Finite Group Databasebeta

← #34003494

Subgroups — #34003494

Order 18688.

518 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 124 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
1C4×#14556651C211yesmaximal, characteristic
2#22625170C211yesmaximal, characteristic
3#22625140C212yesmaximal, complemented
4#22624776C212yesmaximal, complemented
5C2×C8×C22⋊D73C211yesmaximal, characteristic
6#17989604C2212yes—
7C2×#14556651C412yes—
8C4×#11924037C2212yes—
9#17989467C412yes—
10#17989431C412yes—
11#17989430C412yes—
12#17989429—22no—
13#17989428—22no—
14C22×C8×D73C411yescharacteristic
15C2×C4×C22⋊D73C2211yescharacteristic
16C8×C22⋊D73—22no—
17C8×C22⋊D73—22no—
18C2×#14556520C2211yescharacteristic
19C2×C22⋊#10596831C411yescharacteristic
20C2×#14556503C411yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22C23×C8×C73C411yescharacteristic
23C2×#11924037C2×C414yes—
24#14556658—22no—
25#14556651—24no—
26C22⋊#10596833C2×C414yes—
27#14556646—22no—
28C4×#10596833—22no—
29C4×#10596833—22no—
30C22×C22⋊D73C2×C411yescharacteristic
31C22×C4×D73C2×C411yescharacteristic
32C22×#10596831C2×C411yescharacteristic
33C22×#10596831C2×C411yescharacteristic
34C2×C8×D73—21no—
35C2×C8×D73—21no—
36C2×C8×D73—42no—
37C4×C22⋊D73—22no—
38C4×C22⋊D73—22no—
39C2×#11924007C2×C411yescharacteristic
40C2×C22⋊Dic73C2×C411yescharacteristic
41C2×#11924005C2×C411yescharacteristic
42C2×C4×Dic73C2311yescharacteristic
43#14556520—22no—
44C22⋊#10596831—22no—
45#14556503—22no—
46C8×Dic73D412yes—
47C8×Dic73D412yes—
48C23×C4×C73C2×C411yescharacteristic
49C22×C8×C73—21no—
50C22×C8×C73—21no—
51C22×C8×C73—42no—
52C22×C8×C73C2×C411yescharacteristic
53C22×C8×C73C2×C411yescharacteristic
54#11924037—24no—
55#11924037—24no—
56C2×#10596833—24no—
57C2×#10596833—24no—
58C23×D73C4211yescharacteristic
59C2×C22⋊D73—22no—
60C2×C22⋊D73—22no—
61C2×C22⋊D73M1614yes—
62C22×Dic73C4211yescharacteristic
63C22×Dic73C22×C411yescharacteristic
64C22×Dic73C22×C411yescharacteristic
65C2×C4×D73—21no—
66C2×C4×D73—21no—
67C2×C4×D73—41no—
68C2×C4×D73—41no—
69C2×#10596831—21no—
70C2×#10596831—41no—
71C2×#10596831—41no—
72C2×#10596831C22⋊C412yes—
73C8×D73—42no—
74C8×D73—42no—
75#11924007—22no—
76C22⋊Dic73—22no—
77#11924005—22no—
78C4×Dic73C2×D412yes—
79C4×Dic73#4112yes—
80C24×C73C4211yescharacteristic
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—41no—
84C22×C4×C73—41no—
85C22×C4×C73C4211yescharacteristic
86C22×C4×C73C22×C411yescharacteristic
87C2×C8×C73—22no—
88C2×C8×C73—21no—
89C2×C8×C73—42no—
90C2×C8×C73—42no—
91C2×C8×C73—41no—
92C2×C8×C73C22⋊C411yescharacteristic
93C2×C8×C73C22⋊C411yescharacteristic
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73#9712yes—
99C23×C73C2×C4211yescharacteristic
100C22×D73—21no—
101C22×D73—21no—
102C22×D73—41no—
103C22×D73—41no—
104C22×D73#9712yes—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73C2×C22⋊C411yescharacteristic
113C2×C4×C73#11711yescharacteristic
114C22⋊D73—28no—
115C2×Dic73—21no—
116C2×Dic73—41no—
117C2×Dic73—41no—
118C2×Dic73C4×D412yes—
119C2×Dic73C4×D412yes—
120C2×Dic73C2×M1612yes—
121C2×Dic73#13112yes—
122C4×D73—42no—
123C4×D73—41no—
124C4×D73—41no—
125#10596833—44no—
126#10596833—44no—
127C8×C73—21no—
128C8×C73—41no—
129C8×C73#10012yescomplemented
130#10596831—22no—
131#10596831—41no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—22no—
135C22×C73—42no—
136C22×C73—42no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73C4×C22⋊C411yescharacteristic
140C22×C73C2×#9711yescharacteristic
141C22×C73#40511yescharacteristic
142C2×D73—22no—
143C2×D73—42no—
144C2×D73—41no—
145C2×D73—41no—
146C2×D73—42no—
147C4×C73—21no—
148C4×C73—41no—
149C4×C73C2×#10011yescharacteristic
150C4×C73#41311yescharacteristic
151Dic73—22no—
