Finite Group Databasebeta

← #14527971

Subgroups — #14527971

Order 2176.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#11763545C211yesmaximal, characteristic
2#11763155C211yesmaximal, characteristic
3#11763095C211yesmaximal, characteristic, complemented
4#11763020C211yesmaximal, characteristic, complemented
5#11762539C211yesmaximal, characteristic, complemented
6#11762539C211yesmaximal, characteristic, complemented
7C17×#331C211yesmaximal, characteristic
8C17×#127C2211yescharacteristic
9#10587622—21no—
10#10587622—21no—
11#10587621—22no—
12D4×Dic17—21no—
13D4×Dic17—21no—
14C2×Dic68—21no—
15#10587555C2211yescharacteristic
16C4×Dic34C2211yescharacteristic
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—
28C22×Dic17—22no—
29C2×Dic34—21no—
30C2×Dic34D411yescharacteristic
31C2×Dic34D411yescharacteristic
32C2×C8×C17—21no—
33C2×C8×C17—21no—
34C42×C17C2311yescharacteristic
35C22⋊Dic17—22no—
36C22⋊Dic17—22no—
37C22⋊Dic17—22no—
38C4⋊Dic17—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17D411yescharacteristic
41C4⋊Dic17D411yescharacteristic
42#63225—21no—
43#63225—21no—
44C4×Dic17—21no—
45C4×Dic17—21no—
46C2×#3622—21no—
47Dic68—42no—
48C23×C17—22no—
49D4×C17—42no—
50D4×C17—41no—
51D4×C17—41no—
52D4×C17—41no—
53D4×C17—41no—
54C2×C4×C17C2×D411yescharacteristic
55C2×C4×C17C2×D411yescharacteristic
56C2×C4×C17C2×D411yescharacteristic
57C2×Dic17—21no—
58C2×Dic17—21no—
59C2×Dic17—21no—
60C2×Dic17—21no—
61C2×Dic17—41no—
62C2×Dic17—82no—
63Dic34—21no—
64Dic34—21no—
65Dic34—21no—
66Dic34—21no—
67Dic34—41no—
68Dic34—41no—
69C8×C17—41no—
70C8×C17—41no—
71#3622—41no—
72#1640—171nomaximal, Sylow
73C22×C17—42no—
74C22×C17—42no—
75C22×C17—42no—
76C22×C17#12011yescharacteristic
77C4×C17—21no—
78C4×C17#13511yescharacteristic
79C4×C17#13511yescharacteristic
80C4×C17AGL(1,ℤ/8)11yescharacteristic
81C4×C17#13711yescharacteristic
82Dic17—41no—
83Dic17—41no—
84Dic17—41no—
85Dic17—41no—
86Dic17—81no—
87#559—171no—
88#486—171no—
89#475—171no—
90#462—171no—
91#331D1711yescharacteristic
92Q8⋊C8—171no—
93Q8⋊C8—171no—
94C2×C17—82no—
95C2×C17#44911yescharacteristic
96C2×C17#45211yescharacteristic
97C2×C17#45511yescharacteristic
98C2×Q16—341no—
99C4⋊Q8—171no—
100#127C2×D1711yescharacteristic
101#124—341no—
102#124—341no—
103#123—342no—
104C4×Q8—171no—
105C4×Q8—171no—
106C4×D4—341no—
107C4×D4—341no—
108#106—341no—
109C4⋊C8C2×D1711yescharacteristic
110C4⋊C8C2×D1711yescharacteristic
111#103—341no—
112#103—341no—
113#102—342no—
114C4×C8—171no—
115C17#164011yescharacteristic, complemented, Sylow
116C2×Q8—171no—
117C2×Q8—171no—
118C2×Q8—341no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D6812yes—
122C22×C4—342no—
123Q16—682no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—341no—
127C4⋊C4—171no—
128C4⋊C4—171no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C4⋊C4—341no—
133C22⋊C4—342no—
134C22⋊C4—342no—
135C22⋊C4—342no—
136C42—341no—
137C42—341no—
138C42C22×D1711yescharacteristic
139C23—22no—
140Q8—341no—
141Q8—341no—
142Q8—341no—
143Q8—341no—
144Q8—681no—
145Q8—681no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—341no—
152C2×C4—341no—
153C2×C4—341no—
154C2×C4—341no—
155C2×C4—681no—
156C2×C4—1362no—
157C2×C4C2×D6811yescharacteristic
158C2×C4D4×D1711yescharacteristic
159C2×C4D4×D1711yescharacteristic
160C8—41no—
161C8—41no—
162C8—681no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1058756811yescharacteristic
167C4—21no—
168C4—681no—
169C4—681no—
170C4—681no—
171C4—681no—
172C4—1361no—
173C4#1058759711yescharacteristic
174C4#1058759811yescharacteristic
175C4#1058760111yescharacteristic
176C4#1058760211yescharacteristic
177C2—82no—
178C2#1176295511yescharacteristic
179C2#1176297211yescharacteristic
180C2#1176298911yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC42×C17C23
FrattiniC42C22×D17
FittingC17×#331C2
socleC22×C17#120
prime radicalC17×#127C22
hypercenter#331D17
hypocenterC17#1640
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #11763095C2semidirect
4. #11763020C2semidirect
5. #11762539C2semidirect
6. #11762539C2semidirect
115. C17#1640semidirect

Maximal subgroups

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

subgroupindex
#117635452
#117631552
#117630952
#117630202
#117625392
C17×#3312
#164017