Finite Group Databasebeta

← C22×#17989673 #34034493

Subgroups — C22×#17989673 #34034493

Order 18688.

1806 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 349 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
1C22×C22⋊#10596842C212yesmaximal
2C2×#17989673C2112yesmaximal, direct product factor
3C23×C4⋊#64408C211yesmaximal, characteristic
4C23×#10596842—22no—
5C2×C22⋊#10596842—26no—
6C2×C22⋊#10596842C22112yes—
7#17989673C22116yesdirect product factor
8C24×#64408C411yescharacteristic
9C24×#64408C2211yescharacteristic
10C22×C4⋊#64408—22no—
11C22×C4⋊#64408—26no—
12C22×C4⋊#64408C416yes—
13C22×C4⋊#64408C2216yes—
14C23×C4×D73C411yescharacteristic
15C22×#10596842—212no—
16C22×#10596842—48no—
17C22⋊#10596842—212no—
18C22⋊#10596842C2318yes—
19C23×#64408—23no—
20C23×#64408—21no—
21C23×#64408—44no—
22C23×#64408—44no—
23C23×#64408C2×C416yes—
24C23×#64408C2×C413yes—
25C23×#64408D412yes—
26C23×#64408D416yes—
27C23×#64408C2313yes—
28C2×C4⋊#64408—212no—
29C2×C4⋊#64408—212no—
30C2×C4⋊#64408—212no—
31C2×C4⋊#64408—212no—
32C2×C4⋊#64408C2×C4112yes—
33C2×C4⋊#64408C2314yes—
34C24×D73C2×C411yescharacteristic
35C23×Dic73C811yescharacteristic, complemented
36C22×C4×D73—23no—
37C22×C4×D73—21no—
38C22×C4×D73—23no—
39C22×C4×D73—21no—
40C22×C4×D73C816yescomplemented
41C22×C4×D73C2×C416yes—
42C23×C4×C73C811yescharacteristic, complemented
43C2×#10596842—28no—
44C2×#10596842—424no—
45C22×#64408—28no—
46C22×#64408—26no—
47C22×#64408—26no—
48C22×#64408—46no—
49C22×#64408—412no—
50C22×#64408—46no—
51C22×#64408—412no—
52C22×#64408—44no—
53C22×#64408—412no—
54C22×#64408C22⋊C416yes—
55C22×#64408C22⋊C416yes—
56C22×#64408C22×C414yes—
57C22×#64408C22×C413yes—
58C22×#64408C2×D416yes—
59C22×#64408C2×D416yes—
60C22×#64408C2411yescharacteristic
61C4⋊#64408—224no—
62C4⋊#64408—28no—
63C23×D73—24no—
64C23×D73—24no—
65C23×D73C22⋊C413yes—
66C23×D73C22⋊C411yescharacteristic
67C23×D73C2×C813yes—
68C23×D73M1613yes—
69C23×D73M1611yescharacteristic
70C23×D73C22×C413yes—
71C22×Dic73—23no—
72C22×Dic73—21no—
73C22×Dic73C2×C816yescomplemented
74C2×C4×D73—26no—
75C2×C4×D73—212no—
76C2×C4×D73—26no—
77C2×C4×D73—26no—
78C2×C4×D73—26no—
79C2×C4×D73—26no—
80C2×C4×D73—26no—
81C2×C4×D73C2×C8112yescomplemented
82C2×C4×D73C22×C414yes—
83C24×C73C2×C811yescharacteristic
84C22×C4×C73—23no—
85C22×C4×C73—21no—
86C22×C4×C73C2×C816yescomplemented
87C23×C73—24no—
88C23×C73C22⋊C813yes—
89C23×C73C22⋊C811yescharacteristic
90C23×C73C22×C813yes—
91C22×D73—24no—
92C22×D73—212no—
93C22×D73—212no—
94C22×D73—26no—
95C22×D73—26no—
96C22×D73—26no—
97C22×D73—26no—
98C22×D73C22⋊C813yes—
99C22×D73C22⋊C816yes—
100C22×D73C22⋊C813yes—
101C22×D73#9914yes—
102C22×D73#10114yes—
103C22×D73C2×C22⋊C413yes—
104C22×D73C2×C22⋊C413yes—
105C22×D73C22×C813yes—