152Dic73—41no—
153Dic73#43514yes—
154#8164—731nomaximal, Sylow
155C2×C73—21no—
156C2×C73—21no—
157C2×C73—42no—
158C2×C73#175211yescharacteristic
159C2×C73C4×#10011yescharacteristic
160C2×C73#175711yescharacteristic
161D73—42no—
162C2×C8×D4—731no—
163C4×#100—731no—
164#1460—731no—
165#1451—732no—
166#1283—732no—
167C73#816411yescharacteristic, complemented, Sylow
168C23×C8—731no—
169C23×C8#6440811yescharacteristic
170C2×C4×D4—731no—
171C8×D4—1462no—
172C8×D4—1462no—
173#431—732no—
174C2×C4⋊C8—731no—
175C2×#100—732no—
176C2×C22⋊C8—731no—
177C2×C22⋊C8—731no—
178C4×M16—732no—
179C2×C4×C8—731no—
180#343—732no—
181#323—732no—
182#323—732no—
183#322—1462no—
184#322—1462no—
185C22×D4—731no—
186C23×C4—731no—
187C23×C4C2×#6440811yescharacteristic
188C2×M16—734no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—42no—
192C22×C8—2922no—
193C22×C8—731no—
194C22×C8—731no—
195C22×C8—1461no—
196C22×C8—1461no—
197C22×C8C2×#6440811yescharacteristic
198C22×C8C2×#6440811yescharacteristic
199C4×D4—1462no—
200C4×D4—1462no—
201C2×C4⋊C4—731no—
202C2×C22⋊C4—731no—
203C2×C22⋊C4—731no—
204C2×C42—731no—
205C4⋊C8—1462no—
206C4⋊C8—1462no—
207#100—1464no—
208C22⋊C8—734no—
209C22⋊C8—1462no—
210C22⋊C8—1462no—
211#97—1462no—
212C4×C8—732no—
213C4×C8—732no—
214C4×C8—1462no—
215C4×C8—1462no—
216C24—731no—
217C24C4×#6440811yescharacteristic
218C2×D4—734no—
219C2×D4—1462no—
220C2×D4—1462no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—2921no—
226C22×C4—2921no—
227C22×C4—731no—
228C22×C4—731no—
229C22×C4—731no—
230C22×C4—1461no—
231C22×C4—1461no—
232C22×C4C4×#6440811yescharacteristic
233C22×C4C22×#6440811yescharacteristic
234M16—1464no—
235M16—1464no—
236C2×C8—22no—
237C2×C8—21no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8—2922no—
242C2×C8—2922no—
243C2×C8—2921no—
244C2×C8—2921no—
245C2×C8—732no—
246C2×C8—1464no—
247C2×C8—1464no—
248C2×C8—1461no—
249C2×C8C22⋊#6440811yescharacteristic
250C2×C8C22⋊#6440811yescharacteristic
251C4⋊C4—1462no—
252C22⋊C4—1462no—
253C22⋊C4—1462no—
254C42—732no—
255C42—732no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—2921no—
261C23—2921no—
262C23—732no—
263C23—1461no—
264C23—1461no—
265C23#1455664512yes—
266C23C2×C4×#6440811yescharacteristic
267D4—1468no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—2922no—
276C2×C4—2921no—
277C2×C4—2921no—
278C2×C4—2921no—
279C2×C4—2921no—
280C2×C4—732no—
281C2×C4—732no—
282C2×C4—732no—
283C2×C4—732no—
284C2×C4—1461no—
285C2×C4C2×C22⋊#6440811yescharacteristic
286C2×C4#1455667011yescharacteristic
287C8—21no—
288C8—41no—
289C8—2924no—
290C8—2924no—
291C8—2921no—
292C8—1462no—
293C8#1455665112yescomplemented
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—2922no—
302C22—2922no—
303C22—2921no—
304C22—2921no—
305C22—1462no—
306C22C4×C22⋊#6440811yescharacteristic
307C22C2×#1455664511yescharacteristic
308C22#1798955011yescharacteristic
309C4—21no—
310C4—41no—
311C4—2921no—
312C4—734no—
313C4—1462no—
314C4C2×#1455665111yescharacteristic
315C4#1798957311yescharacteristic
316C2—21no—
317C2—21no—
318C2—42no—
319C2—2922no—
320C2#2262566711yescharacteristic
321C2C4×#1455665111yescharacteristic
322C2#2262568611yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC23×C73C2×C42
FrattiniC22×C4C22×#64408
FittingC23×C8×C73C4
socleC22×C73C4×C22⋊C4
prime radicalC22×C22⋊D73C2×C4
cosocleC2×C4×Dic73C23
hypercenterC23×C8#64408
hypocenterC73#8164
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
3. #22625140C2semidirect
4. #22624776C2semidirect
129. C8×C73#100semidirect
167. C73#8164semidirect
293. C8#14556651semidirect

Maximal subgroups

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

subgroupindex
C4×#145566512
#226251702
#226251402
#226247762
C2×C8×C22⋊D732
#816473