Finite Group Databasebeta

← #14544159

Subgroups — #14544159

Order 2176.

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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#11763777C211yesmaximal, characteristic, complemented
2C4×C22⋊#595C211yesmaximal, characteristic, complemented
3#11763663C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#11763316C211yesmaximal, characteristic
6C2×#41×C17C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×C4×#595—21no—
10C2×#63242—21no—
11#10587632C411yescharacteristic, complemented
12#10587632C2211yescharacteristic
13#10587624C411yescharacteristic, complemented
14#10587624C2211yescharacteristic
15#10587623—21no—
16#10587621—21no—
17C4×C22⋊D17—21no—
18C4×C22⋊D17—21no—
19#10587612C411yescharacteristic, complemented
20C22⋊#3624—21no—
21(C2×#595)⋊C4C2212yes—
22#10587543C2212yes—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25#41×C17—42no—
26C2×Q8×C17C2×C411yescharacteristic
27C2×D4×C17—21no—
28C2×D4×C17C2×C411yescharacteristic
29C22×C4×C17—21no—
30C22×C4×C17C2×C411yescharacteristic
31C2×C22⋊D17—21no—
32C2×C22⋊D17—21no—
33C22×Dic17D411yescharacteristic
34C2×C4×D17—42no—
35C2×C4×D17C2×C411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×C4×D17C2311yescharacteristic
39C22⋊#595—24no—
40C2×#3624—22no—
41C4×#595—42no—
42#63242—42no—
43C22⋊Dic17—21no—
44C22⋊Dic17—21no—
45C22⋊Dic17—21no—
46#63227—21no—
47#63227—21no—
48C4⋊Dic17C2×C411yescharacteristic
49C4⋊Dic17C2×C411yescharacteristic
50#63225—21no—
51#63225—21no—
52C4×Dic17—21no—
53C23×C17—21no—
54C23×C17C22⋊C411yescharacteristic
55C22×D17—22no—
56C22×D17—22no—
57C22×D17—41no—
58C22×D17C22⋊C411yescharacteristic
59C22×D17C2×D411yescharacteristic
60C2×#595—24no—
61C2×#595—42no—
62C2×#595—42no—
63Q8×C17—41no—
64D4×C17—41no—
65D4×C17—42no—
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—42no—
69C2×C4×C17C22⋊C411yescharacteristic
70C2×C4×C17C22×C411yescharacteristic
71C22⋊D17—42no—
72C22⋊D17—42no—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17—41no—
76C2×Dic17C22⋊C411yescharacteristic
77C2×Dic17C2×D411yescharacteristic
78C4×D17—22no—
79C4×D17—22no—
80C4×D17—41no—
81C4×D17—41no—
82#3624—42no—
83#1501—171nomaximal, Sylow
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—21no—
87C22×C17—41no—
88C22×C17—41no—
89C22×C17—41no—
90C22×C17C2×C22⋊C411yescharacteristic
91C2×D17—21no—
92C2×D17—21no—
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—44no—
96C2×D17#10412yes—
97C2×D17#13512yes—
98#595—44no—
99C4×C17—21no—
100C4×C17—41no—
101C4×C17#9912yes—
102Dic17—21no—
103Dic17—21no—
104Dic17—41no—
105Dic17—41no—
106#525—171no—
107C2×C22⋊C8—171no—
108C4×C22⋊C4—171no—
109#328—172no—
110#327—172no—
111C2×C17—41no—
112C2×C17—81no—
113C2×C17C2×#9911yescharacteristic
114C2×C17#41611yescharacteristic
115C2×C17C2×#10411yescharacteristic
116D17—22no—
117D17—22no—
118C2×#41#59511yescharacteristic, complemented
119C23×C4—171no—
120C22×C8—341no—
121#123—341no—
122C22⋊Q8—171no—
123C22⋊Q8—171no—
124#121—171no—
125#121—171no—
126#120—341no—
127C4×D4—341no—
128C4×D4—341no—
129#117—171no—
130C2×C22⋊C4—341no—
131C2×C42—341no—
132C22⋊C8—172no—
133C22⋊C8—341no—
134#95—172no—
135C17#150111yescharacteristic, complemented, Sylow
136C24—171no—
137#41—42no—
138C2×Q8C2×#59511yescharacteristic
139C2×D4—21no—
140C2×D4—341no—
141C2×D4—341no—
142C2×D4C2×#59511yescharacteristic
143C22×C4—21no—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—171no—
149C22×C4—342no—
150C22×C4—682no—
151C22×C4C2×#59511yescharacteristic
152C2×C8—342no—
153C2×C8—682no—
154C4⋊C4—171no—
155C4⋊C4—171no—
156C4⋊C4—341no—
157C4⋊C4—341no—
158C22⋊C4—341no—
159C22⋊C4—341no—
160C22⋊C4—341no—
161C22⋊C4—341no—
162C22⋊C4—344no—
163C22⋊C4—341no—
164C42—341no—
165C42—682no—
166C23—21no—
167C23—171no—
168C23—171no—
169C23—342no—
170C23—342no—
171C23—681no—
172C23C22⋊#59511yescharacteristic
173Q8—41no—
174D4—41no—
175D4—42no—
176D4—682no—
177D4—682no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—171no—
182C2×C4—171no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—344no—
186C2×C4—341no—
187C2×C4—341no—
188C2×C4—681no—
189C2×C4—681no—
190C2×C4—681no—
191C2×C4—682no—
192C2×C4—682no—
193C2×C4C22⋊#59511yescharacteristic
194C2×C4C22×#59511yescharacteristic
195C8—682no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22—172no—
203C22—172no—
204C22—341no—
205C22—341no—
206C22—341no—
207C22—341no—
208C22—684no—
209C22C2×C22⋊#59511yescharacteristic
210C4—21no—
211C4—41no—
212C4—341no—
213C4—341no—
214C4—684no—
215C4—681no—
216C4—681no—
217C4#1058755112yes—
218C2—41no—
219C2—81no—
220C2—342no—
221C2—342no—
222C2C2×#1058755111yescharacteristic
223C2#1176379911yescharacteristic
224C2D17⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×#41×C17C4
socleC22×C17C2×C22⋊C4
prime radical#11763316C2
cosocleC2×C4×D17C23
hypercenterC2×#41#595
hypocenterC17#1501
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #11763777C2semidirect
2. C4×C22⋊#595C2semidirect
3. #11763663C2semidirect
4. #11763660C2semidirect
6. C2×#41×C17C4semidirect
11. #10587632C4semidirect
13. #10587624C4semidirect
19. #10587612C4semidirect
118. C2×#41#595semidirect
135. C17#1501semidirect

Maximal subgroups

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

subgroupindex
#117637772
C4×C22⋊#5952
#117636632
#117636602
#117633162
#150117