Finite Group Databasebeta

← #28345756

Subgroups — #28345756

Order 13696.

700 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812139C211yesmaximal, characteristic, complemented
2C22×#13133808C211yesmaximal, characteristic, complemented
3#20293198C211yesmaximal, characteristic, complemented
4#20292789C214yesmaximal, complemented
5#20292783C214yesmaximal, complemented
6#20292636C211yesmaximal, characteristic, complemented
7#20292623C211yesmaximal, characteristic, complemented
8#20292594C211yesmaximal, characteristic, complemented
9C2×2−1+4×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11#16812139—24no—
12C2×#13133808—24no—
13C2×#13133808C2214yes—
14C2×#13133808C2212yes—
15C2×#13133806C2212yes—
16C2×#13133802C2211yescharacteristic
17C22×C107⋊C8C2211yescharacteristic
18#16812107—21no—
19#16812096—21no—
20#16812090—22no—
21#16812088C2214yes—
22#16812068C2211yescharacteristic
23#16812033C2212yes—
24#16812033C2214yes—
25#16812030C2212yes—
26C22⋊(C107⋊C8)C2214yes—
272−1+4×C107—24no—
282−1+4×C107—24no—
29C2×#41×C107—22no—
30C2×#41×C107—21no—
31C2×#41×C107C2212yes—
32C2×#41×C107C2211yescharacteristic
33C2×#41×C107C2211yescharacteristic
34C22×Q8×C107—21no—
35C22×Q8×C107C2212yes—
36C22×Q8×C107C2211yescharacteristic
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—21no—
40C2×Dic214C2312yes—
41C22×Dic107—21no—
42#13133808—28no—
43#13133808—24no—
44#13133806—24no—
45#13133802—22no—
46C2×C107⋊C8—22no—
47C2×C107⋊C8C2312yes—
48C2×C107⋊C8C2312yes—
49C4⋊Dic107C2312yes—
50Dic107⋊C4—24no—
51C22⋊Dic107—21no—
52C22⋊Dic107—24no—
53C4×Dic107—21no—
54#41×C107—216no—
55#41×C107—28no—
56#41×C107—24no—
57#41×C107—24no—
58#41×C107—22no—
59#41×C107—22no—
60#41×C107—22no—
61#41×C107D414yes—
62C2×Q8×C107—28no—
63C2×Q8×C107—24no—
64C2×Q8×C107—24no—
65C2×Q8×C107—22no—
66C2×Q8×C107—21no—
67C2×Q8×C107—22no—
68C2×Q8×C107D414yes—
69C2×Q8×C107D411yescharacteristic
70C2×Q8×C107C2311yescharacteristic
71C2×Q8×C107C2312yes—
72C2×D4×C107—22no—
73C2×D4×C107—21no—
74C2×D4×C107D412yes—
75C2×D4×C107D411yescharacteristic
76C2×D4×C107C2311yescharacteristic
77C22×C4×C107—22no—
78C22×C4×C107—21no—
79C22×C4×C107—22no—
80C22×C4×C107—21no—
81C22×C4×C107C2312yes—
82C22×C4×C107C2311yescharacteristic
83C23×C107—21no—
84C23×C107C2×D412yes—
85C23×C107C2×D411yescharacteristic
86Q8×C107—28no—
87Q8×C107—24no—
88Q8×C107—22no—
89Q8×C107—21no—
90Q8×C107—22no—
91Q8×C107C2×D412yes—
92Q8×C107C2×D414yes—
93D4×C107—28no—
94D4×C107—24no—
95D4×C107—24no—
96D4×C107—22no—
97D4×C107—21no—
98D4×C107C2×D412yes—
99C2×C4×C107—28no—
100C2×C4×C107—28no—
101C2×C4×C107—24no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107—24no—
105C2×C4×C107—22no—
106C2×C4×C107—22no—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107C2×D412yes—
110C2×C4×C107C2×D412yes—
111C2×C4×C107C2411yescharacteristic
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—42no—
115Dic214—22no—
116Dic214—22no—
117Dic214—42no—
118C107⋊C8—22no—
119C107⋊C8—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—24no—
124C22×C107—42no—
125C22×C107#12012yes—
126C22×C107C22×D411yescharacteristic
127C4×C107—24no—
128C4×C107—22no—
129C4×C107—22no—
130C4×C107#12012yes—
131C4×C107C22×D411yescharacteristic
132C4×C107C22×D411yescharacteristic
133Dic107—42no—
134Dic107—42no—
135C2×C107—22no—
