Finite Group Databasebeta

← #13127717

Subgroups — #13127717

Order 1664.

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

normal subgroups only

This group has 336 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#11719427C211yesmaximal, characteristic
2Q8×Dic26C211yesmaximal, characteristic
3C13×#477C211yesmaximal, characteristic
4#11718739C211yesmaximal, characteristic
5C4×#6107C211yesmaximal, characteristic
6#11718536C212yesmaximal
7#11718535C212yesmaximal
8#11718511C211yesmaximal, characteristic
9#11718491C211yesmaximal, characteristic
10#11718455C212yesmaximal
11#11718451C212yesmaximal
12C13×C4⋊Q8C2211yescharacteristic
13C4×Q8×C13C2211yescharacteristic
14Q8×Dic13C2212yes—
15#88198—22no—
16C2×#6107C2211yescharacteristic
17#88142—21no—
18#88142—22no—
19#88142C2212yes—
20#88142C2212yes—
21#88141—21no—
22#88141C2212yes—
23#88123—21no—
24#88123C2211yescharacteristic
25C4×Dic26—21no—
26C4×Dic26C2211yescharacteristic
27C4×Dic26C2211yescharacteristic
28C13×#107C2212yes—
29C13×C4⋊C8C2211yescharacteristic
30C13×#103C2212yes—
31#88075C2211yescharacteristic
32#88058C2212yes—
33#88056C2212yes—
34#88054C2212yes—
35C8×Dic13C2212yes—
36#88050C2211yescharacteristic
37#88047C2211yescharacteristic
38#88047C2212yes—
39#88047C2212yes—
40C4×#1055C2211yescharacteristic
41C2×Q8×C13—21no—
42C2×Q8×C13C2311yescharacteristic
43C2×Dic26—21no—
44C2×Dic26—22no—
45C2×Dic26—22no—
46C2×Dic26D411yescharacteristic
47C2×Dic26C2311yescharacteristic
48C2×C8×C13C2312yes—
49C13×C4⋊C4—21no—
50C13×C4⋊C4—21no—
51C13×C4⋊C4C2312yes—
52C13×C4⋊C4C2311yescharacteristic
53C42×C13—21no—
54C42×C13C2311yescharacteristic
55#6107Q814yescomplemented
56C4⋊Dic13—21no—
57C4⋊Dic13—22no—
58C4⋊Dic13D411yescharacteristic
59C4⋊Dic13C2311yescharacteristic
60C4⋊Dic13C2312yes—
61#6101—21no—
62#6101—22no—
63#6101—22no—
64#6101—22no—
65#6101D412yes—
66C4×Dic13—21no—
67C4×Dic13—22no—
68C4×Dic13—22no—
69C4×Dic13C2312yes—
70C2×#1055C2311yescharacteristic
71C2×#1055C2311yescharacteristic
72Q8×Q16—131nomaximal, Sylow
73Q8×C13—21no—
74Q8×C13—81no—
75Q8×C13C2×Q812yes—
76C2×C4×C13—21no—
77C2×C4×C13—22no—
78C2×C4×C13—21no—
79C2×C4×C13C2×D411yescharacteristic
80C2×C4×C13C2×D411yescharacteristic
81C2×C4×C13C2411yescharacteristic
82C2×Dic13—21no—
83C2×Dic13—21no—
84C2×Dic13—22no—
85C2×Dic13C2×D412yes—
86C2×Dic13C2×D412yes—
87Dic26—21no—
88Dic26—82no—
89Dic26—82no—
90Dic26—81no—
91Dic26Q1614yescomplemented
92Dic26C2×Q812yes—
93C8×C13—22no—
94#1055—21no—
95#1055C2×Q812yes—
96Q82—131no—
97Q82—131no—
98#500—131no—
99#500—132no—
100#477—132no—
101#477—131no—
102#477—132no—
103#477D1311yescharacteristic
104C8×Q8—131no—
105C4×Q16—131no—
106C4×Q16—132no—
107C22×C13C22×D411yescharacteristic
