Finite Group Databasebeta

← #33005651

Subgroups — #33005651

Order 18688.

540 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 81 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#22610211C212yesmaximal
2#22610140C211yesmaximal, characteristic
3#22608056C211yesmaximal, characteristic
4#22608049C212yesmaximal
5C73×#2683C211yesmaximal, characteristic
6#17989062C2211yescharacteristic
7#17988627—21no—
8C2×#14556508C2211yescharacteristic
9C2×#14556506C2212yes—
10#17988460—22no—
11#17988353—22no—
12#17988347—21no—
13C2×C22⋊Q8×C73—22no—
14#396×C73—22no—
15#395×C73—21no—
16#390×C73C2211yescharacteristic
17#389×C73—21no—
18#386×C73C2212yes—
19C2×#11924012—21no—
20D4×Dic73—41no—
21D4×Dic73—41no—
22#14556550—21no—
23C2×#11924007—42no—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26#14556524—41no—
27#14556522—41no—
28#14556508—21no—
29#14556508Q812yescomplemented
30#14556506—22no—
31#14556506Q814yescomplemented
32#14556498—42no—
33C22×Q8×C73—21no—
34C22×D4×C73C2311yescharacteristic
35C23×C4×C73D412yes—
36C22⋊Q8×C73—48no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—22no—
39C2×C4⋊C4×C73—22no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73D411yescharacteristic
42C2×C22⋊C4×C73—22no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73D412yes—
45C2×C42×C73D411yescharacteristic
46#95×C73—21no—
47#95×C73—22no—
48C22×Dic73—22no—
49C22×Dic73—22no—
50C22×Dic73—42no—
51#11924012—22no—
52#11924012—41no—
53C2×#10596831—41no—
54C4⋊Dic73—21no—
55C4⋊Dic73—21no—
56#11924007—84no—
57C22⋊Dic73—24no—
58C22⋊Dic73—42no—
59C22⋊Dic73—42no—
60C22⋊Dic73—41no—
61C4×Dic73—41no—
62C24×C73C2×D412yes—
63C2×Q8×C73—44no—
64C2×Q8×C73—81no—
65C2×D4×C73—41no—
66C2×D4×C73—41no—
67C2×D4×C73C2×Q811yescharacteristic
68C2×D4×C73C2×Q812yes—
69C2×D4×C73#4111yescharacteristic
70C22×C4×C73—22no—
71C22×C4×C73—21no—
72C22×C4×C73—22no—
73C22×C4×C73—21no—
74C22×C4×C73—44no—
75C22×C4×C73—42no—
76C22×C4×C73C2×D411yescharacteristic
77C4⋊C4×C73—24no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73—44no—
80C4⋊C4×C73—44no—
81C4⋊C4×C73—81no—
82C22⋊C4×C73—44no—
83C22⋊C4×C73—44no—
84C22⋊C4×C73—42no—
85C42×C73—41no—
86C23×C73—22no—
87C23×C73—22no—
88C23×C73—42no—
89C23×C73—42no—
90C23×C73#12011yescharacteristic
91C23×C73C22⋊Q812yes—
92C23×C73C4⋊Q812yes—
93Q8×C73—84no—
94D4×C73—41no—
95D4×C73—42no—
96D4×C73—41no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—22no—
100C2×C4×C73—24no—
101C2×C4×C73—42no—
102C2×C4×C73—41no—
103C2×C4×C73—42no—
104C2×C4×C73—41no—
105C2×C4×C73—44no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—42no—
111C2×C4×C73—82no—
112C2×C4×C73C22⋊Q811yescharacteristic
113C2×C4×C73C4⋊Q811yescharacteristic
114C2×Dic73—44no—
115C2×Dic73—41no—
116C2×Dic73—41no—
117C2×Dic73—82no—
118C2×Dic73—82no—
119C2×Dic73—82no—
120C2×Dic73—82no—
121#10596831—42no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73—22no—
125C22×C73—22no—
126C22×C73—42no—
127C22×C73—42no—
128C22×C73—42no—
129C22×C73—42no—
130C22×C73—82no—
131C22×C73#39511yescharacteristic
132C22×C73B(4,𝔽2)11yescharacteristic
133C22×C73#45811yescharacteristic
134C4×C73—21no—
135C4×C73—21no—
136C4×C73—42no—
137C4×C73—44no—
138C4×C73—82no—
139C4×C73—81no—
