Finite Group Databasebeta

← #28337340

Subgroups — #28337340

Order 13696.

340 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13133807C211yesmaximal, characteristic
2C2×#16812032C211yesmaximal, characteristic, complemented
3C2×#16812029C211yesmaximal, characteristic
4C2×#16812019C211yesmaximal, characteristic, complemented
5#20292532C211yesmaximal, characteristic, complemented
6#20292528C211yesmaximal, characteristic
7C2×#107×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×D428C2211yescharacteristic
10C2×#13133807—22no—
11C2×#13133807—22no—
12C2×#13133807C414yescomplemented
13C22×C107⋊C8C2211yescharacteristic
14C2×(C2×D107)⋊C4—22no—
15C2×C4×Dic107C2211yescharacteristic
16#16812032—22no—
17#16812029—22no—
18#16812019—22no—
19C107⋊#95—22no—
20C22×C8×C107C2211yescharacteristic
21C2×C4⋊C4×C107C2211yescharacteristic
22C2×C4⋊C4×C107C2211yescharacteristic
23#107×C107—22no—
24C23×D107—21no—
25C2×Dic214—21no—
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214C2×C412yes—
31C2×D428—21no—
32C2×D428—21no—
33C2×D428—21no—
34C2×D428—21no—
35C2×D428—21no—
36C2×D428C2×C412yes—
37C22×Dic107—21no—
38C22×Dic107D412yes—
39#13133807—28no—
40C2×C107⋊C8—21no—
41C2×C107⋊C8—21no—
42C2×C107⋊C8D412yes—
43(C2×D107)⋊C4—44no—
44C4×Dic107—21no—
45C4×Dic107—21no—
46C22×C4×C107—21no—
47C22×C4×C107—21no—
48C22×C4×C107C2311yescharacteristic
49C2×C8×C107C2×C412yes—
50C2×C8×C107D412yes—
51C2×C8×C107D412yes—
52C4⋊C4×C107—21no—
53C4⋊C4×C107—21no—
54C4⋊C4×C107—21no—
55C4⋊C4×C107—21no—
56C23×C107C2×D411yescharacteristic
57C22×D107—22no—
58C22×D107—41no—
59C22×D107—41no—
60C22×D107—42no—
61C22×D107—42no—
62C2×C4×C107—41no—
63C2×C4×C107—41no—
64C2×C4×C107—42no—
65C2×C4×C107—42no—
66C2×C4×C107C22×C411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107#4111yescharacteristic
71C2×C4×C107#4111yescharacteristic
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107—41no—
77C2×Dic107—41no—
78D428—22no—
79D428—22no—
80D428—41no—
81D428—41no—
82Dic214—22no—
83Dic214—22no—
84Dic214—41no—
85Dic214—41no—
86C8×C107C22⋊C414yescomplemented
87C107⋊C8—22no—
88C22×C107C4×D411yescharacteristic
89C22×C107#12111yescharacteristic
90C22×C107#12711yescharacteristic
91C22×C107AGL(1,ℤ/8)12yes—
92C22×C107#13712yes—
93C2×D107—42no—
94C2×D107—44no—
95C2×D107—82no—
96C2×D107—81no—
97C2×D107—81no—
98C4×C107—81no—
99C4×C107—81no—
100C4×C107C2×C22⋊C411yescharacteristic
101C4×C107C4×D411yescharacteristic
102C4×C107#12111yescharacteristic
103C4×C107#12411yescharacteristic
104Dic107—42no—
105Dic107—42no—
106C2×C107#39011yescharacteristic
107C2×C107#44012yes—
108C2×C107#46812yes—
109C2×C107#49611yescharacteristic
110C2×C107#49711yescharacteristic
111D107—82no—
112#1973—1071nomaximal, Sylow
113C107#197311yescharacteristic, complemented, Sylow
114C22×SD16—1071no—
115C2×#107D10711yescharacteristic, complemented
116C2×#103—1071no—
117C2×#102—1071no—
118C2×#97—1071no—
119#391—1071no—
120#390—1071no—
121C22×Q8—1071no—
122C22×D4—1071no—
123C2×SD16—1074no—
124C2×SD16—2142no—
125C2×SD16—2142no—
126C22×C8—1071no—
127C22×C8C2×D10711yescharacteristic
128C2×C4⋊C4C2×D10711yescharacteristic
129C2×C4⋊C4C2×D10711yescharacteristic
130C2×C22⋊C4—2142no—
131C2×C42—1071no—
132#107—22no—
133#103—2142no—
134#102—2142no—
135#97—2142no—
136#95—2142no—
137C24—2141no—
138C2×Q8—1072no—
139C2×Q8—2141no—
140C2×Q8—2141no—
141C2×Q8—2141no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×D4—1072no—
145C2×D4—2141no—
146C2×D4—2141no—
147C2×D4—2141no—
148C2×D4—2141no—
149C2×D4—2141no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—1072no—
153C22×C4—2141no—
154C22×C4C22×D10711yescharacteristic
155SD16—2148no—
156C2×C8—1072no—
157C2×C8—2141no—
158C2×C8—2141no—
159C2×C8C4×D10712yes—
160C2×C8D42812yes—
161C2×C8C22⋊D10712yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C22⋊C4—4284no—
167C42—2141no—
168C42—2141no—
169C23—4281no—
170C23—4281no—
171C23—4282no—
172C23—4282no—
173C23—2142no—
174C23D4×D10711yescharacteristic
175Q8—4281no—
176Q8—4281no—
177Q8—2142no—
178Q8—2142no—
179D4—4281no—
180D4—4281no—
181D4—2142no—
182D4—2142no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—4281no—
188C2×C4—4281no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4—2142no—
192C2×C4—2142no—
193C2×C4C2×C4×D10711yescharacteristic
194C2×C4D4×D10711yescharacteristic
195C2×C4C2×C22⋊D10711yescharacteristic
196C2×C4C2×D42811yescharacteristic
197C2×C4#1313381911yescharacteristic
198C2×C4#1313381911yescharacteristic
199C8—2142no—
200C8(C2×D107)⋊C414yescomplemented
201C22—8562no—
202C22—8561no—
203C22—8561no—
204C22—4284no—
205C22—4282no—
206C22#1681207111yescharacteristic
207C22#1681208411yescharacteristic
208C22#1681210511yescharacteristic
209C22#1681213312yes—
210C22#1681214112yes—
211C4—81no—
212C4—81no—
213C4—4282no—
214C4—4282no—
215C4C2×(C2×D107)⋊C411yescharacteristic
216C4#1681207111yescharacteristic
217C4#1681208411yescharacteristic
218C4#1681209811yescharacteristic
219C2—8562no—
220C2#2029252811yescharacteristic
221C2#2029272612yes—
222C2#2029287412yes—
223C2#2029301511yescharacteristic
224C2#2029302011yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#107×C107C2
socleC23×C107C2×D4
prime radicalC22×D428C22
cosocleC22×C4×C107C23
hypercenterC2×#107D107
hypocenterC107#1973
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#16812032C2semidirect
4. C2×#16812019C2semidirect
5. #20292532C2semidirect
7. C2×#107×C107C2semidirect
12. C2×#13133807C4semidirect
86. C8×C107C22⋊C4semidirect
113. C107#1973semidirect
115. C2×#107D107semidirect
200. C8(C2×D107)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#131338072
C2×#168120322
C2×#168120292
C2×#168120192
#202925322
#202925282
C2×#107×C1072
#1973107