Finite Group Databasebeta

← #32997495

Subgroups — #32997495

Order 18688.

308 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 85 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#22620518C211yesmaximal, characteristic
2#22611685C211yesmaximal, characteristic
3#22608688C211yesmaximal, characteristic
4#22607986C211yesmaximal, characteristic, complemented
5#22607981C211yesmaximal, characteristic, complemented
6#22607979C211yesmaximal, characteristic, complemented
7C73×#1364C211yesmaximal, characteristic, complemented
8#17989142—21no—
9C2×#14556530—21no—
10#17988967C2211yescharacteristic
11#17988939—21no—
12#17988829—21no—
13#17988829C412yes—
14#17988681—21no—
15#17988681C412yes—
16#17988609—21no—
17C4×C8×D73C2211yescharacteristic
18C4⋊#11924003C2211yescharacteristic
19#17988405—21no—
20#17988340C2211yescharacteristic
21#17988339—21no—
22#500×C73C2211yescharacteristic
23C4⋊C16×C73C2211yescharacteristic
24#335×C73C2211yescharacteristic
25C2×Dic292—21no—
26C2×Dic292C2×C412yes—
27C2×#11924011—41no—
28C2×C8×D73C2×C411yescharacteristic
29C2×C8×D73D411yescharacteristic
30#14556591—21no—
31#14556582—41no—
32#14556571—22no—
33#14556567—41no—
34Q8×Dic73—41no—
35#14556556—22no—
36#14556549—41no—
37C42×D73D411yescharacteristic
38#14556530—42no—
39C2×#11924003—21no—
40C4⋊#10596831—21no—
41#14556520—41no—
42#14556514—22no—
43#14556514—22no—
44C4×#10596831D411yescharacteristic
45C8×Dic73C2×C411yescharacteristic
46C8×Dic73D411yescharacteristic
47C4⋊Q8×C73—21no—
48C2×C16×C73—21no—
49#107×C73—21no—
50#107×C73C2×C412yes—
51C4⋊C8×C73—21no—
52C4×C8×C73C2311yescharacteristic
53C2×Dic146—22no—
54C2×C4×D73C22⋊C411yescharacteristic
55C2×C4×D73D811yescharacteristic
56C2×C4×D73SD1611yescharacteristic
57Dic292—42no—
58Dic292—41no—
59Dic292—41no—
60#11924011—82no—
61C2×#10596831—41no—
62C2×#10596831C22⋊C411yescharacteristic
63C2×#10596831C22⋊C411yescharacteristic
64C8×D73—21no—
65C8×D73—21no—
66C8×D73—41no—
67C4⋊Dic73—41no—
68C4⋊Dic73—41no—
69C4⋊Dic73—42no—
70#11924007—42no—
71#11924005—42no—
72C4×Dic73—42no—
73C4×Dic73C22⋊C411yescharacteristic
74C4×Dic73D811yescharacteristic
75C4×Dic73SD1611yescharacteristic
76#11924003—41no—
77C2×Q8×C73—41no—
78C2×C8×C73—41no—
79C2×C8×C73C22×C411yescharacteristic
80C2×C8×C73C2×D411yescharacteristic
81C4⋊C4×C73—22no—
82C4⋊C4×C73—41no—
83C42×C73C2×D411yescharacteristic
84C16×C73—41no—
85C22×D73#10211yescharacteristic
86Q8×C73—81no—
87Q8×C73—81no—
88C2×C4×C73—42no—
89C2×C4×C73C2×C22⋊C411yescharacteristic
90C2×C4×C73C2×D811yescharacteristic
91C2×C4×C73C2×SD1611yescharacteristic
92C2×Dic73—42no—
93C2×Dic73#10211yescharacteristic
94C2×Dic73#10211yescharacteristic
95C2×Dic73#10211yescharacteristic
96C4×D73—21no—
97C4×D73—21no—
98C4×D73—21no—
99C4×D73—21no—
100C4×D73—22no—
101Dic146—42no—
102Dic146—42no—
103C8×C73—21no—
104C8×C73—81no—
105C8×C73AGL(1,ℤ/8)11yescharacteristic
106C8×C73#13711yescharacteristic
107#10596831—21no—
108#10596831—21no—
109#10596831—81no—
110C22×C73C2×#10211yescharacteristic
111C2×D73—21no—
112C2×D73—21no—
113C2×D73—21no—
114C4×C73—21no—
115C4×C73—82no—
116C4×C73#41311yescharacteristic
117C4×C73#41711yescharacteristic
118C4×C73#50911yescharacteristic
119C4×C73#51011yescharacteristic
120Dic73—21no—
121Dic73—21no—
122Dic73—82no—
