Finite Group Databasebeta

← #14538264

Subgroups — #14538264

Order 2176.

584 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#10587635C212yesmaximal, complemented
2C2×#10587560C211yesmaximal, characteristic, complemented
3#11763009C214yesmaximal, complemented
4#11763004C214yesmaximal, complemented
5#11762833C211yesmaximal, characteristic, complemented
6#11762813C212yesmaximal, complemented
7C4×#41×C17C211yesmaximal, characteristic, complemented
8C2×#41×C17C2211yescharacteristic
9C22×D68—22no—
10C22×D68C2212yes—
11C4×Q8×C17C2212yescomplemented
12C4×D4×C17—22no—
13C4×D4×C17C2212yescomplemented
14C17×#117—21no—
15C17×#117C2211yescharacteristic
16C2×C42×C17—21no—
17C2×C42×C17C2211yescharacteristic
18#10587635—28no—
19C2×#63230C2214yescomplemented
20C2×#63228C2214yescomplemented
21C2×#63223C2212yes—
22#10587560—21no—
23#10587560—21no—
24#10587560C2214yes—
25#10587480C2218yescomplemented
26C4⋊#3622C2214yescomplemented
27C23×D17—22no—
28#41×C17—22no—
29#41×C17D414yescomplemented
30C2×Q8×C17C2311yescharacteristic
31C2×D4×C17—21no—
32C2×D4×C17C2311yescharacteristic
33C22×C4×C17—21no—
34C22×C4×C17—22no—
35C22×C4×C17D412yes—
36C22×C4×C17C2311yescharacteristic
37C2×D68—24no—
38C2×D68—24no—
39C2×D68—24no—
40C2×D68—24no—
41C2×D68—22no—
42C2×D68—48no—
43C2×D68C2314yes—
44C4⋊C4×C17—22no—
45C4⋊C4×C17C2312yes—
46C22⋊C4×C17—21no—
47C22⋊C4×C17—22no—
48C42×C17—21no—
49C42×C17—22no—
50C42×C17D411yescharacteristic
51C42×C17D411yescharacteristic
52C42×C17C2312yes—
53#63230—28no—
54#63228—28no—
55#63223—24no—
56C2×#3622C2314yes—
57C23×C17—21no—
58C23×C17C2×D411yescharacteristic
59C22×D17—24no—
60C22×D17—48no—
61C22×D17—44no—
62Q8×C17—21no—
63Q8×C17C2×D412yescomplemented
64D4×C17—24no—
65D4×C17—21no—
66D4×C17C2×D412yescomplemented
67C2×C4×C17—21no—
68C2×C4×C17—24no—
69C2×C4×C17—24no—
70C2×C4×C17—22no—
71C2×C4×C17—44no—
72C2×C4×C17C2×D412yes—
73C2×C4×C17C2×D412yes—
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D412yes—
76C2×C4×C17#4112yes—
77C2×C4×C17C2411yescharacteristic
78D68—24no—
79D68—24no—
80D68—48no—
81D68—44no—
82D68—48no—
83#3622—24no—
84#3038—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17—22no—
87C22×C17—42no—
88C22×C17#12112yes—
89C22×C17C22×D411yescharacteristic
90C2×D17—44no—
91C2×D17—48no—
92C2×D17—88no—
93C4×C17—22no—
94C4×C17—22no—
95C4×C17—42no—
96C4×C17#12112yes—
97C4×C17AGL(1,ℤ/8)14yescomplemented
98C4×C17C22×D411yescharacteristic
99C4×C17C2×#4111yescharacteristic
100C2×AGL(1,ℤ/8)—172no—
101C2×#127—171no—
102C4×#41D1711yescharacteristic, complemented
103#460—174no—
104#459—174no—
105#423—171no—
106#418—172no—
107C2×C17—22no—
108C2×C17—42no—
109C2×C17C2×#12111yescharacteristic
110C2×C17C2×AGL(1,ℤ/8)12yes—
111D17—84no—
112C2×#41C2×D1711yescharacteristic
113C22×D4—172no—
114C22×D4—342no—
115AGL(1,ℤ/8)—348no—
116C2×SD16—174no—
117C2×D8—174no—
