Finite Group Databasebeta

← #28345947

Subgroups — #28345947

Order 13696.

480 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#16812140C211yesmaximal, characteristic
3C2×#16812107C211yesmaximal, characteristic, complemented
4#20293095C211yesmaximal, characteristic
5#20292836C214yesmaximal, complemented
6#20292831C214yesmaximal, complemented
7#20292650C211yesmaximal, characteristic, complemented
8#20292635C211yesmaximal, characteristic, complemented
9#419×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×Dic214C2211yescharacteristic
12#16812142—24no—
13#16812140—24no—
14C2×Dic428C2212yes—
15C2×#13133808C2212yes—
16C2×#13133807C2212yes—
17C2×#13133805C2212yes—
18C2×#13133802C2211yescharacteristic
19#16812107—21no—
20#16812107—21no—
21#16812107C2214yes—
22#16812074C2212yes—
23#16812071C2212yes—
24(C22⋊D107)⋊C4—22no—
25#16812066—21no—
26C2×Dic107⋊C4—22no—
27C2×C4×Dic107C2211yescharacteristic
28C42×D107—21no—
29Dic107⋊C8C2214yes—
30#16812032C2214yes—
31C22×Q8×C107C2211yescharacteristic
32C2×M16×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34#103×C107C2214yes—
35#13133818—22no—
36#13133818D414yes—
37C2×Dic214—22no—
38C2×Dic214—22no—
39C2×Dic214—21no—
40C2×Dic214C2312yes—
41C2×Dic214C2311yescharacteristic
42C2×D428C2311yescharacteristic
43C2×C22⋊D107—21no—
44C22×Dic107—21no—
45C22×Dic107D412yes—
46C2×C4×D107—21no—
47C2×C4×D107—22no—
48Dic428—24no—
49#13133808—24no—
50#13133807—24no—
51#13133805—24no—
52#13133802—22no—
53C2×C107⋊C8C2312yes—
54Dic107⋊C4—24no—
55Dic107⋊C4—24no—
56Dic107⋊C4—22no—
57(C2×D107)⋊C4—22no—
58C4×Dic107—22no—
59C4×Dic107D411yescharacteristic
60C4×Dic107D411yescharacteristic
61C4×Dic107C2312yes—
62C2×Q8×C107—21no—
63C2×Q8×C107—22no—
64C2×Q8×C107—22no—
65C2×Q8×C107C2312yes—
66C22×C4×C107—21no—
67C22×C4×C107C2311yescharacteristic
68M16×C107—22no—
69C2×C8×C107C2312yes—
70C4⋊C4×C107C2312yes—
71C22⋊C4×C107—21no—
72C42×C107—21no—
73C23×C107C2×D411yescharacteristic
74C22×D107—21no—
75Q8×C107—22no—
76Q8×C107—22no—
77Q8×C107—42no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—42no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D412yes—
83C2×C4×C107#4112yes—
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—24no—
86C2×Dic107—21no—
87C2×Dic107—24no—
88C2×Dic107—22no—
89C2×Dic107—42no—
90C2×Dic107C2×D412yes—
91C2×Dic107C2×D412yes—
92D428—21no—
93D428C2×D412yes—
94C4×D107—24no—
95C4×D107—44no—
96Dic214—21no—
97Dic214—22no—
98Dic214—22no—
99Dic214—42no—
100Dic214C2×D412yes—
101C8×C107—22no—
102C107⋊C8—22no—
103C22×C107—22no—
104C22×C107#12112yes—
105C22×C107C22×D411yescharacteristic
106C2×D107—22no—
107C2×D107—42no—
108C4×C107—42no—
109C4×C107—42no—
110C4×C107#12112yes—
111C4×C107C22×D411yescharacteristic
112C4×C107C2×#4111yescharacteristic
113Dic107—22no—
114Dic107—22no—
115Dic107—42no—
116Dic107#13714yescomplemented
117C2×C107—22no—
