Finite Group Databasebeta

← #34010895

Subgroups — #34010895

Order 18688.

346 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 82 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#22626295C211yesmaximal, characteristic
2C4×#14556659C211yesmaximal, characteristic
3#22624919C214yesmaximal
4#22620515C211yesmaximal, characteristic
5C2×#14556659—21no—
6C4×C4⋊#64408—21no—
7C4⋊#11924031—21no—
8#17989482C2212yes—
9#17989476—21no—
10#17989461C2212yes—
11#17989142—21no—
12#17988992C411yescharacteristic, complemented
13#17988937—21no—
14#17988825—21no—
15#17988825C2212yes—
16Q16×Dic73—21no—
17Q16×Dic73C412yescomplemented
18C4×C8×D73C2211yescharacteristic
19#494×C73C411yescharacteristic, complemented
20C2×C4⋊#64408—22no—
21C2×C4×#64408—41no—
22C2×#11924031—22no—
23#14556659—24no—
24(C2×#64408)⋊C4—41no—
25C2×C8×D73D411yescharacteristic
26C2×C8×D73C2311yescharacteristic
27#14556591—21no—
28#14556582—41no—
29#14556571—41no—
30#14556567—22no—
31Q8×Dic73—22no—
32#14556556—41no—
33#14556549—41no—
34C42×D73D411yescharacteristic
35#14556525—21no—
36#14556525C2×C412yes—
37#14556515—22no—
38#14556515—22no—
39C4×#10596831D411yescharacteristic
40C8×Dic73C2×C411yescharacteristic
41C8×Dic73D411yescharacteristic
42C2×Q16×C73—21no—
43C2×Q16×C73C2×C412yes—
44C4⋊Q8×C73—21no—
45C4×C8×C73C2×C411yescharacteristic
46C22×#64408—42no—
47C4⋊#64408—24no—
48C4⋊#64408—42no—
49C4×#64408—82no—
50#11924031—42no—
51C2×Dic146—41no—
52C2×C4×D73D811yescharacteristic
53C2×C4×D73D811yescharacteristic
54C2×C4×D73C2×D411yescharacteristic
55C2×#10596831C22⋊C411yescharacteristic
56C2×#10596831C2×D411yescharacteristic
57C8×D73—22no—
58C8×D73—41no—
59C4⋊Dic73—22no—
60C4⋊Dic73—41no—
61C4⋊Dic73—42no—
62#11924007—42no—
63#11924005—42no—
64C4×Dic73—42no—
65C4×Dic73C22⋊C411yescharacteristic
66C4×Dic73SD1611yescharacteristic
67C4×Dic73SD1611yescharacteristic
68C2×Q8×C73—22no—
69Q16×C73—42no—
70Q16×C73—42no—
71C2×C8×C73C22⋊C411yescharacteristic
72C2×C8×C73C22×C411yescharacteristic
73C4⋊C4×C73—41no—
74C4⋊C4×C73—41no—
75C42×C73C22⋊C411yescharacteristic
76C22×D73C2×D811yescharacteristic
77C2×#64408—44no—
78C2×#64408—82no—
79C2×#64408—82no—
80Q8×C73—42no—
81Q8×C73—42no—
82C2×C4×C73—42no—
83C2×C4×C73#10211yescharacteristic
84C2×C4×C73#10211yescharacteristic
85C2×C4×C73C2×C22⋊C411yescharacteristic
86C2×Dic73—42no—
87C2×Dic73#10211yescharacteristic
88C2×Dic73#10211yescharacteristic
89C2×Dic73C2×SD1611yescharacteristic
90C4×D73—22no—
91C4×D73—22no—
92C4×D73—21no—
93C4×D73—21no—
94Dic146—82no—
95C8×C73—21no—
96C8×C73#10412yes—
97#10596831—21no—
98#10596831—21no—
99C22×C73C2×#10211yescharacteristic
100C2×D73—21no—
101C2×D73—21no—
102C2×D73#51012yes—
103#64408—84no—
104C4×C73—21no—
105C4×C73—82no—
106C4×C73#36012yes—
107C4×C73#41011yescharacteristic
108C4×C73C2×#10411yescharacteristic
109Dic73—21no—
110Dic73—21no—
111Dic73—21no—
112Dic73—82no—
113#11745—731nomaximal, Sylow
114C2×C73#150211yescharacteristic
