Finite Group Databasebeta

← #33005663

Subgroups — #33005663

Order 18688.

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

normal subgroups only

This group has 484 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×#2804C211yesmaximal, characteristic
6#17989062C2211yescharacteristic
7#17988627—21no—
8C2×#14556508C2211yescharacteristic
9C2×#14556506C2212yes—
10#17988460—22no—
11#17988353—22no—
12#17988347—21no—
13#400×C73—21no—
14#393×C73—22no—
15#391×C73—21no—
16#390×C73C2211yescharacteristic
17#386×C73C2212yes—
18#385×C73—22no—
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—
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73D411yescharacteristic
39C2×C22⋊C4×C73—22no—
40C2×C22⋊C4×C73—22no—
41C2×C22⋊C4×C73D412yes—
42C2×C42×C73D411yescharacteristic
43#95×C73—22no—
44#95×C73—22no—
45#95×C73—21no—
46#95×C73—22no—
47C22×Dic73—22no—
48C22×Dic73—22no—
49C22×Dic73—42no—
50#11924012—22no—
51#11924012—41no—
52C2×#10596831—41no—
53C4⋊Dic73—21no—
54C4⋊Dic73—21no—
55#11924007—84no—
56C22⋊Dic73—24no—
57C22⋊Dic73—42no—
58C22⋊Dic73—42no—
59C22⋊Dic73—41no—
60C4×Dic73—41no—
61C24×C73C2×D412yes—
62C2×Q8×C73—42no—
63C2×Q8×C73—81no—
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—21no—
71C22×C4×C73—22no—
72C22×C4×C73—21no—
73C22×C4×C73—22no—
74C22×C4×C73—42no—
75C22×C4×C73—82no—
76C22×C4×C73C2×D411yescharacteristic
77C4⋊C4×C73—41no—
78C4⋊C4×C73—81no—
79C4⋊C4×C73—81no—
80C22⋊C4×C73—44no—
81C22⋊C4×C73—42no—
82C22⋊C4×C73—82no—
83C42×C73—41no—
84C23×C73—22no—
85C23×C73—22no—
86C23×C73—42no—
87C23×C73—42no—
88C23×C73#12011yescharacteristic
89C23×C73C22⋊Q812yes—
90C23×C73C4⋊Q812yes—
91Q8×C73—84no—
92D4×C73—41no—
93D4×C73—42no—
94D4×C73—41no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—22no—
98C2×C4×C73—41no—
99C2×C4×C73—41no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—44no—
106C2×C4×C73—41no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—82no—
110C2×C4×C73C22⋊Q811yescharacteristic
111C2×C4×C73C4⋊Q811yescharacteristic
112C2×Dic73—44no—
113C2×Dic73—41no—
114C2×Dic73—41no—
115C2×Dic73—82no—
116C2×Dic73—82no—
117C2×Dic73—82no—
118C2×Dic73—82no—
119#10596831—42no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—22no—
123C22×C73—22no—
124C22×C73—42no—
125C22×C73—42no—
126C22×C73—42no—
127C22×C73—42no—
128C22×C73—82no—
129C22×C73#39511yescharacteristic
130C22×C73B(4,𝔽2)11yescharacteristic
131C22×C73#45811yescharacteristic
132C4×C73—21no—
133C4×C73—21no—
134C4×C73—42no—
135C4×C73—82no—
136C4×C73—81no—
137C4×C73—82no—
138Dic73—84no—
139Dic73—81no—
140Dic73—81no—
141#13463—731nomaximal, Sylow
142C2×C73—21no—
143C2×C73—21no—
144C2×C73—42no—
145C2×C73—42no—
146C2×C73#206211yescharacteristic
147C2×C73#219811yescharacteristic
148C2×C73#219911yescharacteristic
149#2804D7311yescharacteristic
150#1867—732no—
151#1856—731no—
152#1393—731no—
153#1390—732no—
154C73#1346311yescharacteristic, complemented, Sylow
155#518—731no—
156#428—1461no—
157C2×#100—731no—
158C2×#99—732no—
159#400—21no—
160#393—22no—
161#391—21no—
162#390C2×D7311yescharacteristic
163#386C2×D7312yes—
164#385—22no—
165#385—1462no—
166#342—1462no—
167#339—1461no—
168C22×Q8—21no—
169C22×D4C22×D7311yescharacteristic
170C23×C4C22⋊D7312yes—
171C2×M16—1461no—
172C4×D4—2921no—
173C4×D4—2921no—
174#117—1461no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—2922no—
178C2×C4⋊C4C22⋊D7311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—1462no—
182C2×C22⋊C4—1462no—
183C2×C22⋊C4C22⋊D7312yes—
184C2×C42C22⋊D7311yescharacteristic
185#107—2921no—
186#106—2921no—
187#100—732no—
188#100—1461no—
189#99—734no—
190#99—1462no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—2922no—
196C24C2×C22⋊D7312yes—
197C2×Q8—42no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4Q8×D7311yescharacteristic
203C2×D4Q8×D7312yes—
204C2×D4#1192402911yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—2922no—
212C22×C4—1462no—
213C22×C4—1462no—
214C22×C4C2×C22⋊D7311yescharacteristic
215M16—2921no—
216M16—1462no—
217C2×C8—2921no—
218C4⋊C4—41no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—5844no—
222C4⋊C4—1461no—
223C4⋊C4—1461no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—2922no—
228C22⋊C4—2922no—
229C22⋊C4—2921no—
230C22⋊C4—1464no—
231C42—41no—
232C42—2921no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1455656011yescharacteristic
238C23#1455656712yes—
239C23#1455658912yes—
240Q8—84no—
241D4—41no—
242D4—42no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—2924no—
260C2×C4—2921no—
261C2×C4—2921no—
262C2×C4—5842no—
263C2×C4—5842no—
264C2×C4—5842no—
265C2×C4—5842no—
266C2×C4#1455656711yescharacteristic
267C2×C4#1455658911yescharacteristic
268C8—2922no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1798850611yescharacteristic
279C22#1798877411yescharacteristic
280C22#1798877711yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—82no—
285C4—81no—
286C4—82no—
287C4—5844no—
288C4—5841no—
289C4—5841no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2261128311yescharacteristic
295C2#2261192711yescharacteristic
296C2#2261193211yescharacteristic
subgroupG/H
centerC22#17988774
commutator, prime radical, cosocleC22×D4×C73C23
FrattiniC22×D4C22×D73
FittingC73×#2804C2
socleC22×C73B(4,𝔽2)
hypercenter#2804D73
hypocenterC73#13463
LCSG ▷ C22×D4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2804

Complements

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

NH ≅ G/Ntype
29. #14556508Q8semidirect
31. #14556506Q8semidirect
154. C73#13463semidirect

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×#28042
#1346373