Finite Group Databasebeta

← C2×#22626968 #34033579

Subgroups — C2×#22626968 #34033579

Order 18688.

598 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#17989673C212yesmaximal
2#22626968C214yesmaximal, direct product factor
3C2×#17989533C211yesmaximal, characteristic
4C2×C22⋊#10596842—22no—
5#17989673—22no—
6#17989673C2214yes—
7C22×C4⋊#64408—22no—
8C22×C4⋊#64408C411yescharacteristic
9C22×C4⋊#64408C2211yescharacteristic
10#17989533—22no—
11#17989533C412yes—
12#17989533C2212yes—
13C2×C42×D73C411yescharacteristic
14C22×#10596842—42no—
15C22⋊#10596842—24no—
16C22⋊#10596842—42no—
17C23×#64408—21no—
18C23×#64408D412yes—
19C2×C4⋊#64408—24no—
20C2×C4⋊#64408—24no—
21C2×C4⋊#64408—21no—
22C2×C4⋊#64408—41no—
23C2×C4⋊#64408—41no—
24C2×C4⋊#64408—48no—
25C2×C4⋊#64408—41no—
26C2×C4⋊#64408—41no—
27C2×C4⋊#64408C2×C412yes—
28C2×C4⋊#64408C2×C411yescharacteristic
29C2×C4⋊#64408D412yes—
30C2×C4⋊#64408C2311yescharacteristic
31C22×C4×D73—21no—
32C22×C4×D73C2×C411yescharacteristic
33C2×C4×Dic73—21no—
34C2×C4×Dic73C811yescharacteristic, complemented
35C42×D73—21no—
36C42×D73—21no—
37C42×D73C812yescomplemented
38C42×D73C2×C412yes—
39C2×C42×C73C811yescharacteristic, complemented
40C2×#10596842—44no—
41C2×#10596842—84no—
42C22×#64408—22no—
43C22×#64408—22no—
44C22×#64408—22no—
45C22×#64408—44no—
46C22×#64408—44no—
47C22×#64408—41no—
48C22×#64408—41no—
49C22×#64408C22⋊C412yes—
50C22×#64408C2×D412yes—
51C4⋊#64408—41no—
52C4⋊#64408—42no—
53C4⋊#64408—41no—
54C4⋊#64408—42no—
55C4⋊#64408—48no—
56C4⋊#64408—48no—
57C4⋊#64408—42no—
58C23×D73C22⋊C411yescharacteristic
59C22×Dic73—21no—
60C22×Dic73M1611yescharacteristic
61C2×C4×D73—24no—
62C2×C4×D73—24no—
63C2×C4×D73—22no—
64C2×C4×D73—21no—
65C2×C4×D73—21no—
66C2×C4×D73—21no—
67C2×C4×D73—21no—
68C2×C4×D73—22no—
69C2×C4×D73C22⋊C411yescharacteristic
70C2×C4×D73C2×C811yescharacteristic
71C2×C4×D73M1611yescharacteristic
72C2×C4×D73C22×C411yescharacteristic
73C4×Dic73—21no—
74C4×Dic73—22no—
75C4×Dic73—22no—
76C4×Dic73C2×C812yescomplemented
77C22×C4×C73—21no—
78C22×C4×C73C2×C811yescharacteristic
79C42×C73—21no—
80C42×C73C2×C812yescomplemented
81C23×C73C22⋊C811yescharacteristic
82C22×D73—22no—
83C22×D73—22no—
84C22×D73C22⋊C811yescharacteristic
85C22×D73#9911yescharacteristic
86C22×D73#9911yescharacteristic
87C22×D73#10111yescharacteristic
88C22×D73#10111yescharacteristic
89C22×D73C2×C22⋊C411yescharacteristic
90C2×#64408—22no—
91C2×#64408—22no—
92C2×#64408—44no—
93C2×#64408—42no—
94C2×#64408—42no—
95C2×#64408—44no—
96C2×#64408—42no—
97C2×#64408—82no—
98C2×#64408—82no—
99C2×#64408—82no—
100#10596842—84no—
101C2×C4×C73—24no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—22no—
105C2×C4×C73C22⋊C811yescharacteristic
106C2×C4×C73C22×C811yescharacteristic
107C2×Dic73—24no—
108C2×Dic73—21no—
109C2×Dic73—21no—
110C2×Dic73—22no—
111C2×Dic73C22⋊C811yescharacteristic
112C2×Dic73C2×M1611yescharacteristic
113C4×D73—28no—
114C4×D73—24no—
115C4×D73—24no—
116C4×D73—21no—
117C4×D73—21no—
118C4×D73—21no—
119C4×D73—21no—
120C4×D73—21no—
121C4×D73—21no—
122C4×D73—21no—
123C4×D73—21no—
124C22×C73—22no—
125C22×C73#32411yescharacteristic
126C22×C73#32411yescharacteristic
127C22×C73C2×C22⋊C811yescharacteristic
128C2×D73—22no—
129C2×D73—22no—
130C2×D73—22no—
131C2×D73—21no—
132C2×D73—21no—
133C2×D73—21no—
134C2×D73—21no—
135C2×D73#32411yescharacteristic
136C2×D73#32411yescharacteristic
137C2×D73#35312yes—
138C2×D73#35612yes—
139C2×D73C2×#9911yescharacteristic
140C2×D73C2×#10111yescharacteristic
141#64408—42no—
142#64408—42no—
143#64408—82no—
144C4×C73—24no—
145C4×C73—21no—
146C4×C73—21no—
147Dic73—24no—
148Dic73—21no—
149Dic73—21no—
150C2×#1325—731nomaximal, Sylow
151C2×C73—21no—
152C2×C73—21no—
153C2×C73#132512yescomplemented
154C2×C73C2×#32411yescharacteristic
