Finite Group Databasebeta

← #28337365

Subgroups — #28337365

Order 13696.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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×#13133805C211yesmaximal, characteristic
2C2×#16812044C211yesmaximal, characteristic, complemented
3#20292636C211yesmaximal, characteristic, complemented
4#20292633C211yesmaximal, characteristic
5C2×#16812025C211yesmaximal, characteristic
6#20292585C211yesmaximal, characteristic, complemented
7C2×#101×C107C211yesmaximal, characteristic, complemented
8C22×D428C2211yescharacteristic
9C2×#13133805—22no—
10C2×#13133805—21no—
11C2×#13133805—21no—
12C2×#13133805C414yescomplemented
13C2×#13133802C2211yescharacteristic
14C22×C107⋊C8C2211yescharacteristic
15#16812068C2211yescharacteristic
16#16812044—22no—
17#16812033—22no—
18#16812031—22no—
19#16812025—22no—
20#16812019—21no—
21C8×Dic107—21no—
22C22×Q8×C107C2211yescharacteristic
23C2×M16×C107C2211yescharacteristic
24C2×M16×C107C2211yescharacteristic
25#101×C107—22no—
26C23×D107—21no—
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428—21no—
31C2×D428—21no—
32C2×D428C2×C412yes—
33#13133805—24no—
34#13133805—24no—
35#13133802—21no—
36#13133802—21no—
37C2×C107⋊C8—21no—
38C2×C107⋊C8C2×C412yes—
39C2×C107⋊C8D412yes—
40C2×C107⋊C8D411yescharacteristic
41C2×C107⋊C8D411yescharacteristic
42C4⋊Dic107D412yes—
43C22⋊Dic107—21no—
44C4×Dic107—21no—
45C2×Q8×C107—21no—
46C2×Q8×C107—21no—
47C2×Q8×C107—21no—
48C2×Q8×C107—21no—
49C2×Q8×C107—21no—
50C2×Q8×C107C2×C412yes—
51C22×C4×C107—21no—
52C22×C4×C107C2311yescharacteristic
53M16×C107—21no—
54M16×C107—21no—
55M16×C107—21no—
56M16×C107D412yes—
57C2×C8×C107—21no—
58C2×C8×C107—21no—
59C23×C107#4111yescharacteristic
60C22×D107—22no—
61C22×D107—41no—
62C22×D107—41no—
63C22×D107—82no—
64Q8×C107—22no—
65Q8×C107—22no—
66Q8×C107—41no—
67Q8×C107—41no—
68C2×C4×C107—22no—
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×C107C2×D411yescharacteristic
76C2×C4×C107#4111yescharacteristic
77C2×Dic107—22no—
78D428—22no—
79D428—22no—
80D428—41no—
81D428—41no—
82C8×C107—22no—
83C8×C107—41no—
84C107⋊C8—41no—
85C107⋊C8C22⋊C412yes—
86C107⋊C8C22⋊C412yes—
87C22×C107—22no—
88C22×C107C4×D411yescharacteristic
89C22×C107#12111yescharacteristic
90C22×C107#12411yescharacteristic
91C2×D107—42no—
92C2×D107—82no—
93C2×D107—82no—
94C2×D107—82no—
95C4×C107—42no—
96C4×C107C2×C22⋊C411yescharacteristic
97C4×C107C4×D411yescharacteristic
98C4×C107#12111yescharacteristic
99C4×C107#12711yescharacteristic
100Dic107—42no—
101C2×C107—21no—
102C2×C107—21no—
103C2×C107#39011yescharacteristic
104C2×C107#47112yes—
105D107—82no—
106#1977—1071nomaximal, Sylow
107C107#197711yescharacteristic, complemented, Sylow
108C22×SD16—1071no—
109C2×#108—1071no—
110#419—1071no—
111#418—1071no—
112C2×#101D10711yescharacteristic, complemented
113C2×#100—1071no—
114#405—1071no—
115C22×Q8C2×D10711yescharacteristic
116C22×D4—1071no—
117C2×SD16—1074no—
118C2×SD16—2141no—
119C2×SD16—2141no—
120C2×SD16—2142no—
121C2×M16—1071no—
122C2×M16C2×D10711yescharacteristic
123C2×M16C2×D10711yescharacteristic
124C22×C8—1071no—
125#117—1071no—
126#108—2142no—
127#103—2142no—
128#102—2142no—
129#101—22no—
130#100—2142no—
131#97—2141no—
132C4×C8—2141no—
133C24—2141no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C4×D10712yes—
140C2×D4—1072no—
141C2×D4—2141no—
142C2×D4—2141no—
143C2×D4—2141no—
144C2×D4—2141no—
145C2×D4—2141no—
146C22×C4—21no—
147C22×C4C22×D10711yescharacteristic
148SD16—2144no—
149SD16—2144no—
150M16—21no—
151M16—21no—
152M16—21no—
153M16—2141no—
154M16—2141no—
155M16C22⋊D10712yes—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—1072no—
159C2×C8—1071no—
160C2×C8—1071no—
161C2×C8—1072no—
162C2×C8—2141no—
163C4⋊C4—1072no—
164C22⋊C4—2141no—
165C42—2141no—
166C23—8562no—
167C23—4281no—
168C23—4281no—
169C23—2142no—
170C23#1313381811yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—4281no—
176D4—4281no—
177D4—2142no—
178D4—2142no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—2142no—
183C2×C4C2×C4×D10711yescharacteristic
184C2×C4D4×D10711yescharacteristic
185C2×C4D4×D10711yescharacteristic
186C2×C4D4×D10711yescharacteristic
187C2×C4C2×C22⋊D10711yescharacteristic
188C2×C4#1313381711yescharacteristic
189C8—22no—
190C8—41no—
191C8—1072no—
192C8—1072no—
193C8—4281no—
194C22—22no—
195C22—8562no—
196C22—8562no—
197C22—8562no—
198C22—4282no—
199C22C4×C22⋊D10711yescharacteristic
200C22#1681208211yescharacteristic
201C22#1681209511yescharacteristic
202C4—42no—
203C4—4282no—
204C4C22⋊C4×D10711yescharacteristic
205C4(C22⋊D107)⋊C411yescharacteristic
206C4#1681208111yescharacteristic
207C4#1681210511yescharacteristic
208C2—21no—
209C2—21no—
210C2—8562no—
211C2#2029252711yescharacteristic
212C2#2029289312yes—
subgroupG/H
centerC22#16812082
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#101×C107C2
socleC22×C107#121
prime radicalC22×D428C22
cosocleC22×C4×C107C23
hypercenterC2×#101D107
hypocenterC107#1977
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#16812044C2semidirect
3. #20292636C2semidirect
6. #20292585C2semidirect
7. C2×#101×C107C2semidirect
12. C2×#13133805C4semidirect
107. C107#1977semidirect
112. C2×#101D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338052
C2×#168120442
#202926362
#202926332
C2×#168120252
#202925852
C2×#101×C1072
#1977107