Finite Group Databasebeta

← #34033354

Subgroups — #34033354

Order 18688.

410 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 97 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#22627174C211yesmaximal, characteristic
2(C2×#10596842)⋊Q8C211yesmaximal, characteristic
3#22627056C212yesmaximal
4#22626938C212yesmaximal
5#101×#64408C211yesmaximal, characteristic
6Q8×#10596842—22no—
7Q8×#10596842—22no—
8#17989700—22no—
9#17989700C2212yes—
10C2×C4⋊#10596842C2211yescharacteristic
11C2×C4×#10596842C2211yescharacteristic
12(C2×#10596842)⋊C4—22no—
13#17989677—22no—
14#17989676—22no—
15C2×Q8×#64408C2211yescharacteristic
16D73⋊#101C411yescharacteristic
17M16×#64408C2212yes—
18#17989466C412yes—
19#101×D73C411yescharacteristic
20C22×#10596842—21no—
21C22×#10596842—21no—
22C4⋊#10596842—22no—
23C4⋊#10596842—22no—
24C4⋊#10596842—22no—
25C4⋊#10596842D412yes—
26C4⋊#10596842D412yes—
27C22⋊#10596842—22no—
28C4×#10596842—24no—
29C4×#10596842—22no—
30C4×#10596842—22no—
31D73⋊M16C2×C412yes—
32Q8×#64408—22no—
33Q8×#64408—22no—
34C2×C4⋊#64408C2×C411yescharacteristic
35C2×C4⋊#64408Q812yes—
36C2×C4×#64408D412yes—
37C2×C4×#64408C2311yescharacteristic
38#14556652—22no—
39#14556645—22no—
40C8×#64408—22no—
41C2×Q8×D73C2×C411yescharacteristic
42M16×D73C2×C412yes—
43#14556510C811yescharacteristic, complemented
44#14556509C812yescomplemented
45#101×C73C811yescharacteristic, complemented
46C2×#10596842—22no—
47C2×#10596842—22no—
48C2×#10596842—22no—
49C2×#10596842—41no—
50C2×#10596842—41no—
51C2×#10596842—42no—
52C2×#10596842—41no—
53C22×#64408C4⋊C412yes—
54C22×#64408C2×D411yescharacteristic
55C22×#64408C2×Q811yescharacteristic
56#11924038—22no—
57#11924037—22no—
58#11924036—22no—
59C4⋊#64408—24no—
60C4⋊#64408—22no—
61C4×#64408—22no—
62C4×#64408—22no—
63C4×#64408C2×D412yes—
64C4×#64408#4112yes—
65C2×Dic146C2×C811yescharacteristic
66C2×Dic146M1612yes—
67Q8×D73—22no—
68Q8×D73—22no—
69C2×C4×D73C4⋊C412yes—
70C2×C4×D73C22×C411yescharacteristic
71#11924012C2×C812yes—
72#11924011—22no—
73C8×D73—22no—
74C2×Q8×C73C2×C811yescharacteristic
75M16×C73C2×C812yes—
76C22×D73C2×C4⋊C411yescharacteristic
77C2×#64408—21no—
78C2×#64408—21no—
79C2×#64408—22no—
80C2×#64408—41no—
81C2×#64408—42no—
82C2×#64408#12011yescharacteristic
83C2×#64408C22⋊Q812yes—
84C2×#64408#12311yescharacteristic
85#10596842—42no—
86#10596842—42no—
87#10596842—42no—
88#10596842—81no—
89Q8×C73—22no—
90C2×C4×C73C4⋊C812yes—
91C2×C4×C73C22×C811yescharacteristic
92C2×Dic73C4⋊C812yes—
93C2×Dic73C2×M1611yescharacteristic
94C4×D73—22no—
95C4×D73—22no—
96C4×D73C4×D412yes—
97C4×D73#13112yes—
98Dic146—22no—
99Dic146—24no—
100#10596833—42no—
101C8×C73—22no—
102#10596831—22no—
103C22×C73C2×C4⋊C811yescharacteristic
104C2×D73—21no—
105C2×D73—21no—
106C2×D73#38511yescharacteristic
107C2×D73#42411yescharacteristic
108#64408—21no—
109#64408—22no—
110#64408—42no—
111#64408#45411yescharacteristic
112#64408#45611yescharacteristic
113C4×C73—22no—