106C22×D73C2×M1613yes—
107C22×D73C2×M1613yes—
108C22×D73C23×C411yescharacteristic
109C2×#64408—212no—
110C2×#64408—212no—
111C2×#64408—24no—
112C2×#64408—46no—
113C2×#64408—43no—
114C2×#64408—46no—
115C2×#64408—41no—
116C2×#64408—412no—
117C2×#64408—412no—
118C2×#64408—44no—
119C2×#64408C2×C22⋊C416yes—
120C2×#64408C22×D412yes—
121#10596842—48no—
122C2×C4×C73—26no—
123C2×C4×C73—26no—
124C2×C4×C73C22×C814yescomplemented
125C2×Dic73—26no—
126C2×Dic73—26no—
127C2×Dic73C22×C814yescomplemented
128C4×D73—24no—
129C4×D73—24no—
130C4×D73—212no—
131C4×D73—212no—
132C22×C73—26no—
133C22×C73—26no—
134C22×C73#32414yescomplemented
135C22×C73C2×C22⋊C813yes—
136C22×C73C2×C22⋊C813yes—
137C22×C73C23×C811yescharacteristic
138C2×D73—26no—
139C2×D73—26no—
140C2×D73—23no—
141C2×D73—21no—
142C2×D73—23no—
143C2×D73—21no—
144C2×D73—212no—
145C2×D73—26no—
146C2×D73—26no—
147C2×D73#324112yescomplemented
148C2×D73C2×C22⋊C813yes—
149C2×D73C2×C22⋊C813yes—
150C2×D73C2×#9916yes—
151C2×D73C2×#10116yes—
152C2×D73C22×C22⋊C411yescharacteristic
153C2×D73C22×M1611yescharacteristic
154#64408—22no—
155#64408—26no—
156#64408—44no—
157C4×C73—24no—
158Dic73—24no—
159C22×#324—731nomaximal, Sylow
160C2×C73—21no—
161C2×C73—23no—
162C2×C73C2×#32416yescomplemented
163C2×C73C22×C22⋊C811yescharacteristic
164D73—21no—
165D73—23no—
166D73C2×#32416yescomplemented
167D73C22×#9911yescharacteristic
168D73C22×#10111yescharacteristic
169C23×C4⋊C4—731no—
170C22×C22⋊C8—732no—
171C2×#324—7312no—
172C73C22×#32411yescharacteristic, complemented, Sylow
173C24×C4—731no—
174C24×C4—731no—
175C24×C4—731no—
176C23×C8—1462no—
177C22×C4⋊C4—736no—
178C22×C4⋊C4—736no—
179C22×C4⋊C4—1466no—
180C22×C4⋊C4—1462no—
181C2×C22⋊C8—7312no—
182C2×C22⋊C8—1466no—
183#324—7316no—
184C25—731no—
185C23×C4—2924no—
186C23×C4—2924no—
187C23×C4—733no—
188C23×C4—733no—
189C23×C4—736no—
190C23×C4—736no—
191C23×C4—732no—
192C23×C4—731no—
193C23×C4—736no—
194C23×C4—736no—
195C23×C4—1463no—
196C23×C4—1461no—
197C23×C4—1463no—
198C23×C4—1461no—
199C23×C4—1461no—
200C23×C4—1463no—
201C23×C4#1059684211yescharacteristic, complemented
202C22×C8—2928no—
203C22×C8—14612no—
204C2×C4⋊C4—734no—
205C2×C4⋊C4—7312no—
206C2×C4⋊C4—14612no—
207C2×C4⋊C4—14612no—
208C2×C4⋊C4—14612no—
209C2×C4⋊C4—14612no—
210C22⋊C8—738no—
211C22⋊C8—14612no—
212C24—731no—
213C24—733no—
214C24—733no—
215C24—731no—
216C24—733no—
217C24—733no—
218C24—1464no—
219C24—1464no—
220C24C2×#1059684211yescharacteristic
221C22×C4—23no—
222C22×C4—21no—
223C22×C4—2924no—
224C22×C4—29212no—
225C22×C4—29212no—
226C22×C4—2926no—
227C22×C4—29212no—
228C22×C4—2926no—
229C22×C4—736no—
230C22×C4—733no—
231C22×C4—736no—
232C22×C4—731no—
233C22×C4—734no—
234C22×C4—736no—
235C22×C4—736no—
236C22×C4—736no—
237C22×C4—7312no—
