Finite Group Databasebeta

← #32977743

Subgroups — #32977743

Order 18688.

738 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 127 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×#17989310C211yesmaximal, characteristic
2#22608613C212yesmaximal, complemented
3#22608355C212yesmaximal
4#22608255C211yesmaximal, characteristic, complemented
5C73×#1801C211yesmaximal, characteristic, complemented
6#17989310—24no—
7#17989310—24no—
8C2×#14556591C2211yescharacteristic
9C2×#14556586—22no—
10C2×#14556579C412yes—
11C2×#14556567—21no—
12C2×C4×Dic146C412yes—
13C2×C4×D292C2211yescharacteristic
14C2×#14556549—21no—
15#17988640C2212yes—
16#17988335—24no—
17#17988314—24no—
18#17988313—22no—
19C2×C4×Q8×C73C2211yescharacteristic
20#423×C73C2212yes—
21C2×#101×C73—21no—
22#343×C73—21no—
23C22×Dic146C2×C412yes—
24C22×D292C2×C411yescharacteristic
25C22×C4×D73—21no—
26#14556591—22no—
27#14556591D414yes—
28#14556586—216no—
29#14556579—24no—
30#14556579—24no—
31#14556567—28no—
32C4×Dic146—24no—
33C4×Dic146—24no—
34C4×D292—22no—
35C4×D292D414yes—
36#14556549—28no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73C2×C411yescharacteristic
40C2×#11924007—22no—
41C2×#11924005—22no—
42C2×#11924005—22no—
43C2×#11924005—21no—
44C2×C4×Dic73—22no—
45C4⋊#10596831—22no—
46C22⋊#10596831—22no—
47C4×#10596831—22no—
48C22×Q8×C73D411yescharacteristic
49C2×M16×C73—22no—
50C4×Q8×C73—22no—
51C4×Q8×C73—22no—
52C2×C4⋊C4×C73D412yes—
53C2×C4⋊C4×C73D411yescharacteristic
54C2×C42×C73C2×C412yes—
55C2×C42×C73C2311yescharacteristic
56C4⋊C8×C73—22no—
57C4⋊C8×C73—22no—
58#101×C73—42no—
59C23×D73—21no—
60C2×Dic146—22no—
61C2×Dic146—44no—
62C2×Dic146C22⋊C414yes—
63C2×D292—21no—
64C2×D292—42no—
65C2×D292C22⋊C412yes—
66C22×Dic73—21no—
67C22×Dic73—22no—
68C2×C4×D73—24no—
69C2×C4×D73—42no—
70C2×C4×D73—41no—
71C2×C4×D73—41no—
72C2×#10596831—42no—
73C4⋊Dic73—21no—
74C4⋊Dic73—24no—
75C4⋊Dic73—24no—
76C4⋊Dic73C22⋊C412yes—
77#11924007—28no—
78#11924005—28no—
79#11924005—28no—
80#11924005—24no—
81C4×Dic73—28no—
82C2×Q8×C73—22no—
83C2×Q8×C73—22no—
84C2×Q8×C73—21no—
85C2×Q8×C73—21no—
86C22×C4×C73C22⋊C412yes—
87C22×C4×C73C22⋊C412yes—
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91M16×C73—42no—
92M16×C73—42no—
93C2×C8×C73—42no—
94C4⋊C4×C73—22no—
95C4⋊C4×C73—24no—
96C42×C73—22no—
97C42×C73—22no—
98C42×C73C2×D412yes—
99C42×C73C2×D412yes—
100C23×C73C2×C22⋊C411yescharacteristic
101C22×D73—22no—
102C22×D73—42no—
103C22×D73—42no—
104C22×D73—41no—
105C22×D73—41no—
106Q8×C73—42no—
107Q8×C73—42no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—41no—
114C2×C4×C73—42no—
115C2×C4×C73C2×C22⋊C411yescharacteristic
116C2×C4×C73C2×C22⋊C411yescharacteristic
117C2×C4×C73#12012yes—
118C2×C4×C73#12012yes—
119C2×C4×C73#13512yes—
120C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C73#13711yescharacteristic
122C2×Dic73—22no—
123C2×Dic73—24no—
124C2×Dic73—44no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127D292—24no—
128D292—42no—
129C4×D73—44no—
130C4×D73—44no—
131Dic146—44no—
132Dic146—44no—
133C8×C73—82no—
134#10596831—82no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73#37911yescharacteristic
138C22×C73#41611yescharacteristic
139C22×C73#41711yescharacteristic
140C2×D73—42no—
141C2×D73—42no—
142C2×D73—42no—
143C2×D73—82no—
144C2×D73—82no—
145C4×C73—22no—
146C4×C73—21no—
147C4×C73—42no—
148C4×C73—42no—
149C4×C73#41312yes—
150C4×C73#44912yes—
151C4×C73#45212yes—
152Dic73—44no—
153Dic73—42no—
154#8394—731nomaximal, Sylow
155C2×C73—21no—
156C2×C73—21no—
157C2×C73#178511yescharacteristic
158C2×C73#178811yescharacteristic
159C2×C73#179611yescharacteristic
