Finite Group Databasebeta

← #34033331

Subgroups — #34033331

Order 18688.

642 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 109 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#22627072C212yesmaximal
2#22626940C214yesmaximal
3C2×C4⋊C4×#64408C211yesmaximal, characteristic
4C2×C22⋊#10596842C2214yes—
5(C2×#10596842)⋊C4—22no—
6(C2×#10596842)⋊C4—22no—
7C22×C4⋊#64408C411yescharacteristic
8C22×C4×#64408C412yes—
9C22×C4×#64408C2211yescharacteristic
10C4⋊C4×#64408—22no—
11C4⋊C4×#64408—22no—
12C2×(C2×#64408)⋊C4C2212yes—
13C2×C4⋊C4×D73C411yescharacteristic
14C22×#10596842—24no—
15C22⋊#10596842—28no—
16C23×#64408C2×C411yescharacteristic
17C23×#64408C2×C412yes—
18C23×#64408C2311yescharacteristic
19C2×C4⋊#64408—24no—
20C2×C4⋊#64408—22no—
21C2×C4×#64408—24no—
22C2×C4×#64408—22no—
23C2×C4×#64408—22no—
24C2×C4×#64408—44no—
25C2×C4×#64408D412yes—
26C2×C4×#64408D412yes—
27(C2×#64408)⋊C4—24no—
28C22×C4×D73C2×C411yescharacteristic
29C22×C4×D73C2×C412yes—
30C2×C4⋊Dic73C811yescharacteristic, complemented
31C2×#11924007C812yescomplemented
32C4⋊C4×D73—22no—
33C4⋊C4×D73—22no—
34C2×C4⋊C4×C73C811yescharacteristic, complemented
35C2×#10596842—44no—
36C2×#10596842—44no—
37C2×#10596842—44no—
38C22×#64408—22no—
39C22×#64408—22no—
40C22×#64408—22no—
41C22×#64408—22no—
42C22×#64408—22no—
43C22×#64408—41no—
44C22×#64408—41no—
45C22×#64408—42no—
46C22×#64408—42no—
47C22×#64408C22⋊C411yescharacteristic
48C22×#64408C22⋊C411yescharacteristic
49C22×#64408M1614yes—
50C22×#64408C2×D412yes—
51C22×#64408#4114yes—
52C4⋊#64408—28no—
53C4×#64408—24no—
54C4×#64408—24no—
55C4×#64408—44no—
56C4×#64408—44no—
57C23×D73C22×C411yescharacteristic
58C22×Dic73C2×C811yescharacteristic
59C22×Dic73C2×C812yes—
60C2×C4×D73—22no—
61C2×C4×D73—22no—
62C2×C4×D73—22no—
63C2×C4×D73—22no—
64C2×C4×D73—42no—
65C2×C4×D73—42no—
66C2×C4×D73C22⋊C412yes—
67C2×C4×D73M1612yes—
68C4⋊Dic73—22no—
69#11924007—24no—
70C22×C4×C73C2×C812yes—
71C22×C4×C73C2×C811yescharacteristic
72C4⋊C4×C73—22no—
73C23×C73C22×C811yescharacteristic
74C22×D73—21no—
75C22×D73—21no—
76C22×D73—21no—
77C22×D73—21no—
78C22×D73C2×C22⋊C411yescharacteristic
79C22×D73#11712yes—
80C22×D73C2×M1612yes—
81C22×D73C2×M1611yescharacteristic
82C2×#64408—28no—
83C2×#64408—21no—
84C2×#64408—21no—
85C2×#64408—22no—
86C2×#64408—22no—
87C2×#64408—21no—
88C2×#64408—21no—
89C2×#64408—24no—
90C2×#64408—44no—
91C2×#64408—44no—
92C2×#64408—42no—
93C2×#64408—42no—
94C2×#64408—44no—
95C2×#64408#10012yes—
96C2×#64408#10112yes—
97C2×#64408#12314yes—
98#10596842—84no—
99C2×C4×C73—22no—
100C2×C4×C73—22no—
101C2×C4×C73—42no—
102C2×C4×C73C22⋊C812yes—
103C2×Dic73—22no—
104C2×Dic73—22no—
105C2×Dic73—42no—
106C2×Dic73C22⋊C812yes—
107C4×D73—22no—
108C4×D73—22no—
109C4×D73—24no—
110C4×D73—44no—
111C4×D73—42no—
112C4×D73—42no—
113C22×C73—21no—
114C22×C73—21no—
115C22×C73C2×C22⋊C811yescharacteristic
116C22×C73#43112yes—
117C2×D73—21no—
118C2×D73—21no—
119C2×D73—22no—
120C2×D73—22no—
121C2×D73—22no—
122C2×D73—22no—
123C2×D73#38111yescharacteristic
124C2×D73#40711yescharacteristic
125C2×D73#41011yescharacteristic
126C2×D73#41011yescharacteristic
127C2×D73C2×#10011yescharacteristic
128C2×D73C2×#10111yescharacteristic
129C2×D73#43212yes—
130#64408—24no—
131#64408—22no—
132#64408—22no—
133#64408—44no—
134C4×C73—22no—
135C4×C73—42no—
136Dic73—22no—
137Dic73—42no—
138#8701—731nomaximal, Sylow
139C2×C73—22no—
140C2×C73#146011yescharacteristic
141C2×C73#146111yescharacteristic
