Finite Group Databasebeta

← #14530004

Subgroups — #14530004

Order 2176.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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#11763385C211yesmaximal, characteristic
2#11762852C211yesmaximal, characteristic, complemented
3C2×#10587476C211yesmaximal, characteristic
4#11762811C211yesmaximal, characteristic
5C2×#10587495C211yesmaximal, characteristic, complemented
6#11762715C211yesmaximal, characteristic, complemented
7C17×#339C211yesmaximal, characteristic, complemented
8C2×#63252C2211yescharacteristic
9C22×Dic34C2211yescharacteristic
10C2×M16×C17C2211yescharacteristic
11C17×#117C2211yescharacteristic
12C2×C42×C17C2211yescharacteristic
13C2×C4⋊Dic17C2211yescharacteristic
14C2×#63223C2211yescharacteristic
15#10587583—21no—
16#10587583—21no—
17#10587572—22no—
18#10587561—21no—
19#10587555—21no—
20#10587502—22no—
21#10587495—22no—
22#10587492—21no—
23#10587487—21no—
24#10587479—21no—
25#10587478—21no—
26#10587476—22no—
27C22×C4×C17—21no—
28C22×C4×C17C2311yescharacteristic
29C2×C22⋊D17—21no—
30C22×Dic17—21no—
31C22×Dic17D412yes—
32#63252—21no—
33#63252—21no—
34#63252—21no—
35#63252—21no—
36C2×D68D411yescharacteristic
37C2×C4×D17—21no—
38C2×Dic34—21no—
39C2×Dic34—21no—
40C2×Dic34—21no—
41C2×Dic34—21no—
42C2×Dic34—21no—
43C2×Dic34—21no—
44C2×Dic34D411yescharacteristic
45M16×C17—21no—
46M16×C17—21no—
47C2×C8×C17—21no—
48C4⋊C4×C17D411yescharacteristic
49C4⋊C4×C17D411yescharacteristic
50C22⋊C4×C17—21no—
51C42×C17—21no—
52C42×C17—21no—
53C42×C17—21no—
54#63227—22no—
55#63227—22no—
56C4⋊Dic17—22no—
57C4⋊Dic17—22no—
58C4⋊Dic17—21no—
59C4⋊Dic17—21no—
60#63223—21no—
61#63223D412yes—
62C2×#3622—21no—
63C23×C17C2×D411yescharacteristic
64C22×D17—21no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—41no—
69C2×C4×C17—41no—
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17#4111yescharacteristic
74C2×C4×C17#4111yescharacteristic
75C2×C4×C17#4111yescharacteristic
76C22⋊D17—41no—
77C22⋊D17—41no—
78C2×Dic17—22no—
79C2×Dic17—21no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—42no—
83C2×Dic17—41no—
84D68—21no—
85D68—21no—
86C4×D17—41no—
87C4×D17—41no—
88Dic34—21no—
89Dic34—21no—
90Dic34—41no—
91Dic34—41no—
92Dic34—41no—
93Dic34—41no—
94C8×C17—41no—
95#3622—22no—
96#2038—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—21no—
99C22×C17#12011yescharacteristic
100C22×C17#12111yescharacteristic
101C22×C17#12311yescharacteristic
102C2×D17—41no—
103C2×D17—41no—
104C2×D17—41no—
105C4×C17—42no—
106C4×C17—41no—
107C4×C17—41no—
108C4×C17#12111yescharacteristic
109C4×C17#12111yescharacteristic
110C4×C17#12311yescharacteristic
111C4×C17#12411yescharacteristic
112Dic17—41no—
113Dic17—42no—
114Dic17—81no—
115#536—171no—
116#428—171no—
117C2×#104—171no—
118#417—171no—
119C2×#101—171no—
120#391—171no—
121#339D1711yescharacteristic, complemented
122C2×C17—21no—
123C2×C17—21no—
124C2×C17#39411yescharacteristic
125C2×C17#45411yescharacteristic
126C2×C17#45611yescharacteristic
127D17—81no—
128C2×#41—171no—
129C22×Q8—171no—
130C2×M16—171no—
131C2×M16C2×D1711yescharacteristic
132C4⋊Q8—341no—
133#124—341no—
134#123—342no—
135C22⋊Q8—341no—
136C22⋊Q8—341no—
137#117C2×D1711yescharacteristic
138C2×C4⋊C4—171no—
139C2×C42C2×D1711yescharacteristic
140#107—341no—
141#106—341no—
142#104—342no—
143#103—341no—
144#102—341no—
145#101—342no—
146#95—342no—
147C17#203811yescharacteristic, complemented, Sylow
148#41—341no—
149#41—341no—
150#41—341no—
151#41—341no—
152C2×Q8—171no—
153C2×Q8—341no—
154C2×Q8—341no—
155C2×Q8—341no—
156C2×Q8—341no—
157C2×Q8—341no—
158C2×Q8—341no—
159C2×D4—171no—
160C2×D4—341no—
161C22×C4—21no—
162C22×C4—172no—
163C22×C4—341no—
164C22×C4—341no—
165C22×C4C22×D1711yescharacteristic
166M16—21no—
167M16—21no—
168M16—172no—
169M16—341no—
170C2×C8—21no—
171C2×C8—341no—
172C4⋊C4—342no—
173C4⋊C4—342no—
174C4⋊C4—341no—
175C4⋊C4—341no—
176C4⋊C4D6811yescharacteristic
177C4⋊C4D6811yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—342no—
180C22⋊C4—342no—
181C42—21no—
182C42—21no—
183C42—21no—
184C23—341no—
185C23D4×D1711yescharacteristic
186Q8—341no—
187Q8—341no—
188Q8—681no—
189Q8—681no—
190Q8—681no—
191Q8—681no—
192D4—341no—
193D4—341no—
194D4—681no—
195D4—681no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—342no—
202C2×C4—341no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—681no—
206C2×C4—681no—
207C2×C4—682no—
208C2×C4—681no—
209C2×C4C2×D6811yescharacteristic
210C2×C4#6325211yescharacteristic
211C2×C4#6325211yescharacteristic
212C2×C4D4×D1711yescharacteristic
213C2×C4D4×D1711yescharacteristic
214C2×C4#6325611yescharacteristic
215C8—41no—
216C8—342no—
217C22—21no—
218C22—21no—
219C22—681no—
220C22—681no—
221C22—681no—
222C22#1058756811yescharacteristic
223C22#1058757011yescharacteristic
224C22#1058758011yescharacteristic
225C4—42no—
226C4—41no—
227C4—41no—
228C4—682no—
229C4—681no—
230C4—1361no—
231C4#1058756111yescharacteristic
232C4#1058757011yescharacteristic
233C4#1058758011yescharacteristic
234C4#1058758111yescharacteristic
235C2—21no—
236C2—21no—
237C2—1361no—
238C2#1176272711yescharacteristic
239C2#1176298311yescharacteristic
240C2#1176299311yescharacteristic
subgroupG/H
centerC22#10587568
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#339C2
socleC22×C17#120
prime radicalC2×#63252C22
hypercenter#339D17
hypocenterC17#2038
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #11762852C2semidirect
5. C2×#10587495C2semidirect
6. #11762715C2semidirect
7. C17×#339C2semidirect
121. #339D17semidirect
147. C17#2038semidirect

Maximal subgroups

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

subgroupindex
#117633852
#117628522
C2×#105874762
#117628112
C2×#105874952
#117627152
C17×#3392
#203817