114C4×C73C8×D412yes—
115Dic73—22no—
116Dic73#43512yes—
117#8836—731nomaximal, Sylow
118C2×C73—21no—
119C2×C73#185011yescharacteristic
120D73—21no—
121D73#185711yescharacteristic
122D73#185811yescharacteristic
123#2982—731no—
124#1984—731no—
125C4×#101—731no—
126#1565—732no—
127#1278—732no—
128C73#883611yescharacteristic, complemented, Sylow
129C2×C4×Q8—731no—
130C8×Q8—1462no—
131C8×Q8—1462no—
132#431—732no—
133#431—1462no—
134C2×C4⋊C8—731no—
135C2×#101—731no—
136C2×#101—731no—
137C4×M16—732no—
138C2×C4×C8—731no—
139#343—732no—
140#336—1462no—
141#335—1462no—
142#334—1462no—
143C22×Q8—731no—
144C2×M16—732no—
145C2×M16—732no—
146C22×C8—1461no—
147C22×C8—1461no—
148C4×Q8—1462no—
149C4×Q8—1462no—
150C2×C4⋊C4—731no—
151C2×C4⋊C4—732no—
152C2×C42—731no—
153C2×C42—732no—
154C4⋊C8—732no—
155C4⋊C8—732no—
156C4⋊C8—1462no—
157C4⋊C8—1462no—
158C4⋊C8—1462no—
159#101—731no—
160#101—732no—
161#101—1462no—
162#101#1059684211yescharacteristic, complemented
163C22⋊C8—1462no—
164#97—1462no—
165C4×C8—1462no—
166C4×C8—1462no—
167C4×C8—1462no—
168C4×C8—1464no—
169C2×Q8—732no—
170C2×Q8—731no—
171C2×Q8—1462no—
172C2×Q8—1462no—
173C2×Q8C2×#1059684211yescharacteristic
174C22×C4—731no—
175C22×C4—732no—
176C22×C4—731no—
177C22×C4—731no—
178C22×C4—732no—
179M16—732no—
180M16—1462no—
181M16—1462no—
182M16—1462no—
183M16C2×#1059684212yes—
184C2×C8—2921no—
185C2×C8—2921no—
186C2×C8—2921no—
187C2×C8—2922no—
188C2×C8—1462no—
189C2×C8—1462no—
190C2×C8—1462no—
191C2×C8—1462no—
192C2×C8—1462no—
193C4⋊C4—1462no—
194C4⋊C4—1464no—
195C42—732no—
196C42—732no—
197C42—1462no—
198C42—1462no—
199C23—731no—
200Q8—22no—
201Q8—1464no—
202Q8—1462no—
203C2×C4—2922no—
204C2×C4—2921no—
205C2×C4—731no—
206C2×C4—731no—
207C2×C4—732no—
208C2×C4—732no—
209C2×C4—731no—
210C2×C4—732no—
211C2×C4—732no—
212C2×C4—1462no—
213C2×C4—1462no—
214C2×C4—1462no—
215C2×C4—1461no—
216C2×C4—1461no—
217C2×C4C4⋊#1059684212yes—
218C2×C4C22×#1059684211yescharacteristic
219C8—22no—
220C8—2922no—
221C8—2922no—
222C8—2922no—
223C8—2922no—
224C8—5841no—
225C8—1462no—
226C22—731no—
227C22—731no—
228C22—1461no—
229C22—1461no—
230C22C2×C4⋊#1059684211yescharacteristic
231C4—22no—
232C4—2922no—
233C4—731no—
234C4—731no—
235C4—732no—
236C4—1461no—
237C4—1462no—
238C4—1462no—
239C4D4×#1059684212yes—
240C2—21no—
241C2—731no—
242C2—731no—
243C2—1461no—
244C2#2262707611yescharacteristic
subgroupG/H
centerC2#22627076
commutatorC2×C4×C73C22×C8
FrattiniC2×C4C22×#10596842
Fitting#101×C73C8
socleC2×C73#1850
prime radicalC22×D73C2×C4⋊C4
cosocleC2×C4×#64408C23
hypercenter#101#10596842
hypocenterC73#8836
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
43. #14556510C8semidirect
44. #14556509C8semidirect
45. #101×C73C8semidirect
128. C73#8836semidirect
162. #101#10596842semidirect

Maximal subgroups

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

subgroupindex
#226271742
(C2×#10596842)⋊Q82
#226270562
#226269382
#101×#644082
#883673