Finite Group Databasebeta

← #28344942

Subgroups — #28344942

Order 13696.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812156C211yesmaximal, characteristic
2#20293393C212yesmaximal, complemented
3#20292611C211yesmaximal, characteristic, complemented
4#20292608C214yesmaximal, complemented
5C2×#16812027C212yesmaximal, complemented
6#101×D107C214yesmaximal, complemented
7#413×C107C211yesmaximal, characteristic, complemented
8#16812156—24no—
9#16812156—24no—
10C2×#13133818C412yes—
11C2×#13133818C2211yescharacteristic
12C2×#13133817C414yes—
13C2×#13133817C2212yes—
14C22×Dic214C412yes—
15C22×Dic214C2211yescharacteristic
16C2×Q8×D107C2212yes—
17#16812117—24no—
18#16812115—24no—
19C2×#13133802C2212yes—
20M16×D107C2218yes—
21#16812028C2212yes—
22#16812027C2218yes—
23#16812026C2212yes—
24C2×#41×C107C2211yescharacteristic
25C2×M16×C107C2212yes—
26#101×C107C2212yes—
27#100×C107C2212yes—
28#13133818—24no—
29#13133818—24no—
30#13133818D414yes—
31#13133818D414yes—
32#13133817—216no—
33#13133817—28no—
34C2×Dic214—22no—
35C2×Dic214—22no—
36C2×Dic214—24no—
37C2×Dic214—24no—
38C2×Dic214C2×C414yes—
39C2×Dic214C2×C412yes—
40C2×Dic214C2×C411yescharacteristic
41C2×Dic214C2312yes—
42Q8×D107—28no—
43C2×D428C2×C411yescharacteristic
44C2×C22⋊D107C2×C414yes—
45C2×C22⋊D107C2×C412yes—
46C22×Dic107C2×C414yes—
47C22×Dic107C2×C412yes—
48C2×C4×D107C2×C414yes—
49C2×C4×D107C2312yes—
50#13133802—22no—
51#13133802C2314yes—
52#13133801—28no—
53C2×C107⋊C8—22no—
54C8×D107—28no—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107C2311yescharacteristic
58C2×D4×C107C2×C412yes—
59C2×D4×C107C2311yescharacteristic
60C22×C4×C107C2×C412yes—
61C22×C4×C107C2311yescharacteristic
62M16×C107—22no—
63M16×C107C2314yes—
64C2×C8×C107—22no—
65C23×C107C22×C412yes—
66C23×C107C22×C411yescharacteristic
67C22×D107C22×C412yes—
68Q8×C107—22no—
69D4×C107—24no—
70D4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107C22×C412yes—
74C2×C4×C107C22×C411yescharacteristic
75C2×C4×C107C2×D412yes—
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2411yescharacteristic
78C22⋊D107—28no—
79C22⋊D107C22⋊C418yes—
80C2×Dic107—24no—
81C2×Dic107—28no—
82C2×Dic107C22×C414yes—
83C2×Dic107C22×C412yes—
84D428C22⋊C412yes—
85D428C22⋊C412yes—
86C4×D107—28no—
87C4×D107C2×D418yes—
88Dic214—22no—
89Dic214—22no—
90Dic214—24no—
91Dic214—28no—
92Dic214C22⋊C412yes—
93Dic214C22⋊C412yes—
94C8×C107—24no—
95C107⋊C8—24no—
96C22×C107—22no—
97C22×C107—41no—
98C22×C107—42no—
99C22×C107C2×C22⋊C412yes—
100C22×C107C23×C411yescharacteristic
101C2×D107—42no—
102C2×D107C2×C22⋊C414yes—
103C4×C107—22no—
104C4×C107C2×C22⋊C411yescharacteristic
105C4×C107C2×C22⋊C411yescharacteristic
106C4×C107C22×D412yes—
107Dic107—24no—
108Dic107C2×C22⋊C414yes—
109C2×C107—21no—
110C2×C107—22no—
111C2×C107—42no—
112C2×C107C22×C22⋊C411yescharacteristic
113D107—24no—
