Finite Group Databasebeta

← #14544209

Subgroups — #14544209

Order 2176.

284 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 58 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
2#11763752C211yesmaximal, characteristic, complemented
3#11763662C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#11763319C211yesmaximal, characteristic
6C2×#63254C411yescharacteristic, complemented
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×#63242—21no—
10C17×#121C411yescharacteristic, complemented
11#10587624C2211yescharacteristic
12#10587623—21no—
13D4×Dic17—21no—
14#10587610C411yescharacteristic, complemented
15#10587583C2211yescharacteristic
16#10587578C411yescharacteristic, complemented
17#10587569—21no—
18#10587567—21no—
19C22⋊#3624—21no—
20(C2×#595)⋊C4—21no—
21(C2×#595)⋊C4C2212yes—
22#10587543C2212yes—
23C23×D17D411yescharacteristic
24C22×#595—22no—
25C2×D4×C17—21no—
26C2×D4×C17C2×C411yescharacteristic
27C22×C4×C17C2×C411yescharacteristic
28C2×C22⋊D17—21no—
29C2×C22⋊D17—21no—
30C22×Dic17—21no—
31C22×Dic17D411yescharacteristic
32#63254—42no—
33C2×C4×D17—42no—
34C2×C4×D17C2×C411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17C2311yescharacteristic
38C2×Dic34C2×C411yescharacteristic
39C22⋊#595—42no—
40C2×#3624—22no—
41#63242—42no—
42C4⋊C4×C17C2×C411yescharacteristic
43C22⋊C4×C17—21no—
44C22⋊Dic17—21no—
45C22⋊Dic17—21no—
46#63227—21no—
47#63227—21no—
48C4⋊Dic17—21no—
49C4⋊Dic17C2×C411yescharacteristic
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—
62D4×C17—41no—
63D4×C17—42no—
64C2×C4×C17—21no—
65C2×C4×C17—41no—
66C2×C4×C17C22⋊C411yescharacteristic
67C2×C4×C17C22×C411yescharacteristic
68C22⋊D17—42no—
69C22⋊D17—42no—
70C2×Dic17—21no—
71C2×Dic17—21no—
72C2×Dic17—41no—
73C2×Dic17—42no—
74C2×Dic17C22⋊C411yescharacteristic
75C2×Dic17C2×D411yescharacteristic
76C4×D17—22no—
77C4×D17—22no—
78C4×D17—41no—
79C4×D17—41no—
80Dic34—41no—
81#3624—42no—
82#1516—171nomaximal, Sylow
83C22×C17—21no—
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—41no—
87C22×C17—41no—
88C22×C17—41no—
89C22×C17C2×C22⋊C411yescharacteristic
90C2×D17—21no—
91C2×D17—21no—
92C2×D17—21no—
93C2×D17—21no—
94C2×D17—44no—
95C2×D17#9912yes—
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—41no—
111C2×C17—81no—
112C2×C17C2×#9911yescharacteristic
113C2×C17#41611yescharacteristic
114C2×C17#42111yescharacteristic
115D17—22no—
116D17—22no—
117C2×#41—171no—
118C23×C4—171no—
119C22×C8—341no—
120#123—341no—
121C22⋊Q8—171no—
122C22⋊Q8—171no—
123#121—171no—
124#121#59511yescharacteristic, complemented
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—682no—
137C2×Q8—171no—
138C2×D4—21no—
139C2×D4—341no—
140C2×D4—341no—
141C2×D4C2×#59511yescharacteristic
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—341no—
148C22×C4—342no—
149C22×C4—682no—
150C22×C4C2×#59511yescharacteristic
151C2×C8—342no—
152C2×C8—682no—
153C4⋊C4—171no—
154C4⋊C4—341no—
155C4⋊C4—341no—
156C4⋊C4C2×#59511yescharacteristic
157C22⋊C4—21no—
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—681no—
172D4—41no—
173D4—42no—
174D4—682no—
175D4—682no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—171no—
179C2×C4—171no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—341no—
183C2×C4—341no—
184C2×C4—681no—
185C2×C4—681no—
186C2×C4—681no—
187C2×C4—682no—
188C2×C4—682no—
189C2×C4—682no—
190C2×C4—682no—
191C2×C4C22⋊#59511yescharacteristic
192C2×C4C22×#59511yescharacteristic
193C8—682no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—172no—
201C22—172no—
202C22—341no—
203C22—341no—
204C22—341no—
205C22—341no—
206C22—684no—
207C22C2×C22⋊#59511yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—341no—
212C4—341no—
213C4—681no—
214C4—681no—
215C4—1362no—
216C2—41no—
217C2—81no—
218C2—342no—
219C2—342no—
220C2D17⋊#9911yescharacteristic
221C2#1176379911yescharacteristic
222C2#1176380711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#121C4
socleC22×C17C2×C22⋊C4
prime radical#11763319C2
cosocleC2×C4×D17C23
hypercenter#121#595
hypocenterC17#1516
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #11763777C2semidirect
2. #11763752C2semidirect
3. #11763662C2semidirect
4. #11763660C2semidirect
6. C2×#63254C4semidirect
10. C17×#121C4semidirect
14. #10587610C4semidirect
16. #10587578C4semidirect
124. #121#595semidirect
134. C17#1516semidirect

Maximal subgroups

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

subgroupindex
#117637772
#117637522
#117636622
#117636602
#117633192
#151617