108C4×C13—22no—
109C4×C13—21no—
110C4×C13—41no—
111C4×C13—42no—
112C4×C13C2×Q1612yes—
113C4×C13C22×Q811yescharacteristic
114C4×C132−1+411yescharacteristic
115Dic13—22no—
116Dic13—22no—
117Dic13—41no—
118Dic13—42no—
119Dic13C2×Q1614yes—
120C2×Q16—131no—
121C4⋊Q8—132no—
122C4⋊Q8—132no—
123C4⋊Q8—131no—
124C4⋊Q8—261no—
125C4⋊Q8—262no—
126C4⋊Q8—261no—
127C4⋊Q8—262no—
128C4⋊Q8C2×D1311yescharacteristic
129C4×Q8—131no—
130C4×Q8—132no—
131C4×Q8—132no—
132C4×Q8—131no—
133C4×Q8—261no—
134C4×Q8—261no—
135C4×Q8C2×D1311yescharacteristic
136#107—132no—
137#107—131no—
138#107—132no—
139#107—132no—
140#107C2×D1312yes—
141C4⋊C8—132no—
142C4⋊C8C2×D1311yescharacteristic
143#103—131no—
144#103—132no—
145#103—131no—
146#103C2×D1312yes—
147C4×C8—132no—
148C4×C8—131no—
149C2×C13D4×Q811yescharacteristic
150C2×C13C22×Q1611yescharacteristic
151C2×C13#57611yescharacteristic
152C2×Q8—21no—
153C2×Q8—131no—
154C2×Q8—131no—
155C2×Q8—262no—
156C2×Q8—262no—
157C2×Q8—261no—
158C2×Q8C22×D1311yescharacteristic
159Q16—134no—
160C2×C8—131no—
161C2×C8—131no—
162C2×C8C22×D1312yes—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—132no—
166C4⋊C4—131no—
167C4⋊C4—131no—
168C4⋊C4—132no—
169C4⋊C4—262no—
170C4⋊C4—262no—
171C4⋊C4—262no—
172C4⋊C4—262no—
173C4⋊C4—261no—
174C4⋊C4—261no—
175C4⋊C4C22×D1311yescharacteristic
176C4⋊C4C22×D1312yes—
177C42—21no—
178C42—132no—
179C42—262no—
180C42—262no—
181C42—261no—
182C42C22×D1311yescharacteristic
183C13Q8×Q1611yescharacteristic, complemented, Sylow
184Q8—21no—
185Q8—81no—
186Q8—134no—
187Q8—132no—
188Q8—261no—
189Q8—1041no—
190Q8—1042no—
191Q8—1042no—
192Q8Q8×D1312yes—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—132no—
197C2×C4—132no—
198C2×C4—261no—
199C2×C4—262no—
200C2×C4—261no—
201C2×C4D4×D1311yescharacteristic
202C2×C4D4×D1311yescharacteristic
203C2×C4C23×D1311yescharacteristic
204C8—22no—
205C8—132no—
206C8—261no—
207C22C2×D4×D1311yescharacteristic
208C4—21no—
209C4—22no—
210C4—42no—
211C4—41no—
212C4—134no—
213C4—262no—
214C4—262no—
215C4—522no—
216C4—521no—
217C4Q16×D1312yes—
218C4C2×Q8×D1311yescharacteristic
219C4#8825711yescharacteristic
220C2#1171933711yescharacteristic
221C2C2×Q16×D1311yescharacteristic
222C2#1171947911yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#477C2
socle, prime radicalC22×C13C22×D4
hypercenter#477D13
hypocenterC13Q8×Q16
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
55. #6107Q8semidirect
91. Dic26Q16semidirect
183. C13Q8×Q16semidirect

Maximal subgroups

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

subgroupindex
#117194272
Q8×Dic262
C13×#4772
#117187392
C4×#61072
#117185362
#117185352
#117185112
#117184912
#117184552
#117184512
Q8×Q1613