Finite Group Databasebeta

← #34009919

Subgroups — #34009919

Order 18688.

454 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 108 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#22626156C212yesmaximal, complemented
2#22624868C212yesmaximal, complemented
3#22624851C212yesmaximal, complemented
4C2×C8×C22⋊D73C211yesmaximal, characteristic
5C2×#14556664—22no—
6#17989482—22no—
7C22⋊#11924031C2214yes—
8C22×C8×D73C2211yescharacteristic
9C2×C4×C22⋊D73C411yescharacteristic
10C8×C22⋊D73—22no—
11C8×C22⋊D73—22no—
12C2×#14556520C2211yescharacteristic
13C2×C22⋊#10596831C2211yescharacteristic
14C2×#14556503C411yescharacteristic
15C2×C8×Dic73C411yescharacteristic
16C23×C8×C73C411yescharacteristic
17#14556664—44no—
18C2×#11924031—24no—
19C22×C22⋊D73C811yescharacteristic
20C22×C4×D73C2×C411yescharacteristic
21C22×#10596831C2×C411yescharacteristic
22C22×#10596831C2311yescharacteristic
23C2×C8×D73—42no—
24C2×C8×D73D412yes—
25C2×C8×D73D412yes—
26C4×C22⋊D73—22no—
27C4×C22⋊D73—22no—
28C2×#11924007C811yescharacteristic
29C2×C22⋊Dic73C811yescharacteristic
30C2×#11924005C811yescharacteristic
31C2×C4×Dic73C2×C411yescharacteristic
32#14556520—22no—
33C22⋊#10596831—22no—
34#14556503—22no—
35C8×Dic73—21no—
36C8×Dic73—21no—
37C23×C4×C73C2×C411yescharacteristic
38C22×C8×C73—21no—
39C22×C8×C73—21no—
40C22×C8×C73—42no—
41C22×C8×C73C2×C411yescharacteristic
42C22×C8×C73C2×C411yescharacteristic
43#11924031—44no—
44C23×D73C2×C811yescharacteristic
45C2×C22⋊D73—22no—
46C2×C22⋊D73—22no—
47C2×C22⋊D73C1614yescomplemented
48C22×Dic73C2×C811yescharacteristic
49C22×Dic73C2×C811yescharacteristic
50C22×Dic73C2×C811yescharacteristic
51C2×C4×D73—41no—
52C2×C4×D73—41no—
53C2×C4×D73C22⋊C412yes—
54C2×C4×D73M1612yes—
55C2×#10596831—21no—
56C2×#10596831—41no—
57C2×#10596831—41no—
58C2×#10596831C2×D412yes—
59C8×D73—24no—
60C8×D73—42no—
61#11924007—22no—
62C22⋊Dic73—22no—
63#11924005—22no—
64C4×Dic73—21no—
65C4×Dic73—21no—
66C24×C73C2×C811yescharacteristic
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—41no—
70C22×C4×C73—41no—
71C22×C4×C73C2×C811yescharacteristic
72C22×C4×C73C22×C411yescharacteristic
73C2×C8×C73—22no—
74C2×C8×C73—21no—
75C2×C8×C73—42no—
76C2×C8×C73—42no—
77C2×C8×C73—41no—
78C2×C8×C73C22⋊C411yescharacteristic
79C2×C8×C73C22⋊C411yescharacteristic
80C23×C73—21no—
81C23×C73—21no—
82C23×C73—41no—
83C23×C73—41no—
84C23×C73C2×C1612yescomplemented
85C23×C73C22×C811yescharacteristic
86C22×D73—41no—
87C22×D73—41no—
88C22×D73C22⋊C812yes—
89C22×D73C2×C1612yes—
90C22×D73M3212yes—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—42no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73—41no—
98C2×C4×C73C2×C22⋊C411yescharacteristic
99C2×C4×C73C2×M1611yescharacteristic
100C22⋊D73—28no—
101C2×Dic73—21no—
102C2×Dic73—21no—
103C2×Dic73—21no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73C22⋊C812yes—
107C2×Dic73C2×C1612yescomplemented
108C4×D73—22no—
109C4×D73—22no—
110C4×D73—42no—
111C8×C73—21no—
112C8×C73—41no—
113C8×C73#9912yes—
114#10596831—22no—
115#10596831—41no—
116C22×C73—21no—
117C22×C73—21no—
118C22×C73—22no—
119C22×C73—42no—
120C22×C73—42no—
121C22×C73—41no—
122C22×C73—41no—
123C22×C73C2×C22⋊C811yescharacteristic
124C22×C73C22×C1611yescharacteristic
125C22×C73C2×M3211yescharacteristic
126C2×D73—22no—
127C2×D73—22no—
128C2×D73—42no—
129C2×D73—42no—
130C2×D73C22⋊C1614yes—
131C4×C73—21no—
132C4×C73—41no—
133C4×C73C2×#9911yescharacteristic
134C4×C73#41311yescharacteristic
135Dic73—22no—
136Dic73—22no—
137Dic73—41no—
138#11465—731nomaximal, Sylow