118C2×C107C2×#12111yescharacteristic
119C2×C107C2×#13712yes—
120D107—42no—
121#3039—1071nomaximal, Sylow
122C107#303911yescharacteristic, complemented, Sylow
123C2×#137—1071no—
124C2×#137—1071no—
125C2×C4⋊Q8—1071no—
126C4×#41—1071no—
127#462—1074no—
128#461—1074no—
129#423—1071no—
130#419—1071no—
131#419D10711yescharacteristic, complemented
132C2×#41—1071no—
133C22×Q8—1071no—
134C22×Q8C2×D10711yescharacteristic
135#137—2144no—
136#137—2144no—
137C2×Q16—1072no—
138C2×Q16—1072no—
139C2×SD16—1072no—
140C2×SD16—1072no—
141C2×M16—1071no—
142C2×M16C2×D10711yescharacteristic
143C4⋊Q8—1074no—
144C4⋊Q8—2141no—
145C4⋊Q8—2141no—
146C4×Q8—1072no—
147C4×D4—1072no—
148C4×D4—2142no—
149#117—2141no—
150#117C2×D10711yescharacteristic
151C2×C4⋊C4—2142no—
152C2×C42—1071no—
153C2×C42—2141no—
154C4⋊C8—1074no—
155#103—1074no—
156#103C2×D10714yes—
157#41—1074no—
158#41—2142no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1071no—
163C2×Q8—1072no—
164C2×Q8—2142no—
165C2×Q8—2142no—
166C2×Q8—2141no—
167C2×Q8C22×D10712yes—
168C2×D4—1071no—
169C2×D4—2141no—
170C22×C4—21no—
171C22×C4—1072no—
172C22×C4—2142no—
173C22×C4—2141no—
174C22×C4—2141no—
175C22×C4C22×D10711yescharacteristic
176Q16—2144no—
177Q16—2144no—
178SD16—2144no—
179SD16—2144no—
180M16—22no—
181M16—2142no—
182C2×C8—1072no—
183C2×C8C22×D10712yes—
184C4⋊C4—2144no—
185C4⋊C4—2144no—
186C4⋊C4—2142no—
187C4⋊C4C22×D10712yes—
188C22⋊C4—21no—
189C22⋊C4—2142no—
190C42—21no—
191C42—1071no—
192C42—1071no—
193C42—1072no—
194C42—2142no—
195C23—2141no—
196C23D4×D10711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—1072no—
201Q8—4282no—
202Q8—2142no—
203Q8—2142no—
204Q8—2141no—
205D4—1072no—
206D4—2144no—
207D4—2141no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—1072no—
212C2×C4—1072no—
213C2×C4—4284no—
214C2×C4—4282no—
215C2×C4—2144no—
216C2×C4—2142no—
217C2×C4—2141no—
218C2×C4—2144no—
219C2×C4D4×D10711yescharacteristic
220C2×C4D4×D10712yes—
221C2×C4#1313381812yes—
222C2×C4C23×D10711yescharacteristic
223C8—22no—
224C8—2142no—
225C22—22no—
226C22—4282no—
227C22—2142no—
228C22#1681208212yes—
229C22C2×D4×D10711yescharacteristic
230C4—42no—
231C4—42no—
232C4—1074no—
233C4—4282no—
234C4—2142no—
235C4—2142no—
236C4#1681208212yes—
237C4C2×D4×D10711yescharacteristic
238C4C2×#1313381811yescharacteristic
239C2—22no—
240C2—4282no—
241C2C2×#1681208211yescharacteristic
242C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#419D107
hypocenterC107#3039
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. C2×#16812107C2semidirect
5. #20292836C2semidirect
6. #20292831C2semidirect
7. #20292650C2semidirect
8. #20292635C2semidirect
9. #419×C107C2semidirect
116. Dic107#137semidirect
122. C107#3039semidirect
131. #419D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121402
C2×#168121072
#202930952
#202928362
#202928312
#202926502
#202926352
#419×C1072
#3039107