Finite Group Databasebeta

← #28337367

Subgroups — #28337367

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×#13133806C211yesmaximal, characteristic
2C2×#16812044C211yesmaximal, characteristic, complemented
3#20292635C211yesmaximal, characteristic, complemented
4C2×#16812027C211yesmaximal, characteristic, complemented
5C2×#16812026C211yesmaximal, characteristic
6#20292583C211yesmaximal, characteristic, complemented
7#418×C107C211yesmaximal, characteristic
8C22×Dic214C2211yescharacteristic
9C2×#13133806—22no—
10C2×#13133806—21no—
11C2×#13133806—21no—
12C2×#13133806C414yescomplemented
13C2×#13133802C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15C22×C107⋊C8C2211yescharacteristic
16#16812044—22no—
17#16812032—22no—
18#16812027—22no—
19#16812026—22no—
20#16812020—21no—
21C4×C107⋊C8—21no—
22C22×D4×C107C2211yescharacteristic
23C2×M16×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25#102×C107—22no—
26C2×Dic214—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214C2×C412yes—
32C22×Dic107—21no—
33#13133806—24no—
34#13133806—24no—
35#13133802—21no—
36#13133802—21no—
37#13133802—21no—
38#13133802D412yes—
39C2×C107⋊C8—21no—
40C2×C107⋊C8—21no—
41C2×C107⋊C8C2×C412yes—
42C2×C107⋊C8D412yes—
43C2×C107⋊C8D411yescharacteristic
44C2×C107⋊C8D411yescharacteristic
45C24×C107—21no—
46C2×D4×C107—21no—
47C2×D4×C107—21no—
48C2×D4×C107—21no—
49C2×D4×C107—21no—
50C2×D4×C107—21no—
51C2×D4×C107C2×C412yes—
52C22×C4×C107C2311yescharacteristic
53M16×C107—21no—
54M16×C107—21no—
55C2×C8×C107—21no—
56C4⋊C4×C107D412yes—
57C22⋊C4×C107—21no—
58C42×C107—21no—
59C23×C107—22no—
60C23×C107—41no—
61C23×C107—41no—
62C23×C107—82no—
63C23×C107#4111yescharacteristic
64D4×C107—22no—
65D4×C107—22no—
66D4×C107—41no—
67D4×C107—41no—
68C2×C4×C107—22no—
69C2×C4×C107C22×C411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D411yescharacteristic
74C2×C4×C107#4111yescharacteristic
75C2×Dic107—22no—
76C2×Dic107—41no—
77C2×Dic107—41no—
78Dic214—22no—
79Dic214—22no—
80Dic214—41no—
81Dic214—41no—
82C8×C107—41no—
83C107⋊C8—22no—
84C107⋊C8—41no—
85C107⋊C8C22⋊C412yes—
86C107⋊C8C22⋊C412yes—
87C22×C107—22no—
88C22×C107—42no—
89C22×C107—82no—
90C22×C107—82no—
91C22×C107—82no—
92C22×C107C4×D411yescharacteristic
93C22×C107#12111yescharacteristic
94C22×C107#12411yescharacteristic
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—82no—
104C2×C107#39011yescharacteristic
105C2×C107#47112yes—
106#1977—1071nomaximal, Sylow
107C107#197711yescharacteristic, complemented, Sylow
108C22×SD16—1071no—
109C2×#108—1071no—
110#419—1071no—
111#418D10711yescharacteristic
112C2×#101—1071no—
113C2×#100—1071no—
114#405—1071no—
115C22×Q8—1071no—
116C22×D4C2×D10711yescharacteristic
117C2×SD16—1074no—
118C2×SD16—2141no—
119C2×SD16—2141no—
120C2×SD16—2142no—
121C2×M16—1071no—
122C2×M16—1071no—
123C2×M16C2×D10711yescharacteristic
124C22×C8—1071no—
125#117C2×D10711yescharacteristic
126#108—2142no—
127#103—2142no—
128#102—22no—
129#101—2142no—
130#100—2142no—
131#97—2141no—
132C4×C8—2141no—
133C24—21no—
134C2×Q8—1072no—
135C2×Q8—2141no—
136C2×Q8—2141no—
137C2×Q8—2141no—
138C2×Q8—2141no—
139C2×Q8—2141no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C4×D10712yes—
146C22×C4—2141no—
147C22×C4C22×D10711yescharacteristic
148SD16—2144no—
149SD16—2144no—
150M16—21no—
151M16—21no—
152M16—1072no—
153M16—2141no—
154M16—2141no—
155M16—2141no—
156C2×C8—21no—
157C2×C8—1072no—
158C2×C8—1071no—
159C2×C8—1071no—
160C2×C8—1072no—
161C2×C8—2141no—
162C2×C8—2141no—
163C4⋊C4C22⋊D10712yes—
164C22⋊C4—21no—
165C42—21no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1313381711yescharacteristic
171Q8—4281no—
172Q8—4281no—
173Q8—2142no—
174Q8—2142no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—22no—
180C2×C4—4281no—
181C2×C4—4281no—
182C2×C4—2142no—
183C2×C4C2×C4×D10711yescharacteristic
184C2×C4D4×D10711yescharacteristic
185C2×C4D4×D10711yescharacteristic
186C2×C4D4×D10711yescharacteristic
187C2×C4C2×C22⋊D10711yescharacteristic
188C2×C4#1313381811yescharacteristic
189C8—41no—
190C8—1072no—
191C8—1072no—
192C8—4281no—
193C8—2142no—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22(C22⋊D107)⋊C411yescharacteristic
200C22#1681208111yescharacteristic
201C22#1681209511yescharacteristic
202C4—42no—
203C4—4282no—
204C4C22⋊C4×D10711yescharacteristic
205C4C4×C22⋊D10711yescharacteristic
206C4#1681208211yescharacteristic
207C4#1681210511yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2029252711yescharacteristic
212C2#2029288812yes—
subgroupG/H
centerC22#16812081
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#418×C107C2
socleC22×C107#121
prime radicalC22×D4×C107C22
cosocleC22×C4×C107C23
hypercenter#418D107
hypocenterC107#1977
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#16812044C2semidirect
3. #20292635C2semidirect
4. C2×#16812027C2semidirect
6. #20292583C2semidirect
12. C2×#13133806C4semidirect
107. C107#1977semidirect

Maximal subgroups

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

subgroupindex
C22×#131338062
C2×#168120442
#202926352
C2×#168120272
C2×#168120262
#202925832
#418×C1072
#1977107