155D73—21no—
156D73—21no—
157D73#132512yescomplemented
158D73C2×#35311yescharacteristic
159D73C2×#35611yescharacteristic
160C2×#384—731no—
161C2×#324—732no—
162#1325—734no—
163C73C2×#132511yescharacteristic, complemented, Sylow
164C22×C4⋊C4—731no—
165C22×C4⋊C4—731no—
166C22×C4⋊C4—1462no—
167C22×C42—731no—
168C2×C22⋊C8—1462no—
169#384—732no—
170#384—732no—
171#384—1462no—
172#324—734no—
173#324—1462no—
174C23×C4—732no—
175C23×C4—731no—
176C23×C4—1461no—
177C23×C4—1461no—
178C22×C8—2922no—
179C2×C4⋊C4—2928no—
180C2×C4⋊C4—2921no—
181C2×C4⋊C4—2921no—
182C2×C4⋊C4—2921no—
183C2×C4⋊C4—2921no—
184C2×C4⋊C4—731no—
185C2×C4⋊C4—732no—
186C2×C4⋊C4—731no—
187C2×C4⋊C4—732no—
188C2×C4⋊C4—1461no—
189C2×C4⋊C4—1464no—
190C2×C4⋊C4—1464no—
191C2×C42—731no—
192C2×C42—732no—
193C2×C42—732no—
194C2×C42—1461no—
195C2×C42—1461no—
196C2×C42—1461no—
197C2×C42#1059684211yescharacteristic, complemented
198C22⋊C8—2922no—
199C22⋊C8—1464no—
200C24—731no—
201C22×C4—21no—
202C22×C4—2924no—
203C22×C4—2921no—
204C22×C4—2921no—
205C22×C4—2924no—
206C22×C4—732no—
207C22×C4—732no—
208C22×C4—731no—
209C22×C4—731no—
210C22×C4—731no—
211C22×C4—731no—
212C22×C4—731no—
213C22×C4—1462no—
214C22×C4—1462no—
215C22×C4—1461no—
216C22×C4—1461no—
217C22×C4—1461no—
218C22×C4—1461no—
219C22×C4—1464no—
220C22×C4—1464no—
221C22×C4—1462no—
222C22×C4—1462no—
223C22×C4—1462no—
224C22×C4—1461no—
225C22×C4C2×#1059684211yescharacteristic
226C2×C8—2924no—
227C2×C8—5844no—
228C4⋊C4—2928no—
229C4⋊C4—2928no—
230C4⋊C4—2922no—
231C4⋊C4—2922no—
232C4⋊C4—2921no—
233C4⋊C4—2922no—
234C4⋊C4—2921no—
235C42—21no—
236C42—732no—
237C42—1462no—
238C42—1462no—
239C42—1461no—
240C42C2×#1059684212yescomplemented
241C23—731no—
242C23—731no—
243C23—731no—
244C23—731no—
245C23—731no—
246C23—731no—
247C23—1462no—
248C23—1462no—
249C23C22⋊#1059684211yescharacteristic
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—2924no—
255C2×C4—2922no—
256C2×C4—2922no—
257C2×C4—2922no—
258C2×C4—2924no—
259C2×C4—5842no—
260C2×C4—5842no—
261C2×C4—5842no—
262C2×C4—731no—
263C2×C4—731no—
264C2×C4—1461no—
265C2×C4—1461no—
266C2×C4—1461no—
267C2×C4—1461no—
268C2×C4—1461no—
269C2×C4—1461no—
270C2×C4—1461no—
271C2×C4—1461no—
272C2×C4—1464no—
273C2×C4—1464no—
274C2×C4—1468no—
275C2×C4—1462no—
276C2×C4—1462no—
277C2×C4—1464no—
278C2×C4—1461no—
279C2×C4—1461no—
280C2×C4—1462no—
281C2×C4C22⋊#1059684211yescharacteristic
282C2×C4C22×#1059684211yescharacteristic
283C8—5844no—
284C22—22no—
285C22—732no—
286C22—732no—
287C22—731no—
288C22—731no—
289C22—731no—
290C22—731no—
291C22—1461no—
292C22—1461no—
293C22—1462no—
294C22—1461no—
295C22—1462no—
296C22—1461no—
297C22—1462no—
298C22#1798967311yescharacteristic
299C22#1798967311yescharacteristic
300C22C2×C22⋊#1059684211yescharacteristic
301C4—24no—
302C4—21no—
303C4—21no—
304C4—2922no—
305C4—2922no—
306C4—5842no—
307C4—1464no—
308C4—1461no—
309C4—1461no—
310C2—21no—
311C2—21no—
312C2—731no—
313C2—732no—
314C2—731no—
315C2—1461no—
316C2—1461no—
317C2#2262696812yesdirect product factor
318C2C2×#1798967311yescharacteristic
subgroupG/H
centerC22#17989673
commutatorC2×C4×C73C22×C8
FrattiniC2×C4C22×#10596842
FittingC2×C42×C73C8
socleC22×C73#324
prime radicalC23×D73C22⋊C4
cosocleC2×C4⋊#64408C23
hypercenterC2×C42#10596842
hypocenterC73C2×#1325
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. #22626968C2direct (N × H)
34. C2×C4×Dic73C8semidirect
37. C42×D73C8semidirect
39. C2×C42×C73C8semidirect
76. C4×Dic73C2×C8semidirect
80. C42×C73C2×C8semidirect
153. C2×C73#1325semidirect
157. D73#1325semidirect
163. C73C2×#1325semidirect
197. C2×C42#10596842semidirect
240. C42C2×#10596842semidirect
317. C2#22626968direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179896732
#226269682
C2×#179895332
C2×#132573