Finite Group Databasebeta

← #14544548

Subgroups — #14544548

Order 2176.

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

normal subgroups only

This group has 320 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
1C2×#10587550C212yesmaximal
2C4×#63243C211yesmaximal, characteristic, complemented
3#11763745C211yesmaximal, characteristic, complemented
4#11763679C212yesmaximal, complemented
5C4×#63256C211yesmaximal, characteristic
6C2×#63256C2211yescharacteristic
7C2×C4×#595C2212yes—
8C2×#63243C2212yes—
9C4×Q8×C17C411yescharacteristic, complemented
10Q8×Dic17C411yescharacteristic, complemented
11#10587579—21no—
12#10587579C411yescharacteristic, complemented
13#10587578—21no—
14#10587578C2211yescharacteristic
15C4×D68—21no—
16C4×D68C411yescharacteristic, complemented
17C42×D17—21no—
18C42×D17C2211yescharacteristic
19#10587550C418yescomplemented
20(C2×#595)⋊C4—21no—
21(C2×#595)⋊C4—21no—
22#10587542—21no—
23C4×#3624—21no—
24C22×#595—22no—
25C2×Q8×C17C2×C411yescharacteristic
26#63256—22no—
27#63256C2×C414yes—
28C2×D68—21no—
29C2×D68C2×C411yescharacteristic
30C2×C4×D17—21no—
31C2×C4×D17—21no—
32C2×C4×D17—21no—
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17C2311yescharacteristic
36C2×#3624—22no—
37C4×#595—22no—
38C4×#595C2×C414yescomplemented
39#63243—22no—
40#63243C2×C414yescomplemented
41C4⋊C4×C17—21no—
42C4⋊C4×C17C2×C411yescharacteristic
43C42×C17—21no—
44C42×C17C2×C411yescharacteristic
45#63227—21no—
46#63227—21no—
47#63227—21no—
48C4⋊Dic17—21no—
49C4⋊Dic17C2×C411yescharacteristic
50C4×Dic17—21no—
51C4×Dic17—21no—
52C4×Dic17C2×C411yescharacteristic
53C4×Dic17D411yescharacteristic
54C4×Dic17D411yescharacteristic
55C22×D17—21no—
56C22×D17C2×D411yescharacteristic
57C2×#595—24no—
58C2×#595—42no—
59C2×#595—42no—
60Q8×C17—21no—
61Q8×C17C4212yes—
62C2×C4×C17—21no—
63C2×C4×C17—21no—
64C2×C4×C17C22⋊C411yescharacteristic
65C2×C4×C17C22⋊C411yescharacteristic
66C2×C4×C17C22×C411yescharacteristic
67C2×Dic17—21no—
68C2×Dic17C22⋊C411yescharacteristic
69C2×Dic17C22⋊C411yescharacteristic
70C2×Dic17C2×D411yescharacteristic
71D68—24no—
72D68—21no—
73D68C4212yes—
74C4×D17—24no—
75C4×D17—41no—
76C4×D17—41no—
77C4×D17—42no—
78C4×D17—42no—
79C4×D17C22×C412yes—
80C4×D17#4112yes—
81#3624—24no—
82#1759—171nomaximal, Sylow
83C22×C17C2×C22⋊C411yescharacteristic
84C2×D17—21no—
85C2×D17—21no—
86C2×D17—22no—
87C2×D17—41no—
88C2×D17—41no—
89C2×D17C4×D412yes—
90#595—44no—
91C4×C17—22no—
92C4×C17—21no—
93C4×C17—21no—
94C4×C17—41no—
95C4×C17C2×C4211yescharacteristic
96C4×C17#11711yescharacteristic
97Dic17—21no—
98Dic17—21no—
99Dic17—41no—
100Dic17C4×D412yes—
101C4×#41—171no—
102C2×#104—172no—
103C4×M16—171no—
104#376—171no—
105#339—172no—
106C2×C17C4×C22⋊C411yescharacteristic
107C2×C17#42111yescharacteristic
108C2×C17#42111yescharacteristic
109D17—22no—
110D17—42no—
111C2×#41—171no—
112C2×M16—172no—
113C4×Q8—171no—
114C4×Q8#59511yescharacteristic, complemented
115C4×D4—171no—
116C4×D4—171no—
117C4×D4—341no—
118C4×D4—341no—
119#117—171no—
120#117—341no—
121C2×C42—171no—
122C2×C42—172no—
123C2×C42—341no—
124#104—178no—
125#97—341no—
126C4×C8—341no—
127#95—341no—
128#95—341no—
129C17#175911yescharacteristic, complemented, Sylow
130#41—174no—
131#41—342no—
132C2×Q8C2×#59511yescharacteristic
133C2×D4—171no—
134C2×D4—341no—
135C22×C4—171no—
136C22×C4—171no—
137C22×C4—171no—
138C22×C4—341no—
139C22×C4—341no—
140C22×C4—342no—
141C22×C4—341no—
142M16—174no—
143M16—342no—
144C2×C8—342no—
145C4⋊C4—21no—
146C4⋊C4—171no—
147C4⋊C4—341no—
148C4⋊C4C2×#59511yescharacteristic
149C22⋊C4—341no—
150C22⋊C4—341no—
151C22⋊C4—341no—
152C42—21no—
153C42—171no—
154C42—171no—
155C42—171no—
156C42—174no—
157C42—341no—
158C42—341no—
159C42—342no—
160C42C2×#59511yescharacteristic
161C23—171no—
162C23—341no—
163Q8—21no—
164Q8C4×#59512yes—
165D4—172no—
166D4—344no—
167D4—341no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—171no—
171C2×C4—172no—
172C2×C4—172no—
173C2×C4—171no—
174C2×C4—171no—
175C2×C4—344no—
176C2×C4—344no—
177C2×C4—341no—
178C2×C4—681no—
179C2×C4—681no—
180C2×C4—682no—
181C2×C4—682no—
182C2×C4—682no—
183C2×C4—682no—
184C2×C4C22⋊#59511yescharacteristic
185C2×C4C22⋊#59511yescharacteristic
186C2×C4C22×#59511yescharacteristic
187C8—344no—
188C22—172no—
189C22—341no—
190C22—341no—
191C22—342no—
192C22—681no—
193C22—681no—
194C22C2×C22⋊#59511yescharacteristic
195C4—22no—
196C4—21no—
197C4—21no—
198C4—41no—
199C4—172no—
200C4—341no—
201C4—341no—
202C4—684no—
203C4—681no—
204C4C2×C4×#59511yescharacteristic
205C4#1058767711yescharacteristic
206C2—342no—
207C2—682no—
208C2C4×C22⋊#59511yescharacteristic
209C2#1176380611yescharacteristic
210C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC4×C17C2×C42
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#63256C22
cosocleC2×C4×D17C23
hypercenterC4×Q8#595
hypocenterC17#1759
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. C4×#63243C2semidirect
3. #11763745C2semidirect
4. #11763679C2semidirect
9. C4×Q8×C17C4semidirect
10. Q8×Dic17C4semidirect
12. #10587579C4semidirect
16. C4×D68C4semidirect
19. #10587550C4semidirect
38. C4×#595C2×C4semidirect
40. #63243C2×C4semidirect
114. C4×Q8#595semidirect
129. C17#1759semidirect

Maximal subgroups

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

subgroupindex
C2×#105875502
C4×#632432
#117637452
#117636792
C4×#632562
#175917