Finite Group Databasebeta

← #28346263

Subgroups — #28346263

Order 13696.

376 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 143 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
1#20293180C211yesmaximal, characteristic
2C2×Q8×Dic107C211yesmaximal, characteristic
3#20292936C212yesmaximal, complemented
4#20292932C212yesmaximal, complemented
5#20292920C212yesmaximal, complemented
6#20292910C212yesmaximal, complemented
7#20292670C211yesmaximal, characteristic
8#20292669C211yesmaximal, characteristic
9#20292650C211yesmaximal, characteristic
10#20292636C211yesmaximal, characteristic
11#419×C107C211yesmaximal, characteristic
12C2×#13133802C2211yescharacteristic
13#16812108C2211yescharacteristic
14#16812108C2211yescharacteristic
15#16812100C2211yescharacteristic
16#16812100C2211yescharacteristic
17#16812086—21no—
18#16812086—21no—
19Q8×Dic107—21no—
20Q8×Dic107—21no—
21Q8×Dic107C2212yes—
22Q8×Dic107C2212yes—
23C4×Dic214—21no—
24C4×Dic214—21no—
25#16812068C2211yescharacteristic
26C2×C4⋊Dic107—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×C4×Dic107—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812043C2212yes—
31#16812042C2212yes—
32#16812041C2212yes—
33#16812040C2212yes—
34Dic107⋊C8C2212yes—
35Dic107⋊C8C2212yes—
36#16812033C2212yes—
37#16812033C2212yes—
38C22×Q8×C107C2211yescharacteristic
39C2×M16×C107C2211yescharacteristic
40#117×C107C2211yescharacteristic
41#103×C107C2212yes—
42#103×C107C2212yes—
43C2×Dic214—21no—
44C2×Dic214—21no—
45C22×Dic107—21no—
46C22×Dic107D411yescharacteristic
47C22×Dic107D411yescharacteristic
48#13133802—22no—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—21no—
51C4⋊Dic107—22no—
52C4⋊Dic107—22no—
53C4⋊Dic107—21no—
54C4⋊Dic107—21no—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57C4⋊Dic107C2312yes—
58Dic107⋊C4—21no—
59Dic107⋊C4—21no—
60Dic107⋊C4—21no—
61Dic107⋊C4—21no—
62C22⋊Dic107—21no—
63C4×Dic107—21no—
64C4×Dic107—22no—
65C4×Dic107—22no—
66C4×Dic107D411yescharacteristic
67C4×Dic107D411yescharacteristic
68C4×Dic107C2311yescharacteristic
69C4×Dic107C2311yescharacteristic
70C2×Q8×C107—21no—
71C2×Q8×C107—22no—
72C2×Q8×C107Q814yescomplemented
73C2×Q8×C107C2312yes—
74C22×C4×C107—21no—
75C22×C4×C107C2311yescharacteristic
76M16×C107—22no—
77C2×C8×C107C2312yes—
78C4⋊C4×C107C2311yescharacteristic
79C4⋊C4×C107C2311yescharacteristic
80C22⋊C4×C107—21no—
81C42×C107—21no—
82C23×C107C2×D411yescharacteristic
83Q8×C107—24no—
84Q8×C107—22no—
85Q8×C107C2×Q814yes—
86C2×C4×C107—24no—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×Q812yes—
92C2×C4×C107#4112yes—
93C2×C4×C107C2411yescharacteristic
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—41no—
99C2×Dic107—42no—
100C2×Dic107C2×D411yescharacteristic
101C2×Dic107C2×D411yescharacteristic
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104Dic214—42no—
105Dic214—42no—
106C8×C107—22no—
107C107⋊C8—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107C22⋊Q812yes—
111C22×C107C22×D411yescharacteristic
112C4×C107—24no—
113C4×C107—41no—
114C4×C107—41no—
115C4×C107C22⋊Q812yes—
116C4×C107C22×Q811yescharacteristic
117C4×C107C2×#4111yescharacteristic
118Dic107—21no—
119Dic107—21no—
120Dic107—21no—
121Dic107—41no—
122Dic107—41no—
