Finite Group Databasebeta

← #27205674

Subgroups — #27205674

Order 13184.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#20268422C211yesmaximal, characteristic
2#20268419C211yesmaximal, characteristic
3#20268199C212yesmaximal, complemented
4#20268198C212yesmaximal, complemented
5#20268192C212yesmaximal, complemented
6#20268188C212yesmaximal, complemented
7#20267833C211yesmaximal, characteristic
8#20267827C211yesmaximal, characteristic
9C2×#15692952C211yesmaximal, characteristic
10#20267749C211yesmaximal, characteristic
11#428×C103C211yesmaximal, characteristic
12C22×C103⋊C8C2211yescharacteristic
13#15693021—21no—
14#15693020—21no—
15#15693020C2212yes—
16#15693020C2212yes—
17#15693013—21no—
18#15693012—21no—
19#15693012C2212yes—
20#15693012C2212yes—
21#15693000—22no—
22#15692998—22no—
23#15692980C2211yescharacteristic
24#15692980C2211yescharacteristic
25C2×C4⋊Dic103C2211yescharacteristic
26#15692955C2212yes—
27#15692954C2211yescharacteristic
28#15692954C2211yescharacteristic
29#15692953C2212yes—
30#15692952C2212yes—
31#15692952C2212yes—
32#15692952C2211yescharacteristic
33#15692952C2211yescharacteristic
34#15692931C2212yes—
35C8×Dic103C2212yes—
36C2×M16×C103C2211yescharacteristic
37#117×C103C2211yescharacteristic
38C2×C4⋊C4×C103C2211yescharacteristic
39#107×C103C2212yes—
40#106×C103C2212yes—
41C2×Dic206—22no—
42C2×Dic206—22no—
43C22×Dic103—21no—
44C2×C103⋊C8Q814yescomplemented
45C2×C103⋊C8C2311yescharacteristic
46C2×C103⋊C8C2311yescharacteristic
47C4⋊Dic103—21no—
48C4⋊Dic103—22no—
49C4⋊Dic103—22no—
50C4⋊Dic103D411yescharacteristic
51C4⋊Dic103D411yescharacteristic
52C4⋊Dic103C2312yes—
53C4⋊Dic103C2311yescharacteristic
54C4⋊Dic103C2311yescharacteristic
55Dic103⋊C4—22no—
56Dic103⋊C4—22no—
57Dic103⋊C4—22no—
58Dic103⋊C4—22no—
59C22⋊Dic103—21no—
60C22⋊Dic103D412yes—
61C4×Dic103—21no—
62C4×Dic103C2312yes—
63C22×C4×C103—21no—
64C22×C4×C103C2311yescharacteristic
65M16×C103Q814yescomplemented
66C2×C8×C103C2312yes—
67C4⋊C4×C103—21no—
68C4⋊C4×C103C2311yescharacteristic
69C4⋊C4×C103C2312yes—
70C4⋊C4×C103C2311yescharacteristic
71C22⋊C4×C103—21no—
72C42×C103—21no—
73C23×C103C2×D411yescharacteristic
74C2×C4×C103—21no—
75C2×C4×C103—21no—
76C2×C4×C103—22no—
77C2×C4×C103—81no—
78C2×C4×C103C2×D411yescharacteristic
79C2×C4×C103C2×Q812yes—
80C2×C4×C103C2×Q812yes—
81C2×C4×C103C2411yescharacteristic
82C2×Dic103—21no—
83C2×Dic103—21no—
84C2×Dic103—22no—
85C2×Dic103—81no—
86C2×Dic103C2×D412yes—
87C2×Dic103C2×D412yes—
88Dic206—82no—
89Dic206—82no—
90C8×C103C2×Q814yes—
91C103⋊C8C2×Q812yes—
92C103⋊C8C2×Q812yes—
93C22×C103—21no—
94C22×C103—21no—
95C22×C103C4⋊Q812yes—
96C22×C103C22×D411yescharacteristic
97C4×C103—41no—
98C4×C103—42no—
99C4×C103—41no—
100C4×C103C4⋊Q812yes—
101C4×C103C22×Q811yescharacteristic
102C4×C103C22×Q811yescharacteristic
103Dic103—22no—
104Dic103—22no—
105Dic103—41no—
106Dic103—42no—
107Dic103—41no—
108C2×C103—22no—
109C2×C103C2×C4⋊Q811yescharacteristic