115C2×C73#214111yescharacteristic
116C2×C73C2×#36011yescharacteristic
117D73—22no—
118#3170—731no—
119#2163—731no—
120C4×#106—731no—
121#1348—734no—
122C73#1174511yescharacteristic, complemented, Sylow
123#533—1461no—
124#500—731no—
125#494#6440811yescharacteristic, complemented
126#487—1461no—
127#467—732no—
128#467—1461no—
129C4×Q16—732no—
130C4×Q16—1461no—
131C2×#106—1461no—
132C2×C4×C8—731no—
133C4×C4⋊C4—1461no—
134C4⋊C16—1461no—
135C22⋊C16—732no—
136#346—1461no—
137#340—732no—
138C2×Q16—21no—
139C2×Q16C2×#6440812yes—
140C22×C8—731no—
141C22×C8—731no—
142C4⋊Q8—21no—
143C4⋊Q8—1461no—
144#125—2921no—
145C22⋊Q8—2921no—
146C22⋊Q8—1462no—
147C4×Q8—2921no—
148C4×Q8—1462no—
149#117—2921no—
150C2×C4⋊C4—1462no—
151C2×C42—2921no—
152C2×C42—731no—
153C2×C16—1462no—
154#107—732no—
155#107—1461no—
156#106—1464no—
157#103—1462no—
158#103—1462no—
159C4×C8—731no—
160C4×C8—731no—
161C4×C8—731no—
162C4×C8C2×#6440811yescharacteristic
163#95—2921no—
164C2×Q8—22no—
165C2×Q8—2921no—
166C22×C4—2922no—
167C22×C4—731no—
168C22×C4—731no—
169C22×C4—731no—
170Q16—42no—
171Q16—42no—
172C2×C8—2921no—
173C2×C8—731no—
174C2×C8—731no—
175C2×C8—1462no—
176C2×C8C22⋊#6440811yescharacteristic
177C2×C8C22×#6440811yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—2922no—
181C4⋊C4—2922no—
182C4⋊C4—2922no—
183C4⋊C4—2921no—
184C4⋊C4—1464no—
185C4⋊C4—1462no—
186C22⋊C4—2922no—
187C42—2922no—
188C42—5842no—
189C42—731no—
190C42—731no—
191C42—731no—
192C42C22⋊#6440811yescharacteristic
193C16—2922no—
194C23—731no—
195Q8—42no—
196Q8—42no—
197Q8—5842no—
198C2×C4—42no—
199C2×C4—2924no—
200C2×C4—2922no—
201C2×C4—5842no—
202C2×C4—5842no—
203C2×C4—731no—
204C2×C4—731no—
205C2×C4—731no—
206C2×C4—1462no—
207C2×C4—1462no—
208C2×C4—1461no—
209C2×C4—1461no—
210C2×C4#1455665311yescharacteristic
211C2×C4#1455665311yescharacteristic
212C2×C4C2×C22⋊#6440811yescharacteristic
213C8—21no—
214C8—1461no—
215C8—1461no—
216C8#1455665512yes—
217C22—732no—
218C22—1461no—
219C22—1461no—
220C22C2×#1455665311yescharacteristic
221C4—21no—
222C4—82no—
223C4—5844no—
224C4—5842no—
225C4—1461no—
226C4—1461no—
227C4—1461no—
228C4#1798950412yes—
229C4#1798956711yescharacteristic
230C4C2×#1455665511yescharacteristic
231C2—1462no—
232C2#2262531111yescharacteristic
233C2#2262624111yescharacteristic
234C2C2×#1798950411yescharacteristic
subgroupG/H
centerC22C2×#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#494×C73C4
socleC22×C73C2×#102
prime radicalC22×D73C2×D8
cosocleC2×C8×D73C23
hypercenter#494#64408
hypocenterC73#11745
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #494

Complements

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

NH ≅ G/Ntype
12. #17988992C4semidirect
17. Q16×Dic73C4semidirect
19. #494×C73C4semidirect
122. C73#11745semidirect
125. #494#64408semidirect

Maximal subgroups

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

subgroupindex
#226262952
C4×#145566592
#226249192
#226205152
#1174573