Finite Group Databasebeta

← #14539867

Subgroups — #14539867

Order 2176.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#11763421C211yesmaximal, characteristic, complemented
2#11763204C211yesmaximal, characteristic, complemented
3#11763188C211yesmaximal, characteristic, complemented
4#11763177C211yesmaximal, characteristic, complemented
5#11763163C211yesmaximal, characteristic, complemented
6#11763028C212yesmaximal, complemented
7#11763016C212yesmaximal, complemented
8#11762971C212yesmaximal
9#11762965C212yesmaximal, complemented
10#11762870C211yesmaximal, characteristic, complemented
11C17×#536C211yesmaximal, characteristic
12C2×#41×C17C2211yescharacteristic
13C22×Q8×C17C2211yescharacteristic
14C17×C4⋊Q8C2211yescharacteristic
15C17×C4⋊Q8C2211yescharacteristic
16C17×#124C2211yescharacteristic
17C17×#124C2211yescharacteristic
18C17×#123—22no—
19C17×C22⋊Q8—22no—
20C17×#117C2211yescharacteristic
21C2×#63231C2212yes—
22C2×#63231C2212yes—
23C2×#63231C2212yes—
24C2×#63229C2212yes—
25#10587610C2212yes—
26#10587575—22no—
27#10587565—22no—
28#10587556C2211yescharacteristic
29#10587555C2211yescharacteristic
30C4×Dic34C2212yes—
31#10587506C2212yes—
32#10587506C2212yes—
33#10587506C2212yes—
34#10587503C2212yes—
35C22⋊#3622C2212yes—
36C4⋊#3622C2212yes—
37#10587474C2211yescharacteristic
38C4×#3622C2211yescharacteristic
39#41×C17—42no—
40C2×Q8×C17—21no—
41C2×Q8×C17—42no—
42C2×Q8×C17C2311yescharacteristic
43C2×Q8×C17C2311yescharacteristic
44C2×Q8×C17C2311yescharacteristic
45C2×D4×C17—21no—
46C2×D4×C17C2311yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C2311yescharacteristic
50C2×Dic34C2312yes—
51C4⋊C4×C17—22no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17D412yes—
54C22⋊C4×C17—22no—
55C22⋊C4×C17—22no—
56C22⋊C4×C17D412yes—
57C42×C17C2311yescharacteristic
58C42×C17C2311yescharacteristic
59C22⋊Dic17—22no—
60#63231—42no—
61#63231—42no—
62#63231—42no—
63#63229—42no—
64C4⋊Dic17—22no—
65C4⋊Dic17C2312yes—
66#63225—22no—
67#63225—22no—
68#63225—22no—
69C4×Dic17—22no—
70C2×#3622C2312yes—
71C2×#3622C2312yes—
72C23×C17—21no—
73C23×C17C2×D411yescharacteristic
74Q8×C17—22no—
75Q8×C17—22no—
76Q8×C17—22no—
77Q8×C17—42no—
78D4×C17—22no—
79D4×C17—42no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17C2×D412yes—
87C2×C4×C17C2×D412yes—
88C2×C4×C17C2×D411yescharacteristic
89C2×C4×C17C2411yescharacteristic
90C2×Dic17—22no—
91C2×Dic17—22no—
92Dic34—22no—
93Dic34—22no—
94#3622—22no—
95#3622—22no—
96#3223—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—22no—
99C22×C17—41no—
100C22×C17—42no—
101C22×C17C22×D411yescharacteristic
102C4×C17—21no—
103C4×C17—22no—
104C4×C17—22no—
105C4×C17—41no—
106C4×C17—41no—
107C4×C17—41no—
108C4×C17H(5,𝔽2)12yes—
109Dic17—42no—
110Dic17—42no—
111#541—171no—
112#536D1711yescharacteristic
113#497—171no—
114#494—171no—
115#491—171no—
116#488—171no—
117#464—172no—
118#462—172no—
119#452—172no—
120#451—172no—
121#433—171no—
122C2×C17—41no—
123C2×C17—81no—
124C2×C17#53511yescharacteristic
125C2×C17#57812yes—
126C2×#41C2×D1711yescharacteristic
127C22×Q8C2×D1711yescharacteristic
128C2×Q16—172no—
129C2×Q16—172no—
130C2×Q16—172no—
131C2×SD16—172no—
132C4⋊Q8—171no—
133C4⋊Q8C2×D1711yescharacteristic
134C4⋊Q8C2×D1711yescharacteristic
135#126—342no—
136#125—171no—
137#125—342no—
138#124C2×D1711yescharacteristic
139#124C2×D1711yescharacteristic
140#123—22no—
141C22⋊Q8—22no—
142C22⋊Q8—172no—
143C4×Q8—172no—
144#117C2×D1711yescharacteristic
145C4⋊C8—172no—
146#103—172no—
147#103—172no—
148#103—172no—
149#102—172no—
150C22⋊C8—172no—
151#97—171no—
152C4×C8—171no—
153C17#322311yescharacteristic, complemented, Sylow
154#41—42no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—172no—
158C2×Q8C22×D1711yescharacteristic
159C2×Q8C22×D1711yescharacteristic
160C2×Q8C22×D1711yescharacteristic
161C2×D4—21no—
162C2×D4C22×D1711yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4C22×D1711yescharacteristic
166Q16—682no—
167Q16—682no—
168Q16—682no—
169SD16—682no—
170C2×C8—172no—
171C2×C8—172no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—172no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4C22⋊D1712yes—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—342no—
183C22⋊C4C22⋊D1712yes—
184C42—342no—
185C42C22×D1711yescharacteristic
186C42C22×D1711yescharacteristic
187C23—21no—
188C23C2×C22⋊D1711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—342no—
194Q8—342no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4C2×C22⋊D1712yes—
206C2×C4C2×C22⋊D1712yes—
207C2×C4C2×C22⋊D1711yescharacteristic
208C2×C4C23×D1711yescharacteristic
209C8—342no—
210C8—342no—
211C22—22no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22C22×C22⋊D1711yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—682no—
223C4—682no—
224C4#1058769112yes—
225C2—41no—
226C2—81no—
227C2#1176337611yescharacteristic
228C2#1176362412yes—
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#536C2
socleC22×C17C22×D4
prime radicalC2×#41×C17C22
hypercenter#536D17
hypocenterC17#3223
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #11763421C2semidirect
2. #11763204C2semidirect
3. #11763188C2semidirect
4. #11763177C2semidirect
5. #11763163C2semidirect
6. #11763028C2semidirect
7. #11763016C2semidirect
9. #11762965C2semidirect
10. #11762870C2semidirect
153. C17#3223semidirect

Maximal subgroups

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

subgroupindex
#117634212
#117632042
#117631882
#117631772
#117631632
#117630282
#117630162
#117629712
#117629652
#117628702
C17×#5362
#322317