Finite Group Databasebeta

← #14544295

Subgroups — #14544295

Order 2176.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763814C211yesmaximal, characteristic, complemented
2#11763810C211yesmaximal, characteristic, complemented
3#11763668C212yesmaximal, complemented
4#11763667C212yesmaximal, complemented
5#11763356C211yesmaximal, characteristic
6C2×#63252—21no—
7C2×#63243—21no—
8C2×#63242—21no—
9C4×D4×C17C411yescharacteristic
10#10587626C2211yescharacteristic
11#10587624C411yescharacteristic
12C4×C22⋊D17—21no—
13#10587581C411yescharacteristic
14#10587574C2211yescharacteristic
15#10587571—21no—
16#10587570—21no—
17C4×D68C411yescharacteristic
18C42×D17C2211yescharacteristic
19#10587544—21no—
20#10587544C2212yes—
21#10587544C2212yes—
22C4⋊#3624—21no—
23C2×D4×C17C2×C411yescharacteristic
24C22×C4×C17—21no—
25C2×C22⋊D17—21no—
26C2×C22⋊D17—21no—
27C2×C22⋊D17—21no—
28#63252—42no—
29#63252—42no—
30C2×D68—21no—
31C2×D68C2×C411yescharacteristic
32C2×C4×D17—21no—
33C2×C4×D17C2×C411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×Dic34—21no—
37C2×#3624—22no—
38C2×#3624—22no—
39#63243—42no—
40#63242—42no—
41C4⋊C4×C17C2×C411yescharacteristic
42C22⋊C4×C17—21no—
43C42×C17C2×C411yescharacteristic
44C22⋊Dic17—21no—
45#63227—21no—
46#63227—21no—
47C4⋊Dic17C2×C411yescharacteristic
48#63225—21no—
49C4×Dic17D411yescharacteristic
50C4×Dic17D411yescharacteristic
51C4×Dic17C2311yescharacteristic
52C23×C17—21no—
53C22×D17—21no—
54C22×D17C22⋊C411yescharacteristic
55D4×C17—41no—
56C2×C4×C17—21no—
57C2×C4×C17—42no—
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22⋊C411yescharacteristic
60C2×C4×C17C22×C411yescharacteristic
61C22⋊D17—42no—
62C22⋊D17—41no—
63C22⋊D17—41no—
64C22⋊D17—41no—
65C22⋊D17—41no—
66C2×Dic17—21no—
67C2×Dic17C22⋊C411yescharacteristic
68C2×Dic17C2×D411yescharacteristic
69C2×Dic17C2×D411yescharacteristic
70D68—41no—
71D68—42no—
72C4×D17—21no—
73C4×D17—21no—
74C4×D17—41no—
75C4×D17—42no—
76C4×D17D812yes—
77C4×D17SD1612yes—
78Dic34—41no—
79Dic34—41no—
80#3624—42no—
81#3624—42no—
82#1547—171nomaximal, Sylow
83C22×C17—41no—
84C22×C17—41no—
85C22×C17—41no—
86C22×C17C2×C22⋊C411yescharacteristic
87C2×D17—21no—
88C2×D17—21no—
89C2×D17—41no—
90C2×D17—41no—
91C2×D17—41no—
92C2×D17#10212yes—
93C4×C17—21no—
94C4×C17—21no—
95C4×C17—41no—
96C4×C17#10212yes—
97Dic17—21no—
98Dic17—41no—
99Dic17C2×D811yescharacteristic
100Dic17C2×SD1611yescharacteristic
101Dic17AGL(1,ℤ/8)11yescharacteristic
102Dic17#13711yescharacteristic
103#532—171no—
104#423—171no—
105C2×C4⋊C8—171no—
106#332—172no—
107#331—172no—
108C2×C17—81no—
109C2×C17#41311yescharacteristic
110C2×C17C2×#10211yescharacteristic
111C2×C17#41711yescharacteristic
112D17—22no—
113D17—81no—
114C2×#41—341no—
115C2×M16—341no—
116C22×C8—341no—
117C4⋊Q8—171no—
118#127—171no—
119#124—341no—
120#121—171no—
121#121—171no—
122#121—341no—
123C4×D4—171no—
124C4×D4—341no—
125C4×D4#59511yescharacteristic
126C2×C42—171no—
127C4⋊C8—172no—
128C4⋊C8—172no—
129C4⋊C8—341no—
130C4⋊C8—341no—
131C17#154711yescharacteristic, complemented, Sylow
132#41—682no—
133#41—682no—
134C2×Q8—341no—
135C2×D4—171no—
136C2×D4—341no—
137C2×D4—341no—
138C2×D4—341no—
139C2×D4—341no—
140C2×D4C2×#59511yescharacteristic
141C22×C4—21no—
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—341no—
146M16—682no—
147C2×C8—342no—
148C2×C8—342no—
149C2×C8—682no—
150C4⋊C4—171no—
151C4⋊C4—341no—
152C4⋊C4C2×#59511yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—341no—
155C22⋊C4—341no—
156C22⋊C4—341no—
157C42—171no—
158C42—171no—
159C42—171no—
160C42C2×#59511yescharacteristic
161C23—21no—
162C23—171no—
163C23—341no—
164Q8—681no—
165Q8—681no—
166D4—41no—
167D4—681no—
168D4—682no—
169D4—681no—
170D4—681no—
171D4—681no—
172D4—681no—
173D4—682no—
174C2×C4—21no—
175C2×C4—42no—
176C2×C4—171no—
177C2×C4—171no—
178C2×C4—172no—
179C2×C4—172no—
180C2×C4—171no—
181C2×C4—341no—
182C2×C4—341no—
183C2×C4—341no—
184C2×C4—681no—
185C2×C4—682no—
186C2×C4C22⋊#59511yescharacteristic
187C2×C4C22⋊#59511yescharacteristic
188C2×C4C22×#59511yescharacteristic
189C8—682no—
190C8—682no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—172no—
195C22—341no—
196C22—341no—
197C22—681no—
198C22—681no—
199C22—681no—
200C22C2×C22⋊#59511yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—171no—
205C4—171no—
206C4—171no—
207C4—171no—
208C4—341no—
209C4—681no—
210C4#1058754712yes—
211C2—81no—
212C2—342no—
213C2—1361no—
214C2#1176379511yescharacteristic
215C2C2×#1058754711yescharacteristic
216C2#1176380011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radical#11763356C2
cosocleC4×Dic17C23
hypercenterC4×D4#595
hypocenterC17#1547
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763814C2semidirect
2. #11763810C2semidirect
3. #11763668C2semidirect
4. #11763667C2semidirect
131. C17#1547semidirect

Maximal subgroups

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

subgroupindex
#117638142
#117638102
#117636682
#117636672
#117633562
#154717