Finite Group Databasebeta

← #14533997

Subgroups — #14533997

Order 2176.

468 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#10587574C211yesmaximal, characteristic
2C2×#10587573C211yesmaximal, characteristic
3#11762750C214yesmaximal
4#11762715C211yesmaximal, characteristic
5#11762714C211yesmaximal, characteristic
6#11762713C214yesmaximal
7#11762674C211yesmaximal, characteristic
8C4×#63225C211yesmaximal, characteristic
9C17×#382C211yesmaximal, characteristic
10C22×Dic34C2212yes—
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15C2×C4⋊Dic17C2211yescharacteristic
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×C4×Dic17C2212yes—
20C2×C4×Dic17C2211yescharacteristic
21C2×C4×Dic17C2211yescharacteristic
22#10587574—24no—
23#10587573—24no—
24C17×#95C2214yes—
25#10587502C2214yes—
26#10587502C2214yes—
27#10587502C2214yes—
28#10587502C2212yes—
29C22×C4×C17C2312yes—
30C22×C4×C17C2312yes—
31C22×C4×C17C2312yes—
32C22×C4×C17C2311yescharacteristic
33C22×Dic17C2312yes—
34C22×Dic17C2312yes—
35C22×Dic17C2311yescharacteristic
36C22×Dic17C2312yes—
37C22×Dic17C2311yescharacteristic
38C2×Dic34—22no—
39C2×Dic34—22no—
40C2×Dic34—24no—
41C2×Dic34—28no—
42C2×Dic34Q814yescomplemented
43C4⋊C4×C17—22no—
44C4⋊C4×C17—22no—
45C42×C17—22no—
46C4⋊Dic17—22no—
47#63225—24no—
48#63225—24no—
49#63225—24no—
50C4×Dic17—22no—
51C4×Dic17—24no—
52C4×Dic17D414yes—
53C23×C17C2411yescharacteristic
54C2×C4×C17—22no—
55C2×C4×C17—22no—
56C2×C4×C17—22no—
57C2×C4×C17—24no—
58C2×C4×C17—22no—
59C2×C4×C17—24no—
60C2×C4×C17—22no—
61C2×C4×C17C2×D412yes—
62C2×C4×C17C2×Q812yes—
63C2×C4×C17#4112yes—
64C2×Dic17—22no—
65C2×Dic17—22no—
66C2×Dic17—22no—
67C2×Dic17—22no—
68C2×Dic17—24no—
69C2×Dic17—22no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17C2×D412yes—
73C2×Dic17C2×D412yes—
74C2×Dic17C2×Q814yes—
75C2×Dic17#4114yes—
76Dic34—28no—
77Dic34—48no—
78#2568—171nomaximal, Sylow
79C22×C17C22×D411yescharacteristic
80C22×C17C22×Q811yescharacteristic
81C22×C17C2×#4111yescharacteristic
82C22×C17C2×#4112yes—
83C22×C172−1+412yes—
84C4×C17—42no—
85C4×C17—42no—
86C4×C17—42no—
87C4×C17C22⋊Q814yes—
88Dic17—24no—
89Dic17—22no—
90Dic17—42no—
91Dic17—42no—
92Dic17#12414yes—
93C2×C4⋊Q8—171no—
94C2×C4×Q8—171no—
95#400—174no—
96#391—171no—
97#391—171no—
98#391—174no—
99#382—171no—
100#382D1711yescharacteristic
101C4×C4⋊C4—171no—
102C2×C17C2×C22⋊Q811yescharacteristic
103C2×C17C2×#12411yescharacteristic
104C2×C17#53611yescharacteristic
105C2×C17#55612yes—
106C2×C17#55712yes—
107C22×Q8—171no—
108C22×Q8—172no—
109C4⋊Q8—344no—
110C4×Q8—344no—
111C2×C4⋊C4—172no—
112C2×C4⋊C4—171no—
113C2×C4⋊C4—172no—
114C2×C4⋊C4—172no—
115C2×C4⋊C4C2×D1711yescharacteristic
116C2×C4⋊C4C2×D1711yescharacteristic
117C2×C42—171no—
118C2×C42—172no—
119C2×C42—171no—
120C2×C42C2×D1711yescharacteristic
121#95—172no—
122#95—174no—
123#95—174no—
124#95—174no—
125#95C2×D1714yes—
126C17#256811yescharacteristic, complemented, Sylow
127C2×Q8—174no—
128C2×Q8—344no—
129C2×Q8—348no—
130C2×Q8—342no—
131C2×Q8—342no—
132C22×C4—171no—
133C22×C4—172no—
134C22×C4—171no—
135C22×C4—172no—
136C22×C4—172no—
137C22×C4C22×D1712yes—
138C22×C4C22×D1712yes—
139C22×C4C22×D1712yes—
140C22×C4C22×D1711yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—344no—
144C4⋊C4—344no—
145C4⋊C4—344no—
146C4⋊C4—342no—
147C42—22no—
148C42—174no—
149C42—344no—
150C42—342no—
151C23C23×D1711yescharacteristic
152Q8—348no—
153Q8—688no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—172no—
162C2×C4—174no—
163C2×C4—172no—
164C2×C4—174no—
165C2×C4—342no—
166C2×C4—342no—
167C2×C4—342no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—344no—
172C2×C4—342no—
173C2×C4#6325212yes—
174C2×C4D4×D1712yes—
175C2×C4Q8×D1712yes—
176C22C2×#6325211yescharacteristic
177C22C2×D4×D1711yescharacteristic
178C22C2×Q8×D1711yescharacteristic
179C22#41×D1712yes—
180C22#1058769712yes—
181C4—42no—
182C4—42no—
183C4—42no—
184C4—174no—
185C4—344no—
186C4—342no—
187C4—682no—
188C4—682no—
189C4#1058758214yes—
190C2C2×#1058758211yescharacteristic
191C2#124×D1711yescharacteristic
192C2#1176337911yescharacteristic
193C2#1176352712yes—
194C2#1176353612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#382C2
hypercenter#382D17
hypocenterC17#2568
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
42. C2×Dic34Q8semidirect
126. C17#2568semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875732
#117627502
#117627152
#117627142
#117627132
#117626742
C4×#632252
C17×#3822
#256817