Finite Group Databasebeta

← #32995202

Subgroups — #32995202

Order 18688.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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#22618574C211yesmaximal, characteristic
2#22611731C211yesmaximal, characteristic, complemented
3#22611727C211yesmaximal, characteristic, complemented
4#22607873C211yesmaximal, characteristic, complemented
5#22607856C211yesmaximal, characteristic
6#22607856C211yesmaximal, characteristic
7C73×#1323C211yesmaximal, characteristic, complemented
8#17989055C411yescharacteristic
9#17988647—21no—
10#17988647—21no—
11#17988639—21no—
12#17988639—21no—
13#17988614C411yescharacteristic
14#17988555C411yescharacteristic
15#17988553C2211yescharacteristic
16M16×Dic73C411yescharacteristic
17C2×C8×Dic73C2211yescharacteristic
18#17988405—21no—
19#17988405—21no—
20C22⋊#11924003C2211yescharacteristic
21C22⋊#11924003C2211yescharacteristic
22#17988362—21no—
23C16×Dic73—21no—
24#424×C73C2211yescharacteristic
25C22⋊C16×C73C2211yescharacteristic
26C22⋊C16×C73C2211yescharacteristic
27C2×#11924028C2×C411yescharacteristic
28C2×#11924012C2×C411yescharacteristic
29C22×#10596831C2×C411yescharacteristic
30#14556556C811yescharacteristic
31#14556556C811yescharacteristic
32#14556553C811yescharacteristic
33#14556553C811yescharacteristic
34#14556551—21no—
35#14556551—21no—
36#14556548—21no—
37C2×C4×Dic73C2×C411yescharacteristic
38C42×D73—21no—
39C2×#11924003—21no—
40C2×#11924003—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43#14556503—21no—
44#14556503—21no—
45#14556503—21no—
46#14556503—21no—
47#14556501—21no—
48C8×Dic73—21no—
49C8×Dic73D411yescharacteristic
50C8×Dic73D411yescharacteristic
51C8×Dic73D411yescharacteristic
52C8×Dic73D411yescharacteristic
53C2×M16×C73C2×C411yescharacteristic
54C22×C8×C73C2311yescharacteristic
55#117×C73C2×C411yescharacteristic
56C2×C16×C73—21no—
57C2×C16×C73—21no—
58C4⋊C8×C73—21no—
59C4⋊C8×C73—21no—
60#11924028—41no—
61#11924028—41no—
62C2×Dic146C2×C811yescharacteristic
63C2×D292C2×C811yescharacteristic
64C2×C22⋊D73—21no—
65C22×Dic73M1611yescharacteristic
66C22×Dic73M1611yescharacteristic
67C2×C4×D73—21no—
68C2×C4×D73—21no—
69C2×C4×D73—21no—
70#11924012—41no—
71C2×#10596831—21no—
72C2×#10596831—41no—
73C2×#10596831C22⋊C411yescharacteristic
74C2×#10596831C22⋊C411yescharacteristic
75#11924007—21no—
76#11924007—21no—
77#11924005—21no—
78#11924005—21no—
79C4×Dic73—21no—
80C4×Dic73—21no—
81C4×Dic73C22⋊C411yescharacteristic
82C4×Dic73C22⋊C411yescharacteristic
83C4×Dic73C2×C811yescharacteristic
84C4×Dic73C2×C811yescharacteristic
85#11924003—41no—
86#11924003—41no—
87C22×C4×C73C22×C411yescharacteristic
88M16×C73—41no—
89C2×C8×C73—21no—
90C2×C8×C73—41no—
91C2×C8×C73C2×D411yescharacteristic
92C2×C8×C73C2×D411yescharacteristic
93C4⋊C4×C73C2×C811yescharacteristic
94C4⋊C4×C73C2×C811yescharacteristic
95C22⋊C4×C73—21no—
96C42×C73—21no—
97C16×C73—41no—
98C16×C73—41no—
99C23×C73C2×M1611yescharacteristic
100C22×D73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73C2×C22⋊C411yescharacteristic
106C2×C4×C73C22×C811yescharacteristic
107C22⋊D73—42no—
108C2×Dic73—22no—
109C2×Dic73—21no—
110C2×Dic73—41no—
111C2×Dic73C22⋊C811yescharacteristic
112C2×Dic73C22⋊C811yescharacteristic
113C2×Dic73C22⋊C811yescharacteristic
114C2×Dic73C22⋊C811yescharacteristic
115D292—41no—
116C4×D73—42no—
117C4×D73—41no—
118C4×D73—41no—
119C4×D73—42no—
120Dic146—41no—
121C8×C73—21no—
122C8×C73—21no—
123C8×C73—41no—
124#10596831—21no—
125#10596831—21no—
126#10596831—41no—
127C22×C73—21no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73C2×C22⋊C811yescharacteristic
131C2×D73—41no—
132C2×D73—41no—
133C2×D73—41no—
134C4×C73—21no—
135C4×C73—41no—