139C2×C73—21no—
140C2×C73—21no—
141C2×C73—42no—
142C2×C73#145911yescharacteristic
143C2×C73C2×C22⋊C1611yescharacteristic
144C2×C73#211511yescharacteristic
145D73—24no—
146C2×C8×D4—731no—
147C2×C22⋊C16—732no—
148#1322—732no—
149#1314—732no—
150C73#1146511yescharacteristic, complemented, Sylow
151C23×C8—731no—
152C23×C8#6440811yescharacteristic
153C2×C4×D4—731no—
154C22×C16—1462no—
155C8×D4—1462no—
156C8×D4—1462no—
157C2×C4⋊C8—731no—
158C2×C22⋊C8—731no—
159C2×C22⋊C8—731no—
160C2×C4×C8—731no—
161C22⋊C16—734no—
162C22⋊C16—1462no—
163C22×D4—731no—
164C23×C4—731no—
165C23×C4C2×#6440811yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—2922no—
170C22×C8—731no—
171C22×C8—732no—
172C22×C8—731no—
173C22×C8—732no—
174C22×C8C2×#6440811yescharacteristic
175C22×C8C2×#6440811yescharacteristic
176C4×D4—1462no—
177C4×D4—1462no—
178C2×C4⋊C4—731no—
179C2×C22⋊C4—731no—
180C2×C22⋊C4—731no—
181C2×C42—731no—
182C2×C16—2924no—
183C2×C16—1464no—
184C4⋊C8—1462no—
185C22⋊C8—1462no—
186C22⋊C8—1462no—
187C4×C8—1461no—
188C4×C8—1461no—
189C24—731no—
190C24#1192403611yescharacteristic
191C2×D4—734no—
192C2×D4—1462no—
193C2×D4—1462no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—2921no—
199C22×C4—2921no—
200C22×C4—731no—
201C22×C4—732no—
202C22×C4—731no—
203C22×C4—731no—
204C22×C4—732no—
205C22×C4#1192403611yescharacteristic
206C22×C4C22×#6440811yescharacteristic
207C2×C8—22no—
208C2×C8—21no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—2922no—
213C2×C8—2921no—
214C2×C8—2921no—
215C2×C8—732no—
216C2×C8—1461no—
217C2×C8—1464no—
218C2×C8C22⋊#6440811yescharacteristic
219C2×C8C22⋊#6440811yescharacteristic
220C4⋊C4—1462no—
221C22⋊C4—1462no—
222C22⋊C4—1462no—
223C42—1461no—
224C42—1461no—
225C16—2924no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—2921no—
231C23—2921no—
232C23—732no—
233C23—732no—
234C23—732no—
235C23#1455666412yescomplemented
236C23C2×#1192403611yescharacteristic
237D4—1468no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—2922no—
246C2×C4—2921no—
247C2×C4—2921no—
248C2×C4—732no—
249C2×C4—732no—
250C2×C4—1461no—
251C2×C4—1461no—
252C2×C4—1461no—
253C2×C4—1462no—
254C2×C4—1462no—
255C2×C4C2×C22⋊#6440811yescharacteristic
256C2×C4D73⋊M1611yescharacteristic
257C8—21no—
258C8—41no—
259C8—2921no—
260C8—1462no—
261C8#1455664912yes—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—42no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—2922no—
270C22—2922no—
271C22—734no—
272C22—1462no—
273C22—1462no—
274C22#1798955511yescharacteristic
275C22C2×#1455666411yescharacteristic
276C22#1798964311yescharacteristic
277C4—21no—
278C4—41no—
279C4—2921no—
280C4—1462no—
281C4—1462no—
282C4C2×#1455664911yescharacteristic
283C4#1798957311yescharacteristic
284C2—21no—
285C2—21no—
286C2—42no—
287C2—1464no—
288C2#2262516611yescharacteristic
289C2#2262615611yescharacteristic
290C2#2262616811yescharacteristic
subgroupG/H
centerC2×C8C22⋊#64408
commutatorC22×C73C22×C16
FrattiniC22×C4C22×#64408
FittingC23×C8×C73C4
socleC22×C73C2×C22⋊C8
prime radicalC22×C22⋊D73C8
cosocleC22×#10596831C23
hypercenterC23×C8#64408
hypocenterC73#11465
LCSG ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #22626156C2semidirect
2. #22624868C2semidirect
3. #22624851C2semidirect
47. C2×C22⋊D73C16semidirect
84. C23×C73C2×C16semidirect
107. C2×Dic73C2×C16semidirect
150. C73#11465semidirect
235. C23#14556664semidirect

Maximal subgroups

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

subgroupindex
#226261562
#226248682
#226248512
C2×C8×C22⋊D732
#1146573