Finite Group Databasebeta

← #16954685

Subgroups — #16954685

Order 4352.

312 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 59 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#14539163C211yesmaximal, characteristic
2D17×#356C212yesmaximal
3#14526957C211yesmaximal, characteristic, complemented
4#14526956C212yesmaximal, complemented
5C17×#1413C211yesmaximal, characteristic, complemented
6#11763365C411yescharacteristic
7C4⋊Q8×D17C2211yescharacteristic
8#11763179C411yescharacteristic
9#11763178C411yescharacteristic
10#11763177C2211yescharacteristic
11#11763158—22no—
12#11762816—22no—
13#101×D17—22no—
14#11762761C411yescharacteristic
15#11762608C2212yes—
16#11762607—22no—
17C17×#491C2211yescharacteristic
18C17×#356C2212yes—
19C2×Q8×D17D412yes—
20C17×C4⋊Q8C2311yescharacteristic
21C17×#125C2×C411yescharacteristic
22#10587630D412yes—
23M16×D17—42no—
24C2×#63218—21no—
25#10587582—41no—
26#10587578—21no—
27C4⋊C4×D17—41no—
28#10587574—21no—
29#10587574—41no—
30#10587573—41no—
31C42×D17C2×C411yescharacteristic
32#10587556C2×C411yescharacteristic
33#10587555C2×C411yescharacteristic
34C17×#103—22no—
35C17×#101—22no—
36C17×#97C2×C411yescharacteristic
37#10587507—22no—
38#10587506—22no—
39#10587495—42no—
40#10587487—22no—
41C8×Dic17—21no—
42#10587481—22no—
43#10587474C2×C411yescharacteristic
44C2×Q8×C17C2×D412yes—
45Q8×D17—42no—
46Q8×D17—42no—
47C2×C4×D17—22no—
48C2×C4×D17—21no—
49C2×C4×D17C22⋊C411yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—41no—
52C2×Dic34C22⋊C412yes—
53M16×C17—42no—
54C2×C8×C17—21no—
55C4⋊C4×C17—21no—
56C4⋊C4×C17—41no—
57C4⋊C4×C17—41no—
58C42×C17C22×C411yescharacteristic
59#63227—41no—
60C4⋊Dic17—21no—
61C4⋊Dic17—41no—
62#63225—42no—
63#63225—41no—
64#63225—41no—
65C4×Dic17—21no—
66C4×Dic17—41no—
67C4×Dic17C22⋊C411yescharacteristic
68#63223—42no—
69C2×#3622—21no—
70#63218—42no—
71#63218—82no—
72C8×D17—82no—
73#13599—171nomaximal, Sylow
74C22×D17#9911yescharacteristic
75Q8×C17—42no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—41no—
79C2×C4×C17C2×C22⋊C411yescharacteristic
80C2×Dic17—21no—
81C2×Dic17—22no—
82C2×Dic17—41no—
83C2×Dic17#9911yescharacteristic
84C4×D17—22no—
85C4×D17—42no—
86C4×D17—42no—
87C4×D17—42no—
88Dic34—42no—
89Dic34—82no—
90Dic34—82no—
91C8×C17—41no—
92C8×C17—82no—
93#3622—41no—
94#3622—82no—
95#3139—171no—
96C2×#356—172no—
97#1413—171no—
98#1413—172no—
99#1413D1711yescharacteristic, complemented
100C22×C17C2×#9911yescharacteristic
101C2×D17—41no—
102C2×D17#35412yes—
103C4×C17—21no—
104C4×C17—41no—
105C4×C17—42no—
106C4×C17—81no—
107Dic17—22no—
108Dic17—21no—
109Dic17—42no—
110Dic17—81no—
111#533—171no—
112C2×C4⋊Q8—171no—
113#491—171no—
114#491—171no—
115#491—171no—
116#491C2×D1711yescharacteristic
117#487—342no—
118#419—342no—
119C2×#101—342no—
120C4×M16—171no—
121#356—172no—
122#356—342no—
123#356C2×D1712yes—
124C2×C17—21no—
125C2×C17C2×#35411yescharacteristic
126D17—22no—
127C22×Q8—172no—
128C2×M16—341no—
129C2×M16—682no—
130C4⋊Q8—172no—
131C4⋊Q8—171no—
132C4⋊Q8—341no—
133C4⋊Q8—681no—
134C4⋊Q8C22×D1711yescharacteristic
135#125—171no—
136#125C4×D1711yescharacteristic
137C22⋊Q8—681no—
138C4×Q8—681no—
139#117—341no—
140C2×C4⋊C4—681no—
141C2×C42—171no—
142#103—22no—
143#103—342no—
144#103—342no—
145#103—342no—
146#101—22no—
147#101—342no—
148#101—682no—
149#97—171no—
150#97C4×D1711yescharacteristic
151C4×C8—341no—
152C17#1359911yescharacteristic, complemented, Sylow
153C2×Q8—172no—
154C2×Q8—342no—
155C2×Q8—681no—
156C2×Q8—682no—
157C2×Q8—682no—
158C2×Q8D4×D1712yes—
159C22×C4—171no—
160C22×C4—341no—
161C22×C4—342no—
162M16—42no—
163M16—682no—
164M16—682no—
165M16—1362no—
166C2×C8—21no—
167C2×C8—341no—
168C2×C8—1362no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—341no—
173C4⋊C4—682no—
174C4⋊C4—681no—
175C4⋊C4—681no—
176C4⋊C4—681no—
177C22⋊C4—681no—
178C42—171no—
179C42—341no—
180C42—681no—
181C42C2×C4×D1711yescharacteristic
182C23—171no—
183Q8—42no—
184Q8—682no—
185Q8—1362no—
186Q8—1362no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—171no—
191C2×C4—342no—
192C2×C4—341no—
193C2×C4—342no—
194C2×C4—682no—
195C2×C4—682no—
196C2×C4—682no—
197C2×C4—681no—
198C2×C4C22⋊C4×D1711yescharacteristic
199C8—41no—
200C8—82no—
201C8—681no—
202C8—1362no—
203C22—172no—
204C22—681no—
205C22#99×D1711yescharacteristic
206C4—21no—
207C4—41no—
208C4—42no—
209C4—81no—
210C4—342no—
211C4—341no—
212C4—682no—
213C4—1361no—
214C2—21no—
215C2—342no—
216C2D17×#35411yescharacteristic
subgroupG/H
centerC2D17×#354
commutatorC42×C17C22×C4
FrattiniC4⋊Q8C22×D17
FittingC17×#1413C2
socleC2×C17C2×#354
prime radicalC22×D17#99
cosocleC17×C4⋊Q8C23
hypercenter#1413D17
hypocenterC17#13599
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
3. #14526957C2semidirect
4. #14526956C2semidirect
5. C17×#1413C2semidirect
99. #1413D17semidirect
152. C17#13599semidirect

Maximal subgroups

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

subgroupindex
#145391632
D17×#3562
#145269572
#145269562
C17×#14132
#1359917