Finite Group Databasebeta

← #14541718

Subgroups — #14541718

Order 2176.

326 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic34C211yesmaximal, characteristic
2#11763537C211yesmaximal, characteristic
3#11763218C212yesmaximal
4#11763207C211yesmaximal, characteristic
5#11763117C212yesmaximal
6#11763105C211yesmaximal, characteristic
7#11763095C212yesmaximal
8#11763095C211yesmaximal, characteristic
9#11762917C212yesmaximal
10C4×Dic68C211yesmaximal, characteristic
11C17×#446C211yesmaximal, characteristic
12C4×Q8×C17C2211yescharacteristic
13Q8×Dic17—21no—
14#10587630—21no—
15C2×Dic68C2211yescharacteristic
16#10587575—22no—
17#10587574—21no—
18#10587574—22no—
19#10587573—21no—
20#10587556C2212yes—
21#10587555C2211yescharacteristic
22#10587555C2211yescharacteristic
23#10587555C2212yes—
24C4×Dic34C2212yes—
25C4×Dic34C2211yescharacteristic
26C4×Dic34C2211yescharacteristic
27C4×Dic34C2212yes—
28C17×C4⋊C8C2211yescharacteristic
29C17×C4⋊C8C2212yes—
30C17×#97C2212yes—
31C4×C8×C17C2211yescharacteristic
32#10587489C2211yescharacteristic
33#10587489C2212yes—
34#10587488C2212yes—
35#10587488C2212yes—
36#10587488C2212yes—
37#10587487C2211yescharacteristic
38#10587487C2212yes—
39#10587487C2212yes—
40#10587487C2211yescharacteristic
41C2×Q8×C17D411yescharacteristic
42C2×Dic34—21no—
43C2×Dic34—22no—
44C2×Dic34—21no—
45C2×Dic34C2311yescharacteristic
46C2×Dic34C2311yescharacteristic
47C2×C8×C17C2311yescharacteristic
48C2×C8×C17C2312yes—
49C2×C8×C17C2311yescharacteristic
50C4⋊C4×C17D411yescharacteristic
51C4⋊C4×C17D412yes—
52C42×C17C2311yescharacteristic
53C42×C17C2312yes—
54C4⋊Dic17—22no—
55C4⋊Dic17—21no—
56C4⋊Dic17—21no—
57C4⋊Dic17C2312yes—
58C4⋊Dic17C2311yescharacteristic
59C4⋊Dic17C2312yes—
60C4⋊Dic17C2311yescharacteristic
61#63225—22no—
62#63225—21no—
63#63225—21no—
64#63225—22no—
65#63225—22no—
66#63225—22no—
67C4×Dic17—22no—
68C4×Dic17—21no—
69C4×Dic17—22no—
70C4×Dic17—21no—
71Dic68Q814yescomplemented
72Q8×C17—22no—
73C2×C4×C17C2×D412yes—
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D412yes—
76C2×C4×C17C2×D411yescharacteristic
77C2×C4×C17C2411yescharacteristic
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—21no—
84Dic34—21no—
85Dic34—21no—
86Dic34—42no—
87Dic34—82no—
88Dic34—81no—
89Dic34C2×Q812yes—
90Dic34C2×Q812yes—
91C8×C17—21no—
92C8×C17—22no—
93C8×C17C2×Q812yes—
94#3387—171nomaximal, Sylow
95C22×C17C22×D411yescharacteristic
96C4×C17—22no—
97C4×C17—22no—
98C4×C17—21no—
99C4×C17#13712yes—
100C4×C17C22×Q811yescharacteristic
101C4×C172−1+411yescharacteristic
102Dic17—22no—
103Dic17—22no—
104Dic17—42no—
105Dic17—41no—
106Dic17—42no—
107Dic17—41no—
108Q82—171no—
109#557—171no—
110#501—172no—
111#498—171no—
112#479—172no—
113#477—171no—
114#475—172no—
115#475—171no—
116#446D1711yescharacteristic
117#441—172no—
118C4×Q16—171no—
119C2×C17D4×Q811yescharacteristic
120C2×C17C2×#13711yescharacteristic
121C2×C17#57711yescharacteristic
122C2×Q16—171no—
123C4⋊Q8—171no—
124C4⋊Q8—172no—
125C4⋊Q8—171no—
126C4⋊Q8—341no—
127C4⋊Q8—341no—
128C4⋊Q8—342no—
129#125—172no—
130#125—342no—
131C4×Q8—171no—
132C4×Q8—172no—
133C4×Q8—171no—
134C4×Q8—172no—
135C4×Q8—341no—
136C4×Q8—341no—
137C4×Q8C2×D1711yescharacteristic
138#107—171no—
139#107—172no—
140#106—172no—
141#106—172no—
142#106—172no—
143C4⋊C8C2×D1711yescharacteristic
144C4⋊C8C2×D1712yes—
145#103—171no—
146#103—171no—
147#103—172no—
148#103—172no—
149#97C2×D1712yes—
150C4×C8C2×D1711yescharacteristic
151C17#338711yescharacteristic, complemented, Sylow
152C2×Q8—171no—
153C2×Q8—171no—
154C2×Q8—341no—
155C2×Q8—342no—
156C2×Q8—341no—
157C2×Q8D6811yescharacteristic
158Q16—174no—
159C2×C8C22×D1711yescharacteristic
160C2×C8C22×D1712yes—
161C2×C8C22×D1711yescharacteristic
162C4⋊C4—172no—
163C4⋊C4—171no—
164C4⋊C4—172no—
165C4⋊C4—171no—
166C4⋊C4—341no—
167C4⋊C4—342no—
168C4⋊C4—342no—
169C4⋊C4—341no—
170C4⋊C4—342no—
171C4⋊C4—342no—
172C4⋊C4—341no—
173C4⋊C4—342no—
174C4⋊C4—341no—
175C4⋊C4D6811yescharacteristic
176C4⋊C4D6812yes—
177C42—342no—
178C42—341no—
179C42—342no—
180C42—341no—
181C42C22×D1711yescharacteristic
182C42C22×D1712yes—
183Q8—22no—
184Q8—172no—
185Q8—172no—
186Q8—341no—
187Q8—341no—
188Q8—682no—
189Q8—1362no—
190Q8—1361no—
191C2×C4—341no—
192C2×C4—341no—
193C2×C4—341no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—341no—
197C2×C4C2×D6812yes—
198C2×C4C2×D6812yes—
199C2×C4C2×D6811yescharacteristic
200C2×C4C2×D6811yescharacteristic
201C2×C4C23×D1711yescharacteristic
202C8—21no—
203C8—22no—
204C8Q8×D1712yes—
205C22C22×D6811yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—342no—
210C4—342no—
211C4—682no—
212C4—682no—
213C4—681no—
214C4—681no—
215C4#1058759512yes—
216C4C2×Q8×D1711yescharacteristic
217C4#1058769711yescharacteristic
218C2Q8×D6811yescharacteristic
219C2C2×#1058759511yescharacteristic
220C2#1176362111yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#446C2
socle, prime radicalC22×C17C22×D4
hypercenter#446D17
hypocenterC17#3387
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
71. Dic68Q8semidirect
151. C17#3387semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic342
#117635372
#117632182
#117632072
#117631172
#117631052
#117630952
#117629172
C4×Dic682
C17×#4462
#338717