Finite Group Databasebeta

← #34004684

Subgroups — #34004684

Order 18688.

690 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 92 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#22625832C211yesmaximal, characteristic
2#22625828C211yesmaximal, characteristic
3#22625341C214yesmaximal
4C2×#17989198C211yesmaximal, characteristic
5#17989559C2212yes—
6#17989530C2212yes—
7#17989467—22no—
8#17989465—22no—
9#17989441—28no—
10#17989198—28no—
11C2×#14556586—22no—
12C2×#14556583C411yescharacteristic, complemented
13C2×#14556581—22no—
14C2×#14556567C2212yes—
15C2×Q8×Dic73C412yescomplemented
16C2×#14556548C2211yescharacteristic
17C2×C4⋊Q8×C73C411yescharacteristic, complemented
18C2×#11924037—22no—
19C2×C22⋊#64408—22no—
20#14556648—24no—
21(C2×#64408)⋊C4—24no—
22C22×C4×D73C2311yescharacteristic
23#14556586—216no—
24#14556583—24no—
25#14556581—216no—
26#14556567—24no—
27#14556567D418yes—
28Q8×Dic73—24no—
29Q8×Dic73—24no—
30#14556548—24no—
31C2×C4⋊Dic73—22no—
32C2×C4⋊Dic73C2×C412yes—
33C2×#11924007—22no—
34C2×#11924007—22no—
35C2×#11924007D412yes—
36C2×#11924005—22no—
37C2×#11924005D412yes—
38C2×C4×Dic73—22no—
39C2×C4×Dic73C2×C411yescharacteristic
40C22×Q8×C73C2×C412yes—
41C4⋊Q8×C73—24no—
42C2×C4⋊C4×C73—22no—
43C2×C42×C73C2×C411yescharacteristic
44C22×#64408—42no—
45#11924037—44no—
46C2×#10596833—42no—
47C22⋊#64408—44no—
48C23×D73C2×D411yescharacteristic
49C22×Dic73—22no—
50C22×Dic73C22⋊C412yes—
51C22×Dic73C2×D411yescharacteristic
52C2×C4×D73—42no—
53C2×C4×D73C2×D414yes—
54C4⋊Dic73—28no—
55C4⋊Dic73—22no—
56C4⋊Dic73C22⋊C414yes—
57#11924007—28no—
58#11924007—28no—
59#11924007—24no—
60#11924005—28no—
61#11924005—24no—
62C4×Dic73—28no—
63C4×Dic73—22no—
64C2×Q8×C73—22no—
65C2×Q8×C73—44no—
66C2×Q8×C73C22⋊C414yes—
67C22×C4×C73—22no—
68C22×C4×C73C22⋊C412yes—
69C22×C4×C73C22×C411yescharacteristic
70C4⋊C4×C73—28no—
71C42×C73—22no—
72C23×C73C2×C22⋊C411yescharacteristic
73C22×D73—22no—
74C22×D73—42no—
75C22×D73#12012yes—
76C2×#64408—82no—
77C2×#64408—84no—
78Q8×C73—44no—
79Q8×C73—44no—
80C2×C4×C73—22no—
81C2×C4×C73—22no—
82C2×C4×C73—24no—
83C2×C4×C73—44no—
84C2×C4×C73—42no—
85C2×C4×C73C2×C22⋊C412yes—
86C2×Dic73—22no—
87C2×Dic73—22no—
88C2×Dic73—24no—
89C2×Dic73—44no—
90C2×Dic73—42no—
91C2×Dic73#12012yes—
92C4×D73—44no—
93#10596833—82no—
94C22×C73—22no—
95C22×C73#37911yescharacteristic
96C22×C73#41012yes—
97C2×D73—24no—
98C2×D73—22no—
99C2×D73—82no—
100#64408—162no—
101C4×C73—22no—
102C4×C73—42no—
103C4×C73—44no—
104Dic73—22no—
105Dic73—42no—
106Dic73—44no—
107#8521—731nomaximal, Sylow
108C2×C73—22no—
109C2×C73#178211yescharacteristic
110C2×C73#179612yes—
111D73—42no—
112C2×#541—731no—
113#1822—731no—
114#1820—731no—
115#1511—734no—
116C73#852111yescharacteristic, complemented, Sylow
117#541—1468no—
118C2×C4⋊Q8#6440811yescharacteristic, complemented
119C2×#126—1462no—
120C2×#125—731no—
121C2×#125—1462no—
122C2×C22⋊Q8—732no—
123C2×C4×Q8—732no—
124C2×#117—731no—
125#407—732no—
126#381—732no—
127#343—1462no—
128#342—1462no—
129#328—1468no—
130C22×Q8C2×#6440812yes—
131C23×C4—731no—
132C2×M16—1462no—
133C4⋊Q8—24no—
134#126—14616no—
135#125—1464no—
136#125—14616no—
137C22⋊Q8—738no—
138C22⋊Q8—1464no—
139C4×Q8—1464no—
140C4×Q8—1464no—
141#117—1464no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—732no—
144C2×C4⋊C4—732no—
145C2×C4⋊C4—1462no—
146C2×C4⋊C4—1462no—
147C2×C4⋊C4—1462no—
148C2×C22⋊C4—732no—
149C2×C22⋊C4—1462no—
150C2×C22⋊C4—1462no—
151C2×C42—731no—
152C2×C42—1462no—
153C2×C42C2×#6440811yescharacteristic
154C22⋊C8—1464no—
155#95—1464no—
156C24—731no—
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊#6440814yes—
160C22×C4—22no—
161C22×C4—2922no—
162C22×C4—2922no—
163C22×C4—734no—
164C22×C4—732no—
165C22×C4—731no—
166C22×C4—1462no—
167C22×C4C22⋊#6440812yes—
168C22×C4C22×#6440811yescharacteristic
169M16—2924no—
170C2×C8—2922no—
171C4⋊C4—28no—
172C4⋊C4—734no—
173C4⋊C4—1464no—
174C4⋊C4—1468no—
175C4⋊C4—1468no—
176C4⋊C4—1468no—
177C4⋊C4—1462no—
178C22⋊C4—2924no—
179C22⋊C4—1464no—
180C22⋊C4—1468no—
181C42—22no—
182C42—1468no—
183C42—1462no—
184C23—2922no—
185C23—732no—
186C23—1462no—
187C23C2×C22⋊#6440811yescharacteristic
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—2924no—
196C2×C4—2924no—
197C2×C4—2922no—
198C2×C4—5842no—
199C2×C4—5844no—
200C2×C4—732no—
201C2×C4—1462no—
202C2×C4—1464no—
203C2×C4—1462no—
204C2×C4C2×C22⋊#6440812yes—
205C8—5842no—
206C22—22no—
207C22—5842no—
208C22—1462no—
209C22—1464no—
210C22#1798952811yescharacteristic
211C22#1798956712yes—
212C4—22no—
213C4—44no—
214C4—42no—
215C4—2924no—
216C4—2922no—
217C4—11682no—
218C4—1462no—
219C2—22no—
220C2—2922no—
221C2#2262576111yescharacteristic
222C2#2262578312yes—
subgroupG/H
centerC22#17989528
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC22×C73#379
prime radicalC23×D73C2×D4
cosocleC22×C4×D73C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#8521
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. C2×#14556583C4semidirect
15. C2×Q8×Dic73C4semidirect
17. C2×C4⋊Q8×C73C4semidirect
116. C73#8521semidirect
118. C2×C4⋊Q8#64408semidirect

Maximal subgroups

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

subgroupindex
#226258322
#226258282
#226253412
C2×#179891982
#852173