Finite Group Databasebeta

← #14544655

Subgroups — #14544655

Order 2176.

292 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 84 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#11763810C211yesmaximal, characteristic
2C2×#10587549C212yesmaximal
3#11763753C211yesmaximal, characteristic
4#11763682C212yesmaximal
5C4×Q8×D17C211yesmaximal, characteristic
6C2×Q8×D17C2211yescharacteristic
7C2×C4⋊#595C2212yes—
8C2×#63242C2212yes—
9C4×Q8×C17C411yescharacteristic, complemented
10Q8×Dic17C411yescharacteristic, complemented
11C4⋊C4×D17—21no—
12C4⋊C4×D17C2211yescharacteristic
13#10587573—21no—
14#10587573C411yescharacteristic, complemented
15C42×D17—21no—
16C42×D17C2211yescharacteristic
17C4×Dic34—21no—
18C4×Dic34C411yescharacteristic, complemented
19#10587549—24no—
20(C2×#595)⋊C4—21no—
21(C2×#595)⋊C4—21no—
22#10587544—21no—
23C4⋊#3624—21no—
24C22×#595—22no—
25C2×Q8×C17C2×C411yescharacteristic
26Q8×D17—21no—
27Q8×D17—21no—
28Q8×D17D412yes—
29Q8×D17Q812yes—
30C2×C4×D17—21no—
31C2×C4×D17—21no—
32C2×C4×D17D411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×C4×D17C2311yescharacteristic
35C2×Dic34—21no—
36C2×Dic34C2×C411yescharacteristic
37C2×#3624—22no—
38C4⋊#595—22no—
39C4⋊#595—22no—
40#63242—22no—
41#63242—22no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17C2×C411yescharacteristic
44C42×C17—21no—
45C42×C17C2×C411yescharacteristic
46C4⋊Dic17—21no—
47C4⋊Dic17C2×C411yescharacteristic
48#63225—21no—
49#63225—21no—
50#63225—21no—
51C4×Dic17—21no—
52C4×Dic17—21no—
53C4×Dic17—21no—
54C4×Dic17C2×C411yescharacteristic
55C4×Dic17D411yescharacteristic
56C4×Dic17D411yescharacteristic
57C22×D17C2×D411yescharacteristic
58C2×#595—42no—
59C2×#595—42no—
60C2×#595—42no—
61Q8×C17—21no—
62Q8×C17C4⋊C412yes—
63C2×C4×C17—21no—
64C2×C4×C17—21no—
65C2×C4×C17C22⋊C411yescharacteristic
66C2×C4×C17C22⋊C411yescharacteristic
67C2×C4×C17C22×C411yescharacteristic
68C2×Dic17—21no—
69C2×Dic17—21no—
70C2×Dic17C22⋊C411yescharacteristic
71C2×Dic17C22⋊C411yescharacteristic
72C2×Dic17C2×D411yescharacteristic
73C4×D17—22no—
74C4×D17—22no—
75C4×D17—22no—
76C4×D17—22no—
77C4×D17—42no—
78C4×D17C2×D411yescharacteristic
79C4×D17C2×Q811yescharacteristic
80C4×D17#4111yescharacteristic
81C4×D17#4111yescharacteristic
82Dic34—21no—
83Dic34—22no—
84Dic34—22no—
85Dic34C4⋊C412yes—
86#3624—42no—
87#1805—171nomaximal, Sylow
88C22×C17C2×C22⋊C411yescharacteristic
89C2×D17#12111yescharacteristic
90C2×D17C22⋊Q811yescharacteristic
91C2×D17#13512yes—
92C2×D17#13712yes—
93#595—82no—
94C4×C17—22no—
95C4×C17—21no—
96C4×C17—21no—
97C4×C17—41no—
98C4×C17C2×C4⋊C411yescharacteristic
99C4×C17#11711yescharacteristic
100Dic17—22no—
101Dic17—21no—
102Dic17—21no—
103Dic17—41no—
104Dic17#12111yescharacteristic
105Dic17C22⋊Q811yescharacteristic
106C2×C4×Q8—171no—
107C2×C4⋊C8—171no—
108C2×#103—172no—
109#382—171no—
110#340—172no—
111C2×C17#38011yescharacteristic
112C2×C17#41611yescharacteristic
113C2×C17#41911yescharacteristic
114D17#46412yes—
115D17#47912yes—
116C22×Q8—171no—
117C22×C8—172no—
118C4×Q8—171no—
119C4×Q8—171no—
120C4×Q8—171no—
121C4×Q8—341no—
122C4×Q8—341no—
123C4×Q8#59511yescharacteristic, complemented
124C2×C4⋊C4—171no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—341no—
127C2×C42—171no—
128C2×C42—341no—
129C4⋊C8—341no—
130C4⋊C8—341no—
131#103—344no—
132#95—341no—
133#95—341no—
134C17#180511yescharacteristic, complemented, Sylow
135C2×Q8—172no—
136C2×Q8—172no—
137C2×Q8—171no—
138C2×Q8—341no—
139C2×Q8—341no—
140C2×Q8—341no—
141C2×Q8C2×#59511yescharacteristic
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—341no—
146C22×C4—341no—
147C22×C4—342no—
148C2×C8—342no—
149C2×C8—342no—
150C2×C8—342no—
151C4⋊C4—21no—
152C4⋊C4—171no—
153C4⋊C4—341no—
154C4⋊C4—341no—
155C4⋊C4—341no—
156C4⋊C4—342no—
157C4⋊C4—342no—
158C4⋊C4—341no—
159C4⋊C4C2×#59511yescharacteristic
160C42—21no—
161C42—171no—
162C42—171no—
163C42—171no—
164C42—341no—
165C42—341no—
166C42—341no—
167C42C2×#59511yescharacteristic
168C23—171no—
169Q8—21no—
170Q8—172no—
171Q8—342no—
172Q8—342no—
173Q8—341no—
174Q8C4⋊#59512yes—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—171no—
178C2×C4—171no—
179C2×C4—171no—
180C2×C4—171no—
181C2×C4—171no—
182C2×C4—171no—
183C2×C4—171no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—341no—
189C2×C4—341no—
190C2×C4—682no—
191C2×C4—682no—
192C2×C4—682no—
193C2×C4—682no—
194C2×C4C22⋊#59511yescharacteristic
195C2×C4C22⋊#59511yescharacteristic
196C2×C4C22×#59511yescharacteristic
197C8—682no—
198C22—171no—
199C22—171no—
200C22—172no—
201C22—172no—
202C22C2×C22⋊#59511yescharacteristic
203C4—22no—
204C4—21no—
205C4—21no—
206C4—41no—
207C4—171no—
208C4—171no—
209C4—342no—
210C4—341no—
211C4—341no—
212C4—681no—
213C4—1362no—
214C4C2×C4⋊#59511yescharacteristic
215C4#1058767711yescharacteristic
216C2—172no—
217C2—172no—
218C2#1176375111yescharacteristic
219C2#1176379911yescharacteristic
220C2#1176380211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C2×D4
cosocleC2×C4×D17C23
hypercenterC4×Q8#595
hypocenterC17#1805
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
9. C4×Q8×C17C4semidirect
10. Q8×Dic17C4semidirect
14. #10587573C4semidirect
18. C4×Dic34C4semidirect
123. C4×Q8#595semidirect
134. C17#1805semidirect

Maximal subgroups

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

subgroupindex
#117638102
C2×#105875492
#117637532
#117636822
C4×Q8×D172
#180517