Finite Group Databasebeta

← #14529957

Subgroups — #14529957

Order 2176.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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#11763366C211yesmaximal, characteristic, complemented
2#11762825C211yesmaximal, characteristic
3C2×#10587476C211yesmaximal, characteristic
4#11762782C211yesmaximal, characteristic
5#11762580C211yesmaximal, characteristic, complemented
6#11762565C211yesmaximal, characteristic, complemented
7C17×#339C211yesmaximal, characteristic, complemented
8C2×#63252C2211yescharacteristic
9C2×M16×C17C2211yescharacteristic
10C17×#117C2211yescharacteristic
11C2×C42×C17C2211yescharacteristic
12#10587612C2211yescharacteristic
13#10587612C2211yescharacteristic
14#10587610—22no—
15C2×#63223C2211yescharacteristic
16#10587556—21no—
17#10587555—21no—
18C4×Dic34—22no—
19#10587496—22no—
20#10587477—22no—
21#10587476—21no—
22#10587476—21no—
23C22×C4×C17—21no—
24C22×C4×C17C2311yescharacteristic
25C2×C22⋊D17—21no—
26#63252—21no—
27#63252—21no—
28#63252—21no—
29#63252—21no—
30C2×D68Q811yescharacteristic, complemented
31C2×C4×D17—21no—
32C2×Dic34—22no—
33C2×Dic34Q811yescharacteristic, complemented
34M16×C17—21no—
35M16×C17—21no—
36C2×C8×C17—21no—
37C4⋊C4×C17—21no—
38C22⋊C4×C17—21no—
39C42×C17—21no—
40C42×C17—21no—
41C42×C17D411yescharacteristic
42C42×C17D411yescharacteristic
43C22⋊Dic17—21no—
44C22⋊Dic17D412yes—
45C4⋊Dic17—22no—
46C4⋊Dic17—21no—
47C4⋊Dic17—21no—
48#63225—24no—
49C4×Dic17—21no—
50C4×Dic17D412yes—
51#63223—21no—
52#63223—21no—
53C2×#3622—21no—
54C23×C17C2×D411yescharacteristic
55C22×D17—21no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17—42no—
59C2×C4×C17—41no—
60C2×C4×C17C2×D411yescharacteristic
61C2×C4×C17C2×D411yescharacteristic
62C2×C4×C17C2×Q811yescharacteristic
63C2×C4×C17#4111yescharacteristic
64C2×C4×C17#4111yescharacteristic
65C2×C4×C17#4111yescharacteristic
66C22⋊D17—41no—
67C22⋊D17—41no—
68C2×Dic17—22no—
69C2×Dic17—22no—
70C2×Dic17—21no—
71D68—21no—
72D68—21no—
73C4×D17—41no—
74C4×D17—41no—
75Dic34—21no—
76Dic34—21no—
77Dic34—44no—
78C8×C17—41no—
79#3622—41no—
80#2031—171nomaximal, Sylow
81C22×C17—41no—
82C22×C17#12011yescharacteristic
83C22×C17C22⋊Q811yescharacteristic
84C22×C17#12411yescharacteristic
85C2×D17—41no—
86C2×D17—81no—
87C4×C17—22no—
88C4×C17—41no—
89C4×C17—81no—
90C4×C17C22⋊Q811yescharacteristic
91C4×C17C22⋊Q811yescharacteristic
92C4×C17#12411yescharacteristic
93C4×C17#12411yescharacteristic
94Dic17—42no—
95Dic17—42no—
96Dic17—41no—
97#533—171no—
98#421—171no—
99C2×#104—171no—
100#410—171no—
101#344—171no—
102#339D1711yescharacteristic, complemented
103H(3,ℤ/4)—171no—
104C2×C17—21no—
105C2×C17—21no—
106C2×C17—21no—
107C2×C17#39311yescharacteristic
108D17—81no—
109C2×#41—171no—
110C2×M16—171no—
111C2×M16C2×D1711yescharacteristic
112C4⋊Q8—341no—
113#125—341no—
114C22⋊Q8—342no—
115C4×Q8—342no—
116#117—171no—
117#117—171no—
118#117C2×D1711yescharacteristic
119C2×C42C2×D1711yescharacteristic
120#104—342no—
121#104—341no—
122#104—341no—
123#99—342no—
124C17#203111yescharacteristic, complemented, Sylow
125#41—341no—
126#41—341no—
127#41—341no—
128#41—341no—
129C2×Q8—171no—
130C2×Q8—342no—
131C2×D4—171no—
132C2×D4—341no—
133C22×C4—21no—
134C22×C4—341no—
135C22×C4C22×D1711yescharacteristic
136M16—21no—
137M16—21no—
138M16—341no—
139M16—341no—
140C2×C8—21no—
141C2×C8—341no—
142C4⋊C4—21no—
143C4⋊C4—344no—
144C4⋊C4—342no—
145C4⋊C4—341no—
146C4⋊C4—341no—
147C22⋊C4—21no—
148C22⋊C4—172no—
149C22⋊C4—341no—
150C42—21no—
151C42—21no—
152C42—172no—
153C42—341no—
154C42D6811yescharacteristic
155C42D6811yescharacteristic
156C23—341no—
157C23D4×D1711yescharacteristic
158Q8—341no—
159Q8—341no—
160Q8—684no—
161D4—341no—
162D4—341no—
163D4—681no—
164D4—681no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—341no—
172C2×C4—681no—
173C2×C4—681no—
174C2×C4C2×D6811yescharacteristic
175C2×C4#6325211yescharacteristic
176C2×C4#6325211yescharacteristic
177C2×C4D4×D1711yescharacteristic
178C2×C4#6325411yescharacteristic
179C2×C4Q8×D1711yescharacteristic
180C8—41no—
181C8—681no—
182C22—41no—
183C22—681no—
184C22—1361no—
185C22#1058756811yescharacteristic
186C22#1058757111yescharacteristic
187C22#1058758211yescharacteristic
188C4—22no—
189C4—41no—
190C4—81no—
191C4—682no—
192C4—682no—
193C4—681no—
194C4#1058756111yescharacteristic
195C4#1058757111yescharacteristic
196C4#1058758211yescharacteristic
197C4#1058758311yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—1361no—
202C2#1176272211yescharacteristic
subgroupG/H
centerC4#10587571
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#339C2
socleC2×C17#393
prime radicalC2×#63252C22
hypercenter#339D17
hypocenterC17#2031
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #11763366C2semidirect
5. #11762580C2semidirect
6. #11762565C2semidirect
7. C17×#339C2semidirect
30. C2×D68Q8semidirect
33. C2×Dic34Q8semidirect
102. #339D17semidirect
124. C17#2031semidirect

Maximal subgroups

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

subgroupindex
#117633662
#117628252
C2×#105874762
#117627822
#117625802
#117625652
C17×#3392
#203117