118C2×M16—172no—
119#127—174no—
120#127—341no—
121#127—341no—
122C4×Q8C2×D1712yescomplemented
123C4×D4—22no—
124C4×D4C2×D1712yescomplemented
125#117—21no—
126#117C2×D1711yescharacteristic
127C2×C42—21no—
128C2×C42C2×D1711yescharacteristic
129C4⋊C8—174no—
130#102—178no—
131C17#303811yescharacteristic, complemented, Sylow
132C24—342no—
133#41—22no—
134#41D6814yescomplemented
135C2×Q8C22×D1711yescharacteristic
136C2×D4—21no—
137C2×D4—174no—
138C2×D4—344no—
139C2×D4—344no—
140C2×D4—344no—
141C2×D4—344no—
142C2×D4—342no—
143C2×D4—688no—
144C2×D4C22×D1711yescharacteristic
145C22×C4—21no—
146C22×C4—22no—
147C22×C4C22⋊D1712yes—
148C22×C4C22×D1711yescharacteristic
149SD16—348no—
150D8—348no—
151M16—344no—
152C2×C8—174no—
153C4⋊C4—22no—
154C4⋊C4C22×D1712yes—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C42—21no—
158C42—22no—
159C42C22⋊D1711yescharacteristic
160C42C22⋊D1711yescharacteristic
161C42C22×D1712yes—
162C23—21no—
163C23—344no—
164C23—688no—
165C23—684no—
166C23C2×C22⋊D1711yescharacteristic
167Q8—21no—
168Q8C2×D6812yescomplemented
169D4—24no—
170D4—21no—
171D4—344no—
172D4—344no—
173D4—688no—
174D4—684no—
175D4—688no—
176D4C2×D6812yescomplemented
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—44no—
182C2×C4C2×D6812yes—
183C2×C4#6325212yes—
184C2×C4C2×C22⋊D1712yes—
185C2×C4C2×C22⋊D1711yescharacteristic
186C2×C4C2×C22⋊D1712yes—
187C2×C4C23×D1711yescharacteristic
188C8—344no—
189C22—22no—
190C22—22no—
191C22—42no—
192C22—684no—
193C22—688no—
194C22—1368no—
195C22#1058761612yes—
196C22C22×C22⋊D1711yescharacteristic
197C4—22no—
198C4—22no—
199C4—42no—
200C4#1058761612yes—
201C4#1058763514yescomplemented
202C4C22×D6811yescharacteristic
203C4C2×#6325211yescharacteristic
204C2—22no—
205C2—42no—
206C2—1364no—
207C2C2×#1058761611yescharacteristic
208C2C2×#1058763512yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC4×#41×C17C2
socleC22×C17C22×D4
hypercenterC4×#41D17
hypocenterC17#3038
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#10587635C2semidirect
2. C2×#10587560C2semidirect
3. #11763009C2semidirect
4. #11763004C2semidirect
5. #11762833C2semidirect
6. #11762813C2semidirect
7. C4×#41×C17C2semidirect
11. C4×Q8×C17C22semidirect
13. C4×D4×C17C22semidirect
19. C2×#63230C22semidirect
20. C2×#63228C22semidirect
25. #10587480C22semidirect
26. C4⋊#3622C22semidirect
29. #41×C17D4semidirect
63. Q8×C17C2×D4semidirect
66. D4×C17C2×D4semidirect
97. C4×C17AGL(1,ℤ/8)semidirect
102. C4×#41D17semidirect
122. C4×Q8C2×D17semidirect
124. C4×D4C2×D17semidirect
131. C17#3038semidirect
134. #41D68semidirect
168. Q8C2×D68semidirect
176. D4C2×D68semidirect
201. C4#10587635semidirect

Maximal subgroups

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

subgroupindex
C2×#105876352
C2×#105875602
#117630092
#117630042
#117628332
#117628132
C4×#41×C172
#303817