Finite Group Databasebeta

← #14527985

Subgroups — #14527985

Order 2176.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#11763479C211yesmaximal, characteristic
2#11763155C211yesmaximal, characteristic, complemented
3#11763095C211yesmaximal, characteristic, complemented
4#11763015C211yesmaximal, characteristic, complemented
5#11762539C211yesmaximal, characteristic, complemented
6#11762536C211yesmaximal, characteristic, complemented
7C17×#331C211yesmaximal, characteristic, complemented
8C2×#63254—21no—
9C17×#127C2211yescharacteristic
10#10587626—21no—
11#10587625—22no—
12#10587624—21no—
13D4×Dic17—21no—
14C2×#63219—21no—
15C4×D68C2211yescharacteristic
16#10587555C2211yescharacteristic
17C4×Dic34C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C17×C4⋊C8C2211yescharacteristic
20#10587503—22no—
21#10587488—21no—
22#10587487—21no—
23#10587487—21no—
24C4×#3622C2211yescharacteristic
25C2×D4×C17—21no—
26C2×D4×C17—21no—
27C2×D4×C17D412yes—
28C2×C22⋊D17—22no—
29C2×C22⋊D17—22no—
30C22×Dic17—22no—
31#63254—42no—
32#63254—42no—
33C2×D68D411yescharacteristic
34C2×C4×D17—21no—
35C2×Dic34—21no—
36C2×Dic34D411yescharacteristic
37C2×C8×C17—21no—
38C2×C8×C17—21no—
39C42×C17C2311yescharacteristic
40C22⋊Dic17—22no—
41#63227—21no—
42C4⋊Dic17—21no—
43C4⋊Dic17—21no—
44C4⋊Dic17D411yescharacteristic
45C4⋊Dic17D411yescharacteristic
46#63225—21no—
47C4×Dic17—21no—
48C2×#3622—21no—
49#63219—42no—
50C23×C17—22no—
51C22×D17—21no—
52D4×C17—42no—
53D4×C17—41no—
54D4×C17—41no—
55D4×C17—41no—
56D4×C17—41no—
57C2×C4×C17C2×D411yescharacteristic
58C2×C4×C17C2×D411yescharacteristic
59C2×C4×C17C2×D411yescharacteristic
60C22⋊D17—44no—
61C22⋊D17—82no—
62C2×Dic17—21no—
63C2×Dic17—21no—
64C2×Dic17—21no—
65C2×Dic17—44no—
66C2×Dic17—41no—
67D68—21no—
68D68—21no—
69C4×D17—41no—
70C4×D17—41no—
71Dic34—21no—
72Dic34—41no—
73Dic34—41no—
74Dic34SD1612yescomplemented
75C8×C17—41no—
76C8×C17—41no—
77#3622—41no—
78#1642—171nomaximal, Sylow
79C22×C17—42no—
80C22×C17—42no—
81C22×C17—42no—
82C22×C17#12011yescharacteristic
83C2×D17—41no—
84C2×D17—41no—
85C2×D17—41no—
86C4×C17—21no—
87C4×C17C2×SD1611yescharacteristic
88C4×C17#13511yescharacteristic
89C4×C17AGL(1,ℤ/8)11yescharacteristic
90C4×C17AGL(1,ℤ/8)11yescharacteristic
91Dic17—22no—
92Dic17—41no—
93Dic17—41no—
94Dic17—81no—
95#550—171no—
96#486—171no—
97#475—171no—
98#461—171no—
99#331D1711yescharacteristic, complemented
100Q8⋊C8—171no—
101D4⋊C8—171no—
102C2×C17—82no—
103C2×C17#44911yescharacteristic
104C2×C17#45011yescharacteristic
105C2×C17#45511yescharacteristic
106D17—81no—
107C2×#41—341no—
108C2×SD16—341no—
109C4⋊Q8—171no—
110#127—341no—
111#127C2×D1711yescharacteristic
112#121—341no—
113#121—342no—
114C4×Q8—171no—
115C4×D4—171no—
116C4×D4—341no—
117#106—341no—
118C4⋊C8C2×D1711yescharacteristic
119C4⋊C8C2×D1711yescharacteristic
120#103—341no—
121#103—341no—
122#102—342no—
123C4×C8—171no—
124C17#164211yescharacteristic, complemented, Sylow
125#41—682no—
126#41—682no—
127C2×Q8—171no—
128C2×Q8—341no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—171no—
132C2×D4—342no—
133C2×D4—342no—
134C2×D4D6812yes—
135C22×C4—342no—
136C22×C4—341no—
137SD16—682no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—341no—
141C4⋊C4—171no—
142C4⋊C4—171no—
143C4⋊C4—341no—
144C4⋊C4—341no—
145C4⋊C4—341no—
146C22⋊C4—342no—
147C22⋊C4—341no—
148C42—341no—
149C42C22×D1711yescharacteristic
150C23—22no—
151C23—341no—
152Q8—172no—
153Q8—341no—
154Q8—681no—
155Q8—681no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—341no—
162D4—341no—
163D4—684no—
164D4—1362no—
165C2×C4—341no—
166C2×C4—341no—
167C2×C4—341no—
168C2×C4—684no—
169C2×C4—681no—
170C2×C4—681no—
171C2×C4—681no—
172C2×C4C2×D6811yescharacteristic
173C2×C4D4×D1711yescharacteristic
174C2×C4D4×D1711yescharacteristic
175C8—41no—
176C8—41no—
177C8—681no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—681no—
182C22—681no—
183C22—681no—
184C22#1058756811yescharacteristic
185C4—21no—
186C4—342no—
187C4—681no—
188C4—681no—
189C4—1361no—
190C4#1058759711yescharacteristic
191C4#1058759711yescharacteristic
192C4SD16×D1711yescharacteristic
193C4#1058760211yescharacteristic
194C2—82no—
195C2—1361no—
196C2#1176295511yescharacteristic
197C2#1176296111yescharacteristic
198C2#1176298911yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#331C2
socleC22×C17#120
prime radical#11763479C2
hypercenter#331D17
hypocenterC17#1642
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #11763155C2semidirect
3. #11763095C2semidirect
4. #11763015C2semidirect
5. #11762539C2semidirect
6. #11762536C2semidirect
7. C17×#331C2semidirect
74. Dic34SD16semidirect
99. #331D17semidirect
124. C17#1642semidirect

Maximal subgroups

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

subgroupindex
#117634792
#117631552
#117630952
#117630152
#117625392
#117625362
C17×#3312
#164217