Finite Group Databasebeta

← #14544222

Subgroups — #14544222

Order 2176.

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

normal subgroups only

This group has 288 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
1C2×#10587543C211yesmaximal, characteristic
2#11763751C211yesmaximal, characteristic
3#11763663C212yesmaximal, complemented
4#11763662C212yesmaximal, complemented
5C22⋊Q8×D17C211yesmaximal, characteristic
6C2×Q8×D17C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C22×#3624—21no—
10C2×C4⋊#595—21no—
11C17×C22⋊Q8C411yescharacteristic, complemented
12#10587632C2211yescharacteristic
13#10587610C411yescharacteristic, complemented
14#10587583C2211yescharacteristic
15#10587582—21no—
16C4⋊C4×D17—21no—
17C4⋊C4×D17C411yescharacteristic, complemented
18C22⋊C4×D17—21no—
19#10587564—21no—
20(C2×#595)⋊C4C2212yes—
21#10587543—21no—
22#10587543C2212yes—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25C2×Q8×C17C2×C411yescharacteristic
26C22×C4×C17C2×C411yescharacteristic
27C22×Dic17D411yescharacteristic
28Q8×D17—42no—
29C2×C4×D17—21no—
30C2×C4×D17—21no—
31C2×C4×D17—42no—
32C2×C4×D17C2×C411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C4×D17D411yescharacteristic
35C2×C4×D17C2311yescharacteristic
36C2×Dic34—21no—
37C2×Dic34C2×C411yescharacteristic
38C22⋊#595—42no—
39C2×#3624—22no—
40C2×#3624—42no—
41C4⋊#595—42no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17C2×C411yescharacteristic
44C22⋊C4×C17—21no—
45C22⋊Dic17—21no—
46#63227—21no—
47#63227—21no—
48C4⋊Dic17—21no—
49C4⋊Dic17C2×C411yescharacteristic
50#63225—21no—
51#63225—21no—
52#63225—21no—
53C23×C17C22⋊C411yescharacteristic
54C22×D17—41no—
55C22×D17—41no—
56C22×D17C22⋊C411yescharacteristic
57C22×D17SD1612yes—
58C22×D17Q1612yes—
59C22×D17C2×D411yescharacteristic
60C2×#595—42no—
61C2×#595—42no—
62C2×#595—42no—
63Q8×C17—41no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—41no—
67C2×C4×C17C22⋊C411yescharacteristic
68C2×C4×C17C22×C411yescharacteristic
69C2×Dic17—22no—
70C2×Dic17—21no—
71C2×Dic17—21no—
72C2×Dic17C22⋊C411yescharacteristic
73C2×Dic17C2×D411yescharacteristic
74C4×D17—22no—
75C4×D17—22no—
76C4×D17—41no—
77C4×D17—41no—
78C4×D17—42no—
79C4×D17—42no—
80Dic34—42no—
81Dic34—41no—
82#3624—42no—
83#1520—171nomaximal, Sylow
84C22×C17—21no—
85C22×C17—21no—
86C22×C17#10312yes—
87C22×C17C2×C22⋊C411yescharacteristic
88C2×D17—21no—
89C2×D17—21no—
90C2×D17—21no—
91C2×D17—21no—
92C2×D17—42no—
93C2×D17—42no—
94C2×D17#10312yes—
95C2×D17C2×SD1611yescharacteristic
96C2×D17C2×Q1611yescharacteristic
97#595—82no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—41no—
101C4×C17—41no—
102Dic17—21no—
103Dic17—41no—
104Dic17—41no—
105Dic17#9912yes—
106C2×C22⋊Q8—171no—
107C2×C22⋊C8—171no—
108#380—171no—
109#328—172no—
110#328—172no—
111C2×C17—22no—
112C2×C17C2×#9911yescharacteristic
113C2×C17C2×#10311yescharacteristic
114C2×C17#42111yescharacteristic
115D17—21no—
116D17—22no—
117D17—21no—
118C22×Q8—171no—
119C23×C4—171no—
120C22×C8—341no—
121C22⋊Q8—171no—
122C22⋊Q8—171no—
123C22⋊Q8—171no—
124C22⋊Q8—341no—
125C22⋊Q8—341no—
126C22⋊Q8#59511yescharacteristic, complemented
127C2×C4⋊C4—171no—
128C2×C4⋊C4—341no—
129C2×C4⋊C4—341no—
130C2×C22⋊C4—341no—
131C2×C22⋊C4—341no—
132C22⋊C8—172no—
133C22⋊C8—341no—
134#95—172no—
135C17#152011yescharacteristic, complemented, Sylow
136C24—171no—
137C2×Q8—171no—
138C2×Q8—341no—
139C2×Q8—682no—
140C2×Q8C2×#59511yescharacteristic
141C22×C4—171no—
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—341no—
147C22×C4—341no—
148C22×C4—342no—
149C22×C4—682no—
150C22×C4C2×#59511yescharacteristic
151C2×C8—342no—
152C2×C8—682no—
153C4⋊C4—21no—
154C4⋊C4—171no—
155C4⋊C4—341no—
156C4⋊C4—341no—
157C4⋊C4—341no—
158C4⋊C4—341no—
159C4⋊C4—682no—
160C4⋊C4C2×#59511yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—341no—
163C22⋊C4—341no—
164C22⋊C4—341no—
165C22⋊C4—682no—
166C23—171no—
167C23—171no—
168C23—172no—
169C23—172no—
170C23—681no—
171C23—681no—
172C23C22⋊#59511yescharacteristic
173Q8—41no—
174Q8—681no—
175Q8—682no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—171no—
180C2×C4—171no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—341no—
185C2×C4—341no—
186C2×C4—681no—
187C2×C4—681no—
188C2×C4—682no—
189C2×C4—682no—
190C2×C4—682no—
191C2×C4—682no—
192C2×C4—682no—
193C2×C4C22⋊#59511yescharacteristic
194C2×C4C22×#59511yescharacteristic
195C8—682no—
196C22—21no—
197C22—21no—
198C22—171no—
199C22—172no—
200C22—171no—
201C22—341no—
202C22—341no—
203C22—341no—
204C22—341no—
205C22—682no—
206C22—682no—
207C22#1058754912yes—
208C22C2×C22⋊#59511yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—172no—
214C4—341no—
215C4—681no—
216C4—681no—
217C4—1362no—
218C2—22no—
219C2—341no—
220C2—342no—
221C2—341no—
222C2D17⋊#9911yescharacteristic
223C2C2×#1058754911yescharacteristic
224C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C22⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radicalC23×D17D4
cosocleC2×C4×D17C23
hypercenterC22⋊Q8#595
hypocenterC17#1520
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #11763663C2semidirect
4. #11763662C2semidirect
6. C2×Q8×D17C4semidirect
11. C17×C22⋊Q8C4semidirect
13. #10587610C4semidirect
17. C4⋊C4×D17C4semidirect
126. C22⋊Q8#595semidirect
135. C17#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#105875432
#117637512
#117636632
#117636622
C22⋊Q8×D172
#152017