114#2889—1071nomaximal, Sylow
115C107#288911yescharacteristic, complemented, Sylow
116C2×2−1+4—1071no—
117#568—1072no—
118#413—1071no—
119#413—1074no—
120#413D10711yescharacteristic, complemented
121C2×#101—1072no—
122C2×#101—1074no—
1232−1+4—2144no—
1242−1+4—2144no—
125C2×#41—1074no—
126C2×#41—1072no—
127C2×#41—1072no—
128C2×#41—1071no—
129C2×#41C2×D10711yescharacteristic
130C22×Q8—1072no—
131C22×Q8—1072no—
132C22×Q8—1071no—
133#131—2144no—
134#131—2144no—
135C2×M16—1072no—
136C2×M16—1078no—
137C2×M16C2×D10712yes—
138#101—1072no—
139#101—1078no—
140#101C2×D10712yes—
141#100—1072no—
142#100C2×D10712yes—
143#41—22no—
144#41—22no—
145#41—1074no—
146#41—1074no—
147#41—2144no—
148#41—2144no—
149#41—2148no—
150#41—21416no—
151C2×Q8—1071no—
152C2×Q8—1074no—
153C2×Q8—1072no—
154C2×Q8—1072no—
155C2×Q8—2144no—
156C2×Q8—2144no—
157C2×Q8—2148no—
158C2×Q8—2142no—
159C2×Q8—2142no—
160C2×Q8C22×D10711yescharacteristic
161C2×D4—1072no—
162C2×D4—1074no—
163C2×D4—1071no—
164C2×D4C4×D10712yes—
165C2×D4C22×D10711yescharacteristic
166C22×C4—1072no—
167C22×C4—1074no—
168C22×C4—1072no—
169C22×C4—1074no—
170C22×C4C4×D10712yes—
171C22×C4C22×D10711yescharacteristic
172M16—22no—
173M16—1074no—
174M16—2148no—
175M16—2142no—
176M16C22×D10714yes—
177C2×C8—22no—
178C2×C8—2148no—
179C2×C8—2142no—
180C23—1072no—
181C23C2×C4×D10712yes—
182C23C2×C4×D10711yescharacteristic
183Q8—22no—
184Q8—1072no—
185Q8—1072no—
186Q8—2148no—
187Q8—2142no—
188Q8—2142no—
189Q8—2144no—
190D4—24no—
191D4—22no—
192D4—1078no—
193D4—1072no—
194D4—1072no—
195D4—2148no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—1072no—
199C2×C4—1078no—
200C2×C4—1074no—
201C2×C4—2148no—
202C2×C4—2148no—
203C2×C4—2144no—
204C2×C4C2×C4×D10712yes—
205C2×C4C2×C4×D10711yescharacteristic
206C2×C4D4×D10712yes—
207C2×C4D4×D10712yes—
208C2×C4C23×D10711yescharacteristic
209C8—24no—
210C8—2144no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—1074no—
215C22—4282no—
216C22C22⋊C4×D10712yes—
217C22C22×C4×D10711yescharacteristic
218C4—22no—
219C4—1074no—
220C4—2144no—
221C4C22⋊C4×D10711yescharacteristic
222C4C22⋊C4×D10711yescharacteristic
223C4C2×D4×D10712yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—2144no—
228C2C2×C22⋊C4×D10711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D107
commutatorC22×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#413×C107C2
socleC2×C107C22×C22⋊C4
prime radicalC2×#16812156C2
cosocleC2×C4×C107C24
hypercenter#413D107
hypocenterC107#2889
LCSG ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20293393C2semidirect
3. #20292611C2semidirect
4. #20292608C2semidirect
5. C2×#16812027C2semidirect
6. #101×D107C2semidirect
7. #413×C107C2semidirect
115. C107#2889semidirect
120. #413D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
#202933932
#202926112
#202926082
C2×#168120272
#101×D1072
#413×C1072
#2889107