Finite Group Databasebeta

← #28336800

Subgroups — #28336800

Order 13696.

338 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#13133808C211yesmaximal, characteristic
2C2×Q8×Dic107C211yesmaximal, characteristic
3C2×#16812034C211yesmaximal, characteristic
4#20292532C211yesmaximal, characteristic
5#20292389C211yesmaximal, characteristic
6#20292381C211yesmaximal, characteristic
7C2×#103×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic
9C2×#13133808—22no—
10C2×#13133808—22no—
11C2×#13133808C414yescomplemented
12C22×C107⋊C8C2211yescharacteristic
13Q8×Dic107—22no—
14Q8×Dic107—22no—
15C2×C4⋊Dic107—21no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×C4×Dic107—21no—
18C2×C4×Dic107C2211yescharacteristic
19#16812034—22no—
20C107⋊#95—21no—
21C107⋊#95—21no—
22C22×Q8×C107C2211yescharacteristic
23C22×C8×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25#103×C107—22no—
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214C2×C412yes—
32C22×Dic107—21no—
33C22×Dic107—21no—
34C22×Dic107D411yescharacteristic
35C22×Dic107D411yescharacteristic
36#13133808—28no—
37C2×C107⋊C8—21no—
38C2×C107⋊C8—21no—
39C2×C107⋊C8C2×C412yes—
40C4⋊Dic107—24no—
41C4⋊Dic107—21no—
42C4⋊Dic107D412yes—
43C4×Dic107—24no—
44C4×Dic107—21no—
45C4×Dic107—21no—
46C2×Q8×C107—21no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C2×Q8×C107C2×C412yes—
50C2×Q8×C107D412yes—
51C2×Q8×C107D412yes—
52C22×C4×C107—21no—
53C22×C4×C107—21no—
54C22×C4×C107C2311yescharacteristic
55C2×C8×C107—21no—
56C2×C8×C107—21no—
57C2×C8×C107—21no—
58C4⋊C4×C107—21no—
59C4⋊C4×C107—21no—
60C23×C107C2×D411yescharacteristic
61Q8×C107—22no—
62Q8×C107—22no—
63Q8×C107—22no—
64Q8×C107C22⋊C414yes—
65C2×C4×C107—22no—
66C2×C4×C107—22no—
67C2×C4×C107—22no—
68C2×C4×C107—41no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107C22×C411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107#4111yescharacteristic
76C2×C4×C107#4111yescharacteristic
77C2×Dic107—21no—
78C2×Dic107—21no—
79C2×Dic107—22no—
80C2×Dic107—21no—
81C2×Dic107—22no—
82C2×Dic107—21no—
83C2×Dic107—21no—
84C2×Dic107—41no—
85C2×Dic107—41no—
86C2×Dic107—41no—
87C2×Dic107—41no—
88C2×Dic107Q1612yescomplemented
89Dic214—22no—
90Dic214—22no—
91Dic214—41no—
92Dic214—41no—
93C8×C107—41no—
94C107⋊C8—22no—
95C22×C107C4×D411yescharacteristic
96C22×C107#12011yescharacteristic
97C22×C107#12111yescharacteristic
98C22×C107C2×Q1611yescharacteristic
99C22×C107#13511yescharacteristic
100C22×C107#13711yescharacteristic
101C22×C107#13711yescharacteristic
102C4×C107—24no—
103C4×C107—81no—
104C4×C107C2×C22⋊C411yescharacteristic
105C4×C107C4×D411yescharacteristic
106C4×C107#12111yescharacteristic
107C4×C107#12311yescharacteristic
108Dic107—22no—
109Dic107—42no—
110Dic107—41no—
111Dic107—42no—
112C2×C107#38611yescharacteristic
113C2×C107C4×Q1611yescharacteristic
114C2×C107#44111yescharacteristic
115C2×C107#45111yescharacteristic
116C2×C107#45211yescharacteristic
117C2×C107#46211yescharacteristic
118C2×C107#46411yescharacteristic
119#1878—1071nomaximal, Sylow