136C4×C73—41no—
137C4×C73#41011yescharacteristic
138C4×C73#41311yescharacteristic
139Dic73—21no—
140Dic73—21no—
141Dic73—21no—
142Dic73—41no—
143Dic73#35012yes—
144#11464—731nomaximal, Sylow
145C2×C73—41no—
146C2×C73#145811yescharacteristic
147C2×C73C2×#35011yescharacteristic
148C2×C73#211711yescharacteristic
149D73—81no—
150#2922—731no—
151#2163—731no—
152#2162—731no—
153#1323—731no—
154#1323D7311yescharacteristic, complemented
155#1315—731no—
156#1315—731no—
157C73#1146411yescharacteristic, complemented, Sylow
158C4×#41—731no—
159#433—1461no—
160#433—1461no—
161#431—1461no—
162#431—1461no—
163#424—731no—
164#424C2×D7311yescharacteristic
165#408—731no—
166#408—731no—
167C4×M16—731no—
168C2×C4×C8—731no—
169C4⋊C16—1461no—
170C4⋊C16—1461no—
171C22⋊C16—731no—
172C22⋊C16—731no—
173C22⋊C16C2×D7311yescharacteristic
174C22⋊C16C2×D7311yescharacteristic
175#346—1461no—
176C4×C16—1461no—
177C2×#41—731no—
178C2×M16—731no—
179C2×M16C4×D7311yescharacteristic
180C22×C8—731no—
181C22×C8C22×D7311yescharacteristic
182C4×Q8—731no—
183C4×Q8—731no—
184C4×D4—731no—
185C4×D4—731no—
186C4×D4—1461no—
187C4×D4—1461no—
188#117—1461no—
189#117C4×D7311yescharacteristic
190C2×C42—731no—
191C2×C42—1461no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—1461no—
195C2×C16—1461no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—1461no—
199C4⋊C8—1461no—
200C22⋊C8—1461no—
201C22⋊C8—1461no—
202C22⋊C8—1461no—
203C22⋊C8—1461no—
204#97—1461no—
205C4×C8—731no—
206C4×C8—731no—
207C4×C8—731no—
208C4×C8—731no—
209C4×C8—1461no—
210#41—2921no—
211#41—2921no—
212C2×Q8—731no—
213C2×D4—731no—
214C2×D4—1461no—
215C22×C4—731no—
216C22×C4—731no—
217C22×C4—1461no—
218C22×C4—1461no—
219C22×C4—1461no—
220C22×C4C2×C4×D7311yescharacteristic
221M16—41no—
222M16—2921no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—2921no—
226C2×C8—731no—
227C2×C8—731no—
228C2×C8—1461no—
229C2×C8D4×D7311yescharacteristic
230C2×C8D4×D7311yescharacteristic
231C4⋊C4—1461no—
232C4⋊C4—1461no—
233C4⋊C4C8×D7311yescharacteristic
234C4⋊C4C8×D7311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—1461no—
237C22⋊C4—1461no—
238C42—21no—
239C42—731no—
240C42—731no—
241C42—731no—
242C42—731no—
243C42—1461no—
244C42—1461no—
245C16—41no—
246C16—41no—
247C16—2921no—
248C16—2921no—
249C23—1461no—
250C23M16×D7311yescharacteristic
251Q8—2921no—
252D4—2921no—
253D4—2922no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—2922no—
259C2×C4—2922no—
260C2×C4—2921no—
261C2×C4—2921no—
262C2×C4—2921no—
263C2×C4—731no—
264C2×C4—731no—
265C2×C4—731no—
266C2×C4—731no—
267C2×C4—1461no—
268C2×C4—1462no—
269C2×C4C22⋊C4×D7311yescharacteristic
270C2×C4C2×C8×D7311yescharacteristic
271C8—21no—
272C8—21no—
273C8—41no—
274C8—2921no—
275C8—1461no—
276C8—1461no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—2921no—
281C22—2921no—
282C22—2921no—
283C22C22⋊C8×D7311yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—2921no—
288C4—732no—
289C4—1461no—
290C4—1461no—
291C4—1461no—
292C4#1798855911yescharacteristic
293C4#1798857011yescharacteristic
294C2—41no—
295C2—5841no—
296C2#2260824411yescharacteristic
297C2#350×D7311yescharacteristic
298C2#2261155711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1323C2
socleC22×C73C2×C22⋊C8
prime radicalC2×#11924028C2×C4
cosocleC22×C8×C73C23
hypercenter#1323D73
hypocenterC73#11464
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #22611731C2semidirect
3. #22611727C2semidirect
4. #22607873C2semidirect
7. C73×#1323C2semidirect
154. #1323D73semidirect
157. C73#11464semidirect

Maximal subgroups

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

subgroupindex
#226185742
#226117312
#226117272
#226078732
#226078562
C73×#13232
#1146473