238C22×C4—734no—
239C22×C4—1466no—
240C22×C4—1466no—
241C22×C4—1466no—
242C22×C4—14612no—
243C22×C4—1466no—
244C22×C4—1466no—
245C22×C4—1466no—
246C22×C4—1466no—
247C22×C4—1466no—
248C22×C4—1468no—
249C22×C4—1463no—
250C22×C4—1461no—
251C22×C4C2×#1059684216yescomplemented
252C2×C8—29224no—
253C2×C8—1468no—
254C4⋊C4—1468no—
255C4⋊C4—14624no—
256C23—24no—
257C23—734no—
258C23—734no—
259C23—733no—
260C23—733no—
261C23—733no—
262C23—736no—
263C23—733no—
264C23—733no—
265C23—733no—
266C23—731no—
267C23—733no—
268C23—1466no—
269C23—1466no—
270C23—14612no—
271C23—1466no—
272C23—14612no—
273C23—1466no—
274C23—1464no—
275C23C22⋊#1059684213yes—
276C23C22⋊#1059684211yescharacteristic
277C23C22×#1059684213yes—
278C2×C4—26no—
279C2×C4—26no—
280C2×C4—29212no—
281C2×C4—29212no—
282C2×C4—2924no—
283C2×C4—2921no—
284C2×C4—2926no—
285C2×C4—2926no—
286C2×C4—2923no—
287C2×C4—732no—
288C2×C4—736no—
289C2×C4—734no—
290C2×C4—1464no—
291C2×C4—14612no—
292C2×C4—1464no—
293C2×C4—14612no—
294C2×C4—1464no—
295C2×C4—14612no—
296C2×C4—14612no—
297C2×C4—1466no—
298C2×C4—1466no—
299C2×C4C22×#1059684214yescomplemented
300C8—2928no—
301C22—26no—
302C22—26no—
303C22—731no—
304C22—733no—
305C22—736no—
306C22—7312no—
307C22—736no—
308C22—731no—
309C22—733no—
310C22—1463no—
311C22—1461no—
312C22—1463no—
313C22—14612no—
314C22—1466no—
315C22—1461no—
316C22—1466no—
317C22—1466no—
318C22—1466no—
319C22#1798967314yesdirect product factor
320C22C2×C22⋊#1059684213yes—
321C22C2×C22⋊#1059684213yes—
322C22C23×#1059684211yescharacteristic
323C4—24no—
324C4—2924no—
325C4—1462no—
326C4—1466no—
327C4—1464no—
328C2—23no—
329C2—21no—
330C2—731no—
331C2—731no—
332C2—736no—
333C2—1461no—
334C2—1463no—
335C2C2×#1798967316yesdirect product factor
336C2C22×C22⋊#1059684211yescharacteristic
subgroupG/H
centerC23C22⋊#10596842
commutatorC22×C73C23×C8
FrattiniC22C23×#10596842
FittingC23×C4×C73C8
socleC23×C73C22⋊C8
prime radicalC24×D73C2×C4
cosocleC22×#64408C24
hypercenterC23×C4#10596842
hypocenterC73C22×#324
LCSG ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#17989673C2direct (N × H)
7. #17989673C22direct (N × H)
35. C23×Dic73C8semidirect
40. C22×C4×D73C8semidirect
42. C23×C4×C73C8semidirect
73. C22×Dic73C2×C8semidirect
81. C2×C4×D73C2×C8semidirect
86. C22×C4×C73C2×C8semidirect
124. C2×C4×C73C22×C8semidirect
127. C2×Dic73C22×C8semidirect
134. C22×C73#324semidirect
147. C2×D73#324semidirect
162. C2×C73C2×#324semidirect
166. D73C2×#324semidirect
172. C73C22×#324semidirect
201. C23×C4#10596842semidirect
251. C22×C4C2×#10596842semidirect
299. C2×C4C22×#10596842semidirect
319. C22#17989673direct (N × H)
335. C2C2×#17989673direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊#105968422
C2×#179896732
C23×C4⋊#644082
C22×#32473