142C2×C73#181711yescharacteristic
143D73—22no—
144D73#146611yescharacteristic
145D73#146711yescharacteristic
146D73#182011yescharacteristic
147D73#182111yescharacteristic
148C2×C4×C4⋊C4—731no—
149#1817—732no—
150#1280—734no—
151C73#870111yescharacteristic, complemented, Sylow
152C22×C4⋊C4—731no—
153C22×C4⋊C4—731no—
154C22×C42—732no—
155C22×C42—731no—
156C2×C22⋊C8—734no—
157C4×C4⋊C4—1462no—
158C4×C4⋊C4—1462no—
159C2×#95—732no—
160#336—1462no—
161#336—1462no—
162C23×C4—731no—
163C23×C4—732no—
164C23×C4—732no—
165C23×C4—731no—
166C23×C4—731no—
167C22×C8—1464no—
168C2×C4⋊C4—732no—
169C2×C4⋊C4—731no—
170C2×C4⋊C4—1462no—
171C2×C4⋊C4—1462no—
172C2×C4⋊C4—1462no—
173C2×C4⋊C4—1464no—
174C2×C4⋊C4#1059684211yescharacteristic, complemented
175C2×C42—2924no—
176C2×C42—732no—
177C2×C42—732no—
178C2×C42—1462no—
179C2×C42—1462no—
180C2×C42—1464no—
181C22⋊C8—1468no—
182#95—1464no—
183C24—731no—
184C22×C4—2922no—
185C22×C4—2922no—
186C22×C4—2922no—
187C22×C4—2922no—
188C22×C4—2921no—
189C22×C4—2921no—
190C22×C4—734no—
191C22×C4—734no—
192C22×C4—732no—
193C22×C4—731no—
194C22×C4—731no—
195C22×C4—732no—
196C22×C4—731no—
197C22×C4—732no—
198C22×C4—732no—
199C22×C4—1462no—
200C22×C4—1462no—
201C22×C4—1462no—
202C22×C4—1462no—
203C22×C4—1462no—
204C22×C4—1462no—
205C22×C4—1462no—
206C22×C4—1462no—
207C22×C4—1462no—
208C22×C4C2×#1059684212yes—
209C22×C4C2×#1059684211yescharacteristic
210C2×C8—2924no—
211C2×C8—2924no—
212C2×C8—2924no—
213C4⋊C4—22no—
214C4⋊C4—1464no—
215C4⋊C4—1468no—
216C4⋊C4—1462no—
217C42—2924no—
218C42—2924no—
219C42—1464no—
220C42—1464no—
221C23—731no—
222C23—731no—
223C23—732no—
224C23—732no—
225C23—1461no—
226C23—1461no—
227C23—1461no—
228C23—1461no—
229C23C22×#1059684211yescharacteristic
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—2922no—
234C2×C4—2922no—
235C2×C4—2924no—
236C2×C4—2922no—
237C2×C4—2922no—
238C2×C4—2924no—
239C2×C4—2922no—
240C2×C4—2924no—
241C2×C4—2924no—
242C2×C4—732no—
243C2×C4—734no—
244C2×C4—732no—
245C2×C4—732no—
246C2×C4—1462no—
247C2×C4—1462no—
248C2×C4—1464no—
249C2×C4—1464no—
250C2×C4—1462no—
251C2×C4—1461no—
252C2×C4—1461no—
253C2×C4—1461no—
254C2×C4—1462no—
255C2×C4—1462no—
256C2×C4—1461no—
257C2×C4—1468no—
258C2×C4—1462no—
259C2×C4C22⋊#1059684212yes—
260C8—5844no—
261C22—21no—
262C22—21no—
263C22—731no—
264C22—731no—
265C22—731no—
266C22—731no—
267C22—731no—
268C22—731no—
269C22—732no—
270C22—1462no—
271C22—1462no—
272C22—1462no—
273C22—1462no—
274C22—1461no—
275C22—1461no—
276C22C2×C22⋊#1059684211yescharacteristic
277C22#1798970012yes—
278C4—22no—
279C4—42no—
280C4—2924no—
281C4—2922no—
282C4—1464no—
283C4—1462no—
284C4—1462no—
285C4—1462no—
286C2—22no—
287C2—731no—
288C2—731no—
289C2—731no—
290C2—731no—
291C2—1462no—
292C2#2262704511yescharacteristic
293C2#2262704611yescharacteristic
294C2#2262707211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10596842
commutatorC23×C73C22×C8
FrattiniC23C22×#10596842
FittingC2×C4⋊C4×C73C8
socleC22×C73C2×C22⋊C8
prime radicalC23×D73C22×C4
cosocleC23×#64408C23
hypercenterC2×C4⋊C4#10596842
hypocenterC73#8701
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
30. C2×C4⋊Dic73C8semidirect
31. C2×#11924007C8semidirect
34. C2×C4⋊C4×C73C8semidirect
151. C73#8701semidirect
174. C2×C4⋊C4#10596842semidirect

Maximal subgroups

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

subgroupindex
#226270722
#226269402
C2×C4⋊C4×#644082
#870173