Finite Group Databasebeta

← #28337600

Subgroups — #28337600

Order 13696.

290 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16812108C211yesmaximal, characteristic
2C2×#16812033C211yesmaximal, characteristic
3#20292534C211yesmaximal, characteristic
4#20292390C211yesmaximal, characteristic
5#20292389C211yesmaximal, characteristic
6#20292381C211yesmaximal, characteristic
7C2×#103×C107C211yesmaximal, characteristic
8C22×Dic214—21no—
9C22×C107⋊C8C2211yescharacteristic
10#16812108—22no—
11#16812108—22no—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×Dic107⋊C4—21no—
15C2×C4×Dic107C2211yescharacteristic
16#16812033—22no—
17C107⋊#95—21no—
18C107⋊#95—21no—
19C22×Q8×C107C2211yescharacteristic
20C22×C8×C107C2211yescharacteristic
21C2×C4⋊C4×C107C2211yescharacteristic
22#103×C107—22no—
23C2×Dic214—24no—
24C2×Dic214—42no—
25C2×Dic214—42no—
26C22×Dic107—21no—
27C22×Dic107—21no—
28C22×Dic107D411yescharacteristic
29C22×Dic107D411yescharacteristic
30C2×C107⋊C8—21no—
31C2×C107⋊C8—21no—
32C2×C107⋊C8—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107D412yes—
37Dic107⋊C4—24no—
38C4×Dic107—21no—
39C4×Dic107—21no—
40C2×Q8×C107—21no—
41C2×Q8×C107—21no—
42C2×Q8×C107—21no—
43C2×Q8×C107—21no—
44C2×Q8×C107—21no—
45C2×Q8×C107Q812yescomplemented
46C22×C4×C107—21no—
47C22×C4×C107—21no—
48C22×C4×C107C2311yescharacteristic
49C2×C8×C107—21no—
50C2×C8×C107—21no—
51C2×C8×C107—21no—
52C4⋊C4×C107—21no—
53C4⋊C4×C107D412yes—
54C23×C107C2×D411yescharacteristic
55Q8×C107—22no—
56Q8×C107—22no—
57Q8×C107—41no—
58Q8×C107—41no—
59C2×C4×C107—22no—
60C2×C4×C107—22no—
61C2×C4×C107—41no—
62C2×C4×C107—41no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×Q811yescharacteristic
68C2×C4×C107#4111yescharacteristic
69C2×C4×C107#4111yescharacteristic
70C2×C4×C107#4111yescharacteristic
71C2×Dic107—21no—
72C2×Dic107—21no—
73C2×Dic107—21no—
74C2×Dic107—21no—
75C2×Dic107—22no—
76C2×Dic107—41no—
77C2×Dic107—41no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81C2×Dic107SD1612yes—
82C2×Dic107Q1612yes—
83Dic214—48no—
84C8×C107—41no—
85C107⋊C8—41no—
86C22×C107#12011yescharacteristic
87C22×C107C22⋊Q811yescharacteristic
88C22×C107#12411yescharacteristic
89C22×C107C2×SD1611yescharacteristic
90C22×C107C2×Q1611yescharacteristic
91C22×C107AGL(1,ℤ/8)11yescharacteristic
92C22×C107#13711yescharacteristic
93C4×C107—42no—
94C4×C107—42no—
95C4×C107C22⋊Q811yescharacteristic
96C4×C107C22⋊Q811yescharacteristic
97C4×C107#12411yescharacteristic
98C4×C107#12411yescharacteristic
99Dic107—22no—
100Dic107—22no—
101Dic107—42no—
102Dic107—81no—
103C2×C107#39311yescharacteristic
104C2×C107#45011yescharacteristic
105C2×C107#45111yescharacteristic
106C2×C107#47511yescharacteristic
107C2×C107#47711yescharacteristic
108C2×C107#48711yescharacteristic
109C2×C107#48911yescharacteristic
110#2024—1071nomaximal, Sylow
111C107#202411yescharacteristic, complemented, Sylow
112C2×C4⋊Q8—1071no—
113C2×#103—1071no—
114C2×#103D10711yescharacteristic
115#391—1071no—
116#340—1071no—
117#340—1071no—
118#336—1071no—
119C22×Q8—2141no—
120C22×Q8C2×D10711yescharacteristic
121C22×C8—1071no—
122C22×C8C2×D10711yescharacteristic
123C4⋊Q8—2142no—
124C4⋊Q8—2142no—
125C2×C4⋊C4—1071no—
126C2×C4⋊C4—1071no—
127C2×C4⋊C4—2141no—
128C2×C4⋊C4C2×D10711yescharacteristic
129C2×C42—1071no—
130#103—22no—
131#103—2142no—
132#95—2141no—
133#95—2141no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—4282no—
140C2×Q8—4282no—
141C2×Q8—2144no—
142C2×Q8Dic21412yescomplemented
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—1071no—
146C22×C4—1071no—
147C22×C4—2141no—
148C22×C4—2141no—
149C22×C4C22×D10711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—2141no—
154C2×C8—2141no—
155C2×C8—2141no—
156C4⋊C4—21no—
157C4⋊C4—1072no—
158C4⋊C4—2144no—
159C4⋊C4—2141no—
160C4⋊C4—2141no—
161C4⋊C4—2141no—
162C4⋊C4C22⋊D10712yes—
163C42—2141no—
164C42—2141no—
165C23D4×D10711yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—4288no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—1072no—
178C2×C4—1072no—
179C2×C4—4281no—
180C2×C4—4281no—
181C2×C4—4281no—
182C2×C4—4281no—
183C2×C4—4281no—
184C2×C4—2142no—
185C2×C4—2141no—
186C2×C4—2141no—
187C2×C4—2141no—
188C2×C4—2141no—
189C2×C4D4×D10711yescharacteristic
190C2×C4C2×C22⋊D10711yescharacteristic
191C2×C4C2×Dic21411yescharacteristic
192C2×C4#1313381711yescharacteristic
193C2×C4#1313381711yescharacteristic
194C2×C4#1313381811yescharacteristic
195C8—41no—
196C8—4281no—
197C22#1681207711yescharacteristic
198C22#1681208611yescharacteristic
199C22#1681209511yescharacteristic
200C22SD16×D10711yescharacteristic
201C22Q16×D10711yescharacteristic
202C22#1681213611yescharacteristic
203C22#1681214111yescharacteristic
204C4—42no—
205C4—42no—
206C4—8561no—
207C4—4282no—
208C4—2142no—
209C4—2142no—
210C4#1681208611yescharacteristic
211C4#1681208811yescharacteristic
212C4#1681209511yescharacteristic
213C4#1681209611yescharacteristic
214C2#2029253911yescharacteristic
215C2#2029277611yescharacteristic
216C2#2029278111yescharacteristic
217C2#2029291011yescharacteristic
218C2#2029292011yescharacteristic
219C2#2029297611yescharacteristic
220C2#2029298811yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socle, prime radicalC23×C107C2×D4
hypercenterC2×#103D107
hypocenterC107#2024
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
45. C2×Q8×C107Q8semidirect
111. C107#2024semidirect
142. C2×Q8Dic214semidirect

Maximal subgroups

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

subgroupindex
C2×#168121082
C2×#168120332
#202925342
#202923902
#202923892
#202923812
C2×#103×C1072
#2024107