Finite Group Databasebeta

← #14544141

Subgroups — #14544141

Order 2176.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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#11763825C211yesmaximal, characteristic, complemented
2#11763812C211yesmaximal, characteristic, complemented
3Q8⋊#3624C212yesmaximal, complemented
4#11763657C212yesmaximal, complemented
5#11763345C211yesmaximal, characteristic
6C2×#63247—21no—
7C17×C22⋊Q8C411yescharacteristic
8#10587633C411yescharacteristic
9Q8×Dic17C2211yescharacteristic
10#10587616C411yescharacteristic
11C2×C4×Dic17C2211yescharacteristic
12#10587584—21no—
13#10587579C2211yescharacteristic
14#10587578—21no—
15#10587576C411yescharacteristic
16#10587572—21no—
17#10587567—21no—
18C22⋊#3624—21no—
19#10587544—21no—
20C4⋊#3624—21no—
21C4⋊#3624C2212yes—
22C4×#3624C2212yes—
23C2×Q8×C17C2×C411yescharacteristic
24C22×C4×C17C2×C411yescharacteristic
25C2×C22⋊D17—21no—
26C22×Dic17D411yescharacteristic
27C22×Dic17D411yescharacteristic
28C2×D68C2×C411yescharacteristic
29C2×C4×D17—21no—
30#63247—42no—
31C2×#3624—22no—
32C2×#3624—22no—
33C4⋊C4×C17—21no—
34C4⋊C4×C17C2×C411yescharacteristic
35C22⋊C4×C17—21no—
36#63227—21no—
37#63227—21no—
38#63227—21no—
39#63227—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17—21no—
42C4⋊Dic17C2×C411yescharacteristic
43#63225—21no—
44C4×Dic17—21no—
45C4×Dic17C2×C411yescharacteristic
46C4×Dic17D411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17C2311yescharacteristic
49C23×C17C22⋊C411yescharacteristic
50C22×D17—21no—
51Q8×C17—22no—
52C2×C4×C17—22no—
53C2×C4×C17—21no—
54C2×C4×C17—21no—
55C2×C4×C17C22⋊C411yescharacteristic
56C2×C4×C17C22×C411yescharacteristic
57C22⋊D17—42no—
58C2×Dic17—21no—
59C2×Dic17—21no—
60C2×Dic17—21no—
61C2×Dic17—41no—
62C2×Dic17C22⋊C411yescharacteristic
63C2×Dic17C22⋊C411yescharacteristic
64C2×Dic17C2×D411yescharacteristic
65C2×Dic17C2×D411yescharacteristic
66D68—22no—
67C4×D17—41no—
68C4×D17—41no—
69#3624—42no—
70#3624—42no—
71#1495—171nomaximal, Sylow
72C22×C17—21no—
73C22×C17—21no—
74C22×C17—21no—
75C22×C17C2×C22⋊C411yescharacteristic
76C2×D17—41no—
77C2×D17—41no—
78C2×D17—41no—
79C4×C17—21no—
80C4×C17—41no—
81C4×C17—41no—
82C4×C17#13112yes—
83Dic17—21no—
84Dic17—21no—
85Dic17—21no—
86Dic17—41no—
87Dic17AGL(1,ℤ/8)11yescharacteristic
88Dic17#13711yescharacteristic
89#529—171no—
90#431—171no—
91#423—171no—
92Q8⋊C8—172no—
93D4⋊C8—172no—
94C2×C17—41no—
95C2×C17#40811yescharacteristic
96C2×C17#41711yescharacteristic
97C2×C17#42111yescharacteristic
98D17—81no—
99C2×M16—341no—
100#126—341no—
101#125—171no—
102#124—171no—
103#123—341no—
104C22⋊Q8#59511yescharacteristic
105#121—171no—
106C4×Q8—171no—
107C4×D4—171no—
108C4×D4—341no—
109#117—341no—
110C2×C42—171no—
111C4⋊C8—172no—
112C4⋊C8—341no—
113C4⋊C8—341no—
114C22⋊C8—341no—
115C4×C8—172no—
116C17#149511yescharacteristic, complemented, Sylow
117C2×Q8C2×#59511yescharacteristic
118C2×D4—171no—
119C2×D4—341no—
120C22×C4—171no—
121C22×C4—171no—
122C22×C4—341no—
123C22×C4C2×#59511yescharacteristic
124M16—682no—
125C2×C8—342no—
126C2×C8—342no—
127C4⋊C4—21no—
128C4⋊C4—171no—
129C4⋊C4—341no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C4⋊C4C2×#59511yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—341no—
135C22⋊C4—341no—
136C22⋊C4—341no—
137C22⋊C4—341no—
138C42—171no—
139C42—171no—
140C42—171no—
141C42—171no—
142C42—341no—
143C23—341no—
144C23C22⋊#59511yescharacteristic
145Q8—22no—
146D4—342no—
147D4—682no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—171no—
152C2×C4—171no—
153C2×C4—171no—
154C2×C4—171no—
155C2×C4—341no—
156C2×C4—341no—
157C2×C4—341no—
158C2×C4—681no—
159C2×C4—681no—
160C2×C4—681no—
161C2×C4C22⋊#59511yescharacteristic
162C2×C4C22×#59511yescharacteristic
163C8—682no—
164C8—682no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—681no—
169C22—681no—
170C22—681no—
171C22C2×C22⋊#59511yescharacteristic
172C4—21no—
173C4—41no—
174C4—41no—
175C4—171no—
176C4—171no—
177C4—341no—
178C4—341no—
179C4—341no—
180C4—681no—
181C4#1058767812yes—
182C2—41no—
183C2—1361no—
184C2#1176378211yescharacteristic
185C2#1176380011yescharacteristic
186C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C22⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radical#10587616C4
cosocleC4×Dic17C23
hypercenterC22⋊Q8#595
hypocenterC17#1495
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #11763825C2semidirect
2. #11763812C2semidirect
3. Q8⋊#3624C2semidirect
4. #11763657C2semidirect
116. C17#1495semidirect

Maximal subgroups

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

subgroupindex
#117638252
#117638122
Q8⋊#36242
#117636572
#117633452
#149517