Finite Group Databasebeta

← C2×#22624858 #34009786

Subgroups — C2×#22624858 #34009786

Order 18688.

726 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 120 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×#17989430C212yesmaximal
2#22624858C214yesmaximal, direct product factor
3C2×#17988448C211yesmaximal, characteristic
4C2×C22⋊#10596833—22no—
5#17989430—22no—
6#17989430C2214yes—
7C24×Dic73C411yescharacteristic
8C22×C22⋊Dic73C2211yescharacteristic
9#17988448—22no—
10#17988448C412yes—
11#17988448C2212yes—
12C2×#14556498—22no—
13C22×C22⋊C4×C73C411yescharacteristic
14C22×#10596833—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73—21no—
18C23×Dic73—21no—
19C23×Dic73—22no—
20C23×Dic73—41no—
21C23×Dic73—41no—
22C23×Dic73—42no—
23C23×Dic73C814yescomplemented
24C23×Dic73C2×C412yes—
25C23×Dic73D412yes—
26C2×C22⋊Dic73—41no—
27C2×C22⋊Dic73—41no—
28C2×C22⋊Dic73C2×C411yescharacteristic
29C2×C22⋊Dic73D412yes—
30C2×C22⋊Dic73C2311yescharacteristic
31#14556498—24no—
32#14556498—24no—
33C25×C73C2×C411yescharacteristic
34C23×C4×C73—21no—
35C2×C22⋊C4×C73—21no—
36C2×C22⋊C4×C73—42no—
37C2×C22⋊C4×C73C2×C412yes—
38C2×#10596833—44no—
39C2×#10596833—84no—
40C22×Dic73—24no—
41C22×Dic73—24no—
42C22×Dic73—22no—
43C22×Dic73—22no—
44C22×Dic73—22no—
45C22×Dic73—22no—
46C22×Dic73—44no—
47C22×Dic73—44no—
48C22×Dic73—42no—
49C22×Dic73—42no—
50C22×Dic73—41no—
51C22×Dic73—41no—
52C22×Dic73—42no—
53C22×Dic73—44no—
54C22×Dic73—44no—
55C22×Dic73—42no—
56C22×Dic73—42no—
57C22×Dic73—44no—
58C22×Dic73C22⋊C412yes—
59C22×Dic73C2×C814yescomplemented
60C22×Dic73C2×D412yes—
61C22⋊Dic73—41no—
62C22⋊Dic73—42no—
63C22⋊Dic73—41no—
64C24×C73—21no—
65C24×C73—21no—
66C24×C73—22no—
67C24×C73—44no—
68C24×C73C22⋊C411yescharacteristic
69C24×C73C22⋊C411yescharacteristic
70C24×C73C2×C812yes—
71C24×C73M1612yes—
72C24×C73C22×C411yescharacteristic
73C22×C4×C73—22no—
74C22×C4×C73—44no—
75C22×C4×C73—42no—
76C22⋊C4×C73—42no—
77C22⋊C4×C73—42no—
78C23×C73—24no—
79C23×C73—22no—
80C23×C73—22no—
81C23×C73—22no—
82C23×C73—22no—
83C23×C73—21no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—21no—
87C23×C73—42no—
88C23×C73—44no—
89C23×C73—44no—
90C23×C73—42no—
91C23×C73—48no—
92C23×C73—44no—
93C23×C73—44no—
94C23×C73C22⋊C812yes—
95C23×C73C22⋊C812yes—
96C23×C73#9911yescharacteristic
97C23×C73#9911yescharacteristic
98C23×C73#10012yes—
99C23×C73C2×C22⋊C411yescharacteristic
100C23×C73C22×C811yescharacteristic
101C23×C73C2×M1611yescharacteristic
102C2×C4×C73—44no—
103C2×C4×C73—42no—
104C2×C4×C73—84no—
105C2×Dic73—24no—
106C2×Dic73—22no—
107C2×Dic73—22no—
108C2×Dic73—48no—
109C2×Dic73—44no—
110C2×Dic73—44no—
111C2×Dic73—44no—
112C2×Dic73—44no—
113C2×Dic73—42no—
114C2×Dic73—42no—
115C2×Dic73—44no—
116C2×Dic73—42no—
117C2×Dic73—42no—
118C2×Dic73—82no—
119#10596833—84no—
120C22×C73—24no—
121C22×C73—24no—
122C22×C73—22no—
123C22×C73—22no—
124C22×C73—22no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—42no—
128C22×C73—41no—
129C22×C73—41no—
130C22×C73—42no—
131C22×C73—42no—
132C22×C73—44no—
133C22×C73—44no—
134C22×C73—44no—
135C22×C73—44no—
136C22×C73—42no—
137C22×C73—42no—
138C22×C73#32312yes—
139C22×C73#32312yes—
140C22×C73#35214yes—
141C22×C73C2×C22⋊C811yescharacteristic
142C22×C73C2×#9911yescharacteristic
143C22×C73C2×#10011yescharacteristic
144C4×C73—82no—
145Dic73—42no—
146Dic73—42no—
147Dic73—44no—
148C2×#1317—731nomaximal, Sylow
149C2×C73—21no—
