Finite Group Databasebeta

← #14527503

Subgroups — #14527503

Order 2176.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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#11763357C211yesmaximal, characteristic
2#11762865C211yesmaximal, characteristic, complemented
3#11762831C211yesmaximal, characteristic, complemented
4#11762556C211yesmaximal, characteristic, complemented
5#11762554C211yesmaximal, characteristic, complemented
6#11762553C211yesmaximal, characteristic, complemented
7C17×#331C211yesmaximal, characteristic, complemented
8C2×#63254—21no—
9C17×#127C2211yescharacteristic
10#10587624—21no—
11#10587624C412yes—
12#10587622—21no—
13D4×Dic17—21no—
14D4×Dic17C412yes—
15C2×#63218—21no—
16C42×D17C2211yescharacteristic
17#10587555C2211yescharacteristic
18C17×C4⋊C8C2211yescharacteristic
19C17×C4⋊C8C2211yescharacteristic
20#10587491—21no—
21#10587486—21no—
22#10587485—21no—
23C4⋊#3622C2211yescharacteristic
24C4⋊#3622C2211yescharacteristic
25C2×D4×C17—21no—
26C2×D4×C17—21no—
27C2×D4×C17C2×C412yes—
28C2×C22⋊D17—22no—
29C22×Dic17—22no—
30#63254—42no—
31#63254—42no—
32C2×C4×D17C2×C411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×Dic34—21no—
36C2×C8×C17—21no—
37C2×C8×C17—21no—
38C42×C17C2311yescharacteristic
39C22⋊Dic17—22no—
40C22⋊Dic17—22no—
41C4⋊Dic17—21no—
42C4⋊Dic17C2×C412yes—
43C4×Dic17C2×C411yescharacteristic
44C4×Dic17D411yescharacteristic
45C4×Dic17D411yescharacteristic
46C2×#3622—21no—
47C2×#3622—21no—
48#63218—42no—
49C23×C17—22no—
50C22×D17C22⋊C411yescharacteristic
51D4×C17—42no—
52D4×C17—41no—
53D4×C17—41no—
54D4×C17—41no—
55D4×C17—41no—
56C2×C4×C17C22×C411yescharacteristic
57C2×C4×C17C2×D411yescharacteristic
58C2×C4×C17C2×D411yescharacteristic
59C22⋊D17—44no—
60C2×Dic17—22no—
61C2×Dic17—44no—
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C22⋊C411yescharacteristic
64C2×Dic17C22⋊C411yescharacteristic
65C4×D17—21no—
66C4×D17—21no—
67C4×D17—21no—
68C4×D17—21no—
69C4×D17—41no—
70Dic34—41no—
71Dic34—41no—
72C8×C17—41no—
73C8×C17—41no—
74#3622—41no—
75#3622—41no—
76#1557—171nomaximal, Sylow
77C22×C17—42no—
78C22×C17—42no—
79C22×C17—42no—
80C22×C17C2×C22⋊C411yescharacteristic
81C2×D17—21no—
82C2×D17—21no—
83C2×D17—21no—
84C4×C17—21no—
85C4×C17AGL(1,ℤ/8)11yescharacteristic
86C4×C17AGL(1,ℤ/8)11yescharacteristic
87C4×C17AGL(1,ℤ/8)11yescharacteristic
88C4×C17#13711yescharacteristic
89Dic17—21no—
90Dic17—21no—
91Dic17—21no—
92Dic17—42no—
93#532—171no—
94#432—171no—
95#423—171no—
96#332—171no—
97#332—171no—
98#331—171no—
99#331D1711yescharacteristic, complemented
100C2×C17—82no—
101C2×C17#41311yescharacteristic
102C2×C17#41711yescharacteristic
103C2×C17#41811yescharacteristic
104D17—41no—
105C2×#41—341no—
106C2×M16—341no—
107C4⋊Q8—171no—
108#127C2×D1711yescharacteristic
109#124—341no—
110#121—172no—
111#121—341no—
112C4×D4—172no—
113C4×D4—341no—
114C2×C42—171no—
115C4⋊C8—171no—
116C4⋊C8—171no—
117C4⋊C8—341no—
118C4⋊C8C2×D1711yescharacteristic
119C4⋊C8C2×D1711yescharacteristic
120C22⋊C8—341no—
121#97—341no—
122C17#155711yescharacteristic, complemented, Sylow
123#41—682no—
124#41—682no—
125C2×Q8—341no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—342no—
129C2×D4C4×D1712yes—
130C22×C4—171no—
131C22×C4—171no—
132C22×C4—171no—
133C22×C4—342no—
134M16—682no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—341no—
138C2×C8—341no—
139C4⋊C4—172no—
140C4⋊C4—341no—
141C22⋊C4—342no—
142C22⋊C4—342no—
143C42—171no—
144C42—171no—
145C42—171no—
146C42C22×D1711yescharacteristic
147C23—22no—
148C23—171no—
149Q8—681no—
150Q8—681no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—684no—
157C2×C4—171no—
158C2×C4—171no—
159C2×C4—171no—
160C2×C4—341no—
161C2×C4—341no—
162C2×C4—341no—
163C2×C4—341no—
164C2×C4—342no—
165C2×C4—681no—
166C2×C4—684no—
167C2×C4C2×C4×D1711yescharacteristic
168C2×C4D4×D1711yescharacteristic
169C2×C4D4×D1711yescharacteristic
170C8—41no—
171C8—41no—
172C8—681no—
173C8—681no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—341no—
178C22—341no—
179C22—341no—
180C22C22⋊C4×D1711yescharacteristic
181C4—21no—
182C4—341no—
183C4—341no—
184C4—341no—
185C4—682no—
186C4#1058759711yescharacteristic
187C4#1058759711yescharacteristic
188C4#1058760011yescharacteristic
189C4#1058760111yescharacteristic
190C2—82no—
191C2—681no—
192C2#1176279011yescharacteristic
193C2#1176280711yescharacteristic
194C2#1176281211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D17
commutatorC2×C4×C17C22×C4
FrattiniC42C22×D17
FittingC17×#331C2
socleC22×C17C2×C22⋊C4
prime radical#11763357C2
cosocleC42×C17C23
hypercenter#331D17
hypocenterC17#1557
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #11762865C2semidirect
3. #11762831C2semidirect
4. #11762556C2semidirect
5. #11762554C2semidirect
6. #11762553C2semidirect
7. C17×#331C2semidirect
99. #331D17semidirect
122. C17#1557semidirect

Maximal subgroups

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

subgroupindex
#117633572
#117628652
#117628312
#117625562
#117625542
#117625532
C17×#3312
#155717