136C2×C107—42no—
137C2×C107—42no—
138C2×C107C2×#12011yescharacteristic
139C2×C107#57812yes—
140#3011—1071nomaximal, Sylow
141C107#301111yescharacteristic, complemented, Sylow
142C2×2−1+4D10711yescharacteristic
143C2×#137—1071no—
144C22×Q16—1071no—
145#536—1071no—
146#452—1074no—
147#451—1074no—
148#419—1071no—
149#416—1071no—
150#408—1071no—
1512−1+4—24no—
1522−1+4—24no—
153C2×#41—22no—
154C2×#41—21no—
155C2×#41C2×D10712yes—
156C2×#41C2×D10711yescharacteristic
157C2×#41C2×D10711yescharacteristic
158C22×Q8—21no—
159C22×Q8—1071no—
160C22×Q8C2×D10712yes—
161C22×Q8C2×D10711yescharacteristic
162#137—2144no—
163C2×Q16—1074no—
164C2×Q16—1072no—
165C2×Q16—2144no—
166C2×SD16—1072no—
167C2×M16—1071no—
168C22×C8—1071no—
169C4⋊Q8—2141no—
170#124—2141no—
171#123—2142no—
172C22⋊Q8—1074no—
173#117—1071no—
174#103—1072no—
175#103—1074no—
176#102—1072no—
177C22⋊C8—1074no—
178#41—216no—
179#41—28no—
180#41—24no—
181#41—24no—
182#41—22no—
183#41—22no—
184#41—22no—
185#41C22⋊D10714yes—
186C2×Q8—28no—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—22no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—1072no—
193C2×Q8—2142no—
194C2×Q8—2142no—
195C2×Q8—2141no—
196C2×Q8C22⋊D10714yes—
197C2×Q8C22⋊D10711yescharacteristic
198C2×Q8C22×D10711yescharacteristic
199C2×Q8C22×D10712yes—
200C2×D4—22no—
201C2×D4—21no—
202C2×D4C22⋊D10712yes—
203C2×D4C22⋊D10711yescharacteristic
204C2×D4C22×D10711yescharacteristic
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—2141no—
210C22×C4C22×D10712yes—
211C22×C4C22×D10711yescharacteristic
212Q16—2148no—
213Q16—2144no—
214SD16—2144no—
215M16—2142no—
216C2×C8—1072no—
217C2×C8—1072no—
218C2×C8—2142no—
219C4⋊C4—1072no—
220C4⋊C4—2144no—
221C22⋊C4—2141no—
222C22⋊C4—2144no—
223C42—2141no—
224C23—21no—
225C23C2×C22⋊D10712yes—
226C23C2×C22⋊D10711yescharacteristic
227Q8—28no—
228Q8—24no—
229Q8—22no—
230Q8—21no—
231Q8—22no—
232Q8—4282no—
233Q8—2142no—
234Q8—2142no—
235Q8C2×C22⋊D10712yes—
236Q8C2×C22⋊D10714yes—
237D4—28no—
238D4—24no—
239D4—24no—
240D4—22no—
241D4—21no—
242D4C2×C22⋊D10712yes—
243C2×C4—28no—
244C2×C4—28no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—4282no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4C2×C22⋊D10712yes—
255C2×C4C2×C22⋊D10711yescharacteristic
256C2×C4C2×C22⋊D10712yes—
257C2×C4C2×C22⋊D10712yes—
258C2×C4C23×D10711yescharacteristic
259C8—2142no—
260C8—2142no—
261C22—22no—
262C22—24no—
263C22—22no—
264C22—22no—
265C22—42no—
266C22#1681207812yes—
267C22C22×C22⋊D10711yescharacteristic
268C4—24no—
269C4—22no—
270C4—22no—
271C4—4282no—
272C4—4282no—
273C4#1681207812yes—
274C4C22×C22⋊D10711yescharacteristic
275C4C22×C22⋊D10711yescharacteristic
276C2—22no—
277C2—42no—
278C2—42no—
279C2C2×#1681207811yescharacteristic
280C2#2029344212yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC2×2−1+4×C107C2
socleC22×C107C22×D4
hypercenterC2×2−1+4D107
hypocenterC107#3011
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C22×#13133808C2semidirect
3. #20293198C2semidirect
4. #20292789C2semidirect
5. #20292783C2semidirect
6. #20292636C2semidirect
7. #20292623C2semidirect
8. #20292594C2semidirect
141. C107#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C22×#131338082
#202931982
#202927892
#202927832
#202926362
#202926232
#202925942
C2×2−1+4×C1072
#3011107