140Dic73—84no—
141Dic73—81no—
142Dic73—81no—
143#13461—731nomaximal, Sylow
144C2×C73—21no—
145C2×C73—21no—
146C2×C73—42no—
147C2×C73—42no—
148C2×C73#206211yescharacteristic
149C2×C73#219611yescharacteristic
150C2×C73#220111yescharacteristic
151#2683D7311yescharacteristic
152#1867—732no—
153#1856—731no—
154#1393—731no—
155#1390—732no—
156C73#1346111yescharacteristic, complemented, Sylow
157C2×C22⋊Q8—22no—
158#518—731no—
159#428—1461no—
160C2×#100—731no—
161C2×#99—732no—
162#396—22no—
163#395—21no—
164#390C2×D7311yescharacteristic
165#389—21no—
166#386C2×D7312yes—
167#385—1462no—
168#342—1462no—
169#339—1461no—
170C22×Q8—21no—
171C22×D4C22×D7311yescharacteristic
172C23×C4C22⋊D7312yes—
173C2×M16—1461no—
174C22⋊Q8—48no—
175C4×D4—2921no—
176C4×D4—2921no—
177#117—1461no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—2922no—
183C2×C4⋊C4C22⋊D7311yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—1462no—
187C2×C22⋊C4—1462no—
188C2×C22⋊C4C22⋊D7312yes—
189C2×C42C22⋊D7311yescharacteristic
190#107—2921no—
191#106—2921no—
192#100—732no—
193#100—1461no—
194#99—734no—
195#99—1462no—
196#95—21no—
197#95—22no—
198#95—2922no—
199C24C2×C22⋊D7312yes—
200C2×Q8—44no—
201C2×Q8—81no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4Q8×D7311yescharacteristic
205C2×D4Q8×D7312yes—
206C2×D4#1192402911yescharacteristic
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—2922no—
214C22×C4—1462no—
215C22×C4—1462no—
216C22×C4C2×C22⋊D7311yescharacteristic
217M16—2921no—
218M16—1462no—
219C2×C8—2921no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—81no—
225C4⋊C4—5844no—
226C4⋊C4—1461no—
227C4⋊C4—1461no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—2922no—
232C22⋊C4—2922no—
233C22⋊C4—2921no—
234C22⋊C4—1464no—
235C42—41no—
236C42—2921no—
237C23—22no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23#1455656011yescharacteristic
242C23#1455656712yes—
243C23#1455658912yes—
244Q8—84no—
245D4—41no—
246D4—42no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—2924no—
264C2×C4—2921no—
265C2×C4—2921no—
266C2×C4—5842no—
267C2×C4—5842no—
268C2×C4—5842no—
269C2×C4—5842no—
270C2×C4#1455656711yescharacteristic
271C2×C4#1455658911yescharacteristic
272C8—2922no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22#1798850611yescharacteristic
283C22#1798877411yescharacteristic
284C22#1798877711yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—44no—
289C4—82no—
290C4—81no—
291C4—5844no—
292C4—5841no—
293C4—5841no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#2261128311yescharacteristic
299C2#2261191511yescharacteristic
300C2#2261194411yescharacteristic
subgroupG/H
centerC22#17988774
commutator, prime radical, cosocleC22×D4×C73C23
FrattiniC22×D4C22×D73
FittingC73×#2683C2
socleC22×C73B(4,𝔽2)
hypercenter#2683D73
hypocenterC73#13461
LCSG ▷ C22×D4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2683

Complements

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

NH ≅ G/Ntype
29. #14556508Q8semidirect
31. #14556506Q8semidirect
156. C73#13461semidirect

Maximal subgroups

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

subgroupindex
#226102112
#226101402
#226080562
#226080492
C73×#26832
#1346173