123Dic73#36212yes—
124#11988—731nomaximal, Sylow
125C2×C73#154711yescharacteristic
126C2×C73#213911yescharacteristic
127C2×C73C2×#36211yescharacteristic
128D73—41no—
129#3170—731no—
130#2150—731no—
131#1567—731no—
132#1364—731no—
133#1364D7311yescharacteristic, complemented
134#1362—731no—
135#1362—731no—
136C73#1198811yescharacteristic, complemented, Sylow
137#533—1461no—
138C2×M32—1461no—
139#500C2×D7311yescharacteristic
140#494—731no—
141#487—1461no—
142#467—732no—
143#467—1461no—
144C4×Q16—732no—
145C4×Q16—1461no—
146#423—1461no—
147C2×C4×C8—731no—
148C4⋊C16—731no—
149C4⋊C16—1461no—
150C4⋊C16C2×D7311yescharacteristic
151#335—731no—
152#335—1461no—
153#335C2×D7311yescharacteristic
154C2×Q16—732no—
155C2×Q16—1461no—
156C2×M16—2921no—
157C22×C8—731no—
158C22×C8—731no—
159C4⋊Q8—21no—
160C4⋊Q8—1461no—
161#125—2921no—
162C22⋊Q8—2921no—
163C22⋊Q8—1462no—
164C4×Q8—2921no—
165C4×Q8—1462no—
166#117—2921no—
167C2×C42—731no—
168M32—2922no—
169C2×C16—21no—
170C2×C16—1461no—
171#107—21no—
172#107C4×D7312yes—
173C4⋊C8—21no—
174C4⋊C8—2921no—
175C4⋊C8—1461no—
176#103—1462no—
177#103—1462no—
178C4×C8—731no—
179C4×C8—731no—
180C4×C8—731no—
181C4×C8C22×D7311yescharacteristic
182C2×Q8—41no—
183C2×Q8—1462no—
184C22×C4—731no—
185C22×C4—731no—
186C22×C4—731no—
187Q16—2922no—
188Q16—2921no—
189Q16—2921no—
190M16—5842no—
191C2×C8—41no—
192C2×C8—2921no—
193C2×C8—2921no—
194C2×C8—731no—
195C2×C8—731no—
196C2×C8—1461no—
197C2×C8—1461no—
198C2×C8C2×C4×D7311yescharacteristic
199C2×C8D4×D7311yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—41no—
202C4⋊C4—2922no—
203C4⋊C4—2922no—
204C4⋊C4—2921no—
205C4⋊C4—2921no—
206C22⋊C4—2922no—
207C42—2922no—
208C42—731no—
209C42—731no—
210C42—731no—
211C42D4×D7311yescharacteristic
212C16—41no—
213C16—2921no—
214C23—731no—
215Q8—81no—
216Q8—81no—
217Q8—2922no—
218Q8—2922no—
219C2×C4—42no—
220C2×C4—2922no—
221C2×C4—731no—
222C2×C4—731no—
223C2×C4—731no—
224C2×C4—1462no—
225C2×C4—1461no—
226C2×C4—1461no—
227C2×C4—1461no—
228C2×C4—1461no—
229C2×C4C22⋊C4×D7311yescharacteristic
230C2×C4D8×D7311yescharacteristic
231C2×C4SD16×D7311yescharacteristic
232C8—21no—
233C8—81no—
234C8—5841no—
235C8—1461no—
236C8—1461no—
237C8#1455661811yescharacteristic
238C8#1455662311yescharacteristic
239C22—1461no—
240C22—1461no—
241C22—1461no—
242C22#102×D7311yescharacteristic
243C4—21no—
244C4—82no—
245C4—5842no—
246C4—732no—
247C4—1461no—
248C4—1461no—
249C4#1798857011yescharacteristic
250C4#1798858711yescharacteristic
251C4#1798902111yescharacteristic
252C4#1798902611yescharacteristic
253C2—2921no—
254C2#2260860711yescharacteristic
255C2#2261164011yescharacteristic
256C2#362×D7311yescharacteristic
subgroupG/H
centerC22#102×D73
commutatorC2×C8×C73C22×C4
FrattiniC4×C8C22×D73
FittingC73×#1364C2
socleC22×C73C2×#102
prime radicalC22×D73#102
cosocleC4×C8×C73C23
hypercenter#1364D73
hypocenterC73#11988
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
4. #22607986C2semidirect
5. #22607981C2semidirect
6. #22607979C2semidirect
7. C73×#1364C2semidirect
133. #1364D73semidirect
136. C73#11988semidirect

Maximal subgroups

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

subgroupindex
#226205182
#226116852
#226086882
#226079862
#226079812
#226079792
C73×#13642
#1198873