110C2×C103#57711yescharacteristic
111C2×C103#57711yescharacteristic
112#3163—1031nomaximal, Sylow
113C103#316311yescharacteristic, complemented, Sylow
114#544—1031no—
115#544—1031no—
116#501—1032no—
117#501—1032no—
118#499—1032no—
119#498—1032no—
120#428—1031no—
121#428D10311yescharacteristic
122#427—1031no—
123C2×#106—1031no—
124#405—1031no—
125C2×M16C2×D10311yescharacteristic
126C22×C8—1031no—
127C4⋊Q8—1032no—
128C4⋊Q8—1032no—
129C4⋊Q8—2061no—
130C4⋊Q8—2061no—
131#125—1032no—
132#125—1032no—
133#125—2061no—
134#125—2061no—
135C22⋊Q8—2062no—
136C22⋊Q8—2062no—
137#117—1031no—
138#117—1031no—
139#117C2×D10311yescharacteristic
140C2×C4⋊C4—1031no—
141C2×C4⋊C4C2×D10311yescharacteristic
142#107—1032no—
143#107—1031no—
144#107—1031no—
145#107C2×D10312yes—
146#106—1032no—
147#106—1032no—
148#106—1032no—
149#106—1031no—
150#106—1031no—
151#106C2×D10312yes—
152#97—1032no—
153C4×C8—1032no—
154C2×Q8—2062no—
155C2×Q8—2062no—
156C22×C4—21no—
157C22×C4—2061no—
158C22×C4C22×D10311yescharacteristic
159M16Dic20614yescomplemented
160C2×C8—1034no—
161C2×C8—1031no—
162C2×C8—1031no—
163C2×C8C22×D10312yes—
164C4⋊C4—21no—
165C4⋊C4—1031no—
166C4⋊C4—1031no—
167C4⋊C4—1032no—
168C4⋊C4—1031no—
169C4⋊C4—1031no—
170C4⋊C4—2062no—
171C4⋊C4—2062no—
172C4⋊C4—2062no—
173C4⋊C4—2062no—
174C4⋊C4—2062no—
175C4⋊C4—2062no—
176C4⋊C4—2061no—
177C4⋊C4C22×D10312yes—
178C4⋊C4C22×D10311yescharacteristic
179C4⋊C4C22×D10311yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—1032no—
182C22⋊C4—2061no—
183C42—21no—
184C42—1032no—
185C42—2061no—
186C23D4×D10311yescharacteristic
187Q8—8242no—
188Q8—8242no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—8241no—
194C2×C4—1032no—
195C2×C4—1032no—
196C2×C4—2061no—
197C2×C4—2061no—
198C2×C4—2062no—
199C2×C4D4×D10311yescharacteristic
200C2×C4Q8×D10312yes—
201C2×C4C2×Dic20612yes—
202C2×C4C23×D10311yescharacteristic
203C8—1032no—
204C8—1032no—
205C8C2×Dic20614yes—
206C22—21no—
207C22—21no—
208C22#1569302012yes—
209C22C2×D4×D10311yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—4122no—
214C4—4121no—
215C4—4121no—
216C4—2062no—
217C4—2062no—
218C4#1569302012yes—
219C4C2×Q8×D10311yescharacteristic
220C4C22×Dic20611yescharacteristic
221C2—22no—
222C2C2×#1569302011yescharacteristic
223C2#2026860011yescharacteristic
224C2#2026860111yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#428×C103C2
socleC22×C103C22×D4
prime radicalC23×C103C2×D4
hypercenter#428D103
hypocenterC103#3163
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20268199C2semidirect
4. #20268198C2semidirect
5. #20268192C2semidirect
6. #20268188C2semidirect
44. C2×C103⋊C8Q8semidirect
65. M16×C103Q8semidirect
113. C103#3163semidirect
159. M16Dic206semidirect

Maximal subgroups

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

subgroupindex
#202684222
#202684192
#202681992
#202681982
#202681922
#202681882
#202678332
#202678272
C2×#156929522
#202677492
#428×C1032
#3163103