Finite Group Databasebeta

← #14540254

Subgroups — #14540254

Order 2176.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#11763426C212yesmaximal, complemented
2#11763097C212yesmaximal, complemented
3#11763093C212yesmaximal, complemented
4#11763067C211yesmaximal, characteristic, complemented
5#11763036C212yesmaximal, complemented
6#11763030C212yesmaximal, complemented
7#11763022C212yesmaximal, complemented
8#11762837C211yesmaximal, characteristic, complemented
9C17×#469C211yesmaximal, characteristic
10C2×D8×C17C2211yescharacteristic
11C2×M16×C17C2211yescharacteristic
12C17×#121C2212yes—
13#10587622—22no—
14#10587622C2212yes—
15#10587621—22no—
16D4×Dic17C2212yes—
17C2×#63229C2212yes—
18#10587612C2211yescharacteristic
19#10587610C2212yes—
20C22×#3622C2211yescharacteristic
21C2×Dic68C2211yescharacteristic
22#10587574C2212yes—
23#10587573C2212yes—
24#10587565—22no—
25#10587564—22no—
26C17×#106C2211yescharacteristic
27C17×#102C2212yes—
28#10587503C2212yes—
29#10587488C2211yescharacteristic
30#10587487C2212yes—
31#10587485C2212yes—
32#10587485C2212yes—
33#10587481C2212yes—
34#10587479C2212yes—
35C2×D4×C17—22no—
36C2×D4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17—22no—
39C2×Dic34—22no—
40C2×Dic34C2312yes—
41D8×C17—41no—
42M16×C17—41no—
43C2×C8×C17C2311yescharacteristic
44C2×C8×C17C2311yescharacteristic
45C4⋊C4×C17C2312yes—
46C22⋊C4×C17—22no—
47C22⋊Dic17—22no—
48C22⋊Dic17—22no—
49C22⋊Dic17—22no—
50C22⋊Dic17—22no—
51C22⋊Dic17D411yescharacteristic
52C22⋊Dic17D411yescharacteristic
53#63229—42no—
54C4⋊Dic17D412yes—
55C4⋊Dic17C2312yes—
56#63225—22no—
57#63225—22no—
58#63225—22no—
59C4×Dic17—22no—
60C4×Dic17C2312yes—
61C2×#3622—41no—
62C2×#3622C2312yes—
63Dic68—41no—
64C23×C17—22no—
65C23×C17C2×D411yescharacteristic
66D4×C17—22no—
67D4×C17—22no—
68D4×C17—82no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—22no—
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2411yescharacteristic
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—82no—
77C2×Dic17C2×D412yes—
78C2×Dic17C2×D412yes—
79Dic34—22no—
80Dic34—22no—
81Dic34—82no—
82C8×C17—21no—
83C8×C17—21no—
84#3622—22no—
85#3263—171nomaximal, Sylow
86C22×C17—21no—
87C22×C17—22no—
88C22×C17—42no—
89C22×C17—42no—
90C22×C17—42no—
91C22×C17C22×D411yescharacteristic
92C4×C17—21no—
93C4×C17—42no—
94C4×C17H(5,𝔽2)11yescharacteristic
95C4×C172−1+411yescharacteristic
96Dic17—22no—
97Dic17—22no—
98Dic17—42no—
99Dic17—42no—
100#542—172no—
101#476—172no—
102#475—172no—
103#470—171no—
104#469D1711yescharacteristic
105#465—172no—
106#464—172no—
107#463—172no—
108#424—171no—
109C2×C17—41no—
110C2×C17—82no—
111C2×C17#53711yescharacteristic
112C2×C17#57712yes—
113C2×Q16—171no—
114C2×SD16—172no—
115C2×D8C2×D1711yescharacteristic
116C2×M16C2×D1711yescharacteristic
117C22×C8—171no—
118C4⋊Q8—172no—
119#126—342no—
120#124—172no—
121#124—342no—
122#123—342no—
123C22⋊Q8—172no—
124C22⋊Q8—342no—
125#121C2×D1712yes—
126C4×Q8—172no—
127C4×D4—172no—
128#117—171no—
129#106—171no—
130#106—172no—
131#106C2×D1711yescharacteristic
132C4⋊C8—172no—
133C4⋊C8—172no—
134#103—172no—
135#103—172no—
136#102—172no—
137#102C2×D1712yes—
138C17#326311yescharacteristic, complemented, Sylow
139C2×Q8—172no—
140C2×Q8—342no—
141C2×D4—22no—
142C2×D4C22×D1712yes—
143C22×C4—342no—
144C22×C4C22×D1711yescharacteristic
145Q16—681no—
146SD16—682no—
147D8—41no—
148M16—41no—
149C2×C8—172no—
150C2×C8—681no—
151C2×C8C22×D1711yescharacteristic
152C2×C8C22×D1711yescharacteristic
153C4⋊C4—172no—
154C4⋊C4—172no—
155C4⋊C4—342no—
156C4⋊C4—342no—
157C4⋊C4—342no—
158C4⋊C4C22×D1712yes—
159C22⋊C4—22no—
160C22⋊C4—171no—
161C22⋊C4—171no—
162C22⋊C4—342no—
163C22⋊C4—342no—
164C22⋊C4—342no—
165C22⋊C4—342no—
166C42—172no—
167C42—342no—
168C23—22no—
169C23D4×D1711yescharacteristic
170Q8—342no—
171Q8—342no—
172Q8—1362no—
173D4—22no—
174D4—22no—
175D4—82no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—172no—
180C2×C4—172no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—1362no—
184C2×C4D4×D1711yescharacteristic
185C2×C4C23×D1711yescharacteristic
186C8—21no—
187C8—21no—
188C8—342no—
189C22—22no—
190C22—21no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22C2×D4×D1711yescharacteristic
195C4—21no—
196C4—42no—
197C4—342no—
198C4—342no—
199C4—682no—
200C4—682no—
201C4#1058769111yescharacteristic
202C4#1058769711yescharacteristic
203C2—41no—
204C2—82no—
205C2#1176338611yescharacteristic
206C2#1176361912yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#469C2
socleC22×C17C22×D4
hypercenter#469D17
hypocenterC17#3263
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #11763426C2semidirect
2. #11763097C2semidirect
3. #11763093C2semidirect
4. #11763067C2semidirect
5. #11763036C2semidirect
6. #11763030C2semidirect
7. #11763022C2semidirect
8. #11762837C2semidirect
138. C17#3263semidirect

Maximal subgroups

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

subgroupindex
#117634262
#117630972
#117630932
#117630672
#117630362
#117630302
#117630222
#117628372
C17×#4692
#326317