Finite Group Databasebeta

← #34034337

Subgroups — #34034337

Order 18688.

450 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 105 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#22627120C211yesmaximal, characteristic, complemented
2C2×#17989688C214yesmaximal, complemented
3C4×#14556686C211yesmaximal, characteristic, complemented
4#22626656C211yesmaximal, characteristic
5C2×#14556688C2212yes—
6C2×#14556686C2212yes—
7C4⋊#11924040—22no—
8#17989688—28no—
9C4×#11924040—22no—
10C2×#14556679C2212yes—
11#17989621—22no—
12#17989615—22no—
13#17989591C411yescharacteristic
14#17989562C412yes—
15C4×#11924036C2211yescharacteristic
16#17989138C411yescharacteristic
17#14556688—24no—
18#14556686—22no—
19#14556686D414yescomplemented
20C2×#11924040—22no—
21C2×#11924040—22no—
22#14556679—24no—
23#14556679—24no—
24C2×#11924038—22no—
25C2×#11924038C2×C412yes—
26C2×#11924036—22no—
27C2×#11924036C2×C411yescharacteristic
28C2×#11924036C2311yescharacteristic
29C4⋊#10596833—22no—
30#14556648—24no—
31C4×#10596833D412yes—
32C4×#10596833D412yes—
33C2×#11924029C2×C412yes—
34#14556588C811yescharacteristic
35#14556580C812yes—
36#14556566—22no—
37#14556553—22no—
38C42×D73C2×C411yescharacteristic
39C4⋊Q8×C73C811yescharacteristic
40#11924040—24no—
41#11924040—42no—
42#11924038—24no—
43#11924038—44no—
44#11924036—22no—
45#11924036—42no—
46#11924036—42no—
47#11924036C2×D412yes—
48#11924036#4112yes—
49C2×#10596833—22no—
50C2×#10596833C22⋊C412yes—
51C2×#10596833C2×D411yescharacteristic
52C2×#10596833C2×D411yescharacteristic
53#11924029—24no—
54#11924029—24no—
55C2×D292—22no—
56C2×D292—22no—
57C2×D292C2×C812yes—
58C2×C4×D73—22no—
59C2×C4×D73M1612yes—
60C2×C4×D73C22×C411yescharacteristic
61#11924005—24no—
62C4×Dic73C22⋊C412yes—
63C4×Dic73C2×C811yescharacteristic
64C2×Q8×C73C2×C812yes—
65C4⋊C4×C73—22no—
66C42×C73C2×C811yescharacteristic
67C22×D73—22no—
68C22×D73C2×M1611yescharacteristic
69Q8×C73—24no—
70C2×C4×C73—22no—
71C2×C4×C73C22⋊C812yes—
72C2×C4×C73C22×C811yescharacteristic
73C2×Dic73C22⋊C812yes—
74C2×Dic73C2×C22⋊C411yescharacteristic
75D292—24no—
76D292—44no—
77D292—44no—
78C4×D73—24no—
79C4×D73—42no—
80C4×D73—42no—
81C4×D73C4×D412yes—
82C4×D73#13112yes—
83#10596833—22no—
84#10596833—42no—
85#10596833#12112yes—
86#10596833#12411yescharacteristic
87#10596833#12711yescharacteristic
88C22×C73C2×C22⋊C811yescharacteristic
89C2×D73—22no—
90C2×D73—44no—
91C2×D73—42no—
92C2×D73#43612yes—
93C4×C73—42no—
94C4×C73#35014yescomplemented
95C4×C73C8×D412yes—
96Dic73—22no—
97Dic73#39011yescharacteristic
98Dic73#40811yescharacteristic
99#13191—731nomaximal, Sylow
100C2×C73#195411yescharacteristic
101C2×C73C2×#35012yes—
102D73—22no—
103D73—82no—
104#2931—731no—
105#2150—731no—
106C2×#350—734no—
107C2×C4×C16—731no—
108C73#1319111yescharacteristic, complemented, Sylow
109C2×#131—732no—
110#532—731no—
111C2×M32—732no—
112C22×C16—732no—
113#436—1462no—
114C8×D4—1462no—
115#423—731no—
116#408—732no—
117C2×C4×C8—731no—
118C4⋊C16—1462no—
119#350—1468no—
120C4×C16—1462no—
121C2×#41—732no—
122#131—1464no—
123#131—1464no—
124C2×M16—732no—
125C2×M16—1462no—
126C22×C8—731no—
127C22×C8—731no—
128C22×C8—1462no—
129C4⋊Q8#1059684211yescharacteristic
130#127—731no—
131#124—732no—
132#121—1462no—
133C4×D4—1462no—
134C2×C42—731no—
135M32—1464no—
136C2×C16—734no—
137C2×C16—1462no—
138C2×C16—1462no—
139C2×C16—1462no—
140C4⋊C8—1462no—
141C22⋊C8—1464no—
142C4×C8—732no—
143C4×C8—732no—
144#41—1464no—
145#41—1464no—
146C2×Q8C2×#1059684212yes—
147C2×D4—732no—
148C2×D4—1462no—
149C2×D4—1462no—
150C22×C4—731no—
151C22×C4—732no—
152C22×C4—1462no—
153M16—2924no—
154M16—1464no—
155C2×C8—2922no—
156C2×C8—2922no—
157C2×C8—731no—
158C2×C8—731no—
159C2×C8—732no—
160C2×C8—732no—
161C2×C8—732no—
162C2×C8—1462no—
163C2×C8—1462no—
164C4⋊C4—22no—
165C22⋊C4—1464no—
166C42—732no—
167C42—731no—
168C42C2×#1059684211yescharacteristic
169C16—2922no—
170C16—1464no—
171C23—731no—
172C23—1462no—
173Q8—24no—
174D4—2924no—
175D4—2924no—
176D4—1464no—
177C2×C4—22no—
178C2×C4—2922no—
179C2×C4—2922no—
180C2×C4—731no—
181C2×C4—732no—
182C2×C4—732no—
183C2×C4—732no—
184C2×C4—1464no—
185C2×C4C22⋊#1059684212yes—
186C2×C4C22×#1059684211yescharacteristic
187C8—2922no—
188C8—731no—
189C8—732no—
190C8—731no—
191C8—1462no—
192C22—2922no—
193C22—2924no—
194C22—732no—
195C22—1462no—
196C22C2×C22⋊#1059684211yescharacteristic
197C4—42no—
198C4—731no—
199C4—731no—
200C4—1462no—
201C4#1798968814yescomplemented
202C4D4×#1059684212yes—
203C2—5842no—
204C2—1462no—
205C2#2262708111yescharacteristic
206C2C2×#1798968812yes—
subgroupG/H
centerC22C2×C22⋊#10596842
commutatorC2×C4×C73C22×C8
FrattiniC2×C4C22×#10596842
FittingC4⋊Q8×C73C8
socleC22×C73C2×C22⋊C8
prime radical#17989138C4
cosocleC2×#11924036C23
hypercenterC4⋊Q8#10596842
hypocenterC73#13191
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22627120C2semidirect
2. C2×#17989688C2semidirect
3. C4×#14556686C2semidirect
19. #14556686D4semidirect
94. C4×C73#350semidirect
108. C73#13191semidirect
201. C4#17989688semidirect

Maximal subgroups

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

subgroupindex
#226271202
C2×#179896882
C4×#145566862
#226266562
#1319173