Finite Group Databasebeta

← #14536739

Subgroups — #14536739

Order 2176.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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#11762751C212yesmaximal
2#11762749C212yesmaximal
3#11762728C212yesmaximal
4#11762707C211yesmaximal, characteristic
5#11762707C211yesmaximal, characteristic
6#11762706C212yesmaximal
7#11762692C212yesmaximal
8#11762690C212yesmaximal
9C17×#391C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C42×C17C2211yescharacteristic
13C2×C4⋊Dic17C2211yescharacteristic
14C2×C4⋊Dic17C2211yescharacteristic
15C2×C4⋊Dic17C2211yescharacteristic
16C2×C4⋊Dic17C2211yescharacteristic
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×C4×Dic17C2212yes—
22C17×#95C2212yes—
23C17×#95C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C2×Q8×C17—22no—
32C2×Q8×C17—41no—
33C2×Q8×C17—41no—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C4⋊C4×C17—41no—
44C42×C17—41no—
45C4⋊Dic17—22no—
46C4⋊Dic17—41no—
47C4⋊Dic17—41no—
48C4⋊Dic17Q814yescomplemented
49#63225—24no—
50#63225—24no—
51#63225—42no—
52#63225—42no—
53C4×Dic17—42no—
54C23×C17C2411yescharacteristic
55Q8×C17—44no—
56C2×C4×C17—21no—
57C2×C4×C17—21no—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17C2×Q812yes—
69C2×C4×C17#4112yes—
70C2×C4×C17#4112yes—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17C2×Q814yes—
83#2805—171nomaximal, Sylow
84C22×C17C22×Q811yescharacteristic
85C22×C17C2×#4111yescharacteristic
86C22×C17C2×#4111yescharacteristic
87C22×C17H(5,𝔽2)11yescharacteristic
88C22×C17H(5,𝔽2)11yescharacteristic
89C22×C172−1+411yescharacteristic
90C22×C172−1+411yescharacteristic
91C4×C17—22no—
92C4×C17—22no—
93C4×C17—22no—
94C4×C17—42no—
95C4×C17—42no—
96Dic17—24no—
97Dic17—42no—
98Dic17—42no—
99Dic17—42no—
100#400—172no—
101#400—172no—
102#395—172no—
103#391D1711yescharacteristic
104#389—171no—
105#389—171no—
106#389—172no—
107#387—172no—
108#387—172no—
109C2×C17#53811yescharacteristic
110C2×C17#53911yescharacteristic
111C2×C17#54411yescharacteristic
112C2×C17#55411yescharacteristic
113C2×C17#55611yescharacteristic
114C2×C17#55711yescharacteristic
115C2×C17#56311yescharacteristic
116C22×Q8C2×D1711yescharacteristic
117C2×C4⋊C4—171no—
118C2×C4⋊C4—172no—
119C2×C4⋊C4—172no—
120C2×C4⋊C4—171no—
121C2×C4⋊C4—172no—
122C2×C4⋊C4—171no—
123C2×C4⋊C4—172no—
124C2×C4⋊C4—171no—
125C2×C4⋊C4C2×D1711yescharacteristic
126C2×C42—172no—
127C2×C42C2×D1711yescharacteristic
128#95—172no—
129#95—172no—
130#95—172no—
131#95—172no—
132#95—172no—
133#95—172no—
134#95—172no—
135#95C2×D1712yes—
136#95C2×D1712yes—
137C17#280511yescharacteristic, complemented, Sylow
138C2×Q8—22no—
139C2×Q8—41no—
140C2×Q8—41no—
141C22×C4—172no—
142C22×C4—172no—
143C22×C4—172no—
144C22×C4—172no—
145C22×C4C22×D1712yes—
146C22×C4C22×D1712yes—
147C22×C4C22×D1711yescharacteristic
148C22×C4C22×D1711yescharacteristic
149C22×C4C22×D1711yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—174no—
152C4⋊C4—344no—
153C4⋊C4—344no—
154C4⋊C4—342no—
155C4⋊C4—682no—
156C4⋊C4—682no—
157C4⋊C4—681no—
158C4⋊C4—681no—
159C42—41no—
160C42—682no—
161C23C23×D1711yescharacteristic
162Q8—44no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—174no—
176C2×C4—342no—
177C2×C4—342no—
178C2×C4—342no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4#6325212yes—
188C2×C4Q8×D1712yes—
189C2×C4#6325612yes—
190C22C2×#6325211yescharacteristic
191C22#1058769111yescharacteristic
192C22#1058769111yescharacteristic
193C22C2×Q8×D1711yescharacteristic
194C22C2×#6325611yescharacteristic
195C22#1058769711yescharacteristic
196C22#1058769711yescharacteristic
197C4—22no—
198C4—22no—
199C4—22no—
200C4—42no—
201C4—42no—
202C4—344no—
203C4—682no—
204C4—682no—
205C4—682no—
206C2#1176339711yescharacteristic
207C2#1176340811yescharacteristic
208C2#1176343811yescharacteristic
209C2#1176352011yescharacteristic
210C2#1176353211yescharacteristic
211C2#1176353711yescharacteristic
212C2#1176355911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#391C2
hypercenter#391D17
hypocenterC17#2805
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
48. C4⋊Dic17Q8semidirect
137. C17#2805semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627492
#117627282
#117627072
#117627062
#117626922
#117626902
C17×#3912
#280517