Finite Group Databasebeta

← #28336933

Subgroups — #28336933

Order 13696.

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

normal subgroups only

This group has 304 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#20293393C211yesmaximal, characteristic
3#20292645C211yesmaximal, characteristic
4#20292635C211yesmaximal, characteristic, complemented
5C2×#16812027C211yesmaximal, characteristic, complemented
6C107⋊#338C211yesmaximal, characteristic, complemented
7#344×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×Dic214C2211yescharacteristic
10#16812142—22no—
11#16812142—22no—
12C2×Dic428C412yescomplemented
13C2×#13133807C412yescomplemented
14#16812117—22no—
15#16812115—22no—
16C2×#13133802C2211yescharacteristic
17C2×#13133802C2211yescharacteristic
18M16×D107—22no—
19#16812037—22no—
20#16812032—22no—
21#16812027—22no—
22C2×M16×C107C2211yescharacteristic
23C2×M16×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25#13133818—21no—
26#13133818—21no—
27#13133818D412yes—
28#13133818D412yes—
29C2×Dic214—21no—
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214—21no—
33C2×Dic214—21no—
34C2×Dic214C2×C412yes—
35C2×Dic214C2×C411yescharacteristic
36C2×D428C2×C411yescharacteristic
37C2×C22⋊D107—21no—
38C22×Dic107—21no—
39C2×C4×D107—21no—
40Dic428—24no—
41#13133807—24no—
42#13133802—21no—
43#13133802—21no—
44#13133802—21no—
45#13133802—21no—
46#13133801—24no—
47C2×C107⋊C8—21no—
48C2×C107⋊C8D412yes—
49C8×D107—24no—
50C22×C4×C107C2311yescharacteristic
51M16×C107—21no—
52M16×C107—21no—
53M16×C107—21no—
54M16×C107D412yes—
55C2×C8×C107—21no—
56C2×C8×C107C2×C412yes—
57C4⋊C4×C107—21no—
58C22⋊C4×C107—21no—
59C42×C107—21no—
60C23×C107#4111yescharacteristic
61C22×D107—21no—
62C2×C4×C107—41no—
63C2×C4×C107C22×C411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107#4111yescharacteristic
69C22⋊D107—22no—
70C22⋊D107—22no—
71C2×Dic107—21no—
72C2×Dic107—22no—
73C2×Dic107—41no—
74C2×Dic107—41no—
75D428—21no—
76D428C22⋊C412yes—
77C4×D107—22no—
78C4×D107—22no—
79Dic214—21no—
80Dic214—22no—
81Dic214—22no—
82Dic214—41no—
83Dic214—41no—
84Dic214C22⋊C412yes—
85C8×C107—22no—
86C8×C107—22no—
87C107⋊C8—22no—
88C107⋊C8—41no—
89C22×C107—41no—
90C22×C107C4×D411yescharacteristic
91C22×C107#12111yescharacteristic
92C22×C107#12311yescharacteristic
93C2×D107—22no—
94C2×D107—81no—
95C4×C107—81no—
96C4×C107C2×C22⋊C411yescharacteristic
97C4×C107C4×D411yescharacteristic
98C4×C107#12011yescharacteristic
99C4×C107#12111yescharacteristic
100Dic107—22no—
101Dic107—42no—
102C2×C107—21no—
103C2×C107—21no—
104C2×C107—21no—
105C2×C107#38611yescharacteristic
106D107—42no—
107#1902—1071nomaximal, Sylow
108C107#190211yescharacteristic, complemented, Sylow
109C2×#137—1071no—
110#568—1071no—
111#421—1071no—
112#419—1071no—
113C2×#101—1071no—
114#344D10711yescharacteristic, complemented
115#338—1071no—
116C2×#41—1071no—
117C22×Q8—1071no—
118#137—2142no—
119#137—2142no—
120C2×Q16—1072no—
121C2×SD16—1072no—
122#131—2142no—
123#131—2142no—
124C2×M16—1071no—
125C2×M16—1071no—
126C2×M16—2142no—
127C2×M16C2×D10711yescharacteristic
128C2×M16C2×D10711yescharacteristic
129#117C2×D10711yescharacteristic
130#104—2142no—
131#103—2142no—
132#101—2142no—
133#41—1072no—
134#41—1072no—
135#41—2141no—
136#41—2141no—
137C2×Q8—1071no—
138C2×Q8—1072no—
139C2×Q8—2141no—
140C2×Q8—2141no—
141C2×Q8—2141no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C2×D4—1071no—
145C2×D4—2141no—
146C22×C4—2141no—
147C22×C4—2141no—
148C22×C4C22×D10711yescharacteristic
149Q16—2144no—
150SD16—2144no—
151M16—21no—
152M16—21no—
153M16—21no—
154M16—2144no—
155M16—2141no—
156M16—2141no—
157M16—2141no—
158M16—2141no—
159M16D42812yes—
160C2×C8—21no—
161C2×C8—1072no—
162C2×C8—2144no—
163C2×C8—2141no—
164C2×C8C4×D10712yes—
165C4⋊C4—21no—
166C22⋊C4—21no—
167C42—21no—
168C23—2141no—
169C23#1313381811yescharacteristic
170Q8—1072no—
171Q8—4281no—
172Q8—4281no—
173Q8—2142no—
174Q8—2142no—
175Q8—2141no—
176D4—1072no—
177D4—2142no—
178D4—2142no—
179D4—2141no—
180C2×C4—41no—
181C2×C4—4281no—
182C2×C4—4281no—
183C2×C4—2142no—
184C2×C4—2141no—
185C2×C4—2142no—
186C2×C4—2142no—
187C2×C4C2×C4×D10711yescharacteristic
188C2×C4D4×D10711yescharacteristic
189C2×C4D4×D10711yescharacteristic
190C2×C4D4×D10711yescharacteristic
191C2×C4C2×D42811yescharacteristic
192C2×C4#1313381911yescharacteristic
193C8—22no—
194C8—22no—
195C8—4281no—
196C8—2142no—
197C22—41no—
198C22—8561no—
199C22—2142no—
200C22C4×D42811yescharacteristic
201C22#1681208211yescharacteristic
202C22#1681209411yescharacteristic
203C4—81no—
204C4—4282no—
205C4—2142no—
206C4C22⋊C4×D10711yescharacteristic
207C4#1681207111yescharacteristic
208C4#1681207911yescharacteristic
209C4#1681208411yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—4282no—
214C2#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#1902
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
4. #20292635C2semidirect
5. C2×#16812027C2semidirect
6. C107⋊#338C2semidirect
7. #344×C107C2semidirect
12. C2×Dic428C4semidirect
13. C2×#13133807C4semidirect
108. C107#1902semidirect
114. #344D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
#202933932
#202926452
#202926352
C2×#168120272
C107⋊#3382
#344×C1072
#1902107