Finite Group Databasebeta

← #28336954

Subgroups — #28336954

Order 13696.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16812142C211yesmaximal, characteristic
2#20293104C211yesmaximal, characteristic, complemented
3#20292643C211yesmaximal, characteristic
4#20292637C211yesmaximal, characteristic, complemented
5#20292600C211yesmaximal, characteristic
6#20292384C211yesmaximal, characteristic, complemented
7#344×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×Dic214C2211yescharacteristic
10#16812142—22no—
11#16812142—22no—
12C2×Dic428C412yescomplemented
13C2×#13133807C412yescomplemented
14C4×Dic214—22no—
15#16812074—22no—
16#16812068C2211yescharacteristic
17#16812068C2211yescharacteristic
18#16812065—22no—
19#16812035—22no—
20#16812034—22no—
21#16812023—22no—
22C2×M16×C107C2211yescharacteristic
23C2×M16×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25#13133818—21no—
26#13133818—21no—
27#13133818—21no—
28#13133818—21no—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214C2×C411yescharacteristic
33C2×Dic214C2×C412yes—
34C2×Dic214D412yes—
35C2×Dic214D412yes—
36C2×D428C2×C411yescharacteristic
37C2×C22⋊D107—21no—
38C22×Dic107—21no—
39C2×C4×D107—21no—
40Dic428—24no—
41#13133807—24no—
42C4⋊Dic107—21no—
43C4⋊Dic107—21no—
44Dic107⋊C4—24no—
45C22⋊Dic107—21no—
46C22⋊Dic107—21no—
47C4×Dic107—24no—
48C4×Dic107—21no—
49C4×Dic107D412yes—
50C22×C4×C107C2311yescharacteristic
51M16×C107—21no—
52M16×C107—21no—
53M16×C107—21no—
54M16×C107—21no—
55C2×C8×C107—21no—
56C2×C8×C107C2×C412yes—
57C4⋊C4×C107—21no—
58C22⋊C4×C107D412yes—
59C42×C107—21no—
60C23×C107C2×D411yescharacteristic
61C22×D107—21no—
62C2×C4×C107—22no—
63C2×C4×C107C22×C411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107#4111yescharacteristic
68C2×C4×C107#4111yescharacteristic
69C22⋊D107—41no—
70C22⋊D107—41no—
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—21no—
75C2×Dic107—22no—
76C2×Dic107—41no—
77D428—21no—
78D428—21no—
79C4×D107—41no—
80C4×D107—41no—
81Dic214—22no—
82Dic214—22no—
83Dic214—22no—
84Dic214—21no—
85Dic214—21no—
86Dic214C22⋊C414yes—
87C8×C107—22no—
88C8×C107—41no—
89C22×C107—41no—
90C22×C107C4×D411yescharacteristic
91C22×C107#12011yescharacteristic
92C22×C107#12111yescharacteristic
93C2×D107—41no—
94C2×D107—81no—
95C4×C107—42no—
96C4×C107C2×C22⋊C411yescharacteristic
97C4×C107C4×D411yescharacteristic
98C4×C107#12111yescharacteristic
99C4×C107#12311yescharacteristic
100Dic107—24no—
101Dic107—42no—
102Dic107—41no—
103Dic107—81no—
104C2×C107—21no—
105C2×C107—21no—
106C2×C107—21no—
107C2×C107#38611yescharacteristic
108D107—81no—
109#1905—1071nomaximal, Sylow
110C107#190511yescharacteristic, complemented, Sylow
111C2×#137—1071no—
112#519—1071no—
113#421—1071no—
114#419—1071no—
115#410—1071no—
116#344D10711yescharacteristic, complemented
117H(3,ℤ/4)—1071no—
118C2×#41—1071no—
119C22×Q8—1071no—
120#137—2142no—
121#137—2142no—
122C2×Q16—1072no—
123C2×SD16—1072no—
124C2×M16C2×D10711yescharacteristic
125C2×M16C2×D10711yescharacteristic
126C4×Q8—2142no—
127C4×Q8—2142no—
128#117—1071no—
129#117—1071no—
130#117—2142no—
131#117C2×D10711yescharacteristic
132#104—2142no—
133#103—2142no—
134#99—2142no—
135#41—2141no—
136#41—2141no—
137#41—2141no—
138#41—2141no—
139C2×Q8—1072no—
140C2×Q8—1071no—
141C2×Q8—1072no—
142C2×Q8—1072no—
143C2×Q8—2141no—
144C2×Q8—2141no—
145C2×Q8—2141no—
146C2×D4—1071no—
147C2×D4—2141no—
148C22×C4—2141no—
149C22×C4—2141no—
150C22×C4C22×D10711yescharacteristic
151Q16—2144no—
152SD16—2144no—
153M16—21no—
154M16—21no—
155M16—21no—
156M16—21no—
157C2×C8—21no—
158C2×C8C4×D10712yes—
159C4⋊C4—21no—
160C4⋊C4—2144no—
161C4⋊C4—2141no—
162C4⋊C4—2141no—
163C22⋊C4—2141no—
164C22⋊C4—2141no—
165C22⋊C4D42812yes—
166C42—21no—
167C42—1072no—
168C42—2144no—
169C42—2141no—
170C23—2141no—
171C23C2×D42811yescharacteristic
172Q8—1074no—
173Q8—2141no—
174Q8—2141no—
175Q8—2142no—
176Q8—2142no—
177Q8—2142no—
178D4—4281no—
179D4—4281no—
180D4—2141no—
181D4—2141no—
182C2×C4—22no—
183C2×C4—4281no—
184C2×C4—4281no—
185C2×C4—4281no—
186C2×C4—2141no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4D4×D10711yescharacteristic
193C2×C4D4×D10711yescharacteristic
194C2×C4D4×D10711yescharacteristic
195C2×C4#1313381811yescharacteristic
196C2×C4#1313381911yescharacteristic
197C8—22no—
198C8—41no—
199C22—41no—
200C22—8561no—
201C22—4281no—
202C22C4×D42811yescharacteristic
203C22#1681207911yescharacteristic
204C22#1681208411yescharacteristic
205C4—42no—
206C4—8561no—
207C4—4282no—
208C4—4281no—
209C4—2144no—
210C4C22⋊C4×D10711yescharacteristic
211C4#1681207111yescharacteristic
212C4#1681208211yescharacteristic
213C4#1681209411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—8561no—
218C2#2029250511yescharacteristic
subgroupG/H
centerC2#20292505
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#344×C107C2
socleC2×C107#386
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenter#344D107
hypocenterC107#1905
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #20293104C2semidirect
4. #20292637C2semidirect
6. #20292384C2semidirect
7. #344×C107C2semidirect
12. C2×Dic428C4semidirect
13. C2×#13133807C4semidirect
110. C107#1905semidirect
116. #344D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
#202931042
#202926432
#202926372
#202926002
#202923842
#344×C1072
#1905107