123Dic107—42no—
124Dic107#13712yes—
125C2×C107—22no—
126C2×C107C2×C22⋊Q811yescharacteristic
127C2×C107C2×#13711yescharacteristic
128C2×C107#57511yescharacteristic
129#3078—1071nomaximal, Sylow
130C107#307811yescharacteristic, complemented, Sylow
131#533—1071no—
132C2×C4×Q8—1071no—
133#479—1072no—
134#479—1072no—
135#477—1072no—
136#475—1072no—
137#428—1071no—
138#428—1071no—
139#423—1071no—
140#419—1071no—
141#419D10711yescharacteristic
142C22×Q8C2×D10711yescharacteristic
143C2×M16—1071no—
144C2×M16C2×D10711yescharacteristic
145C4⋊Q8—1071no—
146C4⋊Q8—1071no—
147#125—1071no—
148#125—1071no—
149C22⋊Q8—2141no—
150C22⋊Q8—2141no—
151C4×Q8—1072no—
152C4×Q8—1072no—
153C4×Q8—2141no—
154C4×Q8—2141no—
155C4×Q8—2141no—
156C4×Q8—2141no—
157#117—1071no—
158#117C2×D10711yescharacteristic
159C2×C4⋊C4—1071no—
160C2×C4⋊C4—2141no—
161C2×C42—1071no—
162C2×C42—2141no—
163#107—1072no—
164#107—1072no—
165#106—1072no—
166#106—1072no—
167C4⋊C8—1072no—
168C4⋊C8—1072no—
169#103—1072no—
170#103—1072no—
171#103C2×D10712yes—
172#103C2×D10712yes—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—2141no—
176C2×Q8—2141no—
177C2×Q8Dic21414yescomplemented
178C2×Q8C22×D10712yes—
179C22×C4—21no—
180C22×C4—1071no—
181C22×C4—1071no—
182C22×C4—2141no—
183C22×C4C22×D10711yescharacteristic
184M16—22no—
185M16—2142no—
186C2×C8—1072no—
187C2×C8C22×D10712yes—
188C4⋊C4—1071no—
189C4⋊C4—1071no—
190C4⋊C4—1072no—
191C4⋊C4—2142no—
192C4⋊C4—2142no—
193C4⋊C4—2141no—
194C4⋊C4—2141no—
195C4⋊C4—2141no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4—2141no—
199C4⋊C4—2141no—
200C4⋊C4C22×D10711yescharacteristic
201C4⋊C4C22×D10711yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—2141no—
204C42—21no—
205C42—1071no—
206C42—1071no—
207C42—1071no—
208C42—1071no—
209C42—2142no—
210C42—2142no—
211C42—2141no—
212C23D4×D10711yescharacteristic
213Q8—24no—
214Q8—22no—
215Q8—4282no—
216Q8—4282no—
217Q8C2×Dic21414yes—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—1071no—
223C2×C4—1071no—
224C2×C4—1071no—
225C2×C4—1071no—
226C2×C4—4282no—
227C2×C4—4281no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4—2141no—
231C2×C4—2141no—
232C2×C4D4×D10711yescharacteristic
233C2×C4C2×Dic21412yes—
234C2×C4#1313381712yes—
235C2×C4C23×D10711yescharacteristic
236C8—22no—
237C8—2142no—
238C22—21no—
239C22—21no—
240C22#1681208612yes—
241C22C2×D4×D10711yescharacteristic
242C4—24no—
243C4—41no—
244C4—41no—
245C4—1072no—
246C4—4281no—
247C4—4282no—
248C4—4281no—
249C4—2141no—
250C4—2141no—
251C4—2141no—
252C4#1681208612yes—
253C4C22×Dic21411yescharacteristic
254C4C2×#1313381711yescharacteristic
255C2—22no—
256C2C2×#1681208611yescharacteristic
257C2#137×D10711yescharacteristic
258C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#419D107
hypocenterC107#3078
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. #20292936C2semidirect
4. #20292932C2semidirect
5. #20292920C2semidirect
6. #20292910C2semidirect
72. C2×Q8×C107Q8semidirect
130. C107#3078semidirect
177. C2×Q8Dic214semidirect

Maximal subgroups

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

subgroupindex
#202931802
C2×Q8×Dic1072
#202929362
#202929322
#202929202
#202929102
#202926702
#202926692
#202926502
#202926362
#419×C1072
#3078107