Finite Group Databasebeta

← #14527739

Subgroups — #14527739

Order 2176.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#11763434C211yesmaximal, characteristic
2#11762965C211yesmaximal, characteristic, complemented
3#11762720C211yesmaximal, characteristic
4#11762547C211yesmaximal, characteristic, complemented
5#11762546C211yesmaximal, characteristic, complemented
6#11762532C211yesmaximal, characteristic
7C17×#451C211yesmaximal, characteristic, complemented
8C22×Q8×C17C2211yescharacteristic
9C2×Q16×C17—21no—
10C17×C22⋊Q8C2211yescharacteristic
11#10587632—21no—
12C2×#63231—21no—
13C2×#63227—21no—
14C2×#63227C2211yescharacteristic
15#10587610C2211yescharacteristic
16#10587583—21no—
17#10587582—21no—
18#10587582—21no—
19C22⋊C4×D17—21no—
20#10587564—21no—
21C17×#103—21no—
22C17×C22⋊C8C2211yescharacteristic
23#10587506—21no—
24#10587502—21no—
25#10587502C2211yescharacteristic
26C22⋊#3622C2211yescharacteristic
27C23×D17D411yescharacteristic
28C2×Q8×C17—21no—
29C2×Q8×C17—21no—
30C2×Q8×C17—42no—
31C2×Q8×C17D411yescharacteristic
32C22×C4×C17—21no—
33C22×C4×C17C2311yescharacteristic
34C22×Dic17—21no—
35C22×Dic17D411yescharacteristic
36C2×C4×D17—21no—
37C2×C4×D17—21no—
38C2×Dic34—21no—
39C2×Dic34D411yescharacteristic
40Q16×C17—42no—
41C2×C8×C17—21no—
42C4⋊C4×C17—21no—
43C4⋊C4×C17D411yescharacteristic
44C22⋊C4×C17—21no—
45C22⋊Dic17—21no—
46#63231—42no—
47#63227—21no—
48#63227—21no—
49#63227—21no—
50#63227—21no—
51#63227—42no—
52C4⋊Dic17—21no—
53C4⋊Dic17D411yescharacteristic
54#63225—21no—
55#63225—21no—
56#63225—21no—
57C2×#3622—21no—
58C23×C17C2×D411yescharacteristic
59C22×D17—21no—
60C22×D17—21no—
61C22×D17—41no—
62C22×D17—41no—
63C22×D17Q1612yescomplemented
64Q8×C17—22no—
65Q8×C17—22no—
66Q8×C17—41no—
67Q8×C17—41no—
68Q8×C17—41no—
69C2×C4×C17—22no—
70C2×C4×C17—21no—
71C2×C4×C17—21no—
72C2×C4×C17—41no—
73C2×C4×C17—41no—
74C2×C4×C17—41no—
75C2×C4×C17C2×D411yescharacteristic
76C2×C4×C17C2×D411yescharacteristic
77C2×Dic17—21no—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—41no—
81C2×Dic17—41no—
82C2×Dic17—41no—
83C2×Dic17—41no—
84C4×D17—81no—
85C4×D17—81no—
86Dic34—41no—
87Dic34—81no—
88C8×C17—41no—
89#3622—41no—
90#1602—171nomaximal, Sylow
91C22×C17—21no—
92C22×C17—21no—
93C22×C17#12011yescharacteristic
94C22×C17C2×Q1611yescharacteristic
95C22×C17#13711yescharacteristic
96C2×D17—22no—
97C2×D17—41no—
98C2×D17—41no—
99C2×D17—42no—
100C2×D17—81no—
101C4×C17—21no—
102C4×C17—21no—
103C4×C17—41no—
104C4×C17—41no—
105C4×C17—42no—
106Dic17—41no—
107Dic17—41no—
108Dic17—41no—
109Dic17—81no—
110#543—171no—
111#451—171no—
112#451D1711yescharacteristic, complemented
113#393—171no—
114#328—171no—
115#328—171no—
116#323—171no—
117C2×C17—21no—
118C2×C17—21no—
119C2×C17#45111yescharacteristic
120C2×C17#45511yescharacteristic
121C2×C17B(4,𝔽2)11yescharacteristic
122D17—42no—
123C22×Q8C2×D1711yescharacteristic
124C2×Q16—21no—
125C2×Q16—341no—
126C22⋊Q8—171no—
127C22⋊Q8—341no—
128C22⋊Q8—341no—
129C22⋊Q8—341no—
130C22⋊Q8—341no—
131C22⋊Q8—341no—
132C22⋊Q8C2×D1711yescharacteristic
133C2×C22⋊C4—171no—
134C2×C22⋊C4—341no—
135C2×C22⋊C4—341no—
136#103—21no—
137#103—341no—
138C22⋊C8—171no—
139C22⋊C8C2×D1711yescharacteristic
140#95—171no—
141#95—341no—
142C17#160211yescharacteristic, complemented, Sylow
143C24—171no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—171no—
148C2×Q8—341no—
149C2×Q8C22⋊D1711yescharacteristic
150C22×C4—21no—
151C22×C4—171no—
152C22×C4—341no—
153C22×C4—341no—
154C22×C4—341no—
155C22×C4C22×D1711yescharacteristic
156Q16—42no—
157Q16—682no—
158C2×C8—21no—
159C2×C8—341no—
160C4⋊C4—21no—
161C4⋊C4—171no—
162C4⋊C4—341no—
163C4⋊C4—341no—
164C4⋊C4—341no—
165C4⋊C4—341no—
166C4⋊C4C22⋊D1711yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—341no—
169C22⋊C4—341no—
170C22⋊C4—341no—
171C22⋊C4—341no—
172C22⋊C4—341no—
173C22⋊C4—682no—
174C23—172no—
175C23—341no—
176C23—341no—
177C23—681no—
178C23—681no—
179C23D4×D1711yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—681no—
186Q8—1361no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—341no—
194C2×C4—341no—
195C2×C4—341no—
196C2×C4—681no—
197C2×C4—681no—
198C2×C4—681no—
199C2×C4—681no—
200C2×C4—1361no—
201C2×C4—1361no—
202C2×C4D4×D1711yescharacteristic
203C2×C4C2×C22⋊D1711yescharacteristic
204C8—41no—
205C8—681no—
206C22—21no—
207C22—21no—
208C22—342no—
209C22—682no—
210C22—681no—
211C22—681no—
212C22—1361no—
213C22Q16×D1711yescharacteristic
214C22#1058760411yescharacteristic
215C22#1058762311yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—681no—
222C4—681no—
223C4—681no—
224C4—1361no—
225C2—21no—
226C2—21no—
227C2—682no—
228C2#1176296311yescharacteristic
229C2#1176298511yescharacteristic
230C2#1176299511yescharacteristic
subgroupG/H
centerC22#10587623
commutator, cosocleC22×C4×C17C23
FrattiniC22×C4C22×D17
FittingC17×#451C2
socleC22×C17#120
prime radicalC23×D17D4
hypercenter#451D17
hypocenterC17#1602
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #11762965C2semidirect
4. #11762547C2semidirect
5. #11762546C2semidirect
7. C17×#451C2semidirect
63. C22×D17Q16semidirect
112. #451D17semidirect
142. C17#1602semidirect

Maximal subgroups

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

subgroupindex
#117634342
#117629652
#117627202
#117625472
#117625462
#117625322
C17×#4512
#160217