150C2×C73—22no—
151C2×C73—21no—
152C2×C73—42no—
153C2×C73—41no—
154C2×C73—41no—
155C2×C73#131714yescomplemented
156C2×C73C2×#32311yescharacteristic
157C2×C73C2×#35212yes—
158C2×#381—731no—
159C2×#323—732no—
160#1317—734no—
161C73C2×#131711yescharacteristic, complemented, Sylow
162C24×C4—731no—
163C22×C22⋊C4—731no—
164C22×C22⋊C4#6440811yescharacteristic
165C2×C22⋊C8—1462no—
166#381—732no—
167#381—732no—
168#381—1462no—
169C2×#95—1462no—
170#323—734no—
171#323—1462no—
172C25C2×#6440811yescharacteristic
173C23×C4—21no—
174C23×C4—2921no—
175C23×C4—2921no—
176C23×C4—2922no—
177C23×C4—732no—
178C23×C4—734no—
179C23×C4—732no—
180C23×C4—1461no—
181C23×C4—1461no—
182C23×C4—1462no—
183C22×C8—2922no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—2921no—
187C2×C22⋊C4—2921no—
188C2×C22⋊C4—732no—
189C2×C22⋊C4—731no—
190C2×C22⋊C4—731no—
191C2×C22⋊C4C2×#6440812yes—
192C22⋊C8—2922no—
193C22⋊C8—1464no—
194#95—1464no—
195#95—1464no—
196C24—21no—
197C24—21no—
198C24—22no—
199C24—44no—
200C24C22⋊#6440811yescharacteristic
201C24C22⋊#6440811yescharacteristic
202C24#1192403612yes—
203C24#1192403812yes—
204C24C22×#6440811yescharacteristic
205C22×C4—22no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—2924no—
209C22×C4—2924no—
210C22×C4—2922no—
211C22×C4—2922no—
212C22×C4—2921no—
213C22×C4—2921no—
214C22×C4—2922no—
215C22×C4—2924no—
216C22×C4—2924no—
217C22×C4—2922no—
218C22×C4—2922no—
219C22×C4—2924no—
220C22×C4—734no—
221C22×C4—732no—
222C22×C4—732no—
223C22×C4—1462no—
224C22×C4—1464no—
225C22×C4—1464no—
226C22×C4—1462no—
227C22×C4—1462no—
228C22×C4—1462no—
229C2×C8—2924no—
230C2×C8—5844no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—2921no—
234C22⋊C4—2922no—
235C22⋊C4—2921no—
236C23—24no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—42no—
246C23—44no—
247C23—44no—
248C23—42no—
249C23—48no—
250C23—44no—
251C23—44no—
252C23#1455664812yes—
253C23#1455664812yes—
254C23#1455665011yescharacteristic
255C23#1455665011yescharacteristic
256C23#1455665112yes—
257C23C2×C22⋊#6440811yescharacteristic
258C23C2×#1192403611yescharacteristic
259C23C2×#1192403811yescharacteristic
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—2924no—
264C2×C4—2928no—
265C2×C4—2924no—
266C2×C4—2924no—
267C2×C4—2924no—
268C2×C4—2922no—
269C2×C4—2922no—
270C2×C4—2924no—
271C2×C4—2922no—
272C2×C4—2922no—
273C2×C4—5842no—
274C2×C4—1464no—
275C2×C4—1462no—
276C2×C4—1462no—
277C8—5844no—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22#1798943112yes—
297C22#1798943112yes—
298C22#1798949014yes—
299C22C2×#1455664811yescharacteristic
300C22C2×#1455665011yescharacteristic
301C22C2×#1455665111yescharacteristic
302C4—82no—
303C4—2922no—
304C4—2922no—
305C4—2924no—
306C2—21no—
307C2—22no—
308C2—21no—
309C2—41no—
310C2—41no—
311C2—42no—
312C2#2262485814yesdirect product factor
313C2C2×#1798943111yescharacteristic
314C2C2×#1798949012yes—
subgroupG/H
centerC23#14556650
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC22×C22⋊C4×C73C4
socleC23×C73#99
prime radicalC25×C73C2×C4
cosocleC2×C22⋊Dic73C23
hypercenterC22×C22⋊C4#64408
hypocenterC73C2×#1317
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22624858C2direct (N × H)
23. C23×Dic73C8semidirect
59. C22×Dic73C2×C8semidirect
155. C2×C73#1317semidirect
161. C73C2×#1317semidirect
312. C2#22624858direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179894302
#226248582
C2×#179884482
C2×#131773