Finite Group Databasebeta

← #14544210

Subgroups — #14544210

Order 2176.

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

normal subgroups only

This group has 284 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
1C2×#10587543C211yesmaximal, characteristic, complemented
2#11763752C211yesmaximal, characteristic, complemented
3#11763663C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#11763316C211yesmaximal, characteristic
6C2×#41×C17C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C22×#3624—21no—
10#10587632C411yescharacteristic, complemented
11#10587632C2211yescharacteristic
12#10587624C411yescharacteristic, complemented
13#10587624C2211yescharacteristic
14#10587623—21no—
15#10587621—21no—
16C4×C22⋊D17—21no—
17C4×C22⋊D17—21no—
18#10587612C411yescharacteristic, complemented
19(C2×#595)⋊C4—21no—
20(C2×#595)⋊C4C2212yes—
21#10587543—21no—
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—42no—
40C2×#3624—22no—
41C2×#3624—42no—
42C22⋊Dic17—21no—
43C22⋊Dic17—21no—
44C22⋊Dic17—21no—
45#63227—21no—
46#63227—21no—
47C4⋊Dic17C2×C411yescharacteristic
48C4⋊Dic17C2×C411yescharacteristic
49#63225—21no—
50#63225—21no—
51C4×Dic17—21no—
52C23×C17—21no—
53C23×C17C22⋊C411yescharacteristic
54C22×D17—22no—
55C22×D17—22no—
56C22×D17—41no—
57C22×D17C22⋊C411yescharacteristic
58C22×D17C2×D411yescharacteristic
59C2×#595—42no—
60C2×#595—42no—
61C2×#595—42no—
62Q8×C17—41no—
63D4×C17—41no—
64D4×C17—42no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—42no—
68C2×C4×C17C22⋊C411yescharacteristic
69C2×C4×C17C22×C411yescharacteristic
70C22⋊D17—42no—
71C22⋊D17—42no—
72C2×Dic17—22no—
73C2×Dic17—21no—
74C2×Dic17—21no—
75C2×Dic17C22⋊C411yescharacteristic
76C2×Dic17C2×D411yescharacteristic
77C4×D17—41no—
78C4×D17—41no—
79C4×D17—41no—
80C4×D17—41no—
81#3624—42no—
82#1516—171nomaximal, Sylow
83C22×C17—21no—
84C22×C17—21no—
85C22×C17—41no—
86C22×C17—41no—
87C22×C17—41no—
88C22×C17#9912yes—
89C22×C17C2×C22⋊C411yescharacteristic
90C2×D17—21no—
91C2×D17—21no—
92C2×D17—21no—
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—44no—
96C2×D17#13512yes—
97#595—82no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—41no—
101Dic17—21no—
102Dic17—21no—
103Dic17—41no—
104Dic17—41no—
105#525—171no—
106C2×C22⋊C8—171no—
107#381—171no—
108#328—172no—
109#327—172no—
110C2×C17—22no—
111C2×C17—81no—
112C2×C17C2×#9911yescharacteristic
113C2×C17#41611yescharacteristic
114C2×C17#42111yescharacteristic
115D17—22no—
116D17—41no—
117C2×#41#59511yescharacteristic, complemented
118C23×C4—171no—
119C22×C8—341no—
120#123—341no—
121C22⋊Q8—171no—
122C22⋊Q8—171no—
123#121—171no—
124#121—171no—
125#120—341no—
126C4×D4—341no—
127C4×D4—341no—
128#117—171no—
129C2×C22⋊C4—341no—
130C22⋊C8—172no—
131C22⋊C8—341no—
132#95—172no—
133#95—341no—
134C17#151611yescharacteristic, complemented, Sylow
135C24—171no—
136#41—42no—
137C2×Q8C2×#59511yescharacteristic
138C2×D4—21no—
139C2×D4—341no—
140C2×D4—341no—
141C2×D4C2×#59511yescharacteristic
142C22×C4—21no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—342no—
149C22×C4—682no—
150C22×C4C2×#59511yescharacteristic
151C2×C8—342no—
152C2×C8—682no—
153C4⋊C4—171no—
154C4⋊C4—171no—
155C4⋊C4—341no—
156C4⋊C4—341no—
157C22⋊C4—341no—
158C22⋊C4—341no—
159C22⋊C4—341no—
160C22⋊C4—341no—
161C22⋊C4—341no—
162C22⋊C4—682no—
163C42—341no—
164C23—21no—
165C23—171no—
166C23—171no—
167C23—342no—
168C23—342no—
169C23—681no—
170C23C22⋊#59511yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—682no—
175D4—682no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—171no—
180C2×C4—171no—
181C2×C4—342no—
182C2×C4—341no—
183C2×C4—341no—
184C2×C4—681no—
185C2×C4—681no—
186C2×C4—681no—
187C2×C4—681no—
188C2×C4—682no—
189C2×C4—682no—
190C2×C4—682no—
191C2×C4C22⋊#59511yescharacteristic
192C2×C4C22×#59511yescharacteristic
193C8—682no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—172no—
200C22—341no—
201C22—341no—
202C22—341no—
203C22—341no—
204C22—341no—
205C22—684no—
206C22#1058755112yes—
207C22C2×C22⋊#59511yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—341no—
212C4—341no—
213C4—681no—
214C4—681no—
215C4—1362no—
216C2—22no—
217C2—81no—
218C2—342no—
219C2—681no—
220C2C2×#1058755111yescharacteristic
221C2#1176379911yescharacteristic
222C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#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#1516
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#10587543C2semidirect
2. #11763752C2semidirect
3. #11763663C2semidirect
4. #11763660C2semidirect
6. C2×#41×C17C4semidirect
10. #10587632C4semidirect
12. #10587624C4semidirect
18. #10587612C4semidirect
117. C2×#41#595semidirect
134. C17#1516semidirect

Maximal subgroups

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

subgroupindex
C2×#105875432
#117637522
#117636632
#117636602
#117633162
#151617