120C107#187811yescharacteristic, complemented, Sylow
121C22×Q16—1071no—
122C2×C4×Q8—1071no—
123C2×#103—1071no—
124C2×#103D10711yescharacteristic
125#391—1071no—
126#340—1071no—
127#336—1071no—
128C22×Q8—1071no—
129C22×Q8C2×D10711yescharacteristic
130C2×Q16—1074no—
131C2×Q16—2142no—
132C2×Q16—2142no—
133C22×C8—1071no—
134C22×C8C2×D10711yescharacteristic
135C4×Q8—2142no—
136C4×Q8—2142no—
137C2×C4⋊C4—1071no—
138C2×C4⋊C4—2141no—
139C2×C4⋊C4C2×D10711yescharacteristic
140C2×C42—1071no—
141C2×C42—2141no—
142#103—22no—
143#103—2142no—
144#95—2141no—
145#95—2141no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1072no—
150C2×Q8—2141no—
151C2×Q8—2141no—
152C2×Q8—2141no—
153C2×Q8—2141no—
154C2×Q8—2141no—
155C2×Q8C4×D10712yes—
156C2×Q8D42812yes—
157C2×Q8C22⋊D10712yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1071no—
161C22×C4—1071no—
162C22×C4—2141no—
163C22×C4—2141no—
164C22×C4C22×D10711yescharacteristic
165Q16—2148no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—1072no—
170C2×C8—2141no—
171C2×C8—2141no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1072no—
175C4⋊C4—2144no—
176C4⋊C4—2141no—
177C42—2144no—
178C42—2141no—
179C42—2141no—
180C23D4×D10711yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—22no—
184Q8—4281no—
185Q8—4281no—
186Q8—2142no—
187Q8—2142no—
188Q8(C2×D107)⋊C414yes—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—1072no—
196C2×C4—4281no—
197C2×C4—4281no—
198C2×C4—4281no—
199C2×C4—4281no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2141no—
203C2×C4—2141no—
204C2×C4—2141no—
205C2×C4—2141no—
206C2×C4—2141no—
207C2×C4C2×C4×D10711yescharacteristic
208C2×C4D4×D10711yescharacteristic
209C2×C4C2×C22⋊D10711yescharacteristic
210C2×C4C2×D42811yescharacteristic
211C2×C4#1313381711yescharacteristic
212C2×C4#1313381711yescharacteristic
213C8—41no—
214C8—2142no—
215C22(C22⋊D107)⋊C411yescharacteristic
216C22#1681207911yescharacteristic
217C22#1681208111yescharacteristic
218C22Q16×D10711yescharacteristic
219C22#1681213011yescharacteristic
220C22#1681213811yescharacteristic
221C22#1681214111yescharacteristic
222C4—24no—
223C4—81no—
224C4—4282no—
225C4—4282no—
226C4—4281no—
227C4—2142no—
228C4C2×(C2×D107)⋊C411yescharacteristic
229C4(C22⋊D107)⋊C411yescharacteristic
230C4#1681208111yescharacteristic
231C4#1681209311yescharacteristic
232C2#2029250411yescharacteristic
233C2#2029271911yescharacteristic
234C2#2029273111yescharacteristic
235C2#2029278411yescharacteristic
236C2#2029279211yescharacteristic
237C2#2029283511yescharacteristic
238C2#2029284711yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socle, prime radicalC23×C107C2×D4
cosocleC22×C4×C107C23
hypercenterC2×#103D107
hypocenterC107#1878
LCSG ▷ C2×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
11. C2×#13133808C4semidirect
88. C2×Dic107Q16semidirect
120. C107#1878semidirect

Maximal subgroups

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

subgroupindex
C22×#131338082
C2×Q8×Dic1072
C2×#168120342
#202925322
#202923892
#202923812
C2×#103×C1072
#1878107