no—
148C4×C73—21no—
149C4×C73—82no—
150C4×C73—81no—
151C4×C73—81no—
152C4×C73#45512yes—
153C4×C73#45612yes—
154C4×C73#46412yes—
155C4×C73#49511yescharacteristic
156C4×C73#49711yescharacteristic
157Dic73—44no—
158Dic73—82no—
159#10298—731nomaximal, Sylow
160C2×C73—22no—
161C2×C73#200411yescharacteristic
162C2×C73#200511yescharacteristic
163C2×C73#201511yescharacteristic
164#3229—731no—
165#3209D7311yescharacteristic
166#2862—731no—
167#1994—731no—
168#1548—731no—
169#1546—731no—
170#1276—731no—
171C73#1029811yescharacteristic, complemented, Sylow
172#533C2×D7311yescharacteristic
173C2×C4⋊Q8—731no—
174C2×C4⋊Q8C2×D7311yescharacteristic
175#498—1461no—
176#494—21no—
177#487—21no—
178#487—1461no—
179#483—1462no—
180#475—1462no—
181#462—22no—
182#451—22no—
183#431—731no—
184#431C2×D7311yescharacteristic
185#423—731no—
186#391—1462no—
187#391—1462no—
188#382—1462no—
189C43—731no—
190#339—1462no—
191#333—1462no—
192#332—1462no—
193C22×Q8—1461no—
194C22×Q8C22⋊D7312yes—
195C2×Q16—24no—
196C2×M16—1461no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—1461no—
202C4⋊Q8—1461no—
203C4⋊Q8—1461no—
204C4⋊Q8—1461no—
205C4⋊Q8C22⋊D7311yescharacteristic
206C4⋊Q8C22⋊D7311yescharacteristic
207#125—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212#117—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—732no—
216C2×C4⋊C4—1461no—
217C2×C42—732no—
218C2×C42—1461no—
219C2×C42—1461no—
220C2×C42C22×D7311yescharacteristic
221#106—1464no—
222C4⋊C8—732no—
223C4⋊C8—1461no—
224C4⋊C8—1461no—
225C4⋊C8C22⋊D7312yes—
226#103—24no—
227#103—1464no—
228C22⋊C8—21no—
229C22⋊C8—1461no—
230C4×C8—21no—
231C4×C8—1461no—
232#95—1468no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—2921no—
239C2×Q8—2921no—
240C2×Q8—2922no—
241C2×Q8—2922no—
242C22×C4—22no—
243C22×C4—1462no—
244C22×C4—1462no—
245C22×C4D4×D7311yescharacteristic
246C22×C4D4×D7311yescharacteristic
247C22×C4C2×C22⋊D7311yescharacteristic
248Q16—84no—
249M16—2922no—
250C2×C8—22no—
251C2×C8—2921no—
252C2×C8—1462no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—2921no—
264C4⋊C4—2921no—
265C4⋊C4—1462no—
266C4⋊C4—1462no—
267C22⋊C4—41no—
268C42—41no—
269C42—2922no—
270C42—1464no—
271C42—1462no—
272C42—1462no—
273C42D4×D7311yescharacteristic
274C42C2×C22⋊D7311yescharacteristic
275C42C2×C22⋊D7311yescharacteristic
276C42#1192402711yescharacteristic
277C23#1455655911yescharacteristic
278Q8—42no—
279Q8—44no—
280Q8—42no—
281Q8—82no—
282Q8—5841no—
283Q8—5841no—
284Q8—5842no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—2922no—
292C2×C4—2922no—
293C2×C4—2924no—
294C2×C4—2922no—
295C2×C4—1464no—
296C2×C4#1455655911yescharacteristic
297C2×C4#1455656011yescharacteristic
298C2×C4#1455656211yescharacteristic
299C2×C4#1455656311yescharacteristic
300C2×C4#1455656311yescharacteristic
301C2×C4#1455658711yescharacteristic
302C2×C4#1455661412yes—
303C2×C4#1455662312yes—
304C8—42no—
305C8—2922no—
306C8—5841no—
307C22—21no—
308C22—21no—
309C22#1798849511yescharacteristic
310C22#1798874812yes—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—81no—
315C4—81no—
316C4—2924no—
317C4—5842no—
318C4#1798876512yes—
319C4#1798876812yes—
320C4#1798881112yes—
321C4#1798897211yescharacteristic
322C4#1798898011yescharacteristic
323C2—22no—
324C2#2261095811yescharacteristic
325C2#2261096411yescharacteristic
326C2#2261102211yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#3209C2
socleC22×C73#392
prime radicalC23×C73#120
hypercenter#3209D73
hypocenterC73#10298
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
171. C73#10298semidirect

Maximal subgroups

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

subgroupindex
#226210722
#226183082
#226109092
#226086122
#226086022
#226077632
C73×#32092
#1029873