160D73—82no—
161C2×#556—731no—
162#1801D7311yescharacteristic, complemented
163#1548—732no—
164#1486—732no—
165#1461—731no—
166C73#839411yescharacteristic, complemented, Sylow
167#556—1464no—
168#556—1464no—
169C2×C4⋊Q8—731no—
170C2×#126—1462no—
171C2×#124—732no—
172C2×C22⋊Q8—1461no—
173C2×C4×Q8—732no—
174C2×C4×Q8C2×D7311yescharacteristic
175C2×C4×D4—731no—
176C2×#117—1461no—
177#431—732no—
178#423C2×D7312yes—
179C2×#101—21no—
180#343—21no—
181#333—1464no—
182D4⋊C8—1464no—
183#324—1462no—
184C22×Q8—732no—
185C22×Q8C22⋊D7311yescharacteristic
186C22×D4—731no—
187C23×C4—1461no—
188C2×M16—22no—
189C4⋊Q8—734no—
190C4⋊Q8—1462no—
191#126—14616no—
192#124—1464no—
193#124—1464no—
194C22⋊Q8—1468no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—1464no—
198C4×Q8—1464no—
199C4×D4—734no—
200C4×D4—1462no—
201#117—1468no—
202C2×C4⋊C4—731no—
203C2×C4⋊C4—1462no—
204C2×C4⋊C4—1461no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4D29212yes—
207C2×C4⋊C4C22⋊D7311yescharacteristic
208C2×C22⋊C4—1462no—
209C2×C22⋊C4—1462no—
210C2×C22⋊C4—1461no—
211C2×C42—1462no—
212C2×C42C4×D7312yes—
213C2×C42C22×D7311yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—1462no—
217#101—42no—
218C22⋊C8—1462no—
219C4×C8—1462no—
220C24—1461no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—2924no—
226C2×Q8—734no—
227C2×Q8—1462no—
228C2×D4—2922no—
229C2×D4—732no—
230C2×D4—1461no—
231C22×C4—2922no—
232C22×C4—2921no—
233C22×C4—2921no—
234C22×C4—1461no—
235C22×C4—1462no—
236C22×C4—1464no—
237C22×C4#1192400512yes—
238C22×C4#1192400512yes—
239C22×C4C2×C4×D7311yescharacteristic
240C22×C4C2×C22⋊D7311yescharacteristic
241C22×C4C2×D29211yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—2922no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—732no—
249C4⋊C4—1468no—
250C4⋊C4—1461no—
251C4⋊C4—1464no—
252C4⋊C4—1464no—
253C22⋊C4—1468no—
254C22⋊C4—1468no—
255C22⋊C4—1464no—
256C42—22no—
257C42—22no—
258C42—1468no—
259C42D4×D7312yes—
260C42D4×D7312yes—
261C23—2922no—
262C23—2922no—
263C23—2921no—
264C23—2921no—
265C23—1462no—
266C23C2×#1192400511yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—2924no—
270Q8—2924no—
271D4—2922no—
272D4—1464no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—2924no—
281C2×C4—2921no—
282C2×C4—2921no—
283C2×C4—2924no—
284C2×C4—2924no—
285C2×C4—1462no—
286C2×C4—1464no—
287C2×C4C22⋊C4×D7311yescharacteristic
288C2×C4C22⋊C4×D7311yescharacteristic
289C2×C4#1455655912yes—
290C2×C4#1455656112yes—
291C2×C4#1455661112yes—
292C2×C4#1455661711yescharacteristic
293C2×C4#1455662211yescharacteristic
294C8—82no—
295C8—5842no—
296C22—21no—
297C22—21no—
298C22—2922no—
299C22—2922no—
300C22—2922no—
301C22—5842no—
302C22—5842no—
303C22#1798844211yescharacteristic
304C22#1798858111yescharacteristic
305C22#1798858611yescharacteristic
306C4—21no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—2924no—
311C4—2922no—
312C4#1798857012yes—
313C4#1798873012yes—
314C4#1798874712yes—
315C2—21no—
316C2—21no—
317C2—5842no—
318C2#2260985811yescharacteristic
319C2#2260986911yescharacteristic
320C2#2260990711yescharacteristic
subgroupG/H
centerC22#17988442
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1801C2
socleC22×C73#379
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1801D73
hypocenterC73#8394
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #22608613C2semidirect
4. #22608255C2semidirect
5. C73×#1801C2semidirect
162. #1801D73semidirect
166. C73#8394semidirect

Maximal subgroups

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

subgroupindex
C2×#179893102
#226086132
#226083552
